Aromaticity and Metal Clusters

AROMATICITY and METAL CLUSTERS

ATOMS, MOLECULES, AND CLUSTERS Structure, Reactivity, and Dynamics Series Editor: Pratim Kumar Chattaraj Aromaticity and Metal Clusters Edited by Pratim Kumar Chattaraj Quantum Trajectories Edited by Pratim Kumar Chattaraj

ATOMS, MOLECULES,

AND

CLUSTERS

Structure, Reactivity, and Dynamics

AROMATICITY and METAL CLUSTERS Edited by

Pratim Kumar Chattaraj

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4398-1335-5 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents Foreword....................................................................................................................ix Series Preface.............................................................................................................xi Preface.................................................................................................................... xiii Editor........................................................................................................................ xv Contributors............................................................................................................xvii Chapter 1. Aromaticity: From Benzene to Atomic Clusters...................................1 Sandeep Nigam and Chiranjib Majumder Chapter 2. On the Measures of Aromaticity......................................................... 31 G. Narahari Sastry, P. Venuvanalingam, and P. Kolandaivel Chapter 3. Aromaticity in Metals: From Clusters to Solids................................. 55 Alina P. Sergeeva and Alexander I. Boldyrev Chapter 4. Computational Studies on Molecules with Unusual Aromaticity: What to Expect?............................................................. 69 Ayan Datta, Sairam S. Mallajosyula, and Swapan K. Pati Chapter 5. Using the Electron Localization Function to . Measure Aromaticity........................................................................... 95 Patricio Fuentealba, Elizabeth Florez, and Juan C. Santos Chapter 6. Polarizability of Metal Clusters: A Coarse-Grained Density Functional Theory Approach............................................................ 103 Swapan K. Ghosh Chapter 7. Reactivity of Metal Clusters.............................................................. 119 Julio A. Alonso, Angel Mañanes, and Luis M. Molina Chapter 8. Electronic Shells and Magnetism in Small Metal Clusters............... 137 Prasenjit Sen

v

vi

Contents

Chapter 9. Using Theory in Determining the Properties of Metal Clusters: Sodium as a Case Study..................................................... 161 Violina Tevekeliyska, Yi Dong, Michael Springborg, and Valeri G. Grigoryan Chapter 10. The Induced Magnetic Field and Its Application.............................. 187 Rafael Islas, Gerardo Martínez-Guajardo, Gotthard Seifert, Thomas Heine, and Gabriel Merino Chapter 11. A Density Functional Investigation on the Structures, Energetics, and Properties of Sodium Clusters through Electrostatic Guidelines and Molecular Tailoring............................205 K. V. Jovan Jose, Subodh S. Khire, and Shridhar R. Gadre Chapter 12. Size and Shape-Dependent Structural and Electronic Properties of Metal Chalcogenide Nanoclusters............................... 227 Sougata Pal and Pranab Sarkar Chapter 13. Correlation between Electron Delocalization and Ring Currents in All Metallic “Aromatic” Compounds........................................... 245 Patrick Bultinck, Stijn Fias, Marcos Mandado, and Robert Ponec Chapter 14. Phenomenological Shell Model and Aromaticity in . Metal Clusters................................................................................... 271 Tibor Höltzl, Tamás Veszprémi, Peter Lievens, and Minh Tho Nguyen Chapter 15. Rationalizing the Aromaticity Indexes Used to Describe the Aromatic Behavior of Metal Clusters............................................... 297 Prasenjit Seal and Swapan Chakrabarti Chapter 16. Aromaticity in All-Metal Rings........................................................ 323 Jose M. Mercero, Ivan Infante, and Jesus M. Ugalde Chapter 17. Synthesis and Structure of Aromatic Alkali Metal Clusters Supported by Molybdenum Metalloligands...................................... 339 Manish Bhattacharjee

vii

Contents

Chapter 18. 5f Orbital Successive Aromatic and Antiaromatic Zones in Triangular Uranium Cluster Chemistry............................................ 349 Constantinos A. Tsipis Chapter 19. Bonding, Aromaticity, and Possible Bond-Stretch Isomerism in an “All-Metal” Cluster—[Be6Zn2]2−............................................. 371 Santanab Giri, R. P. S. Abhijith Kumar, Arindam Chakraborty, Debesh R. Roy, Soma Duley, Ramakrishnan Parthasarathi, Munusamy Elango, Ramadoss Vijayaraj, Venkatesan Subramanian, Gabriel Merino, and Pratim Kumar Chattaraj Chapter 20. Are Dicationic Chalcogenid Ring Systems Aromatic?..................... 387 Patrizia Calaminici, Andreas M. Köster, and Zeferino Gómez-Sandoval Chapter 21. Study of Aromaticity in Phosphazenes.............................................409 Prakash Chandra Jha, Y. Anusooya Pati, and S. Ramasesha Index....................................................................................................................... 423

Foreword Michael Faraday, “the best experimentalist in the history of science,” is responsible for the many contributions to this book. He is my candidate for its inspirational pioneer. Similar to the diverse subject matter of the following chapters, Faraday’s career spanned both chemistry and physics. He discovered benzene in 1825, recognized its special properties, and determined the composition of naphthalene. His seminal involvement with the interrelationship of electricity and magnetism formed the basis for the later association of unusual magnetic properties with aromaticity. Faraday observed that an electrical current was induced when a magnet was moved in a coil of wire. He discovered diamagnetism and paramagnetism and invented the “Faraday balance” for their measurement. The exaltation of magnetic susceptibility due to the induced ring currents in aromatic compounds is among the widely employed magnetic criteria, and is highly pertinent to the characterization of metal clusters. Faraday was the first to report what are now described as “nanoparticles.” He noted in 1847 that properties of gold colloids (clusters!) differed from those of the bulk metal. This “might be considered to be the birth of nanoscience.” Gold atom aggregates are prominent among metal clusters of current interest. Charts of the Periodic Table typically categorize the chemical elements as “nonmetallic” or “metallic.” The 80 metals among the first 105 elements are “good conductors of both electricity and heat” in the bulk. They epitomize electron delocalization, the hallmark of aromaticity. However, the typical reactions and chemical behavior of individual metal atoms (particularly those of the s,p-block) are much like those of the nonmetallic elements. How many atoms of a metal are needed before the cluster becomes “metallic”? The onset of “aromaticity” in any kind of system also is a fuzzy matter of viewpoint and interpretation. Physicists disdain concepts incapable of precise definition. Organic chemists, the principal purveyors of aromaticity, were slow to recognize that elements other than carbon can participate (this realization dates from the isolation of thiophene as an impurity in benzene in 1883). Stock’s “inorganic benzene” (Borazine, 1926) culminated the quest for main group element examples lacking carbon completely. Despite their prevalence among the chemical elements, transition metals were accepted into the panoply of aromaticity even more slowly. Ferrocene and dibenzenechromium drew attention to the participation of d-block metals, but the aromaticity of these sandwiches was largely ascribed to their carbon rings. Replacement of a benzene CH by a metal group was another step forward. Many well-known complexes with multiple transition metal atoms obviously benefit from the highly delocalized bonding associated with “aromaticity,” but this sobriquet was seldom applied until recently. Progress toward the title of this book is outlined in its chapters. Physicists were intrigued by the intensity distributions (“magic numbers”) in the mass spectra of sodium and other metal atom clusters. Chemists discussed the relationship of Li3+ ix

x

Foreword

and Li6 rings to H3+ and H6 as σ analogs of the 4 + 2 π electron cyclopropenium ion and benzene. The cyclotrialuminum anion, Al3−, may be the smallest metal cluster with two π electrons, but Robinson’s Ga3R3(−) cyclotrigallium analog of the cyclopropenium ion is the smallest x-ray characterized milestone. It was only recognized recently that ancient amalgams contained aromatic Hg2+ 4 rings. Major credit goes to Boldyrev, Wang, and their associates for bringing aromaticity and antiaromaticity considerations in “all-metal” clusters to the fore and for their many significant contributions to the topic. While the degeneracy patterns of MO’s occupied by “mobile electrons” are codified well by the Hückel (4n + 2) and the Möbius (4n) rules when molecules essentially with two physical dimensional (perhaps with twisting and writhing) are involved, the third dimension typically favored by metal clusters introduces considerable interpretive complexities, Application of Hirsch’s spherical 2(n + 1)2 rule (2,8,18,32,50,72, … electrons) is complicated by degeneracy limitations even for the highest symmetries molecules can adopt, shells divide into subshells and their ordering is interwoven rather than being simple progressions. Physicists use “Jellium” models to explain the “magic numbers” observed in the mass spectra of metal atom cluster sets, but want to ignore the further intricacies due to different structural arrangements (isomers). Especially since earlier collections of reviews on aromaticity (e.g., the 2001 and 2005 Special Issues of Chemical Reviews) did not deal with examples involving metals (except for Boldyrev and Wang’s notable contribution), Pratim Kumar Chattaraj well deserves our appreciation for conceiving the present volume and selecting its instructive and stimulating chapters devoted to the increasingly timely relationship between aromaticity and metal clusters. Paul von Ragué Schleyer

Series Preface Atoms, Molecules, and Clusters: Structure, Reactivity, and Dynamics While atoms and molecules constitute the fundamental building blocks of matter, atomic and molecular clusters lie somewhere between actual atoms and molecules and extended solids. Helping to elucidate our understanding of this unique area with its abundance of valuable applications, this series includes volumes that investigate the structure, property, reactivity, and dynamics of atoms, molecules, and clusters. The scope of the series encompasses all things related to atoms, molecules, and clusters including both experimental and theoretical aspects. The major emphasis of the series is to analyze these aspects under two broad categories approaches and applications. Approaches includes different levels of quantum mechanical theory with various computational tools augmented by available interpretive methods, as well as state-of-the-art experimental techniques for unraveling the characteristics of these systems including ultrafast dynamics strategies. Various simulation and QSAR protocols will also be included in the area of approaches. Applications includes topics such as membranes, proteins, enzymes, drugs, biological systems, atmospheric and interstellar chemistry, solutions, zeolites, catalysis, aromatic systems, materials, and weakly bonded systems. Various devices exploiting electrical, mechanical, optical, electronic, thermal, piezo-electric and magnetic properties of those systems also come under this purview. The first book in the series is Aromaticity and Metal Clusters. Although separate monographs exist on aromaticity and metal clusters, this is the first volume to connect the two. Standard textbooks highlight aromaticity in typical organic molecules like benzene with an occasional mention of aromaticity in inorganic systems like borazine. The advent of all-metal aromaticity, and the applications made possible by the unique bonding and reactivity of extended metallic solids, necessitates a deepened understanding of the whole concept featuring a new perspective. The current book serves to fill this gap. Pratim Kumar Chattaraj Series Editor

xi

Preface Benzene, the prototypical organic aromatic molecule, is much less reactive than the usual unsaturated hydrocarbons. The perfect hexagonal structure envisaged by Kekule is the emblem of aromaticity, a concept that continues to develop nearly two centuries after Faraday’s discovery of benzene. Despite its importance, aromaticity is not a strictly defined quantity and hence cannot be measured directly experimentally. Computational or theoretical indexes ultimately require comparisons with reference compounds. Nevertheless, various geometrical, electronic, magnetic, thermodynamic, reactivity characteristics provide insights into the manifestations of aromaticity. This volume addresses aromaticity of compounds containing atoms from the 80 metallic elements in the periodic table. Metal clusters constitute an emerging class of systems with peculiar bonding and reactivity patterns, varying from simple molecules to extended metallic solids. Chemical bonding in and the structure of clusters are frequently different from that of solids providing a challenge for theoretical chemists. The best-known example is C60: it was difficult to predict the peculiar bucky ball shape of C60 just knowing the structure of graphite and diamond. Although often being three dimensional, clusters also exhibit the manifestations of closed circuits of delocalized electrons. Main group metal clusters can exhibit σ- and π-aromaticity/antiaromaticity. Owing to the more complicated nodal structure of d- and f-atomic orbitals transition-metal clusters can provide a more diverse (σ-, π-, δ-, and ϕ-) array of aromaticity/antiaromaticity combinations. Aromatic compounds are important in industry as well as in living systems and clusters are important in many areas such as photography, catalysis, and quantum dots. Apart from the basic aspects of aromaticity, experts discuss the structures, properties, reactivity, stability, and other consequences of aromaticity of a variety of metal clusters in this volume. The Foreword was written by Professor Paul v. R. Schleyer, an authority in this subject and the originator of the nucleus-independent chemical shift (NICS) index, the most widely employed measure of aromaticity. This book will be useful for graduate students as well as researchers in this field. I am grateful to all the authors and the reviewers who cooperated with me so that the book could be published in time. I would like to thank Professors P. v. R. Schleyer, A. I. Boldyrev and P. Bultinck, Mr. L. Wobus and Mr. D. Fausel (Taylor & Francis), and Mr. S. Giri and Ms. S. Duley for their help in various ways. Finally, I must express my gratitude toward my wife Samhita and my daughter Saparya for their whole-hearted support. Pratim Kumar Chattaraj IIT Kharagpur

xiii

Editor Pratim Kumar Chattaraj joined the faculty of Indian Institute of Technology (IIT) Kharagpur after obtaining his BSc and MSc from Burdwan University and PhD from IIT Bombay. He is now a professor and the head of the Department of Chemistry and also the convener of the Center for Theoretical Studies at IIT Kharagpur. In the meantime, he visited the University of North Carolina (Chapel Hill) as a postdoctoral research associate and several other universities throughout the world as a visiting professor. Apart from teaching, he is involved in research on density functional theory, the theory of chemical reactivity, aromaticity in metal clusters, ab initio calculations, quantum trajectories, and nonlinear dynamics. He has been invited to deliver special lectures at several international conferences and to contribute chapters to many edited volumes. He is a member of the editorial board of Journal of Molecular Structure (Theochem) and Journal of Chemical Sciences, among others. He is a council member of the Chemical Research Society of India and a fellow of the Indian Academy of Sciences (Bangalore), the Indian National Science Academy (New Delhi), the National Academy of Sciences, India (Allahabad), and the West Bengal Academy of Science and Technology. He is a J. C. Bose National Fellow.

xv

Contributors Julio A. Alonso Departamento de Física Téorica Atómica y Óptica Universidad de Valladolid Valladolid, Spain and Universidad del Páıs Vasco and Donostia International Physics Center San Sebastián, Spain Manish Bhattacharjee Department of Chemistry Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India Alexander I. Boldyrev Department of Chemistry and Biochemistry Utah State University Logan, Utah

Arindam Chakraborty Department of Chemistry Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India Pratim Kumar Chattaraj Department of Chemistry and Center for Theoretical Studies Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India Ayan Datta School of Chemistry Indian Institute of Science Education and Research Thiruvananthapuram, Kerala, India Yi Dong Physical and Theoretical Chemistry University of Saarland Saarbrücken, Germany

Patrick Bultinck Department of Inorganic and Physical Chemistry Ghent University Krijgslaan, Ghent, Belgium

Soma Duley Department of Chemistry Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India

Patrizia Calaminici Department of Química CINVESTAV Avenida Instituto Politécnico Nacional Mexico D. F., Mexico

Munusamy Elango Chemical Laboratory Central Leather Research Institute Chennai, Tamil Nadu, India

Swapan Chakrabarti Department of Chemistry University of Calcutta Kolkata, West Bengal, India

Stijn Fias Department of Inorganic and Physical Chemistry Ghent University Krijgslaan, Ghent, Belgium xvii

xviii

Elizabeth Florez Departamento de Física Universidad de Chile Santiago, Chile Patricio Fuentealba Departamento de Física Universidad de Chile Santiago, Chile Shridhar R. Gadre Department of Chemistry University of Pune Pune, Maharashtra, India Swapan K. Ghosh Theoretical Chemistry Section Bhabha Atomic Research Centre Mumbai, India Santanab Giri Department of Chemistry Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India Zeferino Gómez-Sandoval Department of Química CINVESTAV Avenida Instituto Politécnico Nacional Mexico D. F., Mexico Valeri G. Grigoryan Physical and Theoretical Chemistry University of Saarland Saarbrücken, Germany Thomas Heine School of Engineering and Science Jacobs University Bremen Bremen, Germany

Contributors

Tibor Höltzl Department of Chemistry Katholieke Universiteit Leuven Leuven, Belgium and Department of Inorganic and Analytical Chemistry Budapest University of Technology and Economics Budapest, Hungary Ivan Infante Kimika Fakultatea Euskal Herriko Unibertsitatea Donostia, Euskadi, Spain and Donostia International Physics Center (DIPC) Donostia, Euskadi, Spain Rafael Islas Departamento de Química Universidad de Guanajuato Guanajuato, México Prakash Chandra Jha Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore, Karnataka, India and Theoretical Chemistry Royal Institute of Technology Stockholm, Sweden K. V. Jovan Jose Department of Chemistry University of Pune Pune, Maharashtra, India Subodh S. Khire Department of Chemistry University of Pune Pune, Maharashtra, India

xix

Contributors

P. Kolandaivel Department of Physics Bharathiar University Coimbatore, Tamil Nadu, India

Gerardo Martínez-Guajardo Departamento de Química Universidad de Guanajuato Guanajuato, México

Andreas M. Köster Department of Química CINVESTAV Avenida Instituto Politécnico Nacional Mexico D. F., Mexico

Jose M. Mercero Kimika Fakultatea Euskal Herriko Unibertsitatea Donostia, Euskadi, Spain

R. P. S. Abhijith Kumar Department of Chemistry Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India Peter Lievens Laboratory for Solid State Physics and Magnetisms Katholieke Universiteit Leuven Leuven, Belgium Chiranjib Majumder Chemistry Division Bhabha Atomic Research Centre Mumbai, India Sairam S. Mallajosyula School of Pharmacy University of Maryland Baltimore, Maryland Angel Mañanes Departamento de Física Moderna Universidad de Cantabria Santander, Spain Marcos Mandado Department of Physical Chemistry University of Vigo Vigo, Galicia, Spain

and Donostia International Physics Center (DIPC) Donostia, Euskadi, Spain Gabriel Merino Departamento de Química División de Ciencias Naturales y Exactas Universidad de Guanajuato Guanajuato, México Luis M. Molina Departamento de Física Téorica Atómica y Óptica Universidad de Valladolid Valladolid, Spain Minh Tho Nguyen Department of Chemistry Mathematical Modelling and Computational Science Center—LMCC Katholieke Universiteit Leuven Leuven, Belgium Sandeep Nigam Chemistry Division Bhabha Atomic Research Centre Mumbai, India Sougata Pal Department of Chemistry Visva-Bharati University Santiniketan, West Bengal, India

xx

Ramakrishnan Parthasarathi Chemical Laboratory Central Leather Research Institute Chennai, Tamil Nadu, India Swapan K. Pati Theoretical Sciences Unit and New Chemistry Unit Jawaharlal Nehru Center for Advanced Scientific Research Bangalore, Karnataka, India Y. Anusooya Pati Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore, Karnataka, India Robert Ponec Institute of Chemical Process Fundamentals Academy of Sciences of the Czech Republic Prague 6. Suchdol, Czech Republic S. Ramasesha Solid State and Structural Chemistry Unit Indian Institute of Science Bangalore, Karnataka, India Debesh R. Roy Department of Chemistry Indian Institute of Technology Kharagpur Kharagpur, West Bengal, India Juan C. Santos Departamento de Ciencias Químicas Universidad Andres Bello Santiago, Chile Pranab Sarkar Department of Chemistry Visva-Bharati University Santiniketan, West Bengal, India

Contributors

G. Narahari Sastry Molecular Modeling Group Indian Institute of Chemical Technology Andhra Pradesh, India Prasenjit Seal Department of Chemistry University of Calcutta West Bengal, India Gotthard Seifert Physikalische Chemie Technische Universität Dresden, Germany Prasenjit Sen Harish-Chandra Research Institute Allahabad, Uttarpradesh, India Alina P. Sergeeva Department of Chemistry and Biochemistry Utah State University Logan, Utah Michael Springborg Physical and Theoretical Chemistry University of Saarland Saarbrücken, Germany Venkatesan Subramanian Chemical Laboratory Central Leather Research Institute Chennai, Tamil Nadu, India Violina Tevekeliyska Physical and Theoretical Chemistry University of Saarland Saarbrücken, Germany Constantinos A. Tsipis Laboratory of Applied Quantum Chemistry Aristotle University of Thessaloniki Thessaloniki, Greece

xxi

Contributors

Jesus M. Ugalde Kimika Fakultatea Euskal Herriko Unibertsitatea Donostia, Euskadi, Spain and Donostia International Physics Center (DIPC) Donostia, Euskadi, Spain P. Venuvanalingam School of Chemistry Bharathidasan University Thiruchirapalli, Tamil Nadu, India

Tamás Veszprémi Department of Inorganic and Analytical Chemistry Budapest University of Technology and Economics Budapest, Hungary Ramadoss Vijayaraj Chemical Laboratory Central Leather Research Institute Chennai, Tamil Nadu, India

1 Aromaticity From Benzene to Atomic Clusters Sandeep Nigam and Chiranjib Majumder Contents 1.1 Introduction.......................................................................................................1 1.2 Computational Details.......................................................................................7 1.3 Result and Discussion........................................................................................7 1.3.1 Dianionic Clusters: B24 − , Al 24 − , Ga 24 − ....................................................7 1.3.2 Cationic Clusters: Si 24 + , Ge 24 + , Si 2 Ge 22 + , Si3 Al + , Si3 Ga + , Ge3Al+, Ge3Ga+......................................................................................9 1.3.3 Neutral Clusters: [Q2R2 (Q = Al, Ga; R = Si, Ge), A2G2 (A = Be, Mg; G = N, P), AB3 (A = Be, Mg; B = Si, Ge) and X3Y (X = Al, Ga; Y = P, As)].............................................................. 11 1.3.4 Hydrogenation......................................................................................20 1.3.5 Nucleus-Independent Chemical Shift (NICS) Study of Cationic and Neutral Clusters.............................................................. 22 1.4 Summary......................................................................................................... 23 Acknowledgments.....................................................................................................24 References.................................................................................................................24

1.1 Introduction Aromaticity is a key concept in variety of areas of chemistry [1]. This concept ­constitutes the rational basis to understand the structure, stability, and reactivity of many molecules. Aromaticity has a long history dating back to the early nineteenth century. In 1825, Michel Faraday isolated a compound from a oily mixture collected in tanks used to store coal gas at high pressures. Faraday found that the compound has unusually small hydrogen-to-carbon ratio of 1:1 and he named the compound “bicarburet of hydrogen” [2]. Ten years later, Eillardh Mitscherlich synthesized the same compound by heating benzoic acid, isolated from gum benzoin, with lime and he also found the empirical formula as CH. Since the new compound was derived from gum benzoin, he named it as benzin [3], which became benzene when translated into English. He also established that the molecular formula of this compound was C6H6. Later, it was recognized that even though benzene is an unsaturated ­compound, it differs greatly from the corresponding aliphatic compounds. Benzene 1

2

Aromaticity and Metal Clusters

showed higher stability and contained higher percentage of carbon. It does not exhibit the reactivity associated with typical unsaturated compounds such as alkenes and alkynes. Later, it was found that many other compounds that are isolated from coal (e.g., phenol, aniline, benzoic acid, salicylic acid, and anthranilic acid) also had similar properties. The molecular formula of these compounds suggested that the presence of multiple unsaturated bonds but these compounds were not reactive enough in comparison to alkenes. They were collectively grouped as aromatic compounds based on their distinct odor, or aroma, and benzene becomes the prototype of this class of substance. The structure and stability of benzene were a persistent problem throughout most of the nineteenth century. In 1865, F. Kekule suggested that benzene forms a ring structure and consists of six carbon atoms with alternating single and double bonds [4]. Because there are two ways in which these bonds can be generated, Kekule proposed that benzene was a mixture of two compounds in equilibrium [4]. Kekule’s structure explained the formula of benzene but it could not explain the extra stability of benzene. In subsequent decades, many scientists [Dewar (1867), Ladenburg (1869), V. Meyer (1870), Claus (1867–1882), Baeyer (1884–93), Armstrong (1887), Thiele (1899), and many others] have attempted [5] to explain the stability and the exceptional chemical behavior of benzene with respect to its structure and bonding. Aromatic Sextet Theory, which was proposed by Robinson (1925), was apparently the first to correlate the aromaticity with a number of electrons. This theory says that six electrons, one being contributed by each carbon atoms, form a “closed loop,” which gives rise to the “aromatic properties” [6]. Later, valance bond theory was applied to the benzene problem, and the unusual stability of benzene was attributed to the resonating structure [7], which states that benzene is a resonance hybrid of five hypothetical resonating structures (two corresponding to the Kekule structure and three to the Dewar structure), where the Kekule structure contributes about 80% and  the Dewar structure 20%. However, the resonance theory could not explain the  fact that even though cyclobutadiene and cyclooctatetraene have electronic ­conjugation, they do not have extra stability. The molecular orbital diagram of benzene shows three bonding and three anti­ bonding molecular orbital’s (MO) embracing all six carbon atoms and it was explained that benzene is unusually stable due to delocalization of three pairs of electrons. With the notion of π orbitals and σ−π separation, Hückel (1931) attempted to solve the benzene problem [8,9]. In a benzene structure of D6h symmetry, the σ−π separation technique gave six π electrons completely delocalized over the hexagonal σ frame. The deloca­lized π system was found to have a closed shell and higher stability than the sum of  three isolated π bonds. Further applications by Hückel enabled him to generalize his observations and to formulate the 4n + 2 rule for the entire class of aromatic compounds. Later, many developments, namely azulene’s synthesis and structure elucidation [10], synthesis of tropylium [11], and cyclopropenium aromatic cation [12] and so on, supported the Hückel’s (4n + 2)π rule. In 1964, on purely theoretical grounds, Heilbronner predicted that large cyclic molecules with 4nπ electrons, rather than (4n + 2)π-electrons, would be an aromatic system if their constructed annulene have one end half a twist, that is, Möbius topology [13]. In 1965, R. Breslow introduced the “antiaromatic” terms through his quantum mechanical

Aromaticity

3

calculations [14]. He ­proposed that systems with 4nπ electrons ought to have an antiaromatic character and would be destabilized by resonance. The prediction was examined for the case of cyclopropenyl anion, the simplest system having 4nπ electrons (n = 1). Thereafter till date, both these (4n + 2) and 4nπ rules were among the most important criteria (but not the only criteria) for classification of a molecule, respectively, as aromatic or antiaromatic. During the era of post-Hückel theory, several criteria have been put forward to rationalize and measure aromaticity, that is, regarding the quantification of the degree of aromaticity. These criteria can be roughly divided into four categories: energetic, structural or geometrical, magnetic, and reactivity-based measures. Various diagnostic properties resulting from the cyclic delocalization have been used to establish indices of aromaticity. In the many measures of aromaticity, the most widely accepted categories [15–17] are energetic criterion (aromatic stabilization energy [18], HOMO– LUMO energy gap, the absolute and relative hardness [19]), structural criterion (bond length equalization [20], planarity, geometric indices of aromaticity, like- Harmonic Oscillator Model of Aromaticity (HOMA) [21]), chemical reactivity criterion (electrophilic substitution rather than addition [22]), and the most widely used magnetic criterion (exalted magnetic susceptibilities [23], anisotropies [24], displaced chemical shift [25], plotted maps of the ring current density (CD plots) [26,27], nucleusindependent chemical shifts (NICSs) [28], aromatic ring current shielding (ARCS) [29], ring current intensities [30], anisotropy of the current-induced density (ACID) [31]. Apart from the above-mentioned criteria, there are other techniques as well that were proposed to gauge the extent of aromaticity. For example, Santos et al. provided a topological analysis of the σ- and π-contributions to electron localization function (ELF) to quantify aromaticity [32,33] whereas Sola and workers proposed an aromatic fluctuation index (FLU) describing the fluctuation of electronic charge between adjacent atoms in a given ring [34]. Giambiagi and coworkers have used multicenter bond indices as a measure of aromaticity [35–37]. These indices estimate the extension of π-electron delocalization at the centers of the ring, which is expected to be especially large for aromatic systems. More recently, Bultinck et  al. [38–44] and Mandado et al. [45–49] have employed multicenter delocalization indices computed from Mulliken-type calculation to study the local and total aromaticity of polycyclic aromatic hydrocarbons [38–40,42–46,48], aromaticity of heterocycles [47], homoaromaticity [41,44], and concerted reaction mechanisms [49]. However, because of its multiple manifestations, there is, till date, no generally accepted single and unique quantitative measure of aromaticity. For example, the aromatic stabilization energy (ASE) can be determined but it depends on the choice of reference. There is no direct way to measure the aromaticity in a molecule experimentally. In benzene, which is a prototype aromatic molecule, the experimental verification was done only by indirect methods, namely heat of hydrogenation and deshielded nuclear magnetic resonance (NMR) chemical shift. Although different definitions and indices of aromaticity have their own flaws, qualitatively there has never been a real disagreement on the fact that aromatic compounds are characterized by special stability and this additional stabilization is due to the cyclic electron delocalization [15]. In other words, aromaticity is essentially an excess property, that is, a deviation from an additive scheme and it is not a directly measurable experimental quantity.

4

Aromaticity and Metal Clusters

In 2005, Schleyer and coworkers [16, p. 3844] proposed a qualitative definition of aromaticity as Aromaticity is a manifestation of electron delocalization in closed circuits, either in two or in three dimensions. This results in energy lowering, often quite substantial, and a variety of unusual chemical and physical properties. These include a tendency toward bond length equalization, unusual reactivity, and characteristic spectroscopic features. Since aromaticity is related to the induced ring currents, magnetic properties are particularly important for its detection and evaluation.

Despite many controversial arguments regarding the definition and physical origin of aromaticity [1,15–17], the concept of aromaticity has crossed the boundary of benzenoid hydrocarbons [with (4n + 2)π-electrons] to include heterosystems [50] like pyridine, thiophine, cations such as tropylium [12] and cyclopropenium [13], anions like cyclopentadienyl [51], organometallic systems, namely ferrocene [52], purely carbon-free systems [53,54], namely P 5−, [(P5)2Ti]2−. The three-dimensional aromaticity of boron-based clusters [55] and of fullerenes [56], the homoaromaticity of cationic systems [57], aromaticity of triplet state annulenes [58], and pericyclic transition states [59] has enlarged the concept of aromaticity. Extension of ­a romaticity

S

N Benzene

Phenanthrene

Pyridine

Thiophene

Fe

+

Tropyllium ion

Cyclopentadienyl P

P P

(P5)–

H H

Ti P P

P P

[(P5)2Ti]–2

B

HH B BB

B

BB H H

B

B

P

Ferrocene

H

H

P

P

P

P P

P P

Azulene

H

B H

H

[B12H12]–2

C60 Fullerene

Figure 1.1  Enlargement of aromaticity concepts: Different aromatic molecules.

Aromaticity

5

concept (Figure 1.1) has also made it clear that it must be measured with a variety of indices. Hence, aromaticity is not confirmed merely on the basis of π electrons of planar rings. Furthermore, there are an increasing number of powerful methods of assessing different manifestation of cyclic electron delocalization. Very recently, Li et  al. reported aromaticity in purely metallic MAl −4 and Al 2− 4 (M = Li, Na, and Cu) clusters using the experimental and theoretical techniques [60]. These bimetallic anionic clusters were produced in the gas phase and their electronic energy spectra were recorded using photoelectron spectroscopy. Ab initio calculations showed that all MAl −4 possess a C4v pyramidal structure, where an M+ cation is interacting with an Al 2− 4 unit, which shows cyclic planar structure with two delocalized π-electrons, confirming the (4n + 2)π counting rule of aromaticity. These findings further expanded the aromaticity concept into the territories of metal systems and opened up a new area, which suggests that even metal clusters can show aromatic character, well known for organic molecules with significantly higher stability. Subsequently, the aromaticity of the Al 2− 4 dianion has been verified using different techniques [32,61–72]. Since then, aromaticity has been found in a large number of new gaseous all-metal or metalloid clusters [73–135]: NaGa −4 and NaIn −4 [73], Au5Zn+ [74], CunHn (n = 3–6) [75], Al2(CO)2 [76], X 5− (X = N, P, As, Sb, Bi) [77,90], 2− X 2− 4 (X = B, Al, Ga, In, Tl) [60,70,71,73,84,87], sandwich structures of [Al4TiAl4] − 4− − − and Na[Al4TiAl4] [86], X 3 and NaX3 (X = Al, Ga) [89], Hg6 [92], XAl3 (X = C, Si, − Ge, Sn, Pb) [93], XGa 3− (X = Si, Ge) [95], X 2− 4 and NaX 4 (X = N, P, As, Sb, Bi) [101]. − + − C2 N 2 , M(C2N2) M = Li, Na, K, [N(C2N2)] , N = Be, Mg, Ca, [114], Al 2 P 2− 2 , [M(Al2P 2)] , + M = Li, Na, K, Cu, N(Al2P2); N = Be, Mg, Ca, Zn [115], Cu6Sc and Cu5Sc [116], B2XY, X = N, P and Y = O, S, Se [117], Al2MN, M = C, Si, Ge; N = S, Ge [117], Si2BX, X = Li, K, O, S [118] Sb5− and Sb5M (M = Li, Na and K) [119], O24 + , Si 24 + , Se 24 + [120], Re3Cl9, Re3Br9 and their dianions [121], Cu4Na−, Au4Na− [122], BiGa3 [123], Au6Y− [124], Be32 − , Mg32 − [125], Be8− [126], Na6, and K6 [127], Cu7Sc [128], N 6− [129], − P4Mq, (M = S, Se, q = 0; M = Si, Ge, q = 2-) [130], N 5−, SN4, S2 N 3 , and S3 N 2+ 2 [131]. Importantly, the discovery of all-metal aromatic systems has added a new feature in the concept of aromaticity, namely multifold aromaticity (simultaneous presence of π and σ aromaticity). The multifold aromaticity of the Al 2− 4 dianion has been established with a variety of theoretical methods: bifurcation analysis of the electron localization function (ELF) [32], nucleus-independent current shielding (NICS) [67], valence bond (VB) assessment of π and σ aromaticity [69], resonance energy (RE) estimations [70], maps of ring current [73,74], and ARCS [84]. Very recently, for Cu2− 4 cluster, the cyclical delocalization due to d electrons instead of the usual p orbitals has provided the first quantitative evidence for the existence of “d-orbital aromaticity” or “δ-aromaticity” [132]. Later, the possibility of three-dimensional aromatic clusters involving d-orbitals was also explored in the pseudo-octahedral coinage metal cages of M6Li2 (M = Cu, Ag, Au) as well as the tetrahedral coinage metal cages of M′4 Li 4 (M′ = Cu, Ag) [133]. In a recent report, Boldyrev and coworkers showed the first example of triple-fold (σ-, π-, and δ-) aromaticity in the Hf3 cluster [134]. Subsequently, the [Ta3O3]− cluster also showed an unprecedented multiple (π and δ) aromaticity, which was responsible for the multicenter metal–metal bonding and the perfect triangular Ta3 framework [135]. In the recent past, our group has explored the possibility of aromaticity in ­several tetramer clusters comprising of homo- and heteroatomic elements [136–138]. The

6

Aromaticity and Metal Clusters

choice of tetramer clusters was governed by the fact that these are the smallest ­clusters, which can have either planar or nonplanar atomic structure in the ground state. In particular, the interest was to find out neutral clusters that exhibit an aromatic character and are more stable than the charged clusters. In the first part, we have investigated the previously reported dianionic aromatic clusters X 2− 4 (X = B, Al, Ga) and analog neutral cluster Al4Na2, B4Na2, Ga4Na2, in which the double charge of the anionic cluster have been neutralized by the sodium atoms. The objective of this study was to know the change in the geometrical parameters, structural integrity, and aromaticity of the dianion cluster after interaction with two sodium atoms. In the second part of our study, we have explored the concept of aromaticity in few other tetramer clusters formed by alkali earth metals and main group elements. The choice of the constituents in these tetramer clusters was based on the criterion that each cluster should be iso-electronic with the previously studied Al 2− 4 cluster, that is, having a total of 14 valence electrons, out of which 8 are s-type and 6 are p-type. Thus, the objective of this part of the study was to understand the geometry and electronic structure of a series of neutral and charged tetramer clusters having iso-electronic configurations and to explore the possibility of the existence of aromatic character in these clusters. Initially, the ground-state 2+ geometries and energetics of doubly positively charged clusters such as Si 2+ 4 , Ge 4 2+ 2+ 2− Ge 4 , Si 2 Ge 2 , which are isoelectronic to the Al 4 cluster, have been calculated. The singly charged clusters such as Si3Al+, Si3Ga+ Ge3Al+, and Ge3Ga+, which are also isoelectronic to the Al 2− 4 , were also investigated and verified for aromaticity. Further, detailed calculations have also been performed for a number of neutral clusters, namely Al2Si2, Al2Ge2, Ga2Si2, Ga2Ge2, Si3Be, Si3Mg, Ge3Be, Ge3Mg, Al3P, Al3As, Ga3P, Ga3As, Be2N2, Be2P2, Mg2N2, Mg2P2. The interaction of a hydrogen atom with few selective clusters such as Si 2+ 4 and Al 2 Si 2 was investigated to further test the extra stability of these clusters. To confirm the aromatic character of all the attempted clusters, conventional criteria of aromaticity such as geometrical configurations (planarity), spatial orientation of the occupied orbitals, chemical stability (extra stability), and most widely used magnetic criteria [i.e., nucleus independent chemical shift (NICS)] was utilized. NICS index is defined as the negative value of the shielding, computed at the ring center [NICS(0)] or at some other chosen point. In this method, the NMR parameters, that is, absolute magnetic shielding, are calculated for a ghost atom placed on the point of interest. Significantly negative (i.e., magnetically shielded) NICS value indicates the presence of induced diatropic ring currents or “aromaticity,” whereas positive values (i.e., deshielded) denote paratropic ring currents and “antiaromaticity.” The NICS (0.0) values calculated at the center of the ring are influenced by σ-bonds, whereas the NICS (1) values calculated at a distance of 1.0 Å out of the plane (where the π orbitals have their maximum density) better measure the π electron delocalization as the influences of other magnetic shielding contributions different from the π system are significantly reduced [16,28]. However, there are some drawbacks of NICS. It has been found that the NICS values are sensitive to the basis set used [28,139]. Further, Paolo Lazzeretti has pointed out that NICS is a local aromaticity index and should be treated with care in case of polycyclic systems due to the influence of other rings [140].

Aromaticity

7

1.2 Computational Details Ab initio molecular orbital theory based on linear combination of atomic orbitals (LCAO) approach has been widely used to elucidate the ground-state geometries of small molecules and clusters. It is well recognized that the Hartree–Fock (HF) theorybased calculations do not take electron correlation effects into account and therefore inadequately represent systems possessing extended conjugation. On the contrary, theoretical methods like the density functional theory (DFT) and the second-order Moller–Plesset perturbation theory (MP2) include electron correlation effects and are generally considered to yield reliable results for such conjugated systems. For [10] annulene and bismethano[14]annulene, these methods provided an excellent means for the confirmation of aromaticity or antiaromaticity [141]. Based on these facts, we have primarily used DFT-based methodologies and hybrid exchange correlation energy functional commonly known as B3LYP for initial optimization of all our systems, which were further verified using the MP2 level of theory [142,143]. Different types of standard basis sets were used for different series of clusters. It is known that diffusion function is necessary for systems containing lone pair of electrons or anionic systems, as addition of diffuse function leads to more appropriate description of electrons toward the outer region of molecules. Therefore, standard split-valence with polarization and diffuse functions [6 − 311G + (d)] were employed as the basis set for dianionic clusters, namely X 2− (X = B, Al, Ga) and their sodium complexes 4 [6 − 31 + G(d)] in case cluster having Ga atom]. For Be2N2, Be2P2, Mg2N2, Mg2P2 systems, the standard 6 − 311 + G(d) basis set was used because these tetramer clusters are consisting of group II and group V elements. As group II and group V elements have s2 and p3 configuration (filled and half filled) in their outermost orbital, it is expected that the interaction between these atoms will be weak and can be governed by dispersion force. In case of other cationic and neutral metallic tetramer cluster systems, the diffuse function does not play a significant role. Therefore, for all the 2+ 2+ + + + + cationic and metallic systems (Si 2+ 4 , Ge 4 , Si 2 Ge 2 , Si3Al , Si3Ga Ge3Al , and Ge3Ga , Al2Si2, Al2Ge2, Ga2Si2, Ga2Ge2, Si3Be, Si3Mg, Ge3Be, Ge3Mg, Al3P, Al3As, Ga3P, and Ga3As), standard split-valence basis set with polarization only [6 − 31G(d)] was used. The stability of the lowest-energy isomers of all neutral clusters was verified from the vibrational frequency analysis at the MP2 level [144]. All calculations were carried out using the GAMESS software [145]. Molekel software package was used to plot the orientations of molecular orbitals [146]. The NICS values were calculated using the gauge, including atomic orbitals (GIAO) method using Gaussian-98 software [147].

1.3 Result and Discussion 1.3.1  Dianionic Clusters: b24− , Al24− , ga 24− For B24 − , Al 24 − , and Ga 24 − clusters, the lowest-energy structure was square planar with D4h symmetry. Three-dimensional tetrahedron and other geometries resulted in ­significantly higher energy isomers than the planar configurations. Table 1.1 summarizes the geometrical parameters, total energies, binding energies (BEs), and highest occupied molecular orbital (HOMO) energy. From Table 1.1 it is

8

Aromaticity and Metal Clusters

Table 1.1 Total Energies, Interatomic Separations (Å), Binding Energy per Atom (B.E.) Energy Eigenvalue of the Highest Occupied Molecular Orbitals (HOMO) Cluster 2− 4

B

Al 2− 4 Ga 2− 4

Total Energy (a.u.)

M-M

B.E. (eV)

HOMO (eV)

−98.7483711 (−99.010998) −967.8374139 (−969.5387656) −7685.6598872 (−7691.7510659)

1.68 (1.63)

3.16 (2.75)

3.39 (4.74)

2.56 (2.59)

1.49 (1.12)

1.87 (2.98)

2.53 (2.50)

1.87 (1.54)

1.86 (3.02)

Note: The results presented here have been calculated at the MP2/6-311 + G(d) level and B3LYP/6311 + G(d) level (presented within parentheses).

clear that the energy eigenvalues for the HOMO are positive for negatively charged clusters B24 − , Al 24 − , and Ga 24 −, which represent an unbound state or, in other words, it reflects that doubly negatively charged clusters are extremely reactive in gas phase. It is found that four upper-most occupied orbitals (i.e., HOMO, HOMO-1, HOMO-2, and HOMO-3) have positive energy values. This fact was also mentioned by Mercero et al. in their work [86] on sandwich-like complexes of Al 2− 4 . They point out that doubly charged species are expected to be rather unstable in the gas phase due to large intramolecular Coulomb repulsion. The instability of dianions was also predicted by Kuznetsov et al. [88] and Boldyrev et al. [105]. In this context it is worth mentioning that Dixon and coworkers have calculated [70] the resonance energy of Al 2− 4 system and found 72.7 kcal/mol. However, it is also known that the resonance energy depends upon the choice of the reference model structure relative to which resonance energy of the system is calculated. Jung et al. in their work [98] found the molecule N2S2 aromatic but resonance or aromatic stabilization energy was found to be only 6.5 kcal/mol. Hence it is desirable to stabilize these dianionic clusters or, in other words, the clusters in which the charge of these anionic clusters has been neutralized by some metal atoms. For this purpose, we have carried out the geometric and electronic structure optimization for a series of tetramer binary clusters Al4Na2, B4Na2, Ga4Na2. Table 1.2 summarizes the results of the geometrical parameters and electronic properties for Al4Na2, B4Na2, Ga4Na2 cluster. To confirm the stability of M4Na2 species we have calculated the energies for hypothetical M 4 Na 2 → M24 − + 2 Na + reactions. The calculated energies {MP2 (B3LYP)} are as follows:

B4 Na 2 → B24 − + 2 Na + ∆H = 13.90 (14.68) eV



Al 4 Na 2 → Al 24 − + 2 Na + ∆H = 12.51 (13.42) eV



Ga 4 Na 2 → Ga 24 − + 2 Na + ∆H = 13.79 (14.29) eV

9

Aromaticity

Table 1.2 Total Energies, Interatomic Separations [M-M and M-Na(Å)], Binding Energy per Atom (B.E.) Energy Eigenvalue of HOMO Level Cluster B4Na2 Al4Na2 Ga4Na2

Total Energy (a.u.) −422.606178 (−423.647475) −1291.644038 (−1294.129737) −8009.485808 (−8016.361335)

M-M

M-Na

B.E. (eV)

HOMO (eV)

LUMO (eV)

Gap (eV)

1.69 (1.66)

2.70 (2.67)

2.86 (2.61)

−4.77 (−3.19)

−1.48

1.71

2.61 (2.62)

3.18 (3.16)

1.54 (1.31)

−5.14 (−3.85)

−2.01

1.84

2.55 (2.52)

3.20 (3.12)

1.94 (1.74)

−5.11 (−3.83)

−1.94

1.89

Note: The results presented here have been calculated at MP2/6-311 + G(d) level and B3LYP/6311 + G(d) level (presented within parentheses).

All reactions are highly endothermic, indicating that the M4Na2 species are stable toward decomposition. In order to understand the nature of bonding, the spatial ­orientations of five occupied molecular orbitals (HOMO to HOMO-4) have been worked out for M4Na2 clusters studied and the representative results for Al4Na2 are shown in Figure 1.2. It is clear from Figure 1.2 that all these clusters have at least one of the HOMO energy level representing π-type molecular orbital where two electrons are delocalized over all four atoms. After their interaction with sodium atoms, these dianionic clusters retain their structural integrity and, in addition, the spatial orientation of the HOMO energy levels also remains same except the fact that their ordering has been changed. From the results, it is clear that even after interacting with two sodium atoms the structural integrity and the planarity of these dianionic clusters are retained. Aromaticity of all M4Na2 clusters has also been studied using the NICS method developed by Schleyer and coworkers [28]. The results obtained are given in Table 1.3. From Table 1.3 it is clear that the M2− 4 ring in the M4Na 2 species exhibits the characteristics of aromaticity, or in other words, it retains its aromaticity. 2+ 2+ + 1.3.2 Cationic Clusters: si2+ 4 , ge 4 , si2ge 4 , Si3Al , Si3Ga+, Ge3Al+, Ge3Ga+ 2+ For Si 2+ 4 and Ge 4 clusters, the lowest-energy structure is square planar with all Si–Si and Ge–Ge bond lengths as 2.32 and 2.41 Å, respectively, at MP2 level. In this ­context it may be mentioned that for neutral Si4 and Ge4 clusters, the equilibrium geometries favor rhombus structure over the square planar geometry. Another isoelectronic heteroatomic tetramer cluster Si 2 Ge 2+ 4 also shows square planar structure 2+ as the lowest-energy isomer. The trans-isomer of Si 2 Ge 2+ 4 , that is, Si 2 Ge 4 (t) cluster with four equal Si–Ge bond length of 2.38 Å is found to be 0.137 eV lower in energy than the corresponding cis- isomer, that is, Si 2 Ge 2+ 4 (c). In order to understand the

10

Aromaticity and Metal Clusters (a)

(b)

HOMO

HOMO-1

HOMO-2

HOMO-3

HOMO-4

Figure 1.2  Spatial orientation of top five occupied molecular orbitals for (a) Al 2− 4 and (b) Al4Na2 cluster.

nature of bonding, the spatial orientations of five occupied molecular orbitals (HOMO to HOMO-4) have been plotted for these cationic clusters and a representative result 2+ for Si 2+ 4 cluster is shown in Figure 1.3. It is clear from Figure 1.3 that for the Si 4 clusters the HOMO energy level is representing a π-type molecular orbital where two 2+ electrons are delocalized over all four Si atoms. Similar to Al 2− 4 , Si 4 too have fully delocalized σ-type orbital HOMO-1 and HOMO-2 [41,62,66]. The following four MOs are due to filled 3s valence orbital of Si and they do not play an important role in bonding and can be viewed approximately as lone pairs [62]. Analysis of the pictures of the molecular orbitals indicates the presence of multifold aromaticity in these clusters. A similar shape of molecular orbitals is also observed for Ge 2+ 4 and Si 2 Ge 2+ 4 clusters. The energy eigenvalues for the HOMO of positively charged 2+ 2+ ­clusters (Si 2+ 4 , Ge 4 , Si 2 Ge 4 ) are listed in Table 1.4.

11

Aromaticity

Table 1.3 2− Nucleus Independent Chemical Shift (NICS) Values of b2− 4 , B4Na2, Al4 , 2− Al4Na2, ga 4 , Ga4Na2 clusters Cluster

NICS(0.0)

NICS(1.25)

B

−91.54 (−29.53)

−28.29 (−3.04)

B4Na2 Al 2− 4

−76.95 (−25.51) −52.41 (−34.36)

−33.98 (−19.13) −30.73 (−23.15)

Al4Na2 Ga 2− 4

−19.91 (−12.58) −53.09 (−32.16)

−30.41 (−23.03) −32.81 (−20.21)

Ga4Na2

−20.69 (−14.04)

−30.83 (−24.59)

2− 4

Note: The results presented here have been calculated using the gauge-including atomic orbitals (GIAO) method at MP2/6–311 + G(d) and B3LYP/6–311 + G(d) (values presented within parentheses) level. 6–31 + G(d) basis set in case of cluster contains the Ga atom.

Synthesis of doubly positively charged cluster is difficult as it requires larger ionization energy than the first ionization energy. Considering this, we have calculated the ground-state geometries and electronic structures of Si3Al+, Si3Ga+, Ge3Al+, and Ge3Ga+ clusters. The equilibrium geometries of these clusters show a distorted rhombus structure with C2v symmetry. The lowering of the symmetry is due to the unequal composition of heteroatomic elements in these clusters. To understand the geometrical features, the interatomic distances between different atoms are listed in Table 1.5. Although these clusters are asymmetric as compared to the previously discussed doubly positively charged clusters, it is of interest to note that the orientation of the molecular orbitals for both types of cluster is very similar. Representative shapes for the five highest occupied molecular orbitals are shown in Figure 1.2 taking Si3Al+ as a typical example.

1.3.3 Neutral Clusters: [Q2R2 (Q = Al, Ga; R = Si, Ge), A2G2 (A = Be, Mg; G = N, P), AB3 (A = Be, Mg; B = Si, Ge), and X3Y (X = Al, Ga; Y = P, As)] As the charged clusters are relatively unstable in comparison to the neutral clusters, it is desirable to find out stable neutral clusters exhibiting aromatic behavior. For this purpose, we have carried out the geometric and electronic structure optimization of a series of tetramer binary clusters, namely Q2R2 (Q = Al, Ga; R = Si, Ge), A2G2 (A = Be, Mg; G = N, P), AB3 (A = Be, Mg; B = Si, Ge), and X3Y (X = Al, Ga; Y = P, As). The choice of these neutral clusters is based on their isoelectronic configuration with the doubly or singly charged aromatic clusters discussed above. In order to confirm the stability of the ground-state geometries of these neutral clusters, we have carried out the frequency calculation at the MP2/6-31G(d) level [MP2/6-311G + (d) in case of A2G2], and the corresponding list of frequencies along with their intensities are summarized in Table 1.6. It is important to note that no imaginary frequency was found for any of these clusters, suggesting that these

12

Aromaticity and Metal Clusters (a)

(b)

HOMO

HOMO-1

HOMO-2

HOMO-3

HOMO-4

Figure 1.3  Spatial orientation of top five occupied molecular orbitals for (a) Si 2+ and 4 (b) Si3Al+ cluster.

i­somers represent a stationary point on their respective potential energy surface. The high stability of these clusters is also reflected in their large energy gaps bet­ ween the highest occupied and lowest unoccupied molecular orbitals, that is, global hardness, η(η = HOMO − LUMO)

Q2R2 (Q = Al, Ga; R = Si, Ge)

2+ Unlike Al 2− 4 and Si 4 clusters, these clusters are heteroatomic in nature but with the same number of valence electrons (8s and 6p) in their outermost orbital. The 2+ advantage of these clusters over the Al 2− 4 and Si 4 clusters is that these are neutral and therefore expected to be more stable. The equilibrium geometry has been obtained

13

Aromaticity

Table 1.4 Total Energies, Interatomic Separations (Å), Energy Eigenvalue of HOMO Level Cluster

Si

2+ 4

Total Energy (a.u.)

M-M

HOMO (eV)

−1155.184831

2.32

−18.43

−8292.966784

2.41

−17.75

2+ 2

Si 2 Ge (C2 v )

−4724.078594

2.34, 2.37, 2.43

−11.26

Si 2 Ge 22+ (D 4 h )

−4724.083629

2.38

−11.36

Ge 2+ 4

Note: The results presented here have been calculated at MP2/6-31G(d) level.

by considering two planar structures, namely cis- and trans- configurations. The difference in the stability between these two isomers can be estimated from the corresponding total energies, which are listed in Table 1.7. Three-dimensional tetrahedron and other geometries resulted in significantly higher energy as compared to the planar configurations. Table 1.7 summarizes the results of the important geometrical and electronic structural features of Q2R2 clusters. Table 1.7 presents the total energies, interatomic separations (Å), binding energy# (BE, only MP2 results are listed), vertical ionization potentials (VIPs), vertical electron affinity (VEA), and the energy gap between the HOMO and LUMO energy levels, that is, global hardness η for Q2R2 clusters. The results presented here have been calculated at MP2/6-31G(d) level and B3LYP/6-31G(d) level (presented in bracket). For global hardness η only B3LYP results are listed. From the stability point of view, a molecule or cluster is chemically inert if it is resistant to undergo addition reaction with either an electrophile or a nucleophile. Further, it has been reported that for a stable molecule, the hardness often becomes the maximum and the electrophilicity becomes the minimum [148,149]. As the global hardness is the difference between the HOMO and LUMO energy levels (global hardness η = HOMO − LUMO or by using the Koopmans theorem η = IP − EA), Table 1.5 Total Energies, Interatomic Separations (Å), and Energy Eigenvalue of HOMO Levels Cluster +

Si3Al Si3Ga+ Ge3Al+ Ge3Ga+

Total Energy (a.u.)

M-Si/M-Ge

Si-Si/GeGe

HOMO

−1108.630029 −2778.078467 −6461.971240 −8141.416002

2.31 2.41 2.46 2.45

2.31 2.26 2.37 2.37

−12.77 −12.81 −12.39 −12.43

Source: From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission. Note: The results presented here have been calculated at MP2/6–31G(d) level.

14

Aromaticity and Metal Clusters

Table 1.6 Vibrational Frequencies (cm−1) and IR Intensities of Neutral Tetramers Calculated Clusters Al2Si2

Sym. C2V

Vibrational Frequency (IR Intensities) 158.25 (0.14)

347.09 (0.12)

424.00 (0.15)

427.19 (0.92)

501.13 (0.77) Al2Si2

D2h

148.23 (0.14)

391.33 (0.01)

526.48 (4.74)

Ga2Si2

C2V

134.53 (0.07)

226.90 (0.05)

348.80 (0.20)

Ga2Si2

D2h

120.39 (0.04)

329.11 (0.03)

409.32 (1.94)

Al2Ge2

C2V

148.18 (0.05)

288.96 (0.09)

340.65 (0.12)

341.90 (0.13)

245.25 (0.02)

259.77 (0.05)

714.02 (0.29)

813.93 (0.39)

493.42 (0.39)

352.45 (0.23) Al2Ge2

D2h

134.60 (0.08)

409.50 (2.01)

Ga2Ge2

C2V

121.06 (0.03)

220.23 (0.04)

288.09 (0.14) Ga2Ge2

D2h

101.29 (0.02)

291.50 (0.68)

Be2N2

C2V

311.73 (1.12)

500.53 (1.80)

933.80 (1.94)

1418.60 (0.28)

Be2N2

D2h

1115.22 (4.24)

1136.47 (26.35)

Be2P2

C2V

135.01 (0.01)

159.88 (0.07)

1421.61 (1.22) 567.77 (3.06) 737.07 (10.73)

892.75 (0.06)

932.83 (7.24) Be2P2

D2h

689.49 (1.87)

714.40 (10.27)

Mg2N2

D2h

1734.83 (43.74)

825.23 (18.29)

Mg2P2

C2V

342.68 (0.79)

396.11 (4.15)

Mg2P2

D2h

426.24 (1.91)

429.70 (3.54)

Si3Be

188.62 (0.23)

271.15 (0.22)

437.58 (0.03)

501.87 (0.01)

Si3Mg

141.62 (0.26)

229.55 (0.14)

277.83 (0.18)

495.95 (0.03)

160.12 (0.06)

162.02 (0.15)

292.77 (0.01)

525.09 (0.01)

544.56 (0.52)

100.89 (0.18)

138.65 (0.08)

295.91 (0.03)

307.90 (0.01)

130.73 (0.07)

166.26 (0.55)

441.16 (0.08)

594.84 (0.04) 517.22 (0.19) Ge3Be Ge3Mg Al3P

224.40 (0.08)

289.91 (0.04)

285.24 (0.05)

418.77 (0.07)

459.09 (0.62) Al3As

133.31 (0.03)

165.35 (0.32)

358.64 (0.03)

367.00 (0.39)

Ga3P

106.39 (0.03)

117.38 (0.14)

234.04 (0.01)

335.32 (0.11)

Ga3As

99.62 (0.01) 266.57 (0.23)

124.60 (0.09)

234.36 (0.04)

257.98 (0.02)

365.13 (0.51)

C2v

D2h

C2v

D2h

C2v

D2h

C2v

D2h

Al2Si2

Al2Si2

Ga2Si2

Ga2Si2

Al2Ge2

Al2Ge2

Ga2Ge2

Ga2Ge2

−1061.8532136 (−1063.586042) −1061.8574372 (−1063.580272) −4420.7681024 (−4424.700562) −4420.7593000 (−4424.680626) −4630.7511097 (−4634.660153) −4630.7640621 (−4634.665430) −7989.6599894 (−7995.770804) −7989.6590886 (−7995.761284)

Total Energy (a.u.)

10.33

10.36

10.16

9.81

9.48

9.72

9.12

9.00

B.E. (eV)

—

2.36 (2.31)

—

2.26 (2.61)

—

2.36 (2.22)

—

2.26 (2.21)

Si-Si/ Ge-Ge

—

2.40 (2.46)

—

2.57 (2.61)

—

2.47 (2.45)

—

2.56 (2.61)

M-M

2.43 (2.43)

2.42 (2.42)

2.44 (2.42)

2.42 (2.43)

2.40 (2.37)

2.39 (2.41)

2.40 (2.40)

2.40 (2.44)

M-Si/ M-Ge

7.89 (7.62)

8.16 (7.53)

7.92 (7.52)

8.47 (7.42)

8.02 (7.75)

8.53 (7.63)

8.08 (7.61)

9.12 (7.49)

VIP (eV)

−0.57 (−1.28)

0.85 (−0.60)

−0.46 (−1.51)

0.99 (−0.70)

−0.57 (−1.48)

0.98 (−0.64)

−0.40 (−1.71)

1.13 (−0.74)

VEA (eV)

2.76

3.23

2.54

3.08

2.69

3.29

2.43

3.13

η (eV)

Source: From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission. Note: The results presented here have been calculated at MP2/6-31G(d) level and B3LYP/6-31G(d) level (presented within parentheses). For global hardness η only B3LYP results are listed. a Binding energy (A B ) = Total energy (A B ) − 2* {Total energy (A) + Total energy (B)}. 2 2 2 2

Sym.

Cluster

Table 1.7 Total Energies, Interatomic Separations (Å), Binding Energya (B.E., Only MP2 Results are Listed), Vertical Ionization Potentials (VIP), Vertical Electron Affinity (VEA), and the Energy Gap between the HOMO and LUMO Energy Levels, that is, Global Hardness η for Q2R2 Clusters

Aromaticity 15

16

Aromaticity and Metal Clusters

thus a stable molecule/cluster should have a higher ionization potential as well as a low electron affinity. Recently, Chattaraj et al. have used various descriptors based on chemical reactivity to gauge the aromaticity of Al 2− 4 , and suggested that for an aromatic system the global hardness should be greater than zero [72]. In the present case, higher ionization energy and the lower electron affinity of all Q2R2 clusters corroborate the stability of these clusters toward any addition reaction. In this context it needs to be mentioned that for Al2Si2 under B3LYP the cis geometry is slightly favored than trans whereas for Al2Ge2, the opposite is true. However, after the removal or attachments of one electron to the Al 2Si2 cluster, its lowest-­energy structure changed from cis to trans configuration. Under the MP2 level ­calculations, it has been found that even in the neutral form the Al2Si2 favors trans configuration, which suggests that the geometry of the Al 2Si2 cluster needs very small energy to change between cis and trans configuration. Interestingly, for both configurations the electronic wave functions were found to be delocalized over three occupied energy levels (HOMO, HOMO-1, and HOMO-2). A representative orbital distribution is shown in Figure 1.4 for five highest occupied MOs of these clusters.

A2G2 (A = Be, Mg; G = N, P)

The equilibrium geometries all these clusters have been obtained by considering two planar structures, namely cis and trans configurations. Three-dimensional tetrahedron and other geometries resulted in significantly higher energy as compared to the planar configurations. The difference in the stability between cis and trans ­isomers can be estimated from the corresponding total energies as listed in Table 1.8, which summarizes the geometry and energetics of these clusters. In all the cases, the trans isomer was found to be more stable than the cis isomer. In the case of Mg2N2 the cis isomer was found to be very much higher in energy and has not been included in Table 1.8 for comparison. In order to understand the nature of bonding, the spatial orientations of five occupied molecular orbitals (HOMO to HOMO-4) have been plotted for these cluster and representative results for Be2N2 is shown in Figure 1.4. It is clear from Figure 1.4 that Be2N2 has HOMO-2 (which is of π-type) is fully delocalized.

AB3 (A = Be, Mg; B = Si, Ge)

All these clusters are isoelectronic to that of Si 2+ 4 where one Si atom with doubly positively charge is replaced by an alkaline earth metal. For AB3 clusters, the lowestenergy isomer forms planar configuration instead of pyramidal structure. Table 1.9 summarizes the geometrical parameters and electronic properties such as total energies, binding energies (BEs), ionization potentials, electron affinity (VEA), and hardness. The last column in Table 1.9 shows the HOMO–LUMO gap, that is, hardness η from B3LYP calculations. It is clear from Table 9.1 that all these clusters have higher ionization potential with low electron affinity. In fact, under MP2 the vertical electron affinities show positive values indicating that these clusters are extremely stable and may not undergo

17

Aromaticity (a)

(b)

HOMO

HOMO-1

HOMO-2

HOMO-3

HOMO-4

Figure 1.4  Spatial orientation of top five occupied molecular orbitals for (a) Al2Si2 and (b) Be2N2 cluster.

any reactions easily. In order to understand the bonding in these clusters we have analyzed the shape of the occupied molecular orbitals as shown in Figure 1.5. Similar to the other tetramer clusters, the HOMO for AB3 type of clusters is found to be delocalized over all the constituent atoms.

X3Y (X = Al, Ga; Y = P, As)

These tetramer clusters are similar to that of Al 2− 4 , where one of the Al atoms has been replaced by a group V element, thus leading to the formation of a neutral analog of the Al 2− 4 cluster. The prime objective for the study of this series of clusters is to investigate how the molecular orbitals of the Al 2− 4 cluster and, in particular, how the

18

Aromaticity and Metal Clusters

Table 1.8 Total Energies, Interatomic Separations (Å), Binding Energy (B.E.), Vertical Ionization Potentials (VIP), Vertical Electron Affinity (VEA), and Global Hardness η for Tetramer Clusters Cluster

Sym.

Be2N2 Be2N2 Be2P2 Be2P2 Mg2N2 Mg2P2 Mg2P2

C2v D2h C2v D2h D2h C2v D2h

Total Energy (a.u.)

B.E. (eV)

N-N/ P-P

M-M

M-N/ M-P

VIP (eV)

−138.5560547 −138.5655279 −711.0013276 −711.0891579 −508.6571718 −1081.1955769 −1081.2679543

9.00 9.26 3.75 6.14 3.87 1.16 3.13

1.26 — 2.37 — — 2.08 —

1.96 — 2.72 — — 2.73 —

1.57,1.63 1.54 1.91 1.98 1.92 2.39 2.37

8.37 7.44 7.15 6.48 7.73 3.38 7.58

VEA (eV)

η (eV)

0.91

7.46 7.87 9.28 8.05 6.11 5.36 8.08

−0.43 −2.13 −1.57 1.62 −1.99 −0.50

Note: The results presented here have been calculated at MP2/6–311 + G(d) level. Diagonal bond lengths are not included.

Table 1.9 Total Energies, Interatomic Separationsa (Å), and Binding Energyb (B.E., only MP2 Results are Listed), Vertical Ionization Potentials (VIP), Vertical Electron Affinity (VEA) and Energy Gap between the HOMO and LUMO Energy Levels, that is, Global Hardness η for AB3 Clusters Cluster Si3Be Si3Mg Ge3Be Ge3Mg

Total Energy (a.u.)

B. E. (eV)

M-Si/ M-Ge

Si-Si/ Ge-Ge*

VIP (eV)

VEA (eV)

η (eV)

−881.5701208 (−882.958183) −1066.555140 (−1068.292252) −6234.898745 (−6239.555391) −6419.892659 (−6424.897661)

9.59

2.13 (2.13)

2.29 (2.23)

8.21 (8.04)

0.68 (−1.35)

3.19

8.52

2.60 (2.61)

2.26 (2.26)

8.10 (7.48)

0.60 (−1.71)

2.62

10.30

2.16 (2.12)

2.40 (2.39)

8.58 (7.96)

0.34 (−1.27)

3.47

9.47

2.59 (2.57)

2.38 (2.36)

8.13 (7.48)

0.61 (−1.25)

2.81

Source: From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission. Note: The results presented here have been calculated at MP2/6–31G(d) level and B3LYP/6–31G(d) level (­ presented within parentheses). For global hardness η, only B3LYP results are listed. a Diagonal bond length is not included. b Binding energy (AB ) = Total energy (AB ) − {Total energy (A) + 3*Total energy (B)}. 3 3

19

Aromaticity (a)

(b)

HOMO

HOMO-1

HOMO-2

HOMO-3

HOMO-4

Figure 1.5  Spatial orientation of top five occupied molecular orbitals for (a) Si3Be and (b) Al3P cluster.

delocalized π-orbitals affect the electronic properties of these clusters by incorporation of one heteroatom having large electronegativity. The geometries of all X3Y clusters have been optimized using a similar method as discussed above. Two isomeric structures, planar rhombus and Y atom capped X3 triangle (pyramidal configuration), have been considered to optimize all X3Y clusters. After the geometry optimization, it is found that the 3D pyramidal geometry of X3Y clusters relaxed to planar configuration with D3h symmetry (like BF3 molecule). Comparison of the total energies of these two isomers suggested that planar rhombus structure is more favorable than the other isomeric structure. This is in agreement with previously published reports [150–155]. Further, it may be worth mentioning that in an earlier work [93,94] it has been reported that for heteroatomic clusters with

20

Aromaticity and Metal Clusters

Table 1.10 Total Energies, Interatomic Separationsa (Å), Binding Energyb (B.E., only MP2 results are listed), Vertical Ionization Potentials (VIP), Vertical Electron Affinity (VEA), and the Energy Gap between the HOMO and LUMO Energy Levels, that is, Global Hardness η for X3Y Clusters Cluster Al3P Al3As Ga3P Ga3As

Total Energy (a.u.)

B. E. (eV)

M-Al/ M-Ga

Al-Al/ Ga-Ga

VIP (eV)

VEA (eV)

η (eV)

−1066.6824884 (−1068.439842) −2958.0103477 (−2960.829781) −6105.0472944 (−6110.099118) −7996.3732764 (−8002.491269)

7.53

2.28 (2.28)

2.54 (2.66)

8.15 (7.58)

− 0.36 (−1.39)

2.83

8.03

2.37 (2.35)

2.56 (2.65)

7.95 (7.47)

− 0.40 (−1.28)

2.85

8.39

2.31 (2.29)

2.47 (2.51)

8.20 (7.71)

− 0.51 (−1.10)

3.08

8.83

2.39 (2.36)

2.48 (2.52)

8.07 (7.60)

− 0.51 (−1.03)

3.05

Source: From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission. Note: The results presented here have been calculated at the MP2/6–31G(d) level and B3LYP/6–31G(d) level (presented within parentheses). For global hardness η only B3LYP results are listed. a Diagonal bond length is not included. b Binding energy (X Y) = Total energy (X Y) – {Total energy (Y) + 3*Total energy (X)}. 3 3

large electronegativity difference such as CAl3−, cyclic aromatic structure is not the minimum energy structure. Table 1.10 lists the bond lengths and electronic properties of the clusters of this series. From Table 1.10 it is clear that, like other tetramer clusters, these clusters also have high ionization potential and very low electron affinity, suggesting high stability of these clusters. However, it may be noted that under MP2, while AB3 clusters show a positive value of electron affinity, these clusters show a negative value of the VEA. The reason for this is the presence of group V elements, which have higher electronegativity than that of group II or group III elements. In order to underscore the bonding in this series of clusters, the spatial orientation of few occupied molecular orbitals (HOMO to HOMO-4) has been plotted in Figure 1.5. Unlike the previous four cases, it is found that the HOMO-2 state shows π delocalization of the electronic wave function over the molecule.

1.3.4  Hydrogenation It is known that the aromatic molecules possess cyclic and planar structures having high chemical and structural stability. From our previous calculations it is clear that all investigated tetramer clusters have very high IP and low EA, which can be regarded as an important parameter to understand their electronic stability. Other than this, the structural stability of these clusters can be verified from their interactions with any reactive species. Benzene, which is universally accepted as a leading

21

Aromaticity

example of aromatic molecule, retains its structure when it interacts with metal to form M(C6H6)2 type of molecule. In the present case, all constituent elements are either metallic (Al, Ga, Be, Mg) or semiconducting (Si, P, As) and they interact strongly with hydrogen. Therefore, it is of interest to study the interaction of hydro2− gen with these aromatic clusters. For this purpose we have chosen Si 2+ 4 , Al 4 , Al2Si2 clusters as the typical representative examples. Few initial configurations were optimized to find the preferred location of hydrogen atom. In general, it is found that the hydrogen atom after interaction prefers to connect with one of the constituent atom directly by a single bond. For Al2Si2, the cis conformation, which is a higher-energy isomer, becomes more stable than that of the trans isomer after interacting with four hydrogen atoms. The interaction energies of hydrogen molecule with these three clusters are listed below.

Si 24 + + 2H 2 → (Si 4 H 4 )2 + ; ∆H = − 1.86 eV.



Al2Si2 + 2H2 → (Al2Si4H4);  ΔH = −0.85 eV.



Al 24 − + 2H 2 → (Al 4 H 4 )2 − ; ∆H = − 0.61 eV.

Further, it has been noticed that the structural integrity and the planarity of these clusters retained even after interacting with hydrogen atoms. For comparison, the geometrical parameters of these aromatic clusters before and after hydrogenation are listed in Table 1.11. In addition to the structural integrity we have also checked the bonding aspects of these clusters after their interaction with hydrogen. Figure 1.6 shows the spatial orientation of the HOMO energy level. From this figure it is clear that the delocalized π-electron cloud of the HOMO energy levels remained undisturbed even after interacting with four hydrogen atoms. This has been attributed to the extra stability and struc2− tural integrity of these (Si 2+ 4 , Al 4 , Al2Si2) clusters, a signature of aromatic molecules. Further to corroborate this, we have plotted the eigenvalue spectrum of the hydrogenated clusters and compared them to that of free clusters as shown in Figure 1.7. It

Table 1.11 Comparison of Interatomic Separations (Å) after and before Interactions with Hydrogen Atoms Cluster

M-M (hydrogenated)

M-M (nonhydrogentaed)

Al 2− 4

2.54

Si2+ 4

2.28

2.32

Al2Si2 (C2v) Al2Si2 (D2h)

2.22, 2.40, 2.55 2.38

2.26, 2.40, 2.56 2.40

2.58

Source: From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission. Note: The calculations have been carried out at MP2/6–31G(d) level.

22 (a)

Aromaticity and Metal Clusters (b)

(c)

HOMO

HOMO-1

HOMO-2

HOMO-3

HOMO-4

Figure 1.6  Spatial orientation of top five occupied molecular orbitals for (a) Si 2+ 4 , (b) Al2Si2 (D2h), and (c) Al 2− 4 clusters after addition of four H atoms. (From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission.)

is clear from Figure 1.7 that the HOMO energy level, which forms by the delocalized π electron cloud and bears the signature of aromaticity, remained unperturbed even after reacting with four hydrogen atoms.

1.3.5 Nucleus-Independent Chemical Shift (NICS) Study of Cationic and Neutral Clusters The aromaticity of cationic and neutral cluster has been further studied using the NICS method. In this study, we have calculated the NICS (0.0), NICS (0.5), and NICS (1.0) by placing the ghost atom at the ring center of the molecular plane, 0.5 Å

23

Aromaticity –0.6

–0.2 –0.3

–0.8 –0.9

–0.5

–1.0

–0.6

–1.1

–0.7

–1.2 Si42+

(SiH)2+ 4

0.10 0.05 0.00 –0.05 –0.10 –0.15 –0.20 –0.25 –0.30

Energy (a.u.)

–0.4

Energy (a.u.)

Energy (a.u.)

–0.7

Al2Si2(D2h)

Al2Si2H4

Al42–

(AlH)42–

Figure 1.7  Comparison of molecular orbital energy levels for the free and hydrogenated tetramer aromatic clusters. Top seven occupied levels are considered for this purpose. For clear representation, we have chosen three typical examples one each from positive (Si 2+ 4 ), negative (Al 2− 4 ), and neutral (Al 2Si2) clusters. (From S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. With permission.)

and 1.0 Å above the plane, respectively. In particular, we have calculated the NICS + values of Si 2+ 4 , Si3Al , Al2Si2 , Be2N2 , Si3Be, Al3P clusters as typical representative of each series. The NICS (0.0) values calculated at the center of ring is influenced by σ-bonds, whereas the NICS (0.5), NICS (1.0) values calculated 0.5 Å and 1.0 Å out of the plane are more effected by the π-system. The results obtained for different clusters are given in Table 1.12. + These results indicate that all the clusters (Si 2+ 4 , Si3Al , Al 2Si2 , Be2N2, Si3Be, Al3P) studied in this work have highly negative NICS (0.0), NICS (0.5), NICS (1.0) values indicating the evidence of multifold aromaticity (π and σ aromaticity) in these clusters.

1.4 Summary In this article, we have discussed the evolution of aromaticity from benzene to atomic clusters. To start with, the aromatic behavior was limited to few cyclic benzanoid Table 1.12 + Nucleus-Independent Chemical Shift (NICS) Values of si2+ 4 , Si3Al , Al2Si2, Si3Be, Al3P Clusters as Typical Representative of Each Series Cluster Si

2+ 4 +

Si3Al Al2Si2 (D2h) Be2N2(D2h) Al3P Si3Be

NICS(0.0)

NICS(0.5)

NICS(1.0)

−138.26

−113.63

−69.33

−51.17 −47.71 −70.29 −35.10 −33.85

−45.08 −44.71 −30.73 −31.55 −32.79

−32.64 −35.32 −1.59 −23.08 −26.50

Note: The results presented here have been calculated using the gauge-including atomic orbitals (GIAO) method at the MP2/6–31G(d) (MP2/6–311 + G(d) in case of Be2N2) level.

24

Aromaticity and Metal Clusters

compounds, which was governed by the extra stabilization through π-electrons. Over the years, the concept of aromaticity has expanded into the charged systems, hetero systems, sandwich compounds, carbon-free species, three-dimensional systems, that is, boranes and fullerens, and so on. So far, in most of the literature, aromaticity has been described based on four criteria: (1) thermodynamic stability, (2) structural or geometrical, (3) magnetic, and (4) chemical reactivity. In a nutshell, aromaticity is all about delocalization of electrons in a cyclic loop. Recent finding of aromaticity in atomic cluster has given a new flavor of multifold aromaticity, that is, simultaneous presence of σ-, π-, and δ-aromaticity in a single system. In our study we have investigated a number of tetramer clusters with 14 valence electrons (8 s and 6 p) to demonstrate the existence of aromatic character in such metallic tetramer clusters. The results show that M2− 4 clusters favor square planar structure as a global minimum. In spite of their natural instability in terms of chemical reactivity, these dianionic clusters coordinate to sodium atom to form very stable dipyramidal structure M4Na2, without losing their planarity and aromaticity. After testing the aromaticity in anionic clusters, ab initio calculation on tetramer cationic and neutral cluster was carried out. This study revealed that all the tetramer clusters favor planar configuration over the three-dimensional structure irrespective of homoatomic or heteroatomic system. All these clusters having 6-p electrons in the outermost orbitals form at least one π-type molecular orbital where two electrons are delocalized over all four atoms and two σ-type delocalized orbital indicative of multifold aromaticity in these clusters. The multifold aromaticity (π and σ aromaticity) was further demonstrated by the highly negative NICS values. The extra stability of these clusters has been illustrated through their hardness, high IP, low EA, and their structural stability even after reacting with hydrogen atoms. Finally, we have pro2+ 2+ + posed new aromatic clusters like Si 2+ 4 , Ge 4 , Si 2 Ge 2 , Si3Al , which are electronically 2− more stable than that of recently discovered Al 4 . We have further extended the possibility of aromaticity in other neutral heteroatomic clusters like Al2Si2, Be2N2, Si3Be, Al3P, and their homolog, which are electronically stable species.

Acknowledgments We are thankful to the members of the Computer Division of Bhabha Atomic Research Centre (BARC), for their kind cooperation during this work. We gratefully acknowledge Dr. N. Patwari for helping us get the NICS values.

REFERENCES

1. P. V. R. Schleyer, Chem. Rev.: Special Issue on Aromaticity 101, 1115, 2001. 2. M. Faraday, Philos. Trans. Roy. London 115, 440, 1825. 3. E. Mitscherlich Annalenpharm. 9(1), 39, 1834. 4. A. Kekule, Bull. Soc. Chim. Fr. 3, 98, 1865; A. Kekule, Bull. Acad. R. Belg. 119, 551, 1865; A. Kekule, Ann. Chem. 137, 129, 1866; A. Kekule, Ann. Chem. 162, 77, 1872; A. Kekule, Ann. Chem. 162, 309, 1872. 5. J. Dewar, Proc. R. Soc. Edinburgh 1, 84, 1867; A. Ladenburg, Ber. 2, 140–142, 272–274, 1869; A. Ladenburg, Chem. Ber. 5, 322, 1872; A. Claus, Theoretische Betrachtungen und deren Anwendungen zur Systematik der organischen Chemie; Freiburg, Germany,

Aromaticity







25

207, 1867; A. Baeyer, Ber. 19, 1797, 1886; A. Baeyer, Ann. Chem. 245, 103, 1887; A. Baeyer, Ann. Chem. 251, 257, 1888; A. Baeyer, Ann. Chem. 256, 1, 1890; A. Baeyer, Ann. Chem. 258, 145, 1890; A. Baeyer, Ann. Chem 266, 169, 1892; A. Baeyer, Ann. Chem. 269, 145, 1892; A. Baeyer, Ann. Chem. 276, 255, 1893; H. E. Armstrong, J. Chem. Soc. 51, 258, 1887; H. E. Armstrong, Philos. Mag. 23, 73, 1887; I. L. Finar, Organic Chemistry, ELBS edition of sixth edition, 1985. 6. J. W. Armit, R. Robinson, J. Chem. Soc. 127, 1604, 1925. 7. W. Heitler; F. London, Z. Phys. 44, 455, 1927; L. Pauling; G. W. Wheland, J. Chem. Phys. 1, 280(1933); L. Pauling, J. Chem. Phys. 4, 673, 1936. 8. E. Huckel, Z. Phys. 70, 204(1931); E. Huckel, Z. Phys. 72, 310, 1931. 9. E. Huckel, Trans. Faraday Soc. 30, 40, 1934. 10. P. A. Plattner and A. S. Pfau, Helv. Chim. Acta 20, 224, 1937; A. G. Anderson and J. A. Nelson, Jr. J. Am. Chem. Soc. 72, 3824, 1950; J. Monteath, R. Obertson, H. M. M. Shearer, Nature 177, 885, 1956. 11. W. v. E. Doering, L. H. Knox, J. Am. Chem. Soc. 79, 352, 1957. 12. R. Breslow, J. Lockhart, A. Small, J. Am. Chem. Soc. 84, 2793, 1962. 13. E. Heilbronner, Tetrahedron Lett. 29, 1923, 1964. 14. R. Breslow, Chem. Eng. News 43, 90, 1965; R. Breslow, Angew. Chem. Int. Ed. 7, 565, 1968; R. Breslow, J. Brown, J. Gajewski, J. Am. Chem. Soc. 89, 4383, 1967; R. Breslow, Acc. Chem. Res. 6, 393, 1973. 15. C. A. Tsipis, Coord. Chem. Rev. 249(24), 2740, 2005. 16. Z. Chen, C. S. Wannere, C. Corminboeuf, R. Puchta, and Paul Von Ragué Schleyer Chem. Rev. 105, 3842, 2005; P. V. R. Schleyre, H. Jiao, Pure Appl. Chem. 68, 209, 1996. 17. A. I. Boldyrev, L. S. Wang, Chem. Rev. 105, 3716, 2005. 18. M. N. Glukhovtsev, P. V. R. Schleyer, Chem. Phys. Lett. 198, 547, 1992; C. H. Suresh, N. Koga, J. Org. Chem. 67 1965, 2002; P. George, M. Trachtman, C. W. Bock, A. M. Brett, J. Chem. Soc., Perkin Trans. 2, 1036, 1977. 19. F. De Proft, P. Geerlings, Chem. Rev. 101, 1451, 2001; J. Poater, X. Fradera, M. Duran, M. Sola, Chem. Eur. J. 9, 400, 2003; J. Poater, X. Fradera, M. Duran, M. Sola, Chem. Eur. J. 9, 1113, 2003; C. Lepetit, M. B. Nielsen, F. Diederich, R. Chauvin, Chem. Eur. J. 9, 5056, 2003. 20. T. M. Krygowski, M. K. Cyránski, Chem. Rev. 101, 1385, 2001. 21. T. M. Krygowski, J. Chem. Inf. Comp. Sci. 33, 70, 1993; J. Kruszewski, T. M. Krygowski, Tetrahedron Lett. 13, 3839, 1972. 22. D. Amic, N. Trinajsitc, Bull. Soc. Chim. Belg. 100, 527, 1991; R. Contreras, V. S. Safont, P. Per´ez, J. Andres, V. Moliner, O. Tapia, J. Mol. Struct. (THEOCHEM) 426, 277, 1998. 23. H. J. Dauben, J. D. Wilson, L. J. Laity, J. Am. Chem. Soc. 90, 118, 1968. 24. D. H. Hutter, W. H. Flygare, Top. Curr. Chem. 63, 89, 1976; W. H. Flygare, Chem. Rev. 74, 653, 1974; J. A. Pople, J. Chem. Phys. 24, 1111, 1956. 25. C. W. Wannere, P. V. R. Schleyer, Org. Lett. 5, 605, 2003; R. H. Mitchell, Chem. Rev. 101, 1301, 2001. 26. S. Coriani, P. Lazzeretti, M. Malagoli, R. Zanasi, Theor. Chim. Acta 89, 181, 1994; P. W. Fowler, E. Steiner, R. Zanasi, B. Cadioli, Mol. Phys. 96, 1099, 1999. 27. E. Steiner, P. W. Fowler, Chem. Commun. 21, 2220, 2001; E. Steiner, P. W. Fowler, J. Phys. Chem. A 105, 9553, 2001; V. Steiner, P. W. Fowler, Phys. Chem. Chem. Phys. 6, 261, 2004. 28. P. V. R. Schleyer, C. Maerker, A. Dransfeld, H. Jiao, N. J. R. Eikema Hommes, J. Am. Chem. Soc. 118, 6317, 1996; P. V. R. Schleyer, M. Manoharan, Z. Wang, X. B. Kiran, H. Jiao, R. Puchta, N. J. Eikema Hommes, Org. Lett. 3, 2465, 2001; P. v. R. Schleyer, H. Jiao, N. J. R. Eikema Hommes, V. G. Malkin, O. Malkina, J. Am. Chem. Soc. 119, 12669, 1997. 29. J. Juselius, D. Sundholm, Phys. Chem. Chem. Phys. 1, 3429, 1999.

26

Aromaticity and Metal Clusters

30. J. A. N. F. Gomes, R. B. Mallion, Chem. Rev. 101, 1349, 2001; P. Lazzeretti, in: J. W. Emsley, J. Feeney, L. H. Sutcliffe (Eds.), Progress in Nuclear Magnetic Resonance Spectroscopy, Elsevier, Amsterdam, pp. 1–88, 2000; U. Fleischer, W. Kutzelnigg, P. Lazzeretti, V. Muhlenkamp, J. Am. Chem. Soc. 116, 5298, 1994; P. Lazzeretti, Phys. Chem. Chem. Phys. 6, 217, 2004. 31. R. Herges, D. Geuenich, J. Phys. Chem. A 105, 3214, 2001. 32. J. C. Santos, W. Tiznado, Contreras, P. J. Fuentealba, J. Chem. Phys. 120, 1670, 2004. 33. J. C. Santos, J. Andres, A. Aizman; P. J. Fuentealba, Chem. Theory Comput. 1, 83, 2005. 34. E. Matito, M. Duran, M. Sola, J. Chem. Phys. 122, 014109, 2005. 35. C. G. Bollini, M. Giambiagi, M. S. De Giambiagi, A. P. De Figueiredo, J. Mater. Chem. 28, 71, 2000. 36. C. G. Bollini, M. Giambiagi, M. S. De Giambiagi, A. P. De Figueiredo, Struct. Chem. 12, 113, 2001. 37. M. Giambiagi, M. S. De Giambiagi, C. D. Dos Santos Silva, A. P. De Figueiredo, Phys. Chem. Chem. Phys. 2, 3381, 2000. 38. P. Bultinck, M. Rafat, R. Ponec, B. Van Gheluwe, R. Carbo-Dorca, P. L. A. Popelier, J. Phys. Chem. A 110, 7642, 2006. 39. P. Bultinck, M. Mandado, R. A. Mosquera, J. Math. Chem. Published online (DOI: 10.1007/s10910–006–9184–8). 40. P. Bultinck, R. Ponec, S. Van Damme, J. Phys. Org. Chem. 18, 706, 2005. 41. R. Ponec, P. Bultinck, A. Gallegos, J. Phys. Chem. A 109, 6606, 2005. 42. P. Bultinck, R. Ponec, R. Carbó-Dorca, J. Comput. Chem. 28, 152, 2007. 43. P. Bultinck, S. Fias, R. Ponec, Chem. Eur. J. 12, 8813, 2006. 44. G. F. Caramori, K. T. de Oliveira, S. E. Galembeck, P. Bultinck, M. G. Constantino, J. Org. Chem. 72, 76(2007). 45. M. Mandado, M. J. González-Moa, R. A. Mosquera, J. Comput. Chem. 28, 127, 2007. 46. M. Mandado, M. J. González-Moa, R. A. Mosquera, J. Comput. Chem. 28, 1625, 2007. 47. M. Mandado, N. Otero, R. A. Mosquera, Tetrahedron 62, 12204, 2006. 48. M. Mandado, M. J. González-Moa, R. A. Mosquera, Chem. Phys. Chem. 8, 696, 2007. 49. M. Mandado, P. Bultinck, M. J. González-Moa, R. A. Mosquera, Chem. Phys. Lett. 433, 5, 2006. 50. M. J. Cook, P. Linda, A. R. Katritzky, Adv. Heterocycl. Chem. 17, 255, 1974; A. R. Katritzky, M. Karelson, N. Malhotra, Heterocycles, 32, 127, 1991; B. Ya. Simkin, V. I. Minkin, M. N. Glukhovtsev, Adv. Heterocycl. Chem. 56, 303, 1993; G. Raabe, J. Michl, in: S. Patai, Z. Rappoport (Eds.), The Chemistry of Organic Silicon Compounds, Wiley, New York, p. 1102, 1989; A. T. Balaban, D. C. Oniciu, A. R. Katritzky, Chem. Rev. 104, 2777, 2004; A. R. Katritzky, K. Jug, D. C. Oniciu, Chem. Rev. 101, 1421, 2001. 51. F. G. Bordwell, G. E. Drucker, H. t. E. Fried, J. Org. Chem. 46, 632, 1981. 52. X. W. Li, W. T. Pennington, G. H. Robinson, J. Am. Chem. Soc. 117, 7578, 1995; X. W. Li, Y. Xie, P. R. Schreiner, K. D. Gripper, R. C. Crittendon, C. F. Campana, H. F. Schafer, G. H. Robinson, Organometalics 15, 3798, 1996. 53. M. Baudler, D. Duster, D. Z. Ouzounis, Anorg. Allg. Chem. 544, 87, 1987; E. J. Padma Malar, J. Org. Chem. 57, 3694, 1992; A. Dransfeld, L. Nyulaszi, P. v. R. Schleyer, Inorg. Chem. 37, 4413, 1998. 54. E. Urnezius, W. W. Brennessel, C. J. Cramer, J. E. Ellis, P. v. R. Schleyer, Science 295, 832, 2002. 55. J. Aihara, J. Am. Chem. Soc. 100, 3339, 1978. 56. J. Aihara, H. Hosoya, Bull. Chem. Soc. Jpn. 61, 2657, 1988; R. C. Haddon, L. E. Brus, K. Raghavachari, Chem. Phys. Lett. 125, 459, 1986; M. Buhl, A. Hirsch, Chem. Rev. 101, 1153, 2001; Z. Chen, R. B. King, Chem. Rev. 105, 3613, 2005; X. Lu, Z. Chen, Chem. Rev. 105, 3643, 2005.

Aromaticity

27

57. S. Winstein, J. Am. Chem. Soc. 81, 6524, 1959; S. Winstein, J. Sonnenberg; L. deVries, J. Am. Chem. Soc. 81, 6523, 1959; R. V. Williams, Chem. Rev. 101, 1185, 2001. 58. V. Gogonea, P. v. R. Schleyer, P. R. Schreiner, Angew. Chem., Int. Ed. 37, 1945, 1998. 59. H. Jiao, P. v. R. Schleyer, J. Phys. Org. Chem. 11, 655, 1998. 60. X. Li, A. K. Kuznetsov, H. Hai-Feng Zhang, A. I. Boldyrev, L. Wang, Science 291, 859, 2001. 61. P. W. Fowler, R. W. A. Havenith, E. Steiner, Chem. Phys. Lett. 342, 85, 2001. 62. P. W. Fowler, R. W. A. Havenith, E. Steiner, Chem. Phys. Lett. 359, 530, 2002. 63. R. W. A. Havenith, P. W. Folwer, E. Steiner, S. Shetty, D. G. Kanhere, S. Pal, Phys. Chem. Chem. Phys. 6, 285, 2004. 64. R. W. A. Havenith, P. W. Fowler, Phys. Chem. Chem. Phys. 8, 3383, 2006. 65. J. Juselius, M. Straka, D. Sundholm, J. Phys. Chem. A 105, 9939, 2001. 66. Y. C. Lin, J. Juselius, D. Sundholm, J. Gauss, J. Chem. Phys. 122, 214308, 2005. 67. Z. Chen, C. Corminboeuf, T. Heine, J. Bohmann, P. V. R. Schleyer, J. Am. Chem. Soc. 125, 13930, 2003. 68. R. Islas, T. Heine, G. Merino, J. Chem. Theory Comput. 3, 775, 2007. 69. R. W. A. Havenith, J. H. V. Lenthe, Chem. Phys. Lett., 385, 198, 2004. 70. C. G. Zhan, F. Zheng, D. A. Dixon, J. Am. Chem. Soc. 124, 14795, 2002. 71. A. E. Kuznetsov, A. I. Boldyrev, Inorg. Chem. 41, 532, 2002. 72. P. K. Chattaraj, D. R. Roy, M. Elango, V. Subramanian, J. Mol. Struct. (THEOCHEM) 759, 109, 2006. 73. P. P. Power, Multiple bonding between heavier group 13 elements. Struct. Bonding 103, 58, 2002. 74. H. Tanaka, S. Neukermans, E. Janssens, R. E Silverans, P. Lievens, J. Am. Chem. Soc. 125, 2862, 2003; A. N. Alexandrova, A. I. Boldyrev, H.-J. Zhai, L. S. Wang, J. Phys. Chem. A 109, 562, 2005; C. S. Wannere, C. Corminboeuf, Z.-X. Wang, M. D. Wodrich, R. B. King, P. v. R. Schleyer, J. Am. Chem. Soc. 127, 5701, 2005. 75. A. C. Tsipis, C. A. Tsipis, J. Am. Chem. Soc. 125, 1136, 2003. 76. Q. Kong, M. Chen, J. Dong, Z. Li, K. Fan, M. Zhou, J. Phys. Chem. A 106, 11709, 2002. 77. M. Lein, J. Frunzke, G. Frenking, Angew. Chem., Int. Ed. 42, 1303, 2003. 78. F. Kraus, T. Hanauer, N. Korber, Inorg. Chem. 45, 1117, 2006. 79. X. Hu, H. Li, C. Wang, J. Phys. Chem. B 110, 14046, 2006. 80. S. S. Mallajosyula, A. Datta, S. K. Pati, J. Phys. Chem. B 110, 20098, 2006; A. Datta, S. S. Mallajosyula, S. K. Pati, Acc. Chem. Res. 40(3), 213, 2007. 81. P. K. Chattaraj, D. R. Roy, M. Elango, V. Subramanian, J. Phys. Chem. A 109, 9590, 2005. 82. S. Shetty, D. G. Kanhere, S. Pal, J. Phys. Chem. A 108, 628, 2004. 83. A. E. Kuznetsov, A. I. Boldyrev, Chem. Phys. Lett. 388, 452, 2004. 84. J. Selius, M. Straka, D. Sundholm, J. Phys. Chem. A 105, 9939, 2001. 85. A. E. Kuznetsov, K. A. Birch, A. I. Boldyrev, X. Li, H. J. Zhai, L. S. Wang, Science 300, 623, 2003. 86. J. M. Mecero, J. M. Ugalde, J. Am. Chem. Soc. 126, 3380, 2004. 87. P. W. Fowler, A. Rogowska, A. Soncini, M. Lillington, L. P. Olson, J. Org. Chem. 71, 6459, 2006. 88. A. E. Kuznetsov, A. I. Boldyrev, X. Li, L. S. Wang, J. Am. Chem. Soc. 123, 8825, 2001. 89. A. E. Kunznetsov, A. I. Boldyrev, Struct. Chem. 13, 141, 2002. 90. H. Zhai, L. Wang, A. E. Kunznetsov, A. I. Boldyrev, J. Phys. Chem. A 106, 5600, 2002. 91. A. E. Kunznetsov, A. I. Boldyrev, H. Zhai, X. Li, L. Wang, J. Am. Chem. Soc. 124, 11791, 2002. 92. A. E. Kuznetsov, J. D. Corbett, L. S. Wang, A. I. Boldyrev, Angew. Chem. Int. Ed. 40, 3369, 2001.

28

Aromaticity and Metal Clusters

93. X. Li, H. F. Zhang, L. S. Wang, A. E. Kuznetsov, N. A. Cannon, A. I. Boldyrev, Angew. Chem. Int. Ed. 40, 1867, 2001. 94. A. I. Boldyrev, J. Simons, X. Li, W. Chen, L. S. Wang, J. Chem. Phys. 110, 8980, 1999. 95. X. X. Chi, X. H. Li, X. J. Chen, Z. S. Yuang, J. Mol. Struct. (Theochem) 677, 21, 2004. 96. Q. Li, L. Gong, Z. M. Gao, Chem. Phys. Lett. 394, 220, 2004. 97. C. Zhao, K. Balasubramaniam, J. Chem. Phys. 120(22), 10501, 2004. 98. Y. Jung, T. Henine, P. V. R. Schleyer, M. Head-Gordon, J. Am. Chem. Soc. 126, 3132, 2004. 99. H. M. Tuononen, R. Suontamo, J. Valkonen, R. S. Laitinen, J. Phys. Chem. A 108, 5670, 2004. 100. C. A. Tsipis, E. E. Karagiannis, P. F. Kladou, A. C. Tsipis, J. Am. Chem. Soc., 126, 12916, 2004. 101. A. E. Kuznetsov, H. J. Zhai, L. S. Wang, A. I. Boldyrev, Inorg. Chem. 41, 6062, 2002. 102. A. N. Alexandrova, A. I. Boldyrev, H. J. Zhai, L. S. Wang, E. Steiner, P. W. Flower, J. Phys. Chem. A 107, 1359, 2003. 103. X. Hu, H. Li, W. Liang, S. Han, Chem. Phys. Lett. 397, 180, 2004. 104. Q. Jin, B. Jin, W. G. Xu, Chem. Phys. Lett. 396, 398, 2004. 105. A. I. Boldyrev, L. S. Wang, J. Phys. Chem. A 105, 10775, 2001. 106. A. N. Alexandrova, A. I. Boldyrev, J. Phys. Chem. A 107, 554, 2003. 107. A. E. Kuznetsov, A. I. Boldyrev, Inorg. Chem. 41, 3596, 2002. 108. H. J. Zhai, L. S. Wang, A. E. Kuznetsov, A. I. Boldyrev, J. Phys. Chem. A 106, 5600, 2002. 109. J. M. Matxain, J. M. Ugalde M. D. Towler, R. J. Needs, J. Phys. Chem. A 107, 10004, 2003. 110. Q. S. Li, L. P. Cheng, J. Phys. Chem. A 107, 2882, 2003. 111. A. I. Boldyrev, L. S. Wang, J. Phys. Chem. A 109(1), 236, 2005. 112. H. J. Zhai, A. E. Kuznetsov, A. I. Boldyrev, L. S. Wang, Chem. Phys. Chem. 5, 1885, 2004. 113. L. Todorov, S. C. Sevov, Inorg. Chem. 43, 6490, 2004. 114. W. G. Xu, Y. C. Zhang, Q. Xi, J. Mol. Struct. (TheoChem) 909(1–3), 129, 2009. 115. W. G. Xu, Y. C. Zhang, S. X. Lu, R. C. Zhang, J. Mol. Struct. (TheoChem) 900, 44, 2009. 116. T. Höltzl, N. Veldeman, T. Veszprémi, P. Lievens, M. T. Nguyen, Chem. Phys. Lett. 469, 304, 2009. 117. W. G. Xu, R. C. Zhang, S. X. Lu, Y. C. Zhang, J. Mol. Struct. (TheoChem) 894, 1, 2009. 118. W. G. Xu, R. C. Zhang, S. X. Lu, Y. C. Zhang, J. Mol. Struct. (TheoChem) 864, 1, 2008. 119. Z. Li, C. Zhao, L. Chen, J. Mol. Struct. (TheoChem) 807, 17, 2007. 120. B. M. Elliott, A. I. Boldyrev, J. Phys. Chem. A 109, 236, 2005. 121. L. A. Soto, R. R. Tagle, R. A. Pérez, J. Phys. Chem. A 113, 671, 2009. 122. Y. C. Lin, D. Sundholm, J. Juselius, L. Cui, Xi Li, H. J. Zhai, L. S. Wang, J. Phys. Chem. A 110, 4244, 2006. 123. M. A. Sobhy, J. U. Reveles, U. Gupta, S. N. Khanna, A. W. Castleman, Jr. J. Chem. Phys. 130, 054304, 2009. 124. L. Lin, T. Hçltzl, P. Gruene, P. Claes, G. Meijer, R. Fielicke, P. Lievens, M. T. Nguyen, Chem. Phys. Chem., 9, 2471, 2008. 125. P. K. Chattaraj, S. Giri, J. Mol. Struct. (TheoChem) 865, 53, 2008. 126. P. K. Chattaraj, D. R. Roy, S. Duley, Chem. Phys. Lett. 460, 382, 2008. 127. S. Khatua, D. R. Roy, P. Bultinck, M. Bhattacharjee, P. K. Chattaraj, Phys. Chem. Chem. Phys. 10, 2461, 2008.

Aromaticity

29

128. T. Hçltz, E. Janssens, N. Veldeman, T. Veszprmi, P. Lievens, M. T. Nguyen, Chem. Phys. Chem. 9, 833, 2008. 129. H. Duan, Z. Gong, J. Cheng, W. Zhu, K. Chen, H. Jiang J. Phys. Chem. A 110, 12236, 2006. 130. H. Yu, J. Mol. Struct. (TheoChem) 906, 25, 2009. 131. G. H. Zhang, Y. F. Zhao, J. I. Wu, P. v. R. Schleyer, Inorg. Chem., 48, 6773, 2009. 132. S. Chaitanya, W. C. Corminboeuf, Z. X. Wang, M. D. Wodrich, R. B. King, P. v. R. Schleyer, J. Am. Chem. Soc. 127, 5701, 2005. 133. C. Corminboeuf, S. Chaitanya, D. Wannere, D. R. Roy, R. A. King, P. v. R. Schleyer, Inorg. Chem. 45, 214, 2006. 134. B. B. Averkiev, A. I. Boldyrev, J. Phys. Chem. A 111, 12864, 2007. 135. H. J. Zhai, B. B. Averkiev, D. Yu. Zubarev, L. S. Wang, A. I. Boldyrev, Angew. Chem. Int. Ed. 46, 4277, 2007. 136. S. Nigam, C. Majumder, S. K. Kulshreshtha, J. Mol. Struct. (TheoChem) 755, 187, 2005. 137. S. Nigam, C. Majumder, S. K. Kulshreshtha, Comput. Lett. 1(4), 240, 2005. 138. S. Nigam, C. Majumder, S. K. Kulshreshtha J. Chem. Sci. 118, 575, 2006. 139. A. R. Katritzky, K. Jug, D. C. Oniciu, Chem. Rev. 101, 1421, 2001. 140. P. Lazzeretti, Phys. Chem. Chem. Phys. 6, 217, 2004. 141. M. Nendel, K. N. Houk, L. M. Tolbert, E. Vogel, H. Jiao, P. v. R. Schleyer, Angew. Chem., Int. Ed. Engl. 36, 748, 1997; R. A. King, T. D. Crawford, J. F. Stanton, H. F. Schaefer III, J. Am. Chem. Soc. 121, 10788, 1999. 142. A. D. Becke, Phys. Rev. A 38, 3098, 1988. 143. C. Lee, W. Yang, R. G. Parr, Phys. Rev. B 37, 785, 1988. 144. M. Haser, J. AlmLof, G. E. Scuseria, Chem. Phys. Lett. 181, 497, 1991; C. Moller, M.  S.  Plesset, Phys. Rev. 46, 618, 1934; W. J. Hehre, L. Radom, V. P. R. Schleyer, J. A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1985. 145. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, et al., J. Comput. Chem. 14, 1347, 1993. 146. P. Flükiger, H. P. Lüthi, S. Portmann, J. Weber, MOLEKEL 4.0, Swiss Center for Scientific Computing, Manno, Switzerland, 2000. 147. M.J. Frisch et al., Gaussian-98, Gaussian Inc, Pittsburgh, PA, 1998. 148. P. K. Chattaraj, S. Giri, J. Phys. Chem. A 111, 11116, 2007. 149. R. G. Parr, L. v. Szentpaly, S. Liu. J. Am. Chem. Soc. 121, 1922, 1999. 150. P. Y. Feng, K. Balasubramanian, Chem. Phys. Lett. 288, 1, 1998. 151. P. Y. Feng, K. Balasubramanian, Chem. Phys. Lett. 301, 458, 1999. 152. K. Balasubramanian, X. Zhu, J. Chem. Phys. 115, 8858, 2002. 153. E. F. Archibong, A. St-Amant, S. K. Goh, D. C. Marynick, Chem. Phys. Lett. 361, 411, 2002. 154. E. F. Archibong, A. St-Amant, S. K. Goh, D. C. Marynick, J. Phys. Chem. A 106, 5932, 2002. 155. E. F. Archibong, A. St-Amant, J. Phys. Chem. A 106, 7390, 2002.

the Measures of 2 On Aromaticity G. Narahari Sastry, P. Venuvanalingam, and P. Kolandaivel Contents 2.1 Introduction..................................................................................................... 31 2.2 On the Nature of π Electrons of Benzene........................................................ 33 2.3 Measures of Aromaticity................................................................................. 35 2.3.1 Structural............................................................................................. 35 2.3.2 Energetic.............................................................................................. 37 2.3.3 Magnetic.............................................................................................. 41 2.3.4 Reactivity............................................................................................. 42 2.3.5 Other Measures.................................................................................... 43 2.3.5.1 Atoms in Molecules Theory................................................. 43 2.3.5.2 Generalized Polansky Index.................................................44 2.3.5.3 Polarizability......................................................................... 45 2.4 Bottlenecks in Quantifying Aromaticity.........................................................46 2.5 Summary......................................................................................................... 47 Acknowledgments..................................................................................................... 48 References................................................................................................................. 48

2.1 ​Introduction Aromaticity has been deeply rooted in chemical literature, even before the structural and bonding principles have been clearly established [1–6]. It has been one of the most ubiquitous concepts and its origin dates back to the isolation of benzene by Faraday in 1825 illuminating the gas from whale oil [1]. Historically, the molecule benzene is closely associated with aromaticity. The first known use of the word “aromatic” as a chemical term is by Hofmann in 1855 [2]. However, even before that in 1833, Mitscherlich distilled benzene from benzoic acid and lime [3]. The cyclohexatriene structure for benzene was first proposed by Kekulé in 1865 [4] while the cyclic nature of benzene was finally confirmed by the crystallographer Kathleen Lonsdale [5]. The term aromaticity owes its name to the pleasant aroma that some members of this class have and later on it was denoting exceptional stability that this family of compounds exhibit. An explanation for the exceptional stability of benzene is attributed to Sir Robert Robinson, who coined the term aromatic sextet 31

32

Aromaticity and Metal Clusters

as a group of six electrons that resists disruption, in 1925. Historically, despite the central position occupied [7], aromaticity has been hotly debated over the years in the way it is being used in chemical literature. Aromatic compounds undergo electrophilic aromatic substitution and nucleophilic aromatic substitution reactions, but not electrophilic addition reactions. Initially, aromaticity was restricted to all carbon [8−10], coplanar delocalized π systems with alternating single and double bonds having 4n + 2 π electrons and the contributing atoms are covalently bound and arranged in one or more rings. Later, the scope was extended to nonbenzenoid compounds [11−13], cations and anions, hetero-organic molecules [14−19], and carbon-free main group compounds, inorganic [20−23] and organometallic compounds, metal clusters [24−26], Möbius strips [27−30], transition states, and excited states. More recently, three-dimensional aromaticity involving σ-electrons and multidimensional aromaticity involving d-orbitals have been reported. The scope is further expanding. This has led to the development of several related concepts such as antiaromaticity, homoaromaticity [31], Möbius aromaticity, pseudo aromaticity [32], metal aromaticity, and so on. Despite wide usage of the term, ­frequent criticisms have erupted over the past 150 years, and some of them even advocating the abolishment of the term in chemical literature. Interestingly, the studies directed to probe controversies and confusion regarding its definition and applicability led to the extension of this concept to a completely new set of molecules. Thus, the concept is expanding and getting manifested in different forms after each of the controversy often leading to wider usage and covering new areas of molecules [7]. The quantum mechanical basis was initiated with the pioneering work of Hückel [33], which employs the separation of the bonding electrons into σ and π electrons. Aromaticity has been one of the most ubiquitous concepts in chemical literature. Although loosely defined and appears to have no unambiguous way of quantifying it, the concept has been rapidly applied to various areas. The aromatic systems are very often highly symmetric and, in its turn, high symmetry can be a hint that the system is aromatic. High symmetry usually means presence of a multifold rotation axis in case of planar systems or polyhedral cage-like structure of Td, Oh, and Ih symmetry in case of 3D geometries (spherical aromaticity). On the contrary, antiaromatic ­systems lower symmetry (a kind of Jahn–Teller distortion) to reorder their electronic energy levels and decrease destabilization due to antiaromaticity. By that time they become nonaromatic. Cyclobutadiene is a well-known example for the distortive nature of antiaromatic molecules, as its equilibrium geometry is rectangular. Assignment of aromaticity on the basis of analysis of chemical bonding establishes connection between structure of a certain system and its chemical behavior, but this procedure does not rely on any observable features. Coexistence of various approaches to the definition of aromaticity (on the basis of electronic structure, resonance energy, magnetic properties, and chemical behavior) makes it possible to a better understanding of the concept. We make an attempt to provide a glimpse of several measures of aromaticity that are being used in the chapter. First, we touch upon a historical account which discusses the nature of π-electrons in benzene. This is followed by a brief description of the methods that are in usage to estimate aromaticity based on structural, energetic, magnetic, reactivity, and some other properties of the system. Finally, we tried to

On the Measures of Aromaticity

33

illustrate the multidimensional character of the aromaticity, and bottlenecks in the way of obtaining global aromaticity indices. A large number of aromaticity indices are interesting and important in their own right and the popularity and expansion of the aromaticity concept to virtually the whole of chemistry is described at the end. Without any doubt, benzene remains the cornerstone for the concept of aromaticity and so is the π-electron delocalization. In the following section let us probe the nature of π- as well as σ-electrons in benzene.

2.2 ​On the Nature of π Electrons of Benzene Benzene, is the most closely associated molecule with aromaticity and is the first choice as the example for aromatic molecules. The physico-chemical evidences such as bond length equivalization, planar structure, higher than expected stability, magnetic susceptibility exaltation, magnetic anisotropy, ring current, and preference for substitution over addition in benzene are the manifestations of it and observed in several related molecules. Benzene has more than 200 possible isomers and many of them are synthetically feasible [34]. Interestingly, the relative stability ordering of the isomers of heterobenzenes even when one or two of the CH groups are replaced by isoelectronic species are different compared to the parent compound [15]. However, the systematic computational studies unambiguously reveal that on the C6H6 potential energy surface, benzene is the global minimum. The important structural criterion of bond length equivalization is observed in benzene is due to the σ-framework which adequately compensates the distortive nature of the π-electrons. No doubt, π-electrons impart high stability in benzene, but they nevertheless lead to localized double bonds. The famous Hückel [33,35−36] and Breslow [37−39] rules provide a solid platform for electron counting rules of 4n + 2 and 4n and they are the powerful paradigms in chemistry and have become the basis for conventional perspective that benzene, which is a seminal example of aromaticity, leads to a fully delocalized structure. In contrast, cyclobutadiene and other related antiaromatic molecules prefer a distorted structure with localized π-bonds. These ideas lead to a popular belief that π-electrons are responsible for the high delocalization in aromatic systems and localization in antiaromatic systems [40]. However, even half century ago there existed theoretical and spectroscopic studies which revealed that π-electrons had a distortive nature, which was never brought to the mainstream organic chemistry. The pioneering work of Shaik and coworkers [41−43] clearly showed that π-electrons in benzene are distortive and have established (Figures 2.1 and 2.2) that there exist a fine balance between the distortive π-component and symmetrizing σ-component in all cyclic and acyclic-conjugated systems. In that competition, when σ-component dominates symmetric structures results, however when π-component dominates it prefers a localized structure. Both in distorted D3h arrangement and symmetric D6h geometries the π-electrons experience substantial resonance stabilization [44]. As the distortive tendency of π-electrons is difficult to evaluate experimentally, the experimental community at large is wondering how this phenomenon can be established. Haas and Zilberg’s [45] analysis of the excited state vibrational spectra of β-methyl styrene, which displayed a significant exaltation selectively for the bond alternation mode has

34

Aromaticity and Metal Clusters (a)

ψE

(b)

VG QMRE

ψE VG

ψG

QMRE ψG

Figure 2.1  ​Valence bond curve-crossing diagram, for the π-component, interconversion of two Kekulé structural forms along the bond alternation coordinate (b2u mode). VG and QMRE correspond to vertical gap and quantum mechanical resonance energy, respectively. The two putative possibilities represent that in one case (a) the vertical gap is high and therefore a localized D3h structure is preferred; in the other case (b) a small vertical gap results in bringing the stability to a fully delocalized D6h symmetric structure.

(a)

(b) Eπ

E

Eσ

Eπ

E

Eσ

Eσ+X

Eσ+X D3h

D6h

D3h

D2h

D4h

D2h

Figure 2.2  ​The σ and π components of the energy corresponding to (a) the aromatic benzene molecules and (b) the antiaromatic cyclobutadiene molecule. In both cases σ prefers a symmetric and delocalized structure and in contrast π prefers a distortive and localized structure.

35

On the Measures of Aromaticity

nicely correlated with Shaik’s curve crossing model. Similar spectroscopic observations that were made almost half a century ago essentially gave the indications on the p-distortivity [46]. The explanation of selective exaltation of b2υ mode in the excited state of benzene [47] with Kekulé crossing model provided the much needed experimental link to computational model. Interestingly, there was an experimental report on substituted benzene which adopts a localized structure, apparently owing to π-distortivity, upon excitation resulted in a completely delocalized structure. Thus, the distortive nature of the π-electrons in the ground-state geometries of ­a romatic systems has been unambiguously established.

2.3 ​Measures of Aromaticity Aromaticity manifests in several ways namely through electronic structure and geometry, energetics, stability and reactivity, and magnetic properties. Many of these things are strongly interrelated. This has been realized in many systems and based on this several measures of aromaticity have been defined. For example, the circulating π electrons in an aromatic molecule produce ring currents that oppose the applied magnetic field in NMR. The NMR signals of protons in the plane of an aromatic ring are shifted substantially further down-field than those on nonaromatic sp2 carbons. This is an important way of detecting aromaticity. One of the objectives of this chapter is to provide a description of various quantitative measures to estimate aromaticity. There are various measures which were developed to gauge the aromaticity in molecules, which may be defined according to the structural, energetic, reactivity, and magnetic properties. Besides, there are other techniques which were proposed as measures of aromaticity. It is possible, in certain cases, some measures to clearly categorize a measure into one belonging to either structure, energetic, magnetic, or reactivity, while in some other cases such categorization is not straightforward.

2.3.1 Structural A good number of papers have appeared in the literature to deal with the structural aspects in gauging aromaticity [48−56]. The aromatic molecules have bond length in between double and single bonds. Bond length equivalization in benzene has been one of the striking aspects associated with the concept. Various aromaticity indices have been defined based on either bond lengths or bond orders and therefore they have structural basis for aromaticity. In 1967, July and Francois [49] formulated the first aromaticity index AJ as, AJ = 1 −

225 n

∑

2

Rr   1 − R  ,  

(2.1)

where n is the number of peripheral bonds, Rr is the individual bond lengths and R is their mean value. The constant “225” results from the normalization conditions to obtain a zero value for the Kekulé’ structure of benzene and 1 for any system with

36

Aromaticity and Metal Clusters

all bonds of equal length. However this index can be applied to the species with carbons and cannot be applied to the species with heteroatoms. Bird [46] then modified AJ index replacing bond orders for bond lengths (Equation 2.2).  v  I = 100  1 − , vk  



(2.2)

where V = (100 / N ) (∑( N − N )2 )/n and N = a / R 2 − b, R is the bond length, N is the individual bond order, N is the mean bond order, and n is the number of bonds. Another aromaticity index is known as the bond alternation coefficient [52] (BAC) (Equation 2.3). It is based on the sum of squared differences of the sequential bond lengths. BAC =



∑ (R n

n

− Rn + 1 )2 .

(2.3)



This index solves the problem for monocyclic system but becomes complex for polycyclic systems (as it involves the sequential bond lengths). Also it cannot be applied for systems containing heteroatoms and, to some extent, for systems in which the localization of double bonds is not alternating. The HOMA index [53] (Equation 2.4) takes into account the optimal bond length rather than mean bond length. HOMA = 1 −



α n

∑ (R

opt

− Ri )2 ,

(2.4)



where n is the number of bonds, α is an empirical constant and Ri is the individual bond length. HOMA is equal to zero for the hypothetical Kekulé structures of the aromatic systems and HOMA is equal to one for the systems with bonds equal to optimal value, Ropt. The quantity Ropt is defined as a length of the CC bond for which the energy ­(estimated by use of the harmonic potential) of the compression to the length of a double bond and expansion to the length of a single bond in 1,3-butadiene is minimal. The same procedure can be applied to bonds containing heteroatoms, choosing the model single and double bonds as a reference for the above-presented procedure. From the HOMA model, two terms EN and GEO, as shown in Equation 2.5, which give explanation for the contribution to decrease in aromaticity, can be obtained. EN describes the contribution due to the bond elongation and GEO describes the contribution due to the bond length alternation. α  − Ri )2 = 1 − α( Ropt − Rav )2 + n  = 1 − EN − GEO,

HOMA = 1 −

α n

∑ (R

opt

∑ (R

av

 − Ri )2  

(2.5)

37

On the Measures of Aromaticity

where Rav is an averaged bond length, Rav = (1 / n) ∑in = 1 Ri and α(Ropt − Rav)2 and α/n∑(Rav − Ri)2 represent EN and GEO terms, respectively. If Rav is shorter than Ropt, EN must be taken with a negative sign, since the shorter the bond, the greater its energy. Hence, the formula for HOMA can be written as

HOMA = 1 − EN − GEO

(2.6)



EN = f . α(Ropt − Rav)2,

(2.7)



f = 1:Rav > Ropt,

(2.8)



f = −1:Rav  0.01  aromatic



TRE (PE) (−0.01 −0.01)  nonaromatic



TRE (PE) µρµν ρµν ,



(2.23)

45

On the Measures of Aromaticity

where NL is the number of atoms involved in the ring. 2.3.5.3 ​Polarizability Polarizability [100] is proposed as an alternative to magnetic criteria for aromaticity. Electric dipole polarizability is a measure of the linear response of the electron ­density in the presence of an infinitesimal electric field, F, and it represents a secondorder variation in energy  ∂2 E  α a ,b = −   ∂Fa ∂Fb 



a, b = x, y, z.

(2.24)

Polarizability α is calculated, according to the following equation: α =



1 (α + α yy + α zz ). 3 xx

(2.25)

2.3.5.3.1  Anisotropy of the Induced Current Density (ACID) This is another effective approach to estimate the magnetic aspect aromaticity. The current density indicates ring currents in aromatic systems and is more closer to the general definition of delocalization. The anisotropy of the current (induced) density (ACID) is analogous to the anisotropy of the magnetic susceptibility [101]. The (1) induced current density J is given by



T : Tvµ =

  (at B = 0) J (1) = TB.

∂J v ∂Bµ



(2.26)

T is a tensor of second rank, and the anisotropy ΔT(1) is defined as the standard deviation of the eigenvalues ei. 2

∆T (1) =



∑ (e

i

− e )2

i

(2.27)

with



e=

1 n

∑e

i

and

i

∑e

i

= tr T .

i

(2.28)

We obtain 2



T (1) = tr(T 2 ) −

1 tr (T )2 . n

(2.29)

46

Aromaticity and Metal Clusters

Any real matrix T can be decomposed into a symmetric and antisymmetric part T = TS + TA. The anisotropy of T therefore can be written as 2

2

+

∆T(1) = ∆TS(1) + tr TA(1) TA(1) .



(2.30)

If the contribution of the antisymmetric part is neglected, and then an analytical solution for the symmetric contribution is obtained. 2

∆TS(1) =

1 (t − t yy )2 + (t yy − t zz )2 + (t zz − t xx )2   3  xx 1 + (t xy + t yx )2 + (t xz − t zx )2 + (t yz − t zy )2  . 2

(2.31)

2.4 ​Bottlenecks in Quantifying Aromaticity The concepts such as hybridization, bonding, valency, hypervalency, electronegati­ vity, resonance, and so on along with aromaticity have played a profound role in shaping our understanding of chemistry. Many of these concepts have helped to organize the existing experimental data and also lead to several new experiments. The concepts when introduced are of qualitative nature, and these paradigms are based on the experimental synthesis or physico-chemical property measurements and are employed for routine explanations and predictions. Thus, aromaticity is a multifaceted concept and no single property could be used as a direct measure. Employing a combination of various measures, which may be in concordance or in discordance appear to be the only practical option. Thus, absolute values for aromaticity or global aromaticity indices may continue to remain elusive. However, exploring each dimension of this multidimensional phenomenon is interesting in its own right. Because, various criteria for aromaticity can greatly help in patterning different classes of molecules and therefore will remain a powerful concept in chemistry. The usage of the term “aromaticity” in the title and abstracts of the papers continues to rise every year and this term appears to be one of the most popular keywords in chemical literature—and appears to remain so for several years to come. Thus, although poorly understood, the concept of aromaticity is so deeply grounded in the chemical literature and it is almost impossible to think of chemistry without aromaticity! Aromaticity is a concept which has been closely entangled with chemical epistemology, and it is currently transcending barriers within different classes of molecules to cover nearly all chemistry. In addition to the conventional aromatic hydrocarbons, the concept has been very well established in the areas of pyramidal hydrocarbons, carboranes, boran hydrides, fullerenes, ferrocene and other sandwich molecules, metallic clusters, transition structures, and so on. According to Heilbronner, “the amount of confusion caused by the term aromaticity in the student’s mind is not compensated by gains in the understanding of the chemistry and physics of the molecules so classified [102].” The new class of aromat-

On the Measures of Aromaticity

47

ics such as, homo-, pseudo-, Möbius-, hetero-, metallo-, three-dimensional-, multi­ dimensional-aromaticity concepts facilitates our knowledge or leads to further ­confusion. There are several references [103−107], which typically discusses the ­controversial topic of “what is aromaticity?” With a heat of formation of +22 kcal/mol, benzene is unstable with respect to its constituent atoms in their elemental form. Although the term aromaticity came into existence due to the high carbon content or a pleasant aroma associated with the molecules (benzene does not possess odor which can be particularly called pleasant), the lack of verifiable measure has always invited criticism. For a historical account on the criticism one can go through the historical reviews that appeared in pure and applied chemistry by several influential people in early 1980s. However, each time the term is subjected to a debate or discussion usage has been exponentially expanding and often expanding to territories, which are far from the conventional ones. Thus, it has become almost unthinkable to avoid the term aromaticity in chemical literature although the criteria that are used to quantify aromaticity need not result in converging answers. The analysis of thermodynamic and kinetic aspects of the “extra” stability for the molecule in question, may be contrasting with respect to the question of aromaticity. Thus they may yield different quantitative values, based on the reference system chosen. The structural criterion for aromaticity is often based on the assumption that p-electron tends to delocalize, which appear to be far from true. The reactivity criterion appears to be more qualitative and is not usually associated with the intrinsic property of the molecule. Thus, similar to the energetic criterion, the reactivity ­criterion also needs a reference system for quantification. Magnetic criterion has an advantage that we can estimate the paratropic and diatropic ring currents to experimental values derived from NMR. The methods such as NICS and ACID can estimate the nature of the ring currents, when properly used [83]. However, many known nonaromatic molecules display substantial diamagnetic ring currents, precluding it as a sole criterion for aromaticity. Clar’s theory is one of the important theories applied to polycyclic ­a romatic hydrocarbons, which is quite helpful in deciphering the relative aromaticity of various rings [108]. Thus, it enables to probe the local nature of aromaticity. Considering the limitations and purview of the current chapter we have given only a selected criterion and do not claim to give all the available measures of aromaticity.

2.5 Summary The term “aromaticity” was introduced 150 years ago to indicate pleasant aroma in compounds like benzene and later on it was associated with extra stability of certain organic compounds. Since then the scope of the term is ever expanding and various types of aromaticity such as antiaromaticity, pseudoaromaticity, homoaromaticity, heteroaromaticity, Mobius aromaticity, and metal aromaticity have been defined. The measures for assessment of the aromaticity are plentiful and ranges from purely qualitative to virtually quantitative. Although the concept has a profound impact on chemistry and chemical thinking over several decades, it remains conceptual, as it is not an experimentally measurable or verifiable property.

48

Aromaticity and Metal Clusters

Initially, aromaticity was restricted to all carbon, coplanar delocalized π systems with alternating single and double bonds having 4n + 2 π electrons and the contributing atoms are covalently bound and arranged in one or more rings. Later, the scope was extended to nonbenzenoid compounds, cations and anions, hetero-organic ­molecules, and carbon free main group compounds, inorganic and organometallic compounds, metal clusters and Möbius strips. More recently, three-dimensional ­aromaticity involving σ-electrons and multidimensional aromaticity involving d-­orbitals have been reported. Aromaticity has not been confined to ground states but it has been extended to transition states and excited states. This has led to the development of various qualitative and quantitative models to gauge aromaticity in various kinds of systems, and testing and validation of such models have raised certain doubts and questions. This chapter comprehensively presents a total scenario of this starting from the inception of the concept. A brief mention about π-distortivity in the context of aromaticity has been made. Various measures based on structural, energetic, magnetic, reactivity, and other aspects have been briefly presented with examples drawn from the literature. Quantitative estimation of aromaticity is usually dependent upon the criterion based on one or more of the structural, energetic, magnetic, electronic, or reactivity measures. In most cases, the estimation of aromaticity is based on the comparison with a known reference system, and more often than not, choosing the model system is arbitrary. It should be pointed out here that as the ­literature is so vast it is impossible to cite all the references and there are limitations due to space and scope. As exhaustive referencing is unpractical, only a selected set of publications pertaining to various measures of aromaticity have been cited.

Acknowledgments The authors thank Dr. G. Gayatri for critically reading through the manuscript and also for the technical assistance during its preparation.

References

1. Faraday, M. 1825. On new compounds of carbon and hydrogen, and on certain other products obtained during the decomposition of oil by heat. Philos. Trans. Roy. Soc. London 115: 440–466. 2. Hofmann, A. W. 1855. On insolinic acid. Proc. Roy. Soc. 8: 1–3. 3. Mitscherlich, E. 1834. Ueber das Benzol und die Säuren der Oel–und Talgarten. Ann. Pharm. 9: 39–48. 4. Kekulé, F. A. 1865. Sur la Constitution des substances aromatiques. Bull. Soc. Chim. Paris 3: 98–110. 5. Lonsdale, K. 1929. The structure of the benzene ring in hexamethyl benzene. Proc. Roy. Soc. 123A: 494–515. 6. Armit, J. W. and Robinson, R. Polynuclear heterocyclic aromatic types. Part II. Some anhydronium bases. 1925. J. Chem. Soc. Trans. 127: 1604–1618. 7. The special issue Guest Ed. Schleyer, P.v.R. on “Aromaticity” of 2001. Chem. Rev. 10; also the special issue of 2005. Chem. Rev. 105. 8. Kruszewski, J. 1980. Aromatic stability of polycyclic compounds. Pure Appl. Chem. 52: 1525–1540.

On the Measures of Aromaticity



49

9. Randic´, M. 1980. Local aromatic properties of benzenoid hydrocarbons. Pure Appl. Chem. 52: 1587–1596. 10. Hosoya, H. 2005. Aromaticity index can predict and explain the stability of polycyclic conjugated hydrocarbons. Monatshefte Chem. 136: 1037–1054. 11. Hinrichs, H., Fischer, A. K., Jones, P. G., Hopf, H., and Haley, M. M. 2005. [2.2] Paracyclophane/Dehydroannulene hybrids: Probing the aromaticity of the dehydro[14] annulene framework. Org. Lett. 7: 3793–3795. 12. Chen, Z. and King, R. B. 2005. Spherical aromaticity: Recent work on fullerenes, polyhedral boranes, and related structures. Chem. Rev. 105: 3613–3642. 13. Lier, G. V., De Proft, F., and Geerlings, P. 2002. Ab initio study of the aromaticity of hydrogenated [70] and [76]-fullerenes. Chem. Phys. Lett. 366: 311–320. 14 Priyakumar, U. D. and Sastry, G. N. 2003. Measures to evaluate heteroaromaticity and their limitations: Story of skeletally substituted benzenes. Proc. Indian Acad. Sci. (Chem. Sci.). 115: 49–66. 15. Priyakumar, U. D. and Sastry, G. N. 2002. A computational study of the valence isomers of benzene and their group V hetero analogs. New J. Chem. 26: 347–353. 16. Priyakumar, U. D. and Sastry, G. N. 2000. Structures, energetics, relative stabilities, and out-of-plane distortivities of skeletally disubstituted benzenes, (CH)4X2 (X) N, P, C−, Si−, O+, and S+): An ab initio and DFT study. J. Am. Chem. Soc.122: 11173–11181. 17. Priyakumar, U. D. and Sastry, G. N. 2002. A theoretical study of the structures, energetics, stabilities, reactivities, and out-of-plane distortive tendencies of skeletally substituted benzenes (CH)5XH and (CH)4(XH)2 (X ) B−, N+, Al−, Si, P+, Ga−, Ge, and As+). J. Org. Chem. 67: 271–281. 18. Dhevi, D. M., Priyakumar, U. D., and Sastry, G. N. 2003. Silaaromaticity in polycyclic systems: A computational study. J. Org. Chem. 68: 1168–1171. 19. Street, J. M., Clare, B. W., Kepert, D. L., and Taylor, R. 2004. Isolation and character­ ization of C60F38 loss of aromaticity on F2 addition to C60F36. J. Phys. Chem. B 108: 19228–19232. 20. Gimarc, B. M. and Trinajstic´, N. 1980. Aromatic inorganic rings. Pure Appl. Chem. 52: 1443–1458. 21. Ishida, S., Nishinaga, T., Westb, R., and Komatsu, K. 2005. Generation and aromati­ city of 2-silaimidazolium ion, a new π-conjugated silylium ion. Chem. Commun. 778–780. 22. Aihara, J., Kanno, H. and Ishida, T. 2005. Aromaticity of planar boron clusters confirmed. J. Am. Chem. Soc. 127: 13324–13330. 23. Xu, W. G. and Jin, B. 2006. Aromaticity of regular heptagonal P3−7 Trianion in the MP2−7 (M = Li, Na, K, Rb, and Cs) species. Chem. Phys. Lett. 419: 439–443. 24. Datta, A., John, N. S., Kulkarni, G. U., and Pati, S. K. 2005. Aromaticity in stable tiara nickel thiolates: Computational and structural analysis. J. Phys. Chem. A 109: 11647–11649. 25. Wang, J., Jellinek, J., Zhao, J., Chen, Z., King, R. B., and Schleyer, P. v. R. 2005. Hollow cages versus space-filling structures for medium-sized gold clusters: The spherical aromaticity of the Au50 cage. J. Phys. Chem. A 109: 9265–9269. 26. Boldyrev, A. I. and Wang, L. 2005. All-metal aromaticity and antiaromaticity. Chem. Rev. 105: 3716–3757. 27. Rzepa, H. S. 2005. Möbius aromaticity and delocalization. Chem. Rev. 105: 3697–3715. 28. Yoon, Z. S., Osuka, A., and Kim, D. 2009. Möbius aromaticity and antiaromaticity in expanded porphyrins. Nat. Chem. 1: 113–122. 29. Rzepa, H. S. 2005. Double-twist Möbius aromaticity in a 4n + 2 electron electrocyclic reaction. Chem. Commun. 5220–5222.

50

Aromaticity and Metal Clusters

30. Castro, C., Karney, W. L., Valencia, M. A., Vu, C. M. H., and Pemberton R. P. 2005. Möbius aromaticity in [12]annulene: Cis–trans isomerization via twist-coupled bond shifting. J. Am. Chem. Soc. 127: 9704–9705. 31. Herges, R. and Papafilippopoulos, A. 2001. Homoaromaticity in Tris(ethylene) nickel(0) and Tris(ethyne)nickel(0). Angew. Chem. Int. Ed. 40: 4671–4674. 32. Akilandeswari, L. and Venuvanalingam, P. 2007. Aromaticity pericyclic and pseudopericyclic transition states. Computational insights on lone pair induced modulation during the thermal rearrangement of 6-halopyrone. J. Theor. Comp. Chem. 6: 233–243. 33. Hückel, E. 1931. Quantum-theoretical contributions to the benzene problem. i.The electron configuration of benzene and related compounds. Z. Physik. 71: 204–286. 34. Dinadayalane, T. C., Priyakumar, U.D., and Sastry, G. N. 2004. Exploration of C6H6 potential energy surface: A computational effort to unravel the relative stabilities and synthetic feasibility of new benzene isomers. J. Phys. Chem. A. 108: 11433–11448. 35. Hückel, E. 1931. Quanstentheoretische beiträge zum benzolproblem II. quantentheorie der induzierten polaritäten. Z. Phys. 72: 310–337. 36. Hückel, E. 1932. Quantentheoretische beiträge zum problem der aromatischen und ungesättigten verbindungen. III. Z. Phys. 76: 628–648. 37. Breslow, R. and Mohacsi, E. 1963. Studies on d-orbital conjugation. III. Non-aromaticity of a derivative of the 1,3-dithiepinyl anion, a ten π-electron conjugated system J. Am. Chem. Soc. 85: 431–434. 38. Breslow, R. 1965. Aromatic character. Chem. Eng. News. 43: 90–99. 39. Breslow, R. 1973. Antiaromaticity. Acc. Chem. Res. 6: 393–401. 40. Zimmerman, H. E. in: Pericyclic Reactions, Vol. I, March, A. P. and Lehr, R. E., Eds., Academic Press, New York, 1977, 53. 41. Shaik, S., Shurki, A., Danovich, D., and Hiberty, P. C. 2001. A different story of π-delocalizations—the distortivity of π-electrons and its chemical manifestations. Chem. Rev. 101: 1501–1539. 42. Shaik, S., Shurki, A., Danovich, D., and Hiberty, P. C. 1997. A different story of ­benzene. J. Mol. Struc. (Theochem). 398–399: 155–167. 43. Shaik, S., Shurki and Bar, R. 1984. How important is resonance in organic species? Nouv. J. Chim. 8: 411–420. 44. Sastry, G. N. 2001. A revised look at electron delocalization in benzene. Curr. Sci. 81: 1288–1290. 45. Hass, Y. and Zilberg, S. 1995. The nu.14(b2u) mode of benzene in s0 and s1 and the distortive nature of the pi. electron system: Theory and experiment. J. Am. Chem. Soc. 117: 5387–5388. 46. Berry, R. S. 1961. Zero-point vibrations in benzene. J. Chem. Phys. 35: 2253–2254. 47. Shaik, S., Zilberg, S., and Hass, Y. 1996. A Kekulé-crossing model for the “Anomalous” Behavior of the b2u modes of aromatic hydrocarbons in the lowest excited 1B2u state. Acc. Chem. Res. 29: 211–218. 48. Navarro-Vázquez, A. and Schreiner, P. R. 2005. 1,2-Didehydro[10]annulenes: Structures, aromaticity, and cyclizations. J. Am. Chem. Soc. 127: 8150–8159. 49. Julg, A. and Francüois, Ph. 1967. Recherches sur la géométrie de quelques hydrocarbures non-alternants: Son influence sur les énergies de transition, une nouvelle définition de l’aromaticité. Theor. Chim. Acta. 7: 249–259. 50. Bird, C. W. 1985. A new aromaticity index and its application to five-membered ring heterocycles. Tetrahedron 41: 1409–1414. 51. Krygowski, T. M. and Cyran´ski, M. K. 2001. Structural aspects of aromaticity. Chem. Rev. 101: 1385–1419. 52. Krygowski, T. M., Ciesielski, A., and Cyran´ski, M. 1995. Aromatic character and energy of the five- and seven-membered rings in derivatives of penta- and heptafulvene substituted in exocyclic position. Chem. Pap. 49: 128–132.

On the Measures of Aromaticity

51

53. Kruszewski, J. and Krygowski, T. M. 1972. Definition of aromaticity basing on the harmonic oscillator model. Tetrahedron Lett. 13: 3839–3842. 54. Krygowski, T. M., Cyran´ski, M., Ciesielski, A., S´wirska, B., and Leszczyn´ski, P. 1996. Separation of the energetic and geometric contributions to aromaticity. 2. Analysis of the aromatic character of benzene rings in their various topological environments in the benzenoid hydrocarbons. Crystal and molecular structure of coronene. J. Chem. Inf. Comput. Sci. 36: 1135–1141. 55. Shurki, A. and Shaik, S. 1997. The distortive tendency of benzene π electrons: how is it related to structural observables? Angew. Chem. 36: 2205–2208. 56. Nikitina, A. F. and Sheridan, R. S. 2005. Geometry and aromaticity in highly strained heterocyclic allenes: Characterization of a 2,3-didehydro-2H-thiopyran. Org. Lett. 7: 4467–4470. 57. Balakrishnarajan, M. M. and Venuvanalingam, P. 1995. Computer generation of Pauling bond orders using heuristic search. J. Chem. Inf. Comput. Sci. 35: 717–722. 58. Mulliken, R. S. 1965. Molecular scientists and molecular science: Some reminiscences. J. Chem. Phys. 43: S2–S11. 59. Pauling, L. C. 1939. The Nature of the Chemical Bond. Cornell University of Ithaca, New York. 60. Trinajstic, N. 1983. Chemical Graph Theory, Vol. II. CRC Press, FL. 61. Manoharan, M., Balakrishnarajan, M. M., Venuvanalingam, P., and Balasubramanian, K. 1994. Topological resonance energy predictions on stabilities of fullerene clusters. Chem. Phys. Lett. 222: 95–100. 62. Sabljic´, A. and Trinajstic´, N. 1981. Aromatic stability of bridged annulenes. J. Org. Chem. 46: 3457–3461. 63. Gimarc, B. M. and Trinajstic´, N. 1982. Inorganic aromatic rings: The pi. electronic structure of boron sulfide (B8S16). Inorg. Chem. 21: 21–25. 64. Ilic´, P., Sinkovic´, B., and Trinajstic´, N. 1980. Resonance energies of conjugated structures. Isr. J. Chem. 20: 258–269. 65. Gutman, I., Milun, M., and Trinajstic´, N. 1977. Topological resonance energies of ­annlenes. Croat. Chem. Acta 49: 441–452. 66. Hess Jr. B. A. and Schaad, L. J. 1971. Hueckel molecular orbital pi. resonance energies. New approach. J. Am. Chem. Soc. 93: 305–310. 67. Hess Jr. B. A., Schaad, L. J., and Holyoke Jr. C. W. 1972. On the aromaticity of ­heterocycles containing the amine nitrogen or the ether oxygen. Tetrahedron 28: 3657–3667. 68. Hess Jr. B. A., Schaad, L. J., and Holyoke Jr. C. W. 1975. The aromaticity of heterocycles containing the imine nitrogen. Tetrahedron 31: 295–298. 69. Hess Jr. B. A. and Schaad, L. J. 1973. Hueckel molecular orbital pi.-resonance energies. Heterocycles containing divalent sulphur. J. Am. Chem. Soc. 95: 3907–3912. 70. Winstein, S. in Aromaticity, Spec. Publ. No. 21, The Chemical Society, London, 1967, 47. 71. Baird, N. C. 1972. Quantum organic photochemistry. II. Resonance and aromaticity in the lowest 3.pi..pi.* state of cyclic hydrocarbons. J. Am. Chem. Soc. 94: 4941–4948. 72. Roberts, J. D., Streitwieser, A., and Regan, C. M. 1952. Small-ring compounds. X.  Molecular orbital calculations of properties of some small-ring hydrocarbons and free radicals. J. Am. Chem. Soc. 74: 4579–4582. 73. Pauling, L. C. and Wheland, G. W. 1933. The nature of the chemical bond. V. The quantum– mechanical calculation of the resonance energy of benzene and naphthalene and the hydrocarbon free radicals. J. Chem. Phys. 1: 362–374. 74. Dewar, M. J. S. 1971. Aromatic (Huckel) and antiaromatic (antiHuckel) transition states. Angew. Chem., Int. Ed. 10: 761–776. 75. Dewar, M. J. S. and de Llano, C. 1969. Ground states of conjugated molecules. XI. Improved treatment of hydrocarbons. J. Am. Chem. Soc. 91: 789–795.

52

Aromaticity and Metal Clusters

76. Schaad, L. J. and Hess Jr. B. A. 2001. Dewar resonance energy. Chem. Rev. 101: 1465–1476. 77. Mallion, R. B. 1980. Some comments on the use of the “Ring-Current” concept in diagnosing and defining “Aromaticity”. Pure Appl. Chem. 52: 1541–1548. 78. Gomes, J. A. N. F. and Mallion, R. B. 2001. Aromaticity and ring currents. Chem. Rev. 101: 1349–1383. 79. Chen, Z., Wannere, C. S., Corminboeuf, C., Puchta, R., and Schleyer, P. v. R. 2005. Nucleus-independent chemical shifts (NICS) as an aromaticity criterion. Chem. Rev. 105: 3842–3888. 80. Schmalz, T. G., Gierke, T. D., Beak, P. and Flygare, W. H. 1974. Magnetic susceptibility, electron delocalization and aromaticity. Tetrahedron Lett. 15: 2885–2888. 81. Heine, T., Corminboeuf, C., and Seifert, G. 2005. The magnetic shielding function of molecules and pi-electron delocalization. Chem. Rev. 105: 3889–3910. 82. Bird, C. W. 1996. The relationship of classical and magnetic criteria of aromaticity. Tetrahedron 52: 9945–9952. 83. Herges, R., Jiao, H., and Schleyer, P. v. R. 1994. Magnetic properties of aromatic transition states: The Diels–Alder reactions. Angew. Chem., Int. Ed. 33: 1376–1378. 84. Woodward, R. B. and Hoffmann, R. 1965. Stereochemistry of electrocyclic reactions. J. Am. Chem. Soc. 87: 395–397. 85. Haddon, R. C. and Fukunaga, T. 1980. Unified theory of the thermodynamic and kinetic criteria of aromatic character in the [4n + 2]annulenes. Tetrahedron Lett. 21: 1191–1192. 86. Bird, C. W. 1997. Absolute hardness as a convenient criterion of heteroaromaticity. Tetrahedron 53: 3319–3324. 87. Bean, G. 1998. Application of natural bond orbital analysis and natural resonance theory to delocalization and aromaticity in five-membered heteroaromatic compounds. J. Org. Chem. 63: 2497–2506. 88. De Proft, F. and Geerlings, P. 2004. Relative hardness as a measure of aromaticity. Phys. Chem. Chem. Phys. 6: 242–248. 89. Komorowski, L. in: Chemical Hardness (Structure and Bonding,Vol. 80), Sen, K. D., Ed.; Springer-Verlag: Berlin, 1993, 45. 90. Komorowski, L. 1987. Electronegativity and hardness in the chemical approximation. Chem. Phys. 114: 55–71. 91. Sicilia, E., Russo, N., and Mineva, T. 2001. Correlation between energy, polarizability, and hardness profiles in the isomerization reaction of HNO and ClNO. J. Phys. Chem. A 105: 442–450. 92. Chattaraj, P. K., Roy, D. R., Elango, M., and Subramanian, V. 2005. Stability and reactivity of all-metal aromatic and antiaromatic systems in light of the principles of maximum hardness and minimum polarizability. J. Phys. Chem. A 109: 9590–9597. 93. Politzer, P., Abu-Awwad, F., and Murray, J. S. 1998. Comparison of density functional and Hartree–Fock average local ionization energies on molecular surfaces. Int. J. Quantum Chem. 69: 607–613. 94. Chesnut, D. B. and Bartolotti, L. J. 2000. The electron localization function description of aromaticity in five-membered rings. Chem. Phys. 253: 1–11. 95. Giambiagi, M., de Giambiagi, M. S., dos Santos Silva, C. D., and de Figueiredo, A. P. 2000. Multicenter bond indices as a measure of aromaticity. Phys. Chem. Chem. Phys. 2: 3381–3392. 96. Bultinck, P., Rafat, M., Ponec, R., Van Gheluwe, B., Carbó-Dorca, B., and Popelier, P. 2006. Electron delocalization and aromaticity in linear polyacenes: Atoms in molecules multicenter delocalization index. J. Phys. Chem. A 110: 7642–7648.

On the Measures of Aromaticity

53

97. Bultinck, P., Ponec, R., Gallegos, A., Fias, S., Damme, S. v., and Carbó-Dorca, R. 2006. Generalized polansky index as an aromaticity measure in polycyclic aromatic hydrocarbons. Croat. Chem. Acta 79: 363–371. 98. Bultinck, P., Fias, S., and Ponec, R. 2006. Local aromaticity in polycyclic aromatic hydrocarbons: Electron delocalization versus magnetic indices. Chem. Eur. J. 12: 8813–8818. 99. Polansky, O. E. and Derflinger, G. 1967. Clar theory of local benzene-like regions in condensed aromatic compounds. Int. J. Quantum Chem. 1: 379–401. 100. Cohen, H. D. and Roothan, C. C. J. 1965. Electric dipole polarizability of atoms by the Hartree–Fock method. I. Theory for closed-shell systems. J. Chem. Phys. 43: S34–S39. 101. Geuenich, D., Hess, K., Köhler, F., and Herges, R. 2005. Anisotropy of the Induced Current Density (ACID), a general method to quantify and visualize electronic delocalization. Chem. Rev. 105: 3758–3772. 102. Heilbronner, E. 1971. Aromaticity, Pseudo-Aromaticity, Anti-Aromaticity. Bergmann, E. D. and Pullman, B. Eds. Academic Press: New York. 21. 103. Wheland, G. W. 1955. Resonance in Organic Chemistry. John Wiley, New York, 31. 104. Minkin, V. I., Glukhovstev, M. N., and Sinkin, B. Y. 1994. Aromaticity and Anti– Aromaticity: Electron and Structural Aspects. John Wiley & Sons, New York. 105. Laskoski, M., Steffen, W., Smith, M. D., and Bunz, U. H. F. 2001. Is ferrocene more aromatic than benzene? Chem. Commun. 691–692. 106. Balaban, A. R. 1980. Is aromaticity outmoded? Pure Appl. Chem. 52: 1409–1429. 107. Saieswari, A., Priyakumar, U. D., and Sastry, G. N. 2003. On the use of NICS criterion to evaluate aromaticity in heteroaromatics involving III and IV row main group ­elements. J. Mol. Struc. (Theochem) 663: 145–148. 108. Clar, E. 1970. The Aromatic Sextet. John Wiley & Sons, New York.

in Metals 3 Aromaticity From Clusters to Solids Alina P. Sergeeva and Alexander I. Boldyrev Contents 3.1 Introduction..................................................................................................... 55 3.2 Chemical Bonding of Metal Clusters.............................................................. 56 3.3 Extending Aromaticity from Clusters to Alloys.............................................. 58 3.4 Extending Aromaticity from Clusters to Solids..............................................60 3.5 Extending Aromaticity from Solids to Clusters.............................................. 62 3.6 Concluding Remarks....................................................................................... 63 Acknowledgments.....................................................................................................64 References.................................................................................................................64

3.1 Introduction In chemistry, a cluster is an ensemble of bound atoms intermediate in size between a diatomic molecule and a bulk solid. At first, people thought that clusters are simply small pieces of solids possessing the same structures like that of a corresponding solid with a lot of dangling bonds. The breakthrough in understanding peculiarities of clusters occurred in mid-1980s, when Smalley and coworkers realized that C60 is not a piece of graphite, diamond or carbine, but it had a so-called “buckyball” structure [1]. Thus, the C60 cluster was shown to be a high symmetry structure with no dangling bonds at all. Moreover, this discovery opened a new direction in chemistry, the so-called “Chemistry of Fullerenes.” Another spectacular example is the Au20 cluster which was shown to possess a beautiful tetrahedral structure [2] with unexpected chemical bonding based on 10 four center two electron bonds [3] located at the center of every small tetrahedra comprising Au20 cluster. For gold was considered as most inert metal over the centuries, it was used as coinage material as well as to make jewelry. Surprisingly, in 1987 Haruta et al. [4] discovered that nanoscale gold particles have unusual catalytic properties for selective oxidation of CO. Starting from this point it was clear that clusters are neither molecules nor crystals but unique chemical species with yet unknown structures, peculiar chemical bonding, and unexpected chemical reactivity. Since clusters are frequently composed of just a few atoms, their properties, such as structure, stability, and reactivity can be studied computationally using high levels of theory. Experimentally isolated clusters are generated in a molecular beam, where they are free from environmental influence such as solvent, or counterions in a crystalline lattice. That allows one to study their 55

56

Aromaticity and Metal Clusters

intrinsic properties. Although isolated clusters themselves are certainly exotic species, their chemical bonding, structure, stability, and reactivity could help one rationalize the corresponding properties of novel materials, catalysts, and nanoparticles.

3.2 Chemical Bonding of Metal Clusters Properties of bulk metals can be described in terms of metallic bonding. Delocalization of electrons and the availability of a far larger number of delocalized energy states than of delocalized electrons are thought to be responsible for this type of bonding. When people started studying metal clusters it was natural to use similar concept of the delocalized bonding for them. That is why Jellium model [5,6] incorporating the idea of each metal cluster being a structureless ionic jelly permeated by itinerant electrons became the major chemical bonding tool for metal clusters [7,8]. On the contrary, delocalized bonding in organic chemistry is described from the aromaticity point of view. It was surprising that aromaticity was not used as a major tool in interpreting chemical bonding of pure metal clusters until 2001 [9] because aromaticity was used to describe chemical bonding in Ga3 triangular unit of Na2[[(2,4,6Me3C6H2)2C6H3]Ga]3 [10] in 1995, and in Bi 5− cluster in 1994 [11]. Since 2001, aromaticity was used to describe chemical bonding in numerous all-metal clusters: 4− − 2− Al 2− 4 , MAl 4 , and M2Al4 (M = Li, Na, K, Rb, Cs, Cu, Ag, Au) [9,12–38]; Al 4 , M 2 Al 4 , 2− 4− M3Al , and M4Al4 (M = Li, Na, K) [17,18,21,22,24,28–30,33,39–46]; X 4 and NaX −4 (X = B, Ga, In, Tl) [13,47,48]; Si2X2 (X = B, Al, Ga) [13]; XAl3− (X = C, Si, Ge, Sn, Pb) [49,50]; XGa 3− (X = Si, Ge) [51]; X 3− and NaX3 (X = Al, Ga) [52–54]; X 3+ (X = Li, Na, K, Cu) [55–58]; X 32− ; NaX 3− and Na2X3 (X = Be, Mg) [59–62]; Be82− [63]; X 3− (X = Sc, Y, La) [64]; X 32− (X = Zn, Cd, Hg) [65]; Hf3 [66]; Ta 3− [67]; Au5Zn+ [68]; Cu5Sc, Cu6Sc+, and Cu7Sc [69,70]; M4Li2 (M = Cu, Ag, Au) [71]; M4L2 and M4L − (M = Cu, Ag, Au; L = Li, Na) [72]; Al2(CO)2 [73]; cyclo-CunHn (n = 3–6) [74]; cycloMnHn (M = Ag, Au; n = 3–6) [75]; cyclo-Au3L nH3−n (L = CH3, NH2, OH, and Cl; n = 1–3) [76]; cyclo-CunAg3−nHn (n = 1–3), cyclo-CunAg4−nHn (n = 1–4), and cyclo− − CunAg5−nHn (n = 1–5) [77]; X 2− 4 and NaX 4 (X = N, P, As, Sb, Bi) [78]; and X 5 (X = N, 2− − − P, As, Sb, Bi) [79–82]; M3 O9 and M3 O9 (M = W, Mo) [83]; Ta 3 O3 [84,85]; Nb3 O −n and Nb3On (n = 0–2) [86]; Hg6− 4 [87], [M3(CO)12] and [M4(CO)16] (M = Fe, Ru, Os) 6− [88,89]; Re3X9 and Re3 X 2− 9 (X = Cl, Br) [90,91]; Hg 4 [92]; cyclo-UnX n (n = 3, 4; X = O, NH) [93]. There are a few recent reviews on aromaticity and antiaromaticity in all-metal systems [94–98]. The extension of the concept of aromaticity into all-metal clusters occurred in 2000 when Professor Lai-Sheng Wang and coworkers recorded the photoelectron spectra of the CuAl −4 , LiAl −4 , and NaAl −4 clusters [9]. The theoretical search by Kuznetsov and Boldyrev revealed that the global minimum structures of all MAl −4 clusters (M = Cu, Li, Na) are of square pyramidal shape (Figure 3.1a) consisting of a metal atom coordinated to a square planar aluminum unit [9]. The calculated vertical detachment energies (VDEs) of all the determined global minima were found to be in an excellent agreement with the experimental ones, thus, providing proof of establishing structures of the MAl −4 clusters. Since then the combination of photoelectron spectroscopy and ab initio calculations became a major tool for deciphering structure and chemical bonding of isolated clusters. All

57

Aromaticity in Metals (a)

CuAl–4

LiAl–4

C4v, A1

C4v, A1

1

1

NaAl–4 C4v,1A1

Al2– 4 D4h,1A1g

(b)

(c)

HOMO HOMO-1 HOMO-2 HOMO-3 HOMO-4 HOMO-4′ HOMO-5 1a2u 2a1g 1b2g 1b1g 1eu 1eu 1a1g

π

σr

σt

Figure 3.1  (a) The global minimum structures of the MAl −4 clusters (M = Cu, Li, Na) and 2− the isolated Al 2− 4 cluster; (b) valence canonical molecular orbitals (CMOs) of the isolated Al 4 cluster; (c) schematic representation of valence CMOs as linear combinations of four 3pz atomic orbitals (AOs) comprising highest occupied molecular orbital (HOMO), four 3p-radial AOs (HOMO-1), four 3p-tangential (HOMO-2), as well as four different linear combinations of 3s AOs (HOMO-3, HOMO-4, HOMO-4′, HOMO-5).

of the three MAl −4 clusters were viewed as consisting of an M+ cation coordinated to a square planar Al 2− 4 unit in ionic limit. The search for the global minimum of the metastable Al 2− (it is not stable with respect to an electron detachment) revealed that 4 the planar square structure was indeed the lowest in energy. The question was why the Al 2− 4 cluster adopts such a high symmetry structure. Molecular orbital analysis (Figure 3.1b and c) revealed that out of seven valence canonical molecular orbitals of the Al 2− 4 cluster, four represented linear combination of lone pairs on each of the aluminum atoms, the rest of them were completely delocalized σ- and π-CMOs. The occupation of completely delocalized π-CMO of the Al 2− 4 cluster satisfying 4n + 2 Huckel’s Rule (n = 0) was used to assign this species to a family of π-aromatic species. Since π-aromaticity was well established in chemistry it was decided to make a stress on π-aromaticity only in the discussion of chemical bonding of the Al 2− 4 cluster at that moment. In the follow-up publications [47,87], the presence of two types of σ-aromaticity (σ-radial and σ-tangential) in Al 2− 4 , as well as in valence isoelectronic Ga 24 − , In 24 − , and Hg64− clusters was also added to the chemical bonding discussion. The schematic representations of σr- and σt- MOs on an example of Al 2− 4 are given in Figure 3.1c. One can see that radial MOs are composed of orbitals directed toward the center of the cyclic structure, while tangential MOs are composed of orbitals that are perpendicular to the radial ones. Aromaticity/antiaromaticity based on the 4n + 2/4n rule for electrons occupying radial MOs is referred as radial aromaticity/antiaromaticity and those for electrons occupying tangential MOs is referred as tangential aromaticity/antiaromaticity. Thus, planar all-metal clusters have the striking feature of chemical bonding—the possibility of the multifold nature of aromaticity, antiaromaticity, and conflicting aromaticity. When only s-atomic orbitals (AOs) are involved in chemical bonding, one may expect only

58

Aromaticity and Metal Clusters

σ-aromaticity or σ-antiaromaticity. If p-AOs are involved, σ-tangential (σt-), σ-radial (σr-), and π-aromaticity/antiaromaticity could occur. In this case, there can  be multiple (σ- and π-) aromaticity, multiple (σ- and π-) antiaromaticity, and  conflicting aromaticity (simultaneous σ-aromaticity and π-antiaromaticity or σ-antiaromaticity and π-aromaticity). If d-AOs are involved in chemical bonding, σ-tangential (σt-), σ-radial (σr-), π-tangential (πt-), π-radial (πr-), and δ-aromaticity/ antiaromaticity could occur. In this case, there can be multiple (σ-, π-, and δ-) ­a romaticity, multiple (σ-, π-, and δ-) antiaromaticity, and conflicting aromaticity (simultaneous aromaticity and antiaromaticity among the three σ-, π-, and δ-types) 2− 2− 6− [98]. The Al 2− 4 cluster as well its valence isoelectronic Ga 4 , In 4 , and Hg 4 species are examples of double σ- (both σt- and σr- to be precise) and π-aromatic clusters − [47,87]. The Al 4− 4 unit of the Li 3 Al 4 cluster is an example of conflicting aromatic species (it is π-antiaromatic and σ-aromatic) [39,94]. The first d-AO-based σ-aromatic clusters M3 O9− (M = W, Mo) were experimentally discovered in 2005 by Lai-Sheng Wang and coworkers [83]. The X 3− (Sc, Y, La) clusters were theoretically predicted by Chi and Liu [64] to be both d-AO-based σ- and π-aromatic species. The Ta 3 O3− cluster was the first δ-aromatic cluster to be discovered [85]. The Hf 3 cluster in the lowest singlet state was theoretically predicted to be the first example of triple (σ-, π-, and δ-) aromaticity [66]. Tsipis, Kefalidis, and Tsipis showed that the delocalized f electron density in the rings of planar isocyclic and heterocyclic uranium clusters could be associated with cyclic electron delocalization [93], which is a characteristic feature of aromaticity. From the discussion above, it is clear that there are many multiple aromatic, multiple antiaromatic, and clusters with types of conflicting aromaticity yet undiscovered.

3.3 Extending Aromaticity from Clusters to Alloys Understanding chemical bonding of pure clusters may have significantly more ­profound effect in chemistry even though clusters are very exotic species being generated in a molecular beam for a short period of time. In this chapter we would like to address this particular issue. In order to facilitate chemical bonding analysis of cluster species, we used a recently developed tool: the Adaptive Natural Density Partitioning (AdNDP) method [99]. This method leads to partitioning of the charge density into elements with the highest possible degree of localization of electron pairs. If some part of the density cannot be localized in this manner, it is represented as completely delocalized objects, similar to canonical MOs, naturally incorporating the idea of delocalized bonding, that is, n center–two electron (nc–2e) bonds. If one encounters a molecule or a cluster in which AdNDP analysis reveals that σ- or π-electrons cannot be localized into lone pairs or 2c–2e bonds, such a species is considered from the aromaticity/antiaromaticity point of view. If delocalization occurs over the whole molecule and corresponding bonds satisfy the 4n + 2 rule, we consider such species to be globally aromatic. Thus, we assess aromaticity in a ­particular chemical species on the basis of the presence of delocalized bonding recovered by the AdNDP analysis in a cyclic structure. First, we applied the AdNDP analysis to the isolated Hg6− 4 cluster, as well as to the same cluster embedded in a crystalline lattice of Na3Hg2 amalgam (Figure 3.2).

59

Aromaticity in Metals (a)

Hg6– 4

D4h,1A1g 4c–2e 4c–2e 4c–2e Four 1c–2e lone pairs on mercury atoms p-AO based σr-bond p-AO based σt-bond p-AO based π-bond ON = 2.00 |e| ON = 2.00 |e| ON = 1.94 |e| ON = 2.00 |e|

(b)

Na3Hg2 28c–2e p-AO based σr-bond ON = 2.00 |e|

Four 1c–2e lone pairs on mercury atoms ON = 1.84 |e|

28c–2e p-AO based σt-bond ON = 2.00 |e|

28c–2e p-AO based π-bond ON = 2.00 |e|

Figure 3.2  The bonding elements recovered by the AdNDP analysis (a) for the isolated 6− Hg6− 4 cluster and (b) for the Hg 4 cluster embedded in part of the crystalline lattice of Na 3Hg2 amalgam comprised of 24 sodium cations placed at their position in the real crystalline structure.

2− Since the isolated Hg6− 4 cluster is valence isoelectronic to the Al 4 cluster, it was natural to demonstrate that the chemical bonding picture of the isolated Hg6− 4 cluster 6− was similar to that of the Al 2− 4 cluster [87]. Indeed, 14 valence electrons of the Hg 4 cluster form four 1c–2e lone pairs on mercury atoms, one 4c–2e p-AO-based σr-bond, one 4c–2e p-AO-based σt-bond, and, finally, one 4c–2e p-AO-based π-bond (Figure 3.2a). In all the cases, occupation numbers were found to be close to the ideal of two electrons per bond. Therefore, the Hg6− 4 cluster is doubly σ- (both σr- and σt-) and π-aromatic, similar to the Al 2− 4 cluster. Since the results of chemical bonding of the Hg6− 4 cluster were published, the question has frequently arisen whether it is appropriate to apply the chemical bonding of a given isolated cluster to the one in a crystalline lattice environment where countercations are claimed to play a crucial

60

Aromaticity and Metal Clusters

role for the chemical bonding. In order to resolve this issue, we performed the AdNDP analysis for the Hg6− 4 cluster embedded in a crystalline lattice of Na 3Hg2 amalgam (surrounded by 24 sodium cations located at the positions of a real crystal of Na3Hg2). These results are presented in Figure 3.2b. One can see that the seven valence chemical bonding elements recovered by the AdNDP cluster for both isolated and embedded clusters are essentially the same with some distortion imposed by the influence of sodium counterions. Though, delocalized bonds recovered by the AdNDP, namely p-AO-based σr-bond, p-AO-based σt-bond, and p-AO-based π-bond can no longer be presented as 4c–2e bonds but rather as 28c–2e bonds, the nodal structure of those 28c–2e bonds remains the same as there in 4c–2e bonds of isolated Hg6− 4 cluster. If we restrict the AdNDP analysis to search for 4c–2e bonds on mercury atoms instead of 28c–2e bonds in the embedded cluster, the occupation numbers for the recovered delocalized σ- and π-bonds drop significantly from 2.00 |e| to 1.18 and 0.88 |e| for 4c–2e σ-delocalized bonds, and to 0.49 |e| for 4c–2e π-delocalized bond. That clearly shows significant covalent bonding between sodium and mercury 6− + atoms. The embedded Hg6− 4 cluster of formal charge +18 (Hg 4  + 24Na ) is still a doubly σ- (both σr- and σt-) and π-aromatic system. It is customary in inorganic chemistry to divide Zintl phases into building blocks that correspond to multiply charged anions. For instance, In84− , Tl 75 − , and In 95 − multiply charged anions (MCAs) were proposed as building blocks for Zintl phases with the stoichiometry Na2In, Na2K 21Tl19, La3In5, and many others [36–38]. The isolated Hg6− 4 cluster is an MCA that is unstable with respect to an electron detachment. In all our calculations of the isolated Hg6− 4 cluster, we use compact basis sets, which do not allow electrons to escape, since there is no admixture of scattering solutions to the bound-state wave functions taken into account. The usage of compact basis sets assures proper description of the chemical bonding pattern of the isolated and embedded cluster. Although, the Hg4 cluster embedded in a crystalline lattice certainly does not carry the charge of 6−, we believe that aromaticity of the isolated Hg6− 4 cluster allows one to understand why mercury atoms comprise a square planar structure in the Na3Hg2 amalgam even though isolated mercury atoms do not have positive electron affinity. Stability of the negatively charged Hg4 cluster is due to the stabilizing environment of sodium counterions and the presence of aromaticity. Therefore, we proved that the chemical bonding analysis of the isolated MCA ­carrying a very high additional charge could be helpful in understanding chemical bonding of alloys. This was also the first example of extending multiple aromaticity discovered for the gas-phase clusters of LiAl −4 , NaAl −4 , and CuAl −4 into a solid state [87].

3.4 Extending Aromaticity from Clusters to Solids The all-metal Bi 5− and other pnictogen Pn 5− (Pn = P, As, and Sb) clusters are valence isoelectronic to one of the prototypical aromatic hydrocarbon cyclopentadienyl C5 H 5− anion, on that basis those clusters were recognized as being π-aromatic [79−82]. The bridge between isolated all-metal Pn 5− clusters and valence isoelectronic clusters embedded in a crystalline lattice was made in 2004 when Todorov and Sevov reported synthesis of the new Zintl phases: Na8BaPb6, Na8BaSn6, and Na8EuSn6

61

Aromaticity in Metals

[100], despite looking at the chemical formula of those Zintl phases one may think that the building block of those compounds should be either Pb6 or Sn6 clusters. Yet it was shown by Todorov and Sevov that those phases contain isolated flat heavy metal aromatic pentagonal rings of Sn 65 − and Pb65 − , as well as isolated anions of Sn4− and Pb4− (Figure 3.3b). (a) Sn56–

D5h, 1A1′

Five 1c–2e lone pairs on tin atoms ON = 1.94 |e|

Five 2c–2e peripheral Sn-Sn σ-bonds ON = 1.97 ⎜e⎥

Three totally delocalized 5c–2e p-AO based π-bonds ON = 2.00 |e|

(b)

Na8BaSn6

Ba Sn Na

Five 1c–2e lone pairs on tin atoms ON = 1.76–1.86 |e|

Five 2c–2e peripheral Sn-Sn σ-bonds ON = 1.66–1.93 |e|

Three totaly delocalized 5c–2e p-AO based π-bonds ON = 1.62–1.95 |e|

Figure 3.3  The bonding elements recovered by the AdNDP analysis (a) for the isolated 6− Sn 6− 5 cluster and (b) for the Sn 5 cluster embedded in part of the crystalline lattice of the Na8BaSn6 compound comprised of 12 sodium cations and two barium dications placed at their position in the real crystalline structure.

62

Aromaticity and Metal Clusters

In this work, we conducted the AdNDP analysis for both isolated Sn 6− 5 , as well as for the embedded Sn 6− cluster in a Zintl phase Na BaSn (Figure 3.3). 5 8 6 The chemical bonding picture obtained by the AdNDP method in both cases (a and b in Figure 3.3) is the following: 26 valence electrons of the Sn 6− 5 clusters form five 1c–2e lone pairs on tin atoms, five 2c–2e peripheral Sn–Sn σ-bonds, and three totally delocalized 5c–2e p-AO-based π-bonds satisfying the Huckel 4n + 2 rule for aromaticity (n = 1). The occupation numbers vary from 1.94 to 2.00 |e| for all bonding elements in the isolated Sn 6− 5 cluster and from 1.62 to 1.95 |e| for the same elements in the embedded Sn 6− cluster. If we allow the AdNDP analysis to recover the 5 π-delocalized bonds not on five centers (tin atoms only), but as 19c–2e bonds delocalized over the whole model Na12Ba2Sn5 system of formal +10 charge—the occupation numbers of the three 19c–2e π-bonds rises up to 2.00 |e|. This indicates that some covalent bonding occurs between tin, sodium, and barium atoms. Still, the contribution to the chemical bonding that holds tin atoms in an almost perfect pentagon structure comes primarily from tin atoms. Thus, bonding picture recovered by the AdNDP method for the isolated Sn 6− 5 cluster can be used for deciphering chemical bonding in Na8BaSn6 solid.

3.5 Extending Aromaticity from Solids to Clusters Robinson and coworkers in 1995 synthesized remarkable organometallic Na2[[(2,4,6Me3C6H2)2C6H3]Ga]3 compound containing Ga3 core (Figure 3.4a) [10]. On the basis of geometrical and electronic calculations he concluded that this compound was a π-aromatic system. The follow-up quantum chemical calculations of the [M2(GaH)3] (M = Li, Na, K) model systems [101] confirmed aromaticity of cyclogallenes. In this section, we would like to make an analogy between the experimentally observed cyclogallenes and the isolated Ga 3− cluster [52]. The AdNDP analyses of the Na2(GaH)3 system and the isolated Ga 3− are presented in Figure 3.4b and c, respectively. Our AdNDP analysis of the model Na2(GaH)3 system recovered three 2c–2e Ga–H σ-bonds, three 2c–2e Ga–Ga σ-bonds, and one 5c–2e π-bond. If we restrict the AdNDP analysis to search for the π-bond on the three gallium atoms only, disregarding the density from sodium atoms, then the resulting 3c–2e π-bond has lower occupation number of 1.74 |e| compared with 2.00 |e| of that of 5c–2e π-bond (Figure 3.4d). The lower occupation number of 3c–2e π-bond compared to that of 5c–2e π-bond is due to the contribution of sodium atoms to the bonding. For the isolated Ga 3− cluster, the AdNDP analysis revealed a lone pair per gallium atom, 3c–2e σr-bond and 3c–2e π-bond. If one compares the chemical bonding of the isolated Ga 3− cluster and the model cyclogallene (Na2(GaH)3), it is clear that the addition of three hydrogen atoms and an extra electron to the isolated Ga 3− cluster results in substitution of the lone pairs of isolated Ga 3− cluster by 2c–2e Ga–H σ-bonds of Na2(GaH)3 and the formation of three 2c–2e Ga–Ga σ-bonds instead of one 3c–2e σr-bond. Again, understanding of chemical bonding in isolated clusters can be very helpful in rationalizing chemical bonding in complicated organometallic compounds.

63

Aromaticity in Metals (a)

Na Ga C H

Na2[(Mes2 C6H3)Ga]3 (Mes = 2,4,6-Me3C6H2)

(b) Na2(GaH)3

D3h, 1A1′

Three 2c–2e Ga–H σ-bonds ON = 1.99 |e|

3c–2e p-AO based π-bond ON = 1.74 |e|

(d)

Three 2c–2e Ga σ-bonds ON = 1.85 |e|

(c)

5c–2e p-AO based π-bond ON = 2.00 |e|

Ga–3

D3h, 1A1′

Three 1c–2e lone pairs on gallium atoms ON = 1.81 |e|

3c–2e p-AO based σr-bond ON = 2.00 |e|

3c–2e p-AO based π-bond ON = 2.00 |e|

Figure 3.4  (a) The structure of the Na2[[(2,4,6-Me3C6H2)2C6H3]Ga]3 crystal; (b) bonding elements recovered by the AdNDP analysis for the model Na2(GaH)3 system; (c) bonding ­elements recovered by the AdNDP analysis for the isolated Ga 3− cluster; and (d) the 5c–2e π-bond of the model Na2(GaH)3 system.

3.6 Concluding Remarks In the introduction we stated that clusters are neither molecules nor solids but unique chemical species with yet unknown structures, peculiar chemical bonding, and unexpected chemical reactivity. Despite a few decades of theoretical and experimental studies of clusters they continue to be mysterious chemical species that are still poorly understood. Nevertheless, as discussed in this chapter, understanding chemical bonding of clusters can be useful for deciphering chemical bonding of solids, nanoparticles, novel materials, catalysts, active sites of enzymes, and so on. We still do not have a simple chemical bonding model for clusters similar to the Lewis chemical bonding model for organic molecules capable of rationalizing and predicting their structure. Although in recent years some progress has been made in developing such a chemical bonding model, we still discover almost every day remarkable ­unexpected novel structures of clusters with surprising chemical bonding and

64

Aromaticity and Metal Clusters

r­ eactivity. The development of a comprehensive chemical bonding model for clusters would be an important step toward a unified chemical bonding theory in chemistry.

Acknowledgments This work was supported by the National Science Foundation (CHE-0714851). Computer time at the Center for High Performance Computing, Utah State University is gratefully acknowledged. The computational resource, the Uinta cluster supercomputer, was provided through the National Science Foundation under Grant ­CTS-0321170 with supporting funds from the Utah State University.

References

1. Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. C60: Buck­ minsterfullerene. Nature 1985, 318, 162–163. 2. Li, J.; Li, X.; Zhai, H. J.; Wang, L. S. Au20: A Tetrahedral cluster. Science 2003, 299, 864–867. 3. Zubarev, D. Yu.; Boldyrev, A. I. Deciphering chemical bonding in Golden Cages. J. Phys. Chem. A. 2009, 113, 866–868. 4. Haruta, M.; Kobayashi, T.; Sano, H.; Yamada, N. N. Gold catalysts for the oxidation of carbon monoxide at a temperature far below 0°C. Chem. Lett. 1987, 16, 405–408. 5. Knight, W. D.; Clemenger, K.; de Heer, W. A.; Saunders, W. A.; Chou, M. Y.; and Cohen, M. L. Electronic shell structure and abundances of sodium clusters. Phys. Rev. Lett. 1984, 52, 2141–2143. 6. Cohen, M. L.; Chou, M. Y.; Knight, W. D.; de Heer, W. A. Physics of metal clusters. J. Phys. Chem. 1987, 91, 3141–3149. 7. Castleman, A. W.; Jena, P. Clusters: A bridge between disciplines. Proc. Natl. Acad. Sci. USA 2006, 103, 10552–10553. 8. Boldyrev, A. I.; Wang, L. S. Beyond classical stoichiometry: Experiment and theory. J. Phys. Chem. A 2001, 105, 10759–10775. 9. Li, X.; Kuznetsov, A. E.; Zhang, H. F.; Boldyrev, A. I.; Wang, L. S. Observation of ­all-metal aromatic molecules. Science 2001, 291, 859–861. 10. Li, X. W.; Pennington, W. T.; Robinson, G. H. Metallic system with aromatic character. Synthesis and molecular structure of Na2[[(2,4,6-Me3C6H2)2C6H3]Ga]3 the first cyclo­ gallane. J. Am. Chem. Soc. 1995, 117, 7578–7579. 11. Gausa, M.; Kaschner, R.; Lutz, H. O.; Seifert, G.; Weiwes-Broer, K.-H. Photoelectron and theoretical investigations on bismuth and antimony pentamer anions: Evidence for aromatic structure. Chem. Phys. Lett. 1994, 230, 99–102. 12. Fowler, P. W.; Havenith, R. W. A.; Steiner, E. Unconventional ring currents in all-metal aromatic, Al 2− 4 . Chem. Phys. Lett. 2001, 342, 85–90. 13. Juselius, J.; Straka, M.; Sundholm, D. Magnetic-shielding calculations on Al 2− 4 and analogues. A new family of aromatic molecules? J. Phys. Chem. A 2001, 105, 9939–9944. 14. Boldyrev, A. I.; Kuznetsov, A. E. On the resonance energy in new all-metal aromatic molecules. Inorg. Chem. 2002, 41, 532–537. 15. Zhan, C.-G.; Zheng, F.; Dixon, D. A. Electron affinities of Aln clusters and multiple-fold aromaticity of the square Al 2− 4 structure. J. Am. Chem. Soc. 2002, 124, 14795–14803. 16. Fowler, P. W.; Havenith, R. W. A.; Steiner, E. Ring current and electron delocalisation in an all-metal cluster, Al 2− 4 Chem. Phys. Lett. 2002, 359, 530–536. 17. Chen, Z.; Corminboeuf, C.; Heine, T.; Bohmann, J.; Schleyer, P. v. R. Do all-metal antiaromatic clusters exist? J. Am. Chem. Soc. 2003, 125, 13930–13931.

Aromaticity in Metals

65

18. Havenith, R. W. A.; Fowler, P. W.; Steiner, E.; Shetty, S.; Kanhere, D.; Pal, S. Aromaticity and antiaromaticity of LixAl4 clusters: Ring current patterns versus electron counting. Phys. Chem. Chem. Phys. 2004, 6, 285–288. 19. Havenith, R. W. A.; van Lenthe, J. H. A valence bond study of the σ and π aromatic species Al 2− 4 . Chem. Phys. Lett. 2004, 385, 198–201. 20. Mercero, J. M.; Ugalde, J. M. Sandwich-like complexes based on “all-metal” (Al 2− 4 ) aromatic compounds. J. Am. Chem. Soc. 2004, 126, 3380–3381. 21. Lin, Y.-C.; Juselius, J.; Sundholm, D.; Gauss, J. Magnetically induced current densities 4− in Al 2− 4 and Al 4 species studied at the coupled-cluster level. J. Chem. Phys. 2005, 122, 214308. 22. Chattaraj, P. K.; Roy, D. R.; Elango, M.; Subramanian, V. Stability and reactivity of allmetal aromatic and antiaromatic systems on light of the principles of maximum hardness and minimum polarizability. J. Phys. Chem. 2005, 109, 9590–9597. 23. Hu, X.; Li, H.; Liang, W.; Han, S. Theoretical study on all-metal aromatic complexes: interaction with small gas molecule. Chem. Phys. Lett. 2004, 397, 180–184. 24. Santos, J. C.; Andres, J.; Aizman, A.; Fuentealba, P. An aromaticity scale based on the topological analysis of the electron localization function including σ and π contributions. J. Chem. Theory. Comput. 2005, 1, 83–86. 25. Hu, X.; Li, H.; Wang, C.; Han, S. How O2 is activated on all-metal aromatic complex (Al4Ca) surface. Chem. Phys. Lett. 2006, 426, 39–42. 26. Hu, X.; Li, H.; Wang, C. The Reactivity of all-metal aromatic complexes: A theoretical investigation on the methane activation reaction. J. Phys. Chem. B. 2006, 110, 14046–14049. 27. Havenith, R. W. A.; Fowler, P. W. The origin of the ring current in the all-metal aromatic, Al 2− 4 . Phys. Chem. Chem. Phys. 2006, 8, 3383–3386. 28. Chattaraj, P. K.; Roy, D. R.; Elango, M.; Subramanian, V. Chemical reactivity descriptor 4− based aromaticity indices applied to Al 2− 4 and Al 4 systems. J. Mol. Struct. (THEOCHEM) 2006, 759, 109–110. 29. Mallajosyula, S. S.; Datta, A.; Pati, S. K. Aromatic superclusters from all-metal aromatic and antiaromatic monomers, [Al4]2- and [Al4]4-. J. Phys. Chem. B. 2006, 110, 20098–20101. 30. Shetty, S.; Kar, R.; Kanhere, D. G.; Pal, S. Intercluster reactivity of metalloaromatic and antiaromatic compounds and their applications in molecular electronics: A theoretical investigation. J. Phys. Chem. A. 2006, 110, 252–256. 31. Mang, C.; Liu, C.; Zhou, J.; Li, Z.; Wu, K. Relative stabilities of all-metal clusters MAl −4 (M = Li+, Na+, K+, Rb+, Cs+, Cu+, Ag+, Au+). Chem. Phys. Lett. 2007, 438, 20–25. 32. Hu, Z.; Li, H. All-metal aromatic complexes show high reactivity in the oxidation reaction of methane and some hydrocarbons. J. Phys. Chem. A. 2007, 111, 8352–8356. 4− 33. Islas, R.; Heine, T.; Merino, G. Structure and Electron Delocalization in Al 2− 4 and Al 4 . J. Chem. Theory Comput. 2007, 3, 775–781. 34. Yang, L. M.; Ding, Y. H.; Sun, C. C. Sandvich-like Compounds Based on the All-Metal Aromatic unit Al 2− 4 and the Main-Group metals M (M=Li, Na, k, Be, Mg, Ca). Chem. Eur. J. 2007, 13, 2546–2555. 35. Mang, C. Y.; Zhau, X.; He, L. X.; Liu, C. P.; Wu, K. C. Minimum Polarizability Principle Applied to lowest Energy isomers of Some Gaseous All-Metal Clusters. J. Phys. Chem. A. 2008, 112, 1661–1665. 36. Corbett, J. D. Polyanionic Clusters and Networks of the Early p-Element Metals in the Solid State: Beyond the Zintl Boundary. Angew. Chem. Int. Ed. 2000, 39, 670–690. 37. Kauzlarich, S. M. Ed. Chemistry, Structure, and Bonding of Zintl Phases and Ions; VCH: New York, NY, 1996. 38. Greewood, N. N.; Earnshaw, A. Chemistry of the Elements, 2nd ed; ButterworthHeinemann: Oxford, UK, 1998.

66

Aromaticity and Metal Clusters

39. Kuznetsov, A. E.; Birch, K. A.; Boldyrev, A. I.; Li, X.; Zhai, H. J.; Wang, L. S. All-metal − antiaromatic molecule: Rectangular Al 4− 4 in the Li 3 Al 4 anion. Science 2003, 300, 622–625. 40. Shetty, S.; Kanhere, D. G.; Pal, S. Metallo-antiaromatic Al4Na4 and Al 4 Na 3− compounds: A theoretical investigation. J. Phys. Chem. A 2004, 108, 628–631. 41. Datta, A.; Pati, S. K. Charge-transfer induced large nonlinear optical properties of small Al clusters: Al4M4 (M = Li, Na, and K). J. Phys. Chem. A. 2004, 108, 9527–9530. 42. Datta, A.; Pati, S. K. Stable transition metal complexes of an all-metal antiaromatic molecule (Al4Na4): Role of complexations. J. Am. Chem. Soc. 2005, 127, 3496–3500. 43. Datta, A.; Pati, S. K. Rationalization of the (anti)aromaticity in all metal molecular clusters. J. Chem. Theory Comput. 2005, 1, 824–826. 44. Zhang, R. J.; Wang, X. Q.; Cheng, F. L. Comparison of magnetic properties between spin singlet and triplet Li3Al −4 clusters. J. Cluster Sci. 2007, 18, 431–443. 45. Satpati, P.; Sebastian, K. L. Stabilization of an all-metal antiaromatic molecule (Al4Li4) using BH and C as caps. Inorg. Chem. 2008, 47, 2098–2102. 46. Yang, L.; Shao, C.; Ding, Y.; Sun, C. Cluster-assembled compounds comprising an allmetal subunit Li3Al −4 . Phys. Chem. Chem. Phys. 2008, 10, 2020–2025. 47. Kuznetsov, A. E.; Boldyrev, A. I.; Li, X.; Wang, L. S. On the aromaticity of square planar Ga 24 − and In 24 − in gaseous NaGa −4 and NaIn −4 clusters. J. Am. Chem. Soc. 2001, 123, 8825–8831. 48. Zhao, C.; Balasubramanian, K. Spectroscopic properties of novel aromatic metal clusters: NaM4 (M = Al, Ga, In) and their cations and anions. J. Chem. Phys. 2004, 120, 10501–10510. 49. Li, X.; Zhang, H. F.; Wang, L. S.; Kuznetsov, A. E.; Cannon, N. A.; Boldyrev, A. I. Experimental and theoretical observations of aromaticity in heterocyclic XAl3− (X = Si, Ge, Sn, Pb) systems. Angew. Chem., Int. Ed. 2001, 40, 1867–1870. 50. Li, Z.; Zhao, C.; Chen, L. Spectroscopic properties of aromatic heterocyclic systems: XAl3 (X = Si, Ge, Sn, Pb) and their anions and cations. J. Mol. Struct. (THEOCHEM). 2007, 809, 45–54. 51. Chi, X. X.; Li, X. H.; Chen, X. J.; Yuang, Z. S. Ab inition studies on the aromaticity of bimetallic anionic clusters XGa 3− (X = Si, Ge). J. Mol. Struct. (THEOCHEM) 2004, 677, 21–27. 52. Kuznetsov, A. E.; Boldyrev, A. I. Theoretical evidence of aromaticity in X3− (X = B, Al, Ga) species. Struct. Chem. 2002, 13, 141–148. 53. Yang, L.; Ding, Y.; Sun, C. Design of the sandwich-like compounds based on the allmetal aromatic unit Al3−. Chem. Phys. Chem. 2006, 7, 2478–2482. 54. Yang, L.; Li, X.; Ding, Y.; Sun, C. Theoretical study on a class of organometallic complexes based on all-metal aromatic Ga 3− through sandwiching stabilization. Eur. J. Inorg. Chem. 2008, 2099–2106. 55. Alexandrova, A. N.; Boldyrev, A. I. σ-Aromaticity and σ-antiaromaticity in alkali metal and alkaline earth metal small clusters. J. Phys. Chem. A. 2003, 107, 554–560. 56. Havenith, R. W. A.; De Proft, F.; Fowler, P. W.; Geerlings, P. Aromaticity in H3+ and Li3+: Insights from ring-current maps. Chem. Phys. Lett. 2004, 407, 391–396. 57. Yong, L.; Wu, S. D.; Chi, X. X. Theoretical study of aromaticity in small hydrogen and metal cation clusters X3+ (X = H, Li, Na, K, and Cu). Int. J. Quant. Chem. 2007, 107, 722–728. 58. Foroutan, C.; Rashidi-Ranjbar, P. Chemical Bonding in the lightest tri-atomic clusters: H3+ , Li3+ and B3−. J. Mol. Struct. (THEOCHEM) 2009, 901, 243–248. 59. Kuznetsov, A. E.; Boldyrev, A. I. A Single π-bond captures 3, 4 and 5 atoms. Phys. Lett. 2004, 388, 452–456. 60. Chattaraj, P. K.; Giri, S. Variation in aromaticity and bonding patterns in a reaction cycle involving Be32 − and Mg32 − dianions. J. Mol. Struct. (THEOCHEM). 2008, 865, 53–56.

Aromaticity in Metals

67

61. Roy, D. R.; Chattaraj, P. K. Selectivity, and aromaticity of Be32− and its complexes. J. Phys. Chem. A. 2008, 112, 1612–1621. 62. Jimenez-Halla, J. O. C.; Matito, E.; Blancafort, L.; Robles, J.; Sola, M. Tuning aromaticity in trigonal alkaline earth metal clusters and their alkali metal salts. J. Comput. Chem. 2009, 30, 2764–2776. 63. Chattaraj, P. K.; Roy, D. R.; Duley, S. Bonding and aromaticity in an all-metal sandwich-like compound, Be82−. Chem. Phys. Lett. 2008, 460, 382–385. 64. Chi, X. X.; Liu, Y. Theoretical evidence of d-orbital aromaticity in anionic metal X3− (X = Sc, Y, La) clusters. Int. J. Quant. Chem. 2007, 107, 1886–1896. 65. Yong, L.; Chi, X. X. Theoretical study on the aromaticity of dianions X32− 76 76(X = Zn, Cd, Hg). J. Mol. Struct. (THEOCHEM) 2007, 818, 93–99. 66. Averkiev, B. B.; Boldyrev, A. I. Hf3 Cluster is triply (σ-, π-, and δ-) aromatic in the lowest D3h 1A1’ state. J. Phys. Chem. A 2007, 111, 12864–12866. 67. Wang, B.; Zhai, H. J.; Huang, X.; Wang, L. S. On the electronic structure and chemical bonding in the tantalum trimer cluster. J. Phys. Chem. A 2008, 112, 10962–10967. 68. Tanaka, H.; Neukemans, S.; Janssens, E.; Silverans, R. E.; Lievens, P. Aromaticity of the bimetallic Au5Zn+ cluster. J. Am. Chem. Soc. 2003, 125, 2862–2863. 69. Holtzl, T.; Janssens, E.; Veldeman, N.; Veszpremi, T.; Lievens, P.; Nguyen, M. T. The Cu7Sc cluster is a stable σ-aromatic seven-membered ring. Chem. Phys. Chem. 2008, 9, 833–838. 70. Holtzl, T.; Veldeman, N.; Veszpremi, T.; Lievens, P.; Nguyen, M. T. Cu6Sc+ and Cu5Sc: Stable, high symmetry and aromatic scandium-doped coinage metal clusters. Chem. Phys. Lett. 2009, 469, 304–307. 71. Wannere, C. S.; Corminboeuf, C.; Wang, Z. X.; Wodrich, M. D.; King, R. B.; Schleyer, P. v. R. Evidence for d orbital aromaticity in square planar coinage metal clusters. J. Am. Chem. Soc. 2005, 127, 5701–5705. 72. Lin, Y. C.; Sundholm, D.; Juselius, J.; Cui, L. F.; Li, X.; Zhai, H. J.; Wang, L. S. Experimental and computational studies of alkali-metal coinage-metal clusters. J. Phys. Chem. A. 2006, 110, 4244–4250. 73. Kong, Q.; Chen, M.; Dong, J.; Li, Z.; Fan, K.; Zhou, M. Matrix infrared spectrum and aromaticity of the Al2(CO)2 molecule. J. Phys. Chem. A, 2002, 106, 11709–11713. 74. Tsipis, C. A.; Tsipis, A. C. Hydrometal Analogues of aromatic hydrocarbons: A new class of cyclic hydrocoppers(I). J. Am. Chem. Soc. 2003, 125, 1136–1137. 75. Tsipis, C. A.; Karagiannis, E. E.; Kladou, P. F.; Tsipis, A. C. Aromatic gold and silver ‘Rings’: Hydrosilver(I) and hydrogold(I) analogues of aromatic hydrocarbons. J. Am. Chem. Soc. 2004, 126, 12916–12929. 76. Tsipis, C. A.; Tsipis, A. C. Ligand-stabilized aromatic three-membered gold rings and their sandwichlike complexes. J. Am. Chem. Soc. 2005, 127, 10623–10638. 77. Tsipis, A. C.; Stalikas, A. V. A new class of “all-metal” aromatic hydrido-bridged binary coinage metal heterocycles. A DFT study. New J. Chem. 2007, 31, 852–859. 78. Kuznetsov, A. E.; Zhai, H. J.; Wang, L. S.; Boldyrev, A. I. Peculiar antiaromatic inorganic molecules of tetrapnictogen in Na + Pn −4 (Pn = P, As, Sb) and important consequences for hydrocarbons. Inorg. Chem. 2002, 41, 6062–6070. 79. De Proft, F.; Fowler, P. W.; Havenith, R. W. A.; Schleyer, P. v. R.; Van Lier, G.; Geerlings, P. Ring currents as probes of the aromaticity of inorganic monocycles: P5− , As5− , S2 N 2 , S3N3− , S4 N3+ , S4 N32 + , S5N 5+ , S24 + and Se 24 +. Chem. Eur. J. 2004, 10, 940–950. 80. Gausa, M.; Kaschner, R.; Seifert, G.; Faehmann, J. H.; Lutz, H. O.; Weiwes-Broer, K.-H. Photoelectron investigations and density functional calculations of anionic Sb −n and Bi −n clusters. J. Chem. Phys. 1996, 104, 9719–9729. 81. Lein, M.; Frunzke, J.; Frenking, G. A. Novel class of aromatic compounds: Metal-­ centered planar cations [Fe(Sb5)]+ and [Fe(Bi5)]+. Angew. Chem. Int. Ed. 2003, 42, 1303–1306.

68

Aromaticity and Metal Clusters

82. Zhai, H.-J.; Wang, L. S.; Kuznetsov, A. E.; Boldyrev, A. I. Probing the electronic structure and aromaticity of pentapnictogen cluster anions Pn 5− (Pn = P, As, Sb, and Bi) using photoelectron spectroscopy and Ab initio calculations. J. Phys. Chem. 2002, 106, 5600–5606. 83. Huang, X.; Zhai, H. J.; Kiran, B.; Wang, L. S. Observation of d-orbital aromaticity. Angew. Chem. Int. Ed. 2005, 44, 7251–7254. 84. Zhai, H. J.; Averkiev, B. B.; Zubarev, D. Y.; Wang, L. S.; Boldyrev, A. I. δ aromaticity in [Ta3O3]-. Angew. Chem. Int. Ed. 2007, 46, 4277–4280. 85. Li, S. D.; Miao, C. Q.; Guo, J. C. [Ta3O3]A (A = Li, Na, K) and [Ta3O3]B[Ta3O3] (B = Ca, Sr, Ba) : Sandwich-type complexes containing Ta3O3-δ and π double aromatic ligands. Eur. J. Inorg. Chem. 2008, 1205–1209. 86. Zhai, H. J.; Wang, B.; Huang, X.; Wang, L. S. Probing the electronic and structural properties of the Niobium trimer cluster and its mono- and di-oxides: Nb3O −n and Nb3On (n = 0–2). J. Phys. Chem. A 2009, 113, 3866–3875. 87. Kuznetsov, A. E.; Corbett, J. D.; Wang, L. S.; Boldyrev, A. I. Aromatic mercury clusters in ancient amalgams. Angew. Chem. Int. Ed. 2001, 40, 3369–3372. 88. Corminboeuf, C.; Schleyer, P. v. R.; King, R. B. Aromaticity of tri- and tetranuclear metal-carbonyl clusters based on magnetic criteria. Chem. Eur. J. 2006, 12, 978–984. 89. King, R. B. Metal cluster topology. 21. Sigma aromaticity in triangular metal carbonyl clusters. Inorg. Chem. Acta 2003, 350, 126–130. 90. Alvarado-Soto, L.; Ramirez-Tagle, R.; Arratia-Perez, R. Spin-orbit effects on the aromaticity of the Re3Cl9 and Re3Br9 clusters. Chem. Phys. Lett. 2008, 467, 94–96. 91. Alvarado-Soto, L.; Ramirez-Tagle, R.; Arratia-Perez, R. Spin-orbit effects on the aromaticity of the Re3X 2− (X = Cl, Br) cluster ions. J. Phys. Chem. A. 2009, 113, 9 1671–1673. 92. Kuznetsov, A. E.; Corbett, J. D.; Wang, L. S.; Boldyrev, A. I. Aromatic mercury clusters in ancient amalgams. Angew. Chem. Int. Ed. 2001, 40, 3369–3372. 93. Tsipis, A. C.; Kefalidis, C. E.; Tsipis, C. A. The role of the 5f orbitals in bonding, aromaticity, and reactivity of planar isocyclic and heterocyclic uranium clusters. J. Am. Chem. Soc. 2008, 130, 9144–9155. 94. Boldyrev, A. I.; Wang, L. S. All-metal aromaticity and antiaromaticity. Chem. Rev. 2005, 105, 3716–3757. 95. Chen, Z.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; Schleyer, P. v. R. Nucleusindependent chemical shifts (NICS) as an aromaticity criterion. Chem. Rev. 2005, 105, 3716–3757. 96. Tsipis, C. A. DFT study of “all-metal” aromatic compounds. Coord. Chem. Rev. 2005, 249, 2740–2762. 97. Datta, A.; Mallajosyula, S. S.; Pati, S. K. Nonlocal electronic distribution in metallic clusters: A critical examination of aromatic stabilization. Acc. Chem. Res. 2007, 40, 213–221. 98. Zubarev, D. Y.; Averkiev, B. B.; Zhai, H. J.; Boldyrev, A. I.; Wang, L. S. Aromaticity and antiaromaticity in transition-metal systems. Phys. Chem. Chem. Phys. 2008, 10, 257–267. 99. Zubarev, D. Y.; Boldyrev, A. I. Developing paradigm of chemical bonding: Adaptive natural density partitioning. Phys. Chem. Chem. Phys. 2008, 10, 5207–5217. 100. Todorov, I.; Sevov, S. C. Heavy-metal aromatic rings: Cyclopentadienyl anion analogues Sn 65 − and Pb65 − in the zintl phases Na8BaPb6, Na8BaSn6, and Na8EuSn6. Inorg. Chem. 2004, 43, 6490–6494. 101. Xie, Y.; Scheiner, P. R.; Schaefer, H. F. III.; Li, X. W.; Robinson, G. H. Are cyclogallenes [M2(GaH)3] (M = Li, Na, K) aromatic? J. Am. Chem. Soc. 1996, 118, 10635–10639.

Studies 4 Computational on Molecules with Unusual Aromaticity What to Expect? Ayan Datta, Sairam S. Mallajosyula, and Swapan K. Pati Contents 4.1 Introduction..................................................................................................... 69 4.2 Establishing Criterion for Aromaticity in Heteroatomic Compounds................70 4.3 Stability of Tiara Nickel Thiolates: Influence of Aromatic Interactions.................................................................................................... 78 4.4 To What Extent Benzene May Pucker and Yet Remain Aromatic?................ 82 4.5 Hydrogen Adsorption on Octathio[8]circulene: Effect of Aromaticity.............. 86 4.6 Summary.........................................................................................................90 Acknowledgments..................................................................................................... 91 References................................................................................................................. 91

4.1 Introduction Recent syntheses of many new and novel aromatic molecules have challenged the theorists to explain their unusual structural features and stability [1–5]. Over the last few years, there has been a tremendous progress in the area of multiatom inorganic metal complexes that form interesting two-dimensional (cyclic) and three-­ dimensional (cages) structures [6–9]. With the list of such high-resolution crystal structures ever expanding, provides theorists a wonderful starting point for elaborate computational studies, since, within the time-scale of high-quality crystallization, the thermodynamically controlled product is mostly formed. Although aromaticity is a frequently used concept in the chemical literature, it lacks an unambiguous basis. It has no precise quantitative definition and cannot be directly measured experimentally. Aromaticity is thus described by various criteria, which can be broadly classified into three categories: structure, energy, and magnetic properties [10,11]. The structural features manifest in charge delocalization and thereby energy-lowering phenomena with unusual chemical and physical properties relating to reactivity, spectroscopic features, and magnetic ring currents [11]. 69

70

Aromaticity and Metal Clusters

The advances made in quantum chemistry allow one to characterize aromatic molecules and aromaticity in terms of descriptors that cover the three categories described above.





1. Structural Criteria: The advances in density functional theory (DFT) and perturbation methods now allow theorists to calculate the structural ­features of molecules with appreciable match to crystallographic structures [12]. Methods such as MP2 [13] have been proved to be successful to even describe weak interactions such as hydrogen bonds to a reasonable accuracy [14]. For aromaticity one generally looks at the equalization of bond lengths, which acts as a structural evidence for aromatic stabilization. 2. Energy Criteria: It is known that aromaticity leads to the stabilization of the molecular system. Thus aromatic stabilization energies (ASE), have been regularly calculated for aromatic systems using isodesmic, homodesmotic, or hyperhomodesmotic reaction schemes. 3. Magnetic Criteria: This has been the main characterization criterion for the aromatic molecules from the theoretical standpoint. There are various methods, which have been developed that can describe the magnetic properties for aromatic compounds. Some of them are, multiple aromaticity (simultaneous presence of σ- and π-aromaticity) [15], mapping ring currents, aromatic ring current shielding (ARCS) [16], nuclear-independent chemical shift (NICS) [17], bifurcation analysis of the electron localization function (ELF) [18] and many more. For a comprehensive review of the various techniques and the developments in the field of aromaticity, the readers are referred to two volumes of Chemical Reviews that were edited by Professor P. v. R. Schleyer [1,11].

In this chapter, we first discuss the applicability of the quantum chemical c­ alculations in describing the aromatic/antiaromatic behavior in representative ­inorganic compounds such as B3N3H6, N3P3F6, B2N2H4, and N2P2F4. We then use aromaticity as a tool to derive an atomistic understanding of the structures and prop­ erties in three diverse experimentally determined structures, tiara-Ni complexes, puckered benzene rings in hexaferrocenyl benzene, and recently discovered highly symmetric “sulflower” molecule. We note that aromaticity provides a tool for a broad understanding of structural features and stability in these classes of molecules. We also show that aromaticity provides chemically intuitive means to predict advanced electronic and energy storage applications in these new molecular materials.

4.2 Establishing Criterion for Aromaticity in Heteroatomic Compounds Many in the literature have debated the aromatic/antiaromatic characteristics of B–N and P–N analogues of benzene and cyclobutadiene [19]. In this work [20], we have  used established parameters such as NICS [17], charge density at the ring ­critical point (ρRCP) [21], and stabilization energies to quantify the nature of interactions in these molecular systems. All the geometries for the molecular systems were

Computational Studies on Molecules with Unusual Aromaticity

71

fully optimized at the DFT [12] level using the Becke, Lee, Yang, and Parr three-­ parameter correlation functional (B3LYP) at the 6-311G ++(d,p) basis set level [22]. All the calculations were performed using the Gaussian 03 set of programs [23]. Additional calculations at the MP2 level were performed to further verify the groundstate low-energy structures for these molecules. Frequency calculations were also performed to confirm the ground-state geometries [13]. We present the optimized ground-state geometries of C6H6, C4H4, B3N3H6, B2N2H4, N3P3H6, and N2P2F4 in Figure 4.1a–f, respectively. We used the bond-length alteration (BLA), defined as the average difference between the bond lengths of two consecutive bonds, as a criterion to quantify the structural features of the molecular systems under consideration. The BLA for the isocyclic systems C6H6 and C4H4 were found to be 0.00 and 0.24 Å, indicative of the aromatic and antiaromatic features, respectively. It was found that while the ­six-membered heterocyclic clusters, B3N3H6 and N3P3F6, have a 0.00 BLA with ­high-symmetric hexagonal structure, the four-membered rings, B2N2H4 and N2P2F4 are rhombohedral with equal bond lengths and unequal diagonal lengths. For B2N2H4, the shorter and longer diagonals are found to be 1.90 and 2.15 Å, respectively, and for N2P2F4, they are 2.15 and 2.47 Å, respectively. Thus, from a structural analysis, it becomes evident that for the homocyclic C4H4, the static Jahn–Teller (JT) distortions lead to a rectangular geometry from a square geometry, while for the heterocyclic four-membered ring systems, such distortions lead to a rhombohedral geometry. From a structural viewpoint, the case of the four-membered B–N compound, B2N2H4, requires a special mention. We found that the ground-state structure corresponds to a puckering of 17.3° from planarity. However, the bond lengths are all equivalent, suggesting that the lone pair of electrons on the N atom are localized and are not transferred to the nearby B atom, and a resonance form similar to that for C4H4 (two alternate short and long bonds) is not realized. In fact, the planar structure for B2N2H4 is found to be 1.00 kcal/mol higher in energy compared with the puckered structure and has one imaginary frequency corresponding to the out-of-plane bending mode of the atoms. However, a difference of 1 kcal/mol in energies between these two structures is comparable to the thermal energy at room temperature (0.6 kcal/mol). Thus, there is a possibility for such four-membered rings to exist in two different polymorphs: the planar and the puckered forms. On performing a search for such polymorphs in the Cambridge Crystallographic Database [24] (CCSD) for the four-membered B2N2 ring systems, we retrieved 47 crystallographic structures. Of them, two compounds showed polymorphism in the ring structure. The compound, 1,3-di-tert-butyl-2,4-bis(pentafluorophenyl)-1,3,2,4- diazadiboretidine (CCSD code: BFPDZB) crystallizes in a I2/c point group and has a planar B2N2 unit [25], while the molecule, tetrakis(tert-butyl)-1,3,2,4-diazadiboretidine crystallizes (CCSD code: CETTAW) with a point group of Pc and has a puckered B2N2 unit with a puckering angle of 18° [26]. We note that these differences are indicative of the effects of charge transfer (CT) in these systems. In Figure 4.2, we present the highest occupied molecular orbitals (HOMOs) for the above-mentioned molecular systems. For the homocyclic molecular systems C6H6 and C4H4 (Figure 4.2a and b), we find that the nodal plane passes through the bonds and the molecular orbitals (MOs) are delocalized over all the

72

Aromaticity and Metal Clusters

(a)

(b) 1.57

1.40

1.33

1.40

(c)

(d)

1.43

1.43 1.45

1.45

(e) (f)

1.58

1.63

1.63

1.58

Figure 4.1  Ground-state optimized geometries of (a) C6H6, (b) C4H4, (c) B3N3H6, (d) B2N2H4, (e) N3P3F6, and (f) N2P2F4. Bond lengths in Å shown for each structure. (From Rehaman, A. et al. J. Chem. Theory Comput. 2006, 2, 30–36. With permission.)

Computational Studies on Molecules with Unusual Aromaticity (a)

(b)

(c)

(d)

(e)

(f )

73

Figure 4.2  Highest occupied molecular orbitals (HOMOs) of (a) C6H6, (b) C4H4, (c) B3N3H6, (d) B2N2H4, (e) N3P3F6, and (f) N2P2F4. (From Rehaman, A. et al. J. Chem. Theory Comput. 2006, 2, 30–36. With permission.)

atoms. But, for B3N3H6 and N3P3F6 (Figure 4.2c and e), the MOs manifest electronegativity differences of constituent atoms. The effect of electronegativity differences becomes prominent in case of the four-membered ring systems, B2N2H4 and N2P2F4 (Figure 4.2d and f). We find that the contributions from the electronegative N are very large when compared to the negligible contributions from the electropositive atoms, suggesting CT from the N orbital to the vacant orbitals ( pz orbital of B and dxz  and dyz orbitals of P). As expected on the basis of symmetry and relative

74

Aromaticity and Metal Clusters

e­ lectronegativities, for B2N2H4 (Figure 4.2d), the node passes through the less ­electronegative B atoms. In contrast, the node passes through the longer C−C bonds for C4H4 (Figure 4.2b). With an understanding of the structural features, we proceeded with a quantitative estimation of aromaticity/antiaromaticity in these systems. For this, we have calculated the NICS [17], the charge density (ρRCP) and its Laplacian (∇ 2ρRCP) at the ring critical point. Recent studies have established the relationship between these topological descriptors and aromaticity/antiaromaticity for molecules with similar architecture [27]. At the outset, we summarize our results (magnitudes of the BLA, stabilization energies, NICS, ρRCP, and ∇ 2ρRCP) for all the systems in Table 4.1. We calculate the stabilization energies as the difference in energy between the molecules reported and the independent fragments such as Cn H n → nC2 H 2 Bn N nH 2n → nBNH 2 N n Pn F2n → nNPF2





These reported energies are corrected for thermal parameters (zero-point energies and the entropy corrections). Note that, the stabilization energies for the weakly interacting systems are also corrected for basis set superposition errors using the counterpoise corrections scheme [28]. As evident from Table 4.1, all the systems except C4H4 show aromaticity (negative NICS). Among the six-membered rings, the aromaticity in the systems follows the order C6H6 > N3P3F6 > B3N3H6, following the order of decreasing covalency in these systems. For C6H6, the conjugation is most effective because of pπ–pπ overlap, while it decreases for N3P3F6, because of the less-effective pπ–dπ overlap. For borazine, however, such orbital overlap is poor, and Table 4.1 Magnitudes of BLA in Å, Stabilization Energies in kcal/mol (ΔE), NucleusIndependent Chemical Shift (NICS) in ppm, Charge Density at the Ring Critical Point (ρRCP) in e/Å3 Units, and Laplacian of the Charge Density (∇ 2ρRCP) in e/Å5 Units for the Systems Considered Systems

BLA

ΔE

NICS

ρRCP

∇2ρRCP

C6H6 C4H4 B3N3H6 B2N2H4 N3P3F6 N2P2F4

0.00 0.24 0.00 0.00 0.00 0.00

−219.4 −52.3 −142.2 −49.4 −238.5 −105.6

−8.1 35.8

0.022 0.102 0.020 0.099 0.020 0.104

0.161 0.458 0.119 0.369 0.095 0.210

−1.6 −2.9 −6.9 −10.9

Source: From Rehaman, A. et al. J. Chem. Theory Comput. 2006, 2, 30–36. With permission.

Computational Studies on Molecules with Unusual Aromaticity

75

the stabilization in B3N3H6 is primarily due to CT from N to B. In the four-membered ring systems, aromaticity follows the order N2P2F4 > B2N2H4 > C4H4 (antiaromatic). The decrease in aromaticity from N2P2F4 to B2N2H4 arises as a result of stronger covalency in the N–P bond compared to that of the B–N bond. An analysis of the charge density (ρRCP) and the Laplacian of the charge density (∇ 2ρRCP) at the ring critical points for these systems reveal clear distinctions between the nature of ­interactions in the rings. Both ρRCP and ∇ 2ρRCP show maximum localizations for the four-membered rings C4H4, B2N2H4, and N2P2F4, followed by the six-membered rings C6H6, B3N3H6, and N3P3F6. Thus, both NICS and topological aspects suggest ­substantial electronic delocalizations across the rings. However, the delocalization of the π electrons over the cyclic architectures differs for the homocyclic and heterocyclic systems. Unlike carbon, nitrogen and phosphorus do not have a straightforward σ–π separation of their lower energy levels. One of the most direct methods for considering the role of σ and π electrons is to separate the total energy of the system into σ and π components [15]. For realizing the σ contribution to the structure, we consider the highest spin (H. S.) state for the systems and freeze all the π electrons in the H. S. configuration. Thus, for the six-membered rings such as C6H6, B3N3H6, and N3P3F6, the H. S. state corresponds to S = 3, while for the four-membered systems such as C4H4, B2N2H4, and N2P2F4, the H. S. state has a spin of S = 2. The π energy for a system is then calculated as E(π) = E(G. S.) − E(H. S.), where E(G. S.) corresponds to the energy of the ground state which is a singlet (S = 0) state. This method of σ–π separation for both organic and inorganic molecules has been successfully benchmarked by us [29]. In Figure 4.3, we report this σ–π analysis for C6H6, C4H4, B3N3H6, and N3P3F6, as a function of distortion (BLA) in the rings using ΔE(π) = ΔE(G. S.) − ΔE(H. S.), where the energies are scaled so that the most stable structure corresponds to the zero of energy. Note that, we define ΔE(H. S.) = ΔE(σ). For benzene (Figure 4.3a), the symmetric D 6h structure (BLA = 0.00), is associated with the stabilization of the σ energy, while the π energy stabilizes the distorted structure. The energy scale for σ equalization overwhelms the π distortion (by 20 kcal/mol), and thus, the symmetric structure is stabilized in case of benzene. One can clearly observe the role of the σ energies in controlling the structure of benzene. Similar results have also been reported previously [30]. Contrary to the situation for benzene, C4H4 (Figure 4.3b) shows π distortion overwhelming σ equalization. Thus, the distorted D2h structure is stabilized over the undistorted structure. Note that, for these homocyclic systems, we derive results identical to those well-known from π-only electron theories ­claiming benzene to be aromatic (BLA = 0.00) and C4H4 to be JT-distorted antiaromatic. The heterocyclic B–N and P–N systems also show similar electronic features (Figure 4.3c and d). B3N3H6 is identical to benzene in being σ equalized and π distorted. However, compared to benzene, the σ equalization energy is smaller (by 5 kcal/mol), suggesting B3N3H6 to be less aromatic, a result already derived from both NICS and topological analysis. N3P3F6, on the contrary, shows double equalization, and both π and σ energies stabilize the BLA = 0.00 structure. While, σ equalization is expected for a cyclic structure, π equalization suggests the predominating p(π)–d(π) delocalizations.

76

Aromaticity and Metal Clusters

–0.1

BLA 0

0.1

0.2

–0.05

BLA 0

0.05

0.1

(b)

25

20 15 10

Energy

Energy

–0.2 30 (a)

5

10

10

5

0 –0.2

–0.1

–0.2

–0.1

0 BLA BLA 0

(e)

0.1

0.2

–0.12

0.1

0.2

–0.12

0 BLA BLA –0.02 0.08

(f )

Energy

15

(d)

20

25

Energy

(c)

0

0.12

0.18

15 10

15 5

Energy

Energy

30

5

Energy

15

0

(g)

10 5 0 –0.2

–0.1

0 BLA

0.1

0.2

Figure 4.3  σ–π separation of energies for (a) C6H6, (b) C4H4, (c) B3N3H6, (d) N3P3F6, (e) N2P2F4, (f) B2N2H4 (planar), and (g) B2N2H4 (puckered). Energies are reported in kcal/mol and BLA in Å. Circles and squares correspond to σ and π energies, respectively. (From Rehaman, A. et al. J. Chem. Theory Comput. 2006, 2, 30–36. With permission.)

N2P2F4 (Figure 4.3e), on the contrary, is σ distorted but π equalized (with π e­ qualization > σ distortion by 10 kcal/mol), suggesting strong π delocalization ­overwhelming minor JT distortion. Thus, N2P2F4 may be considered as π aromatic. In B2N2H4, for both the planar (Figure 4.3f) and the puckered structures (Figure 4.3g), σ equalization overwhelms π distortion (by 10 kcal/mol). Thus, both the structures correspond to predominantly σ aromatic with BLA = 0.00 geometries.

77

Computational Studies on Molecules with Unusual Aromaticity

Energy

0.1

0.2

–0.12

–0.02

0.08

0.18

(b)

100 (c)

(d)

300

400 200 1500

500 300

300 600

Energy

0

100 (e)

600

(f )

1000

400

500

200

0 –0.2

–0.1

0 BLA

0.1

0.2

Energy

600

–0.1

(a)

–0.12

–0.02 BLA

0.08

0.18

Energy

Energy

–0.2

BLA

Energy

BLA

0

Figure 4.4  Variation of individual energy components (in kcal/mol) Vee (circles), Vnn (squares), Vne (triangles), and VKE (stars) with BLA in Å for (a) C6H6, (b) C4H4, (c) B3N3H6, (d) B2N2H4 (planar), (e) N3P3F6, and (f) N2P2F4. (From Rehaman, A. et al. J. Chem. Theory Comput. 2006, 2, 30–36. With permission.)

From the above σ–π analysis, it is clear that JT distortion in the backbone leads to structures with large BLAs. However, for a more insightful understanding of the stabilization preference, we have performed an analysis of the fragmentation of the total energy into contributions from the nuclear–nuclear (Vnn ), electron–nuclear (Ven ), electron–electron (Vee ), and kinetic energy (K.E.) components as a function of BLA [31]. The results for each of the systems are shown in Figure 4.4. For all cases, the electron–nuclear (Ven ) component favors distortion, while Vnn, Vee, and K.E. have a preference for the undistorted structure. The Vnn, Vee, and K.E. components are ­stabilized in structures with BLA = 0.00 as they are associated with a complete delocalization of electrons across the ring. For large BLAs, electrons are localized in the shorter bonds. Thus, the actual preference for the highly symmetric or distorted structure is governed by the competition between all other components and Ven. In C6H6, Ven is overwhelmed by the other components (Figure 4.4a), while in the case of C4H4, Ven is the major component (Figure 4.4b) and the structure as already discussed, is overall distorted. The preference for the heterocyclic systems such as B3N3H6 (Figure 4.4c), B2N2H4 (planar) (Figure 4.4d), N3P3F6 (Figure 4.4e), and N2P2F4 (Figure 4.4f) adapting to a highly symmetric structure (BLA = 0.00) is also clearly understood from the fact that, for these systems, Ven is only a minor component. Thus, we note that organic molecules such as C6H6 and C4H4 are stabilized through isotropic delocalization of the π electrons over the full perimeter of the rings. The

78

Aromaticity and Metal Clusters

CT and p(π)–d(π) interactions in B3N3H6 and N3P3F6, respectively, lead to aromaticity in these systems although the aromatic character is less than that of benzene. Four-membered heterocyclic systems such as N2P2F4 and B2N2H4 are also found to be aromatic when compared with their antiaromatic homocyclic analogue, C4H4. We observe that the quantum chemical descriptors such as nucleus-independent chemical shifts, charge density, and the laplacian at the ring critical point act as excellent descriptors to quantify aromaticity. We further use these descriptors to study three novel classes of materials in subsequent sections.

4.3 Stability of Tiara Nickel Thiolates: Influence of Aromatic Interactions Metal thiolates exhibiting symmetric cyclic structures are of great interest, not just because of their unusual stability, but also because of their biological importance as active sites in enzymes [32–34]. Tiara Ni-thiolates have a general formula of [Ni(SR)2]n where R is an alkyl group like CH3, C2H5, C3H7, C4H9, and C5H11 and n represents the nuclearity of the cluster. High-resolution crystal structures of the trimer (n = 3) [CSD code: PCNTNI], tetramer (n = 4) [CSD code: CELXAS], ­pentamer (n = 5) [CSD code: FOTVEP], and hexamer (n = 6) [CSD code: JOXMOY] are available in CCSD [24]. Interestingly, the cyclic architectures have a planar Nin backbone, which is perfectly polygonal with the ∠Ni–Ni–Ni very close to the exact (n − 2) × 180/n value. In Table 4.2, we compare the experimental and the gas-phase computed structural parameters for (R = C2H5) Nin rings. The optimized geometries for these molecules were obtained at the B3LYP/LANL2MB level [22,23]. In Figure 4.5, we present the computed structures [35,36]. The differences observed in the Ni–Ni distances between the computed and the experimental structures are mainly due to the ligands that are attached to sulfur in the latter case, in addition to the kinetic factors that are involved in the stabilization of the crystal structures. The trimeric (n = 3) structure represents maximum pyramidal distortion with small Ni–Ni distance that incorporates ring strain analogous to cyclopropane. On the contrary, the Ni6 ring is hexagonal, very similar to benzene. Thus, these systems are available for comparison with their organic counterparts. The trans ∠S–Ni–S measures the extent of pyramidalization, and a value of 180° represents perfect planarity of the Ni atom with the neighboring S4 ring. Thus, the value, (180-trans S–Ni–S)° quantifies the pyramidalization. As can be seen in Table 4.2, the pyramidalization angle decreases with increase in n. Furthermore, with increase in ring size, n, the tendency for distortion in the S4 ring increases, and as a result, the square-like S4 ring in the trimer gets distorted into a rectangular geometry in the case of a hexamer with BLA of 0.73 Å. In fact, such a distortion reduces the pyramidalization of the Ni atom and brings the Ni atoms in plane with the S4 ring. Similar trends have been observed in longer alkyl chain lengths as well. To quantify aromaticity in these systems, we calculated NICS at the center of the Nin rings at the GIAO/RB3LYP LANL2MB level (see Figure 4.6a) [17]. As can be seen from Figure 4.6a, the magnitude of NICS in all these systems is positive. For a given nuclearity (n), NICS increases with an increase in the number of alkyl groups (m). For example, for n = 3, the magnitude of NICS increases from 22

79

Computational Studies on Molecules with Unusual Aromaticity

Table 4.2 Variations of the Ni–Ni Distances (in Å) and Polygonal Angle and trans S–Ni–S Angle (both in deg) for the Optimized and Experimental Molecules of [Ni(SC2H5)2]n n

d(Ni–Ni) Computed (exptl)

∠Ni–Ni–Ni Computed (exptl)

∠Trans S–Ni–S Computed (exptl)

3 4 5 6

2.61 (2.64) 3.02 (2.66) 3.14 (2.82) 3.35 (2.92)

59.9 (59.72) 90.17 (89.25) 107.96 (107.97) 120.02 (119.99)

160.7 (150.04) 167.3 (168.89) 178.48 (171.15) 179.6 (177.50)

Source: From Datta, A. et al. J. Phys. Chem. A. 2005, 109, 11647–11649. With permission.

to 48 ppm for R = H to R = C5H11. This is revealed in the Mulliken charges as well. The charge on the S atom increases from −0.28e to −0.45e (R = H to C5H11), following an increase in the +I (inductive) effects of the alkyl groups resulting in charge localization on the bridged S atoms. This in turn reduces charge transfer between the Ni atoms, thereby reducing the diamagnetic current (aromaticity). Also, very interestingly, NICS tends to decrease with the nuclearity (n), suggesting more facile delocalization for the larger tiara complexes compared to the smaller nuclearity systems.

n=3

n=5

n=4

n=6

Figure 4.5  Ground-state geometries for cyclic [Ni(SR)2]n (R = C2H5) for n = 3, n = 4, n = 5, and n = 6. Note that Ni atoms share the edges of a regular polygon. (From Datta, A. et al. J. Phys. Chem. A. 2005, 109, 11647–11649. With permission.)

80

Aromaticity and Metal Clusters (a)

50

30

NICS

40

20 10 0

5

No 4 .o fa

n3

lky

3

lg

n4

2

ro u

ps

n5

(m

1

)

0

n6

(b) C

C 25.53 kcal/mol

C

C

Ni

Ni

cle Nu

ty ari

of

Ni

) (n ms o at

C

C

C

C

Ni

Ni

Ni

Ni

15.34 kcal/mol

Ni

Ni

Figure 4.6  (a) GIAO/NICS values (in ppm) for the tiara complexes with increasing numbers of methylene groups (m) and increasing nuclearity (n). Note that all NICS values are positive. (b) Representation of favorable JT distortion in C4H4 and favorable D4h structure of Ni4 compared to the distorted D 2h geometry. Note that the H-atoms in C4H4 and S-atoms/ethyl groups in [Ni(SR)2]4 are not shown for clarity. (From Datta, A. et al. J. Phys. Chem. A. 2005, 109, 11647–11649. With permission.)

This is consistent with the smaller pyramidalization angles associated with the Ni atoms in larger-nuclearity rings. Note that, for simple organic molecules, a positive NICS value (paramagnetic ring current) suggests antiaromaticity, and they are expected to be distorted (nonzero BLA). However, the structures for all the Nin rings are perfectly planar polygonal,

Computational Studies on Molecules with Unusual Aromaticity

81

invariably suggesting aromatic characteristics. To have a better understanding of ­aromaticity/antiaromaticity in these systems, we performed a detailed analysis of the contributions of the 3d orbitals and the core orbitals (3s and 3p) of the Ni atoms and the 3p and 3s orbitals of the S atoms to all the MOs. The frontier orbitals (from HOMO to HOMO—5) have very little contributions from the 3d orbitals of the Ni atoms (~0.01), and the major contributions arise from the 3p orbitals of the S atoms. The contributions from the core orbitals for both Ni and S are very small for all the MOs. Thus, these frontier orbitals do not possess extended conjugation across the Ni and S atoms and thus essentially show antiaromaticity. However, as one considers the lower-energy orbitals (HOMO—6 and below), the contributions from the 3d orbitals of the Ni atoms start to increase by more than twofold, and in particular, the HOMO— 10 and HOMO—11 have very large contributions from the 3d orbitals of Ni. Also, very interestingly for these orbitals, the contributions from the 3p orbitals of sulfur atoms are also similar (~0.06). Therefore, it leads to extended delocalization across the Nin ring with almost equal contributions from the Ni atoms and the S atoms. The origin of the antiaromatic nature of the frontier orbitals is traced back to the symmetry of the d orbitals of the Ni atoms and the p orbitals of the S atoms involved in MO formation. For example, in the frontier MOs, the only contributing d orbitals are the d xz and dyz orbitals, while the major contributions from the S atoms arise from the 3py and 3pz orbitals. This results in poor overlap between the Ni and S atoms of the ring. However, for the low-energy MOs, the major contributions from the 3d orbitals are from the dz2 , d x2 − 2y and d xy orbitals. Note that, these orbitals possess the correct symmetry to undergo effective overlap with the S orbitals because of the square planar geometry of Ni. Thus, the low-energy orbitals are highly delocalized. Such multiple aromaticity features (antiaromatic frontier orbitals and aromatic lowenergy orbitals) are also evident from the plots of the highly delocalized orbitals. We have shown the lowest-energy orbitals, which fully share conjugations across the Nin ring for four n values in Figure 4.7. This is in sharp contrast to the pπ-conjugated aromatic systems like C6H6, where the frontier orbitals are highly delocalized. Thus, the highly symmetric structures in Nin arise from the greatly favorable delocalization of the d electrons in low-energy orbitals that lead to equalization of all the Ni–Ni bonds in the polygon. Note that, in the previous section we have discussed this scenario for C6H6, where the low-energy σ orbitals force a highly symmetric structure, while the π electrons of the frontier orbitals have an overwhelming ­tendency for a distorted D3h structure. The overall structure is of course governed by the more predominating σ equalization over the π distortion [37]. It is interesting to observe very similar features in these transition-metal complexes as well. The aromatic nature of these Ni thiolates is also evident from the fact that these structures do not exhibit the conventional antiaromatic JT distorted geometry as that in C4H4. This is schematically shown in Figure 4.6b, by comparing the distortion energies for C4H4 and its analogue, tetranuclear Ni thiolate. JT distortion in the D4h square structure of C4H4 leads to a stabilization of 25.5 kcal/mol and thereby stabilizes the D2h structure with BLA of 0.24 Å. However, the case for the Ni4 ring in [Ni(SR)2]4 (R = C2H5) is exactly opposite. In fact, JT distortion destabilizes the Ni system by 15.3 kcal/mol (for the same BLA as in C4H4). Thereby, for Ni thiolates, the highly symmetric D4h square geometry corresponds to the ground state.

82

Aromaticity and Metal Clusters (a)

(b)

Homo-10 (c)

Homo-10 (d)

Homo-13

Homo-11

Figure 4.7  Highly delocalized MOs for (a) trimer, (b) tetramer, (c) pentamer, and (d) hexamer. (From Datta, A. et al. J. Phys. Chem. A. 2005, 109, 11647–11649. With permission.)

4.4 To What Extent Benzene May Pucker and Yet Remain Aromatic? A very unusual molecule, hexaferrocenylbenzene (HFB) has been recently synthesized by Vollhardt and coworkers [38]. Synthesis of such a molecule has been long sought after because of its possible applications as multidecker single-molecule magnetic (SMM) material and also as a material for nanomechanics. However, from the molecular viewpoint, this represents an extreme example of a sterically crowded system with the six H-atoms of the benzene ring being replaced by six ferrocene molecules. Investigation of the crystal structure (CCSD 603715) reveals that for HFB [24,38], the central benzene ring loses its planarity and has a puckered chair geometry. The average puckering angle (defined as sum of the dihedral angles/6) for the benzene ring is 13.990°. This results in a distortion of the 1 and 4 para-carbon atoms from the plane (defined by atoms 2, 3, 5, and 6) by +0.148 Å and −0.153 Å which leads to a chair-type geometry. The fact that arranging ferrocene molecules around the benzene ring leads to puckering, must have a direct implication on the stability of the benzene ring. It is thus a fundamental question as to how the puckering mode of distortion, arising because of steric interactions around the benzene ring, affects its resonance stability. We have considered two specific modes of distortions: the chair form and the boat form (represented by [+d, −d] and [+d, +d] distortion in the 1,4positions in benzene respectively) [39]. The heat of hydrogenation (defined by the ­reaction: C6H6 + 3H2 = C6H12 ) is calculated for each step of distortion for both the

Computational Studies on Molecules with Unusual Aromaticity

83

cases. The calculations reveal that the heat of hydrogenation undergoes a transition from an endothermic nature to exothermic nature with increase in the distortion |d| > 0.3 Å. Thus, the aromatic stabilization of benzene exists till such critical distortions after which, however, benzene behaves as nonaromatic due to the poor overlap of the pz orbitals. The calculations for aromaticity through NICS suggest that the benzene ring is more aromatic in the boat form rather than in the chair form, for same magnitude of distortion (| d |). The geometry of the benzene ring for HFB is retrieved from the crystallographic information file [38]. The ferrocene molecules around the benzene ring are substituted by H-atoms and a constrained geometry optimization is performed for the positions of the H-atoms by freezing the coordinates for the six carbon atoms of the benzene ring. We use the B3LYP correlational function for nonlocal corrections at the 6-31 + G(d) basis set level. The optimized C–H bond length is 1.086 Å for the benzene ring in HFB structure. Also, apart from exhibiting puckering as discussed above, it also possesses an average BLA of 0.02 Å. The heat of hydrogenation is calculated as ΔE(hydrogenation) = E(C6H12) − [E(C6H6) + 3E(H2)]. The energy for C6H12 is calculated by optimizing the position of the hydrogen atoms while fixing the positions for the ­carbon atoms. ΔE(hydrogenation) for the benzene ring is found to be +2.082 kcal/mol. The positive value of ΔE(hydrogenation) indicates that the process is endothermic and thus the benzene ring in HFB structure retains its ­a romatic nature. For a comprehensive understanding of the variation in the stability of the benzene ring with increase in the puckering distortion in the chair and the boat forms, the ground-state optimized planar geometry of benzene is puckered by translating the positions of the C1 and the C4 atoms (along with the H-atoms associated with them) (above and below)/(and both above) (corresponding to +d/−d and +d/+d distortions). The schematic model is shown in Figure 4.8. The coordinates are varied so that we sample the full energy profile for the potential energy surface (PES) for the puckering distortion to chair and boat forms from the planar form. For each puckering angle, the heat of hydrogenation is calculated using the scheme as discussed above. In Figure 4.8a, the sum of the energies for benzene and 3H2 molecules and that for cyclohexane (in both the boat and chair forms) are plotted. The boat form corresponds to the positive X-axis and the chair form is associated with the negative X-axis. The point corresponding to the 0 on the X-axis is related with the planar undistorted benzene ring. While, cyclohexane (both the chair and the boat forms) is progressively stabilized with increase in the distortion angle, E(C6H6 + 3H2) is progressively destabilized. This can be easily understood from the fact that while the planar form (0 distortion) is the ground-state geometry for benzene, for cyclohexane, the puckered forms correspond to the ground-state geometry (the calculations reveal that the chair form is the true minima and is stabilized over the boat form by 12 kcal/ mol for a distortion of +0.6 Å). Very interestingly, at a puckering distortion of d = ±0.3 Å, the energies for two-states cross. In Figure 4.8b, the variation in the heat of hydrogenation of the benzene rings with the increase in the distortions is shown. Till a puckering distortion of ±0.3 Å, the hydrogenation process is endothermic and thus until such distortions, benzene

84

Aromaticity and Metal Clusters

(a) Distortion 6 5 Distortion

2 3

4

Displacement (d) –0.325 0 0.325

0.65

–50 Energy

1

–0.65 0

–100

0

–150

(c)

(b) 50

–2

0

NICS

–4

ΔE

–50

–6

–100

–8 –10 –0.5

–0.25 0 0.25 Displacement (d)

0.5

–150 –0.65

–0.325 0 0.325 Displacement (d)

0.65

Figure 4.8  Schematic representation of the distortion mode in benzene leading to a puckered chair form and the puckered boat form of 1,3,5-cyclohexatriene. The coordinates of C1 and C4 are varied antisymmetrically (+/−) and symmetrically (+/+) so as to reach the chair and the boat forms, respectively. These structures are hydrogenated to calculate the energy for cyclohexane. (a) Variation in the energies (in kcal/mol) for benzene + 3H2 (square) and cyclohexane (stars) as a function of the distortion axis (in Å) for +d/−d and +d/+d ­distortion modes. (b) Variation in the heat of hydrogenation (in kcal/mol) for the benzene ring with increase in the distortion (in Å). The horizontal line represents the zero of energy. (c) Variation in NICS (in ppm) at the center-of-mass of the benzene ring with increase in the distortions (in Å). (From Datta, A.; Pati, S. K. Chem. Phys. Lett. 2006, 433, 67–70. With permission.)

retains its identity and aromatic stabilization. To quantify the variation in the aromaticity of benzene with increase in puckering angle, the NICS at the center-of-mass of the ring for both the modes of distortions is calculated [17] as shown in Figure 4.8c. The diamagnetic ring current becomes progressively weaker as the benzene ring is puckered. However, for similar puckering distortions |d|, the boat form is more ­diamagnetic than the chair form suggesting larger aromaticity in the boat form ­compared to the chair form. In fact, the larger aromaticity of the boat mode of distortion is traced back to the onset of three-dimensional aromaticity due to effective delocalization of electrons over the entire surface of the molecule because of its more compact structure. The NICS, calculated for the benzene ring in HFB structure is −8.86 ppm which although smaller than NICS for planar benzene (NICS = −9.79 ppm), is substantial, suggesting weak aromaticity. The origin for the decaying aromaticity for benzene with the increase in the puckering in the boat and the chair forms can be qualitatively understood from the plots

Computational Studies on Molecules with Unusual Aromaticity

(d = 0.0, 0.0)

(d = +0.3, +0.3)

(d = +0.3, –0.3)

85

(d = +0.5, +0.5)

(d = +0.5, –0.5)

Figure 4.9  The highest occupied molecular orbital (HOMO) for benzene in various modes of distortions (d in Å). (From Datta, A.; Pati, S. K. Chem. Phys. Lett. 2006, 433, 67–70. With permission.)

of the HOMOs (Figure 4.9). For the planar benzene, the pz orbitals are parallel to each other thus ensuring maximum overlap and thereby high delocalization. With the increase in puckering in the chair form, the σ–π orbitals are mixed and the symmetry of the π orbitals are progressively lost (for d = 0.3 Å, HOMO still has a π character) [20]. For a completely puckered benzene in the chair form (d = 0.6 Å), σ and π orbital energies become comparable with complete quenching of pz delocalizations. However, for the boat mode of distortion, even at d = 0.6 Å, the HOMO shows substantial delocalization which explains large aromaticity for the puckered ring in the boat form. In summary, based on the computations of the stabilization energies for benzene at various puckering distortions, we find that the ring becomes progressively nonaromatic. Substitution of hydrogens by bulky groups are possible by puckering the ­benzene ring up to 0.3 Å for both the chair and the boat mode of distortions. Interestingly, the chair form of distortion in the crystalline form of hexaferrocenylbenzene (HFB) is well within this limit. However, we argue that for bulkier ligands, the loss of resonance energy has to be compensated by various kinetic factors, if such molecules are to exist.

86

Aromaticity and Metal Clusters

4.5 Hydrogen Adsorption on Octathio[8]circulene: Effect of Aromaticity Highly symmetric octathio[8]circulene, popularly called sulflower (sulfur + flower), the first fully heterocyclic circulene, has been synthesized recently. Molecular modeling and x-ray diffraction studies have indicated that this molecule has almost ­planar geometry with nearly D 8h symmetry [40]. Apart from its very appealing chemical structure, this molecule has a very large surface area (56 Å2), and coupled with its planar structure that ensures equal activity on either surface, it should be capable of adsorbing small molecules such as H2. Also, the planar structure (as a consequence of the presence of the S atoms) should ensure similar activity of both the top and bottom surfaces of the rings. The geometries for all of the molecular structures were optimized at the hybrid PW-91 GGA level with a TZP basis set within the ADF package [41,42]. In Figure 4.10, the molecular structures for the optimized geometry for octathio[8]circulene and its organic analogue are shown [43]. Although the heteroatomic analogue is perfectly planar (verified by the absence of any imaginary frequencies in the modes of vibrations), its organic analogue has a bowl-shaped structure. It is important to note that only octathio[8]circulene and nonathio[9]circulene are planar while the smaller and larger (than eight and nine) nuclearity rings are reported to be puckered. However, in its organic analogues, all of the systems are puckered irrespective of the nuclearity [n] [44]. The planarity in octathio[8]circulene is facilitated by the longer C–S bond length of 1.76 Å compared to the shorter 1.42 Å C–C bond length in its organic analogue, releasing the convex stress. This is also clearly understood by comparing the HOMOs for both the molecules. For octathio[8]circulene, the HOMO is predominately composed of the delocalized pz orbitals that are symmetric along the nodal plane, and this along with the planarity and highly symmetric molecular structure is suggestive of aromaticity in this molecule [45,46]. However, the case of its organic analogue is very different. The π orbitals have a greater contribution in the concave side than in the convex side. This is a direct consequence of puckering in the ring that leads to mixing of the σ and π orbitals and thereby quenching of the aromaticity [20,39]. This explicitly suggests that the binding of small molecules such as H2 will be asymmetric in organic octa[8]circulene and its thio derivative (octathio[8]circulene) will be a better candidate for enhanced physiosorption of molecular hydrogen because both surfaces are highly active. To understand the favorable modes of binding of the H2 molecule with octathio[8] circulene, a series of structures with varying orientations of the H2 molecule (at horizontal and vertical positions) above and below the plane of the molecule are considered. The minimum-energy structure corresponds to the vertical orientation of H2 over the molecule. However, determination of the exact position of the vertical H2 in the molecule requires a detailed PES computation. The H2 binding sites in octathio[8] circulene can be categorized into three classes: over the eight S-atoms, over the eight thiophene rings, and over the central C8 ring. The PES for the two most favorable binding sites, namely, over the central C8 ring and above the thiophene unit, are shown in Figure 4.11a and b, respectively. The ground state corresponds to a distance of 2.9 Å for H2 in the central C8 ring and a distance of 3.2 Å from the thiophene ring,

87

Computational Studies on Molecules with Unusual Aromaticity (a) 1.76 1.76 1.76 1.38 1.76 1.76

1.38

(a)

1.76

1.38

1.42

1.42

1.76 1.76

1.76 1.76

1.42

1.42 1.38

1.38 1.42

1.42

1.42 1.76

1.42 1.38 1.76

1.42

1.76 1.76

1.38

1.76

1.76

1.38 1.38 1.45 1.45

1.42 1.42 1.45 1.45

1.42 1.451.42

1.45 1.45

1.42 1.42

1.42

1.42

Figure 4.10  Top panel: Optimized geometries of (a) octathio[8]circulene and (b) octa[8] circulene. Bottom panel: HOMO plots for octathio[8]circulene and octacirculene. (From Datta, A.; Pati, S. K. J. Phys. Chem. C. 2007, 111, 4487–4490. With permission.)

respectively. The binding energy, ΔE [calculated as E(complex) − E(molecule) − E(H 2)] for the complexes are found to be −0.7 and −0.65 kcal/mol, respectively. Binding of the H2 over the S atoms is relatively endothermic compared to the other sites, thereby suggesting that the nature of interactions is essentially physiosorption which leaves the lone pairs of electrons on the S atoms practically inert. An important conclusion from the PES calculations is that binding of H2 distinguishes between the central C8 ring and the thiophene rings and the marginally stronger binding to the C8 ring is a direct consequence of the shorter intermolecular contact. Thus, functionalization of octathio[8]circulene will lead to hydrogenation of the central ring followed by the thiophene rings. Once the favored sites of interactions for the H2 with octathio[8]circulene are known, the energetics of formation of complexes with more H2 molecules are considered. Incorporation of more hydrogens will be controlled by the electrostatic interactions of

88

Aromaticity and Metal Clusters Distance (in angs.)

E (kcal/mol)

(a)

E (kcal/mol)

(b)

3

2

3

2

3

4

5

6

4

5

6

2 1 0 3 2 1 0

Distance (in angs.) (2) (6)

+H2 (3)

(1)

ΔE = –0.65 (4)

ΔE = –0.70

+H2

ΔE = –0.51

(7) ΔE = –1.20

(5) ΔE = –0.11

+3H2

(8)

ΔE = –1.03

(10) ΔE = –2.13

ΔE = –1.13

(9)

ΔE = –1.28

+5H2 (11) ΔE = –2.04

(12) ΔE = –4.5

Figure 4.11  (a) PES for binding of H2 to the central eight-membered carbon ring. (b) PES for binding of H2 to the five-membered thiophene ring. Low-energy structures for the various hydrogen-loading proportions in octathio[8]circulene and reaction pathway for the formation of binding complex of 10 H2 within octathio[8]circulene. ΔE represents the exothermic binding energies in kcal/mol. (From Datta, A.; Pati, S. K. J. Phys. Chem. C. 2007, 111, 4487–4490. With permission.)

Computational Studies on Molecules with Unusual Aromaticity

89

the approaching H2 with the hydrogen molecules already present, due to the effect of cooperativity. Another very important issue regarding the H2 loading ­efficiency will be whether functionalization on the top surface modifies the binding efficiency for hydrogen to the bottom surface. For a critical understanding of the role of cooperative interactions among the H2 molecules, the structures of the molecules are optimized by varying the number of hydrogen molecules. In Figure 4.11 (lower panel), the structures of the molecules with a range of H2 adsorptions are shown. As can be clearly seen from Figure 4.11, the addition of the first H2 molecule leads to preferable binding to the central C8 ring and the thiophene ring (structures 2 and 3). Addition of the second hydrogen molecule leads to binding either to the same (top) surface or the bottom surface of the molecule. Thus, up–up structures (6 and 7) are almost as favored as the up–down structures (8 and 9). It is important to note that the energetics for all of the structures (6–9) are exothermic and the overall population of the H2 adsorption will be Boltzmann-averaged at a given temperature. Further addition of H2 molecules leads to hydrogenation of the alternate thiophene rings in the same plane (as shown in structure 10) or hydrogenation of the alternate rings in the up–down configuration (11). Note that, both processes are equally exothermic. However, interestingly, in no cases does the molecule bind to eight H2 over the thiophene rings. In fact, based on extensive calculations with varying all possible H2 positions, we find that the ground state corresponds to four H2 molecules adsorbed in the alternate positions over eight thiophene rings. Thus, either the top or the bottom plane can bind at most only five H2 molecules (four over the thiophene rings and one over the central C8 ring). Further addition of H2 molecules leads to the stable structure (12) that binds 10 hydrogen molecules symmetrically above and below the circulene ring. Note that, this structure is quite exothermic with the binding energy −4.5 kcal/mol. As mentioned above, all the 18 binding sites (9 on either side) cannot be satisfied and this leads to van der Waals repulsion among the H2 molecules on either surface (ΔE = +23.7 kcal/mol). Importantly, the cooperative interactions are negligible for the formation of 10 H2 at sulflower from two 5 H2 at sulflower, provided the alternate binding sites of the thiophene rings are kept vacant on both the top and the bottom surfaces. The structure of 10 H2 at octathio[8]circulene (12) can be further stabilized through additional π-stacking interactions within the molecules. Such weak intermolecular interactions mediated through the π-stacking forces lead to the stabilization of many technologically important organic conductors like TTF, TCNQ, and chloranil in a quasi-one-dimensional nature [47]. In Figure 4.12a, the geometry optimized structure for the one-dimensional aggregate of a trimer with the voids in the intermolecular cavities being occupied by the H2 molecules are shown. The distance between the octathio[8]circulene is 7.16 Å and the five H2 molecules are intercalated within each layer at a mean interlayer separation of 3.2 Å. The stabilization energy for this π-stacked trimer is −10.2 kcal/mol. The enhanced stabilization of this trimer ­compared to structure 12 arises primarily by the fact that the vertical hydrogen ­molecules for such stacks interact with the circulene rings on either side. Thus, the binding energy for the hydrogen molecules increases. However, it is important to note that removal of the H2 molecules leads to the formation of a slipped parallel π-stacked aggregate with a mean interplanar separation of 3.6 Å, consistent with the established Hunter’s stacking rules wherein simple induced-dipole, induced-dipole

90

Aromaticity and Metal Clusters

Figure 4.12  (a) Structure for the stacked aggregate of octathio[8]circulene with the adsorbed 10 H2 molecules. (b) ELF plot for the trimeric aggregate. (From Datta, A.; Pati, S. K. J. Phys. Chem. C. 2007, 111, 4487–4490. With permission.)

interactions favor a slipped parallel stacking structure [48]. Analysis of the ELF (Figure 4.12b) shows that the H2 molecules lie above the basins of the π electrons of the layers of octathio[8]circulene and no charge localization in the intermolecular region between the sulflower and the adsorbed H2 molecules. This is suggestive of the fact that there is negligible mixing between the σ MOs of the H2 and the π orbitals of the molecule. Thus, the nature of the bonding interaction between the molecule and the H2 in the stack is essentially physical adsorption.

4.6 Summary The discussions in this chapter clearly show that aromaticity is one important single parameter that controls stability, structure, reactivity, and material applicability of various molecular species. From the viewpoint of a theoretical chemist, there exist very few tools that allow such broad generalizations. The ease with which aromaticity bridges the gap between traditional aspects of organic chemistry and latest discoveries in inorganic chemistry is truly fascinating. In most contexts, the general opinion that the concepts of aromaticity are being overstretched to inorganic chemistry is indeed a great challenge to the computational chemist to firmly establish generalized criterion of aromaticity across various disciplines, which we hope would eventually open new and exciting areas in chemistry.* * The authors have wondered how and why a standard textbook in organic chemistry that primarily deals with mainly one atom, namely carbon, has more pages than an inorganic textbook that deals with large number of atoms. However, with the rapid progress in inorganic reaction mechanisms, the inorganic texts would indeed pick up weight soon!

Computational Studies on Molecules with Unusual Aromaticity

91

Acknowledgments SKP acknowledges the Department of Science and Technology, CSIR, Government of India for research funds. AD acknowledges the DST—Fast track scheme for young scientists, Government of India for partial funding. AD also thanks the students of the 1st batch of Integrated Master’s of Science for providing an opportunity to learn and enjoy chemistry afresh and dedicates this chapter to the cause of integrated education in science. The authors thank Professor G. U. Kulkarni, Dr. Neena Susan John, and Abdul Rehaman MS for co-authoring manuscripts discussed in this contribution.

REFERENCES









1. Schleyer, P. v. R. Guest Ed. Special edition on aromaticity. Chem. Rev. 2001, 101, 1115–1118. 2. Katritzky, A. R. Guest Ed. Special edition on heterocycles. Chem. Rev. 2004, 104, 2125–2126. 3. Harvey, R. G. Polycyclic Aromatic Hydrocarbons; Wiley-VCH: New York, 1997. 4. Togni, A.; Haltermann, R. L., Eds. Metallocenes; Wiley-VCH: New York, 1998. 5. Schleyer, P. v. R.; Subramanian, G.; Jiao, H.; Najafian, K.; Hofmann, M. Are boron compounds aromatic? An analysis of their magnetic properties. In Advances in Boron Chemistry, Ed. W. Siebert, 3, The Royal Society of Chemistry: Cambridge, 1997. 6. Fokin, A. A.; Jiao, H.; Schleyer, P. v. R. From Dodecahedrapentaene to the “[n]Trannulenes.” A new in-plane aromatic family. J. Am. Chem. Soc. 1998, 120, 9364–9365. 7. (a) Mauksch, M.; Gogonea, V.; Jiao, H.; Schleyer, P. v. R. Monocyclic (CH)9+-A heilbronner möbius aromatic system revealed. Angew. Chem., Int. Ed. 1998, 37, 2395–2397. (b) Ajami, D.; Oeckler, O.; Simon, A.; Herges, R. Synthesis of a möbius aromatic hydrocarbon. Nature 2003, 426, 819–821. 8. Schleyer, P. v. R.; Kiran, B.; Simion, D. V.; Sorensen, T. S. Does Cr(CO)3 complexation reduce the aromaticity of benzene? J. Am. Chem. Soc. 2000, 122, 510–513. 9. (a) King, R. B. Three-dimensional aromaticity in polyhedral boranes and related molecules. Chem. Rev. 2001, 101, 1119–1152. (b) Zhai, H. J.; Kiran, B.; Li, J.; Wang, L. S. Hydrocarbon analogues of boron clusters–planarity, aromaticity and antiaromaticity. Nat. Mater. 2003, 2, 827–833. 10. (a) Minkin, V. I.; Glukhovtsev, M. N.; Simkin, B. Y. Aromaticity and Antiaromaticity; John Wiley & Sons: New York, 1994. (b) Schleyer, P. v. R.; Jiao, H. What is aromaticity? Pure Appl. Chem. 1996, 68, 209–218. (c) Lloyd, D. What is aromaticity? J. Chem. Inf. Comput. Sci. 1996, 36, 442–447. (d) Krygowski, T. M.; Cyránski, M. K.; Czarnocki, Z.; Hafelinger, G.; Katritzky, A. R. Aromaticity: A theoretical concept of immense practical importance. Tetrahedron 2000, 56, 1783–1793. 11. Schleyer, P. v. R. Guest Ed. Special edition on aromaticity. Chem. Rev. 2005, 105, 3433–3435. 12. (a) Parr, R. G.; Yang, W. Density-Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (b) Koch, W.; Holthausen, M. C. A Chemist’s Guide to Density Functional Theory; Wiley-VCH: Weinheim, 2002. (c) Martin, R. M. Electronic Structure: Basic Theory and Practical Methods; Cambridge University Press: New York, 2004. 13. (a) Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618–622. (b) Head-Gordon, M.; Pople, J. A.; Frisch, M.  J.  MP2 energy evaluation by direct methods. Chem. Phys. Lett. 1988, 153, 503–506.

92

Aromaticity and Metal Clusters

14. (a) Scheiner, S. Hydrogen Bonding: A Theoretical Perspective; Oxford University Press: Oxford, 1997. (b) Desiraju, G. R., Steiner, T. The Weak Hydrogen Bond in Structural Chemistry and Biology; Oxford University Press: Oxford, 1999. 15. (a) Jug, K.; Hiberty, P. C.; Shaik, S. σ−π Energy separation in modern electronic theory for ground states of conjugated systems. Chem. Rev. 2001, 101, 1477–1500. (b) Hilberty, P. C.; Shaik, S. S.; Lefour, J.-M.; Ohanessian, G. Is the delocalized π system of benzene a stable electronic system? J. Org. Chem. 1985, 50, 4657–4659. 16. (a) Juselius, J.; Sundholm, D. Ab initio determination of the induced ring current in aromatic molecules. Phys. Chem. Chem. Phys. 1999, 1, 3429–3436. (b) Fowler, P. W.; Steiner, E.; Zanasi, R.; Cadioli, B. Electric and magnetic properties of hexaethynylbenzene. Mol. Phys. 1999, 96, 1099–1108. (c) Merino, G.; Heine, T.; Seifert, G. Chem. Eur. J. 2004, 10, 4367–4371. 17. (a) Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H. F.; Hommes, N. J. R. V. Nucleus-independent chemical shifts: A simple and efficient aromaticity probe. J. Am. Chem. Soc. 1996, 118, 6317–6318. (b) Chen, Z.; Wannere, C. S.; Corminboeuf, C.; Puchta, R.; Schleyer, P. V. R. Nucleus-independent chemical shifts (NICS) as an aromaticity criterion. Chem. Rev. 2005, 105, 3842–3888. 18. (a) Chesnut, D. B.; Bartolotti, L. The electron localization function description of aromaticity in five-membered rings. Chem. Phys. 2000, 253, 1–11. (b) Fuster, F.; Sevin, A.; Silvi, B. Topological Analysis of the Electron Localization Function (ELF) Applied to the Electrophilic Aromatic Substitution. J. Phys. Chem. A. 2000, 104, 852–858. (c) Silvi, B. How topological partitions of the electron distributions reveal delocalization. Phys. Chem. Phys. Chem. 2004, 6, 256–260. 19. (a) Katritzky, A. R.; Jug, K.; Oniciu, D. C. Quantitative measures of aromaticity for mono-, bi-, and tricyclic penta- and hexaatomic heteroaromatic ring systems and their interrelationships. Chem. Rev. 2001, 101, 1421–1450. (b) Jemmis, E. D.; Kiran, B. Aromaticity in X3Y3H6 (X = B, Al, Ga; Y = N, P, As), X3Z3H3 (Z = O, S, Se), and phosphazenes. Theoretical study of the structures, energetics, and magnetic properties. Inorg. Chem. 1998, 37, 2110–2116. 20. Rehaman, A.; Datta, A.; Mallajosyula, S. S.; Pati, S. K. Quantifying aromaticity at the molecular and supramolecular limits: Comparing homonuclear, heteronuclear, and H-bonded Systems. J. Chem. Theory Comput. 2006, 2, 30–36. 21. (a) Howard, S. T.; Krygowski, T. M. Benzenoid hydrocarbon aromaticity in terms of charge density descriptors. Can. J. Chem. 1997, 75, 1174–1181. (b) Grimme, S. Theoretical bond and strain energies of molecules derived from properties of the charge density at bond critical points. J. Am. Chem. Soc. 1996, 118, 1529–1534. (c) Proft, F. D.; Geerlings, P. Conceptual and computational DFT in the study of aromaticity. Chem. Rev. 2001, 101, 1451–1464. 22. (a) Becke, A. D. A new mixing of Hartree–Fock and local density-functional theories. J. Chem. Phys. 1993, 98, 1372–1377. (b) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785–789. 23. Gaussian 03, revision B.05; Gaussian, Inc.: Pittsburgh, PA, 2003. 24. Conquest, version 1.7; Cambridge Crystallographic Database, Cambridge Crystal­ lographic Data Centre: Cambridge, UK, 2004. 25. Paetzold, P.; Richter, A.; Thijssen, T.; Wurtenberg, S. Bildung, Struktur und Reaktivität von (Pentafluorphenyl)bor-tert-butylimid und seinem Cyclodimeren. Chem. Ber. 1979, 112, 3811–3827. 26. Paetzold, P.; von Plotho, C.; Schmid, G.; Boese, R.; Schrader, B.; Bougeard, D.; Pfeiffer, U.; Gleiter, R.; Schafer, W. Reaktionen und Struktur von tert-Butyl(tert-butylimino) boran. Chem. Ber. 1984, 117, 1089–1102.

Computational Studies on Molecules with Unusual Aromaticity

93

27. (a) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Clarendon Press: Oxford, 1995. (b) Cole, J. M.; Copley, R. C. B.; McInyre, G. J.; Howard, J. A. K.; Szablewski, M.; Cross, G. H. Charge-density study of the nonlinear optical precursor DED-TCNQ at 20 K. Phys. Rev. B 2002, 65, 125107–125118. (c) Luana, V.; Pendas, A. M.; Costales, A.; Carriedo, G. A.; G-Alonson, F. J. Topological analysis of chemical bonding in cyclophosphazenes. J. Phys. hem. A. 2001, 105, 5280–5291. 28. (a) Hobza, P.; Zahradnik, R. Intermolecular interactions between medium-sized systems. Nonempirical and empirical calculations of interaction energies. Successes and failures. Chem. Rev. 1988, 88, 871–897. (b) Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553–566. 29. Datta, A.; Pati, S. K. Rationalization of the π − σ (Anti)aromaticity in all metal molecular clusters. J. Chem. Theory Comput. 2005, 1, 824–826. 30. Shaik, S.; Hilberty, P. C. When does electronic delocalization become a driving force of molecular shape and stability? 1. The aromatic sextet. J. Am. Chem. Soc. 1985, 107, 3089–3095. 31. Mallajosyula, S. S.; Datta, A.; Pati, S. K. Conformational preference in heteroatomic analogues of ethane, H3X−YH3 (X = B, Al; Y = N, P): Implications of charge Transfer. J. Phys. Chem. A. 2006, 110, 5156–5163. 32. (a) Dance, I. G. The structural chemistry of metal thiolate complexes. Polyhedron 1986, 5, 1037–1104. (b) Mahmoudkhani, A. H.; Langer, V. Structural investigation of nickel thiolates: a cyclic hexanuclear nickel thiolate with highly distorted NiS4 units Polyhedron 1999, 18, 3407–3410. (c) Watson, A. D.; Pulla Rao, C.; Dorfman, J. R.; Holm, R. H. Systematic stereochemistry of metal(II) thiolates: synthesis and structures of [M2(SC2H5)6]2– (M = Mn(II), Ni(II), Zn(II), Cd(II)) Inorg. Chem. 1985, 24, 2820–2826. (d) Li, Z.; Ohki, Y.; Tatsumi, K. Dithiolato-bridged dinuclear iron−nickel complexes [Fe(CO)2(CN)2(μ-SCH2CH2CH2S) Ni(S2CNR2)]-Modeling the active site of [NiFe] hydrogenase. J. Am. Chem. Soc. 2005, 127, 8950–8951. 33. Nobusada, K.; Yamaki, T. Electronic properties of palladium—rhiolate complexes with tiara-like structures. J. Phys. Chem. A. 2004, 108, 1813–1817. 34. Alemany, P.; Hoffmann, R. Toroidal nickel thiolates: structure and bonding. J. Am. Chem. Soc. 1993, 115, 8290–8297. 35. Datta, A.; John, N. S.; Kulkarni, G. U.; Pati S. K. Aromaticity in stable tiara nickel ­thiolates: Computational and structural analysis. J. Phys. Chem. A. 2005, 109, 11647–11649. 36. John, N. S.; Kulkarni, G. U.; Datta, A.; Pati, S. K.; Komori, F.; Kavitha, G.; Narayana, C.; Sanyal, M. K. Magnetic interactions in layered nickel alkanethiolates. J. Phys. Chem. C. 2007, 111, 1868–1870. 37. Shaik, S. S.; Hiberty, P. C.; Lefour, J.-M.; Ohanessian, G. Is delocalization a driving force in chemistry? Benzene, allyl radical, cyclobutadiene, and their isoelectronic species. J. Am. Chem. Soc. 1987, 109, 363–374. 38. Yu, Y.; Bond, A. D.; Leonard, P. W.; Lorenz, U. J.; Timofeeva, T. V.; Vollhardt, K. P. C.; Whitener G. D.; Yakovenko, A. A. Hexaferrocenylbenzene. Chem. Commun. 2006, 2572–2574. 39. Datta, A.; Pati, S. K. Limit to puckering of benzene with sterically crowded molecules: Hexaferrocenylbenzene. Chem. Phys. Lett. 2006, 433, 67–70. 40. Chernichenko, K. Y.; Sumerin, V. V.; Shpanchenko, R. V.; Balenkova, E. S.; Nenajdenko, G. “Sulflower”: A new form of carbon sulfide. Angew. Chem., Int. Ed. 2006, 45, 7367–7370. 41. Perdew, J. P.; Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B 1992, 45, 13244–13249.

94

Aromaticity and Metal Clusters

42. te Velde, G.; Bickelhaupt, F. M.; van Gisbergen, S. J. A.; Fonseca Guerra, C.; Baerends, E. J.; Snijders, J. G.; Ziegler, T. Chemistry with ADF. J. Comput. Chem. 2001, 22, 931–967. 43. (a) Datta, A.; Pati, S. K. Computational design of high hydrogen adsorption efficiency in molecular “Sulflower.” J. Phys. Chem. C. 2007, 111, 4487–4490. (b) Mohakud, S.; Pati, S. K. J. Mater. Chem. 2009, 19, 4356. (c) Fujimoto, T.; Matsushita, M. M.; Yoshikaawa, H.; Awaga, K. Electrochemical and electrochromic properties of octa­ thio[8]circulene thin films in Ionic Liquids. J. Am. Chem. Soc. 2008, 130, 15790– 15791. (d) Fujimoto, T.; Suize, R.; Yoshikawa, H.; Awaga, K. Molecular, crystal, and thin-film structures of octathio[8]circulene: Release of antiaromatic molecular distortion and lamellar structure of self-assembling thin films. Chem. Eur. J. 2008, 14, 6053–6056. 44. (a) Sygula, A.; Rabideau, P. W. A practical, large scale synthesis of the coran-nulene ­system. J. Am. Chem. Soc. 2000, 122, 6323–6324. (b) Aprahamian, I.; Preda, D. V.; Bancu, M.; Belanger, A. P.; Sheradsky, T.; Scott, L. T.; Rabino-vitz, M. Reduction of bowl-shaped hydrocarbons: Dianions and tetraanions of annelated corannulenes. J. Org. Chem. 2006, 71, 290–298. (c) Kammermeier, S.; Jones, P. G.; Herges, R. Ring-expanding metathesis of tetradehydro-anthracene-synthesis and structure of a tubelike, fully conjugated ­hydrocarbon. Angew. Chem., Int. Ed. 1996, 35, 2669–2671. (d) Scott, L. T. Methods for the chemical synthesis of fullerenes. Angew. Chem., Int. Ed. 2004, 43, 4994–5007. 45. Poater, J.; Frader, X.; Duran, M.; Sola, M. An insight into the local aromaticities of polycyclic aromatic hydrocarbons and fullerenes. Chem. Eur. J. 2003, 9, 1113–1122. 46. (a) Datta, A.; Pati, S. K. New examples of metalloaromatic Al-clusters: (Al4M4) Fe(CO)3(M = Li, Na and K) and (Al4M4)2Ni: rationalization for possible synthesis. Chem. Commun. 2005, 5032–5034. (b) Datta, A.; Pati, S. K. Stable Transition Metal Complexes of an All-Metal Antiaromatic Molecule (Al4Li4): Role of Complexations. J. Am. Chem. Soc. 2005, 127, 3496–3500. (c) Mallajosyula, S. S.; Datta, A.; Pati, S. K. Aromatic superclusters from all-metal aromatic and antiaromatic monomers, [Al4]2– and [Al4]4-. J. Phys. Chem. B. 2006, 110, 20098–20101. (d) Datta, A.; Mallajosyula, S. S.; Pati, S. K. Nonlocal electronic distribution in metallic clusters: A critical examination of aromatic stabilization. Acc. Chem. Res. 2007, 40, 213–221. (e) Datta, A.; Pati, S. K. Dipolar interactions and hydrogen bonding in supramolecular aggregates: understanding cooperative phenomena for 1st hyperpolarizability. Chem. Soc. Rev. 2006, 35, 1305– 1323. (f) Boldyrev, A. I.; Wang, L. S. All-metal aromaticity and antiaromaticity. Chem. Rev. 2005, 105, 3716–3757. 47. (a) Tsiperman, E.; Becker, J. Y.; Khodorkovsky, V.; Shames, A.; Shapiro, L. A rigid neutral molecule involving TTF and TCNQ moieties with intrinsic charge- and electrontransfer properties that depend on the polarity of the solvent. Angew. Chem., Int. Ed. 2005, 44, 4015–4018. (b) Murata, T.; Morita, Y.; Fukui, K.; Sato, K.; Shiomi, D.; Takui, T.; Maesato, M.; Yamochi, H.; Saito, G.; Nakasuji, K. A purely organic molecular metal based on a hydrogen-bonded charge-transfer complex: Crystal structure and electronic properties of TTF-Imidazole-p-Chloranil. Angew. Chem., Int. Ed. 2004, 43, 6343–6346. 48. (a) Cockroft, S. L.; Hunter, C. A.; Lawson Perkins, J.; Urch, C. J. Electrostatic control of aromatic stacking interactions. J. Am. Chem. Soc. 2005, 127, 8594–8595. (b) Hunter, C. A. Angew. Chem., Int. Ed. 2004, 43, 5310. (c) Hunter, C. A.; Sanders, J. K. M. The nature of π-π interactions. J. Am. Chem. Soc. 1990, 112, 5525–5534.

the Electron 5 Using Localization Function to Measure Aromaticity Patricio Fuentealba, Elizabeth Florez, and Juan C. Santos Contents 5.1 Topological Analysis of ELF to Determine Aromaticity................................ 98 Acknowledgments................................................................................................... 101 References............................................................................................................... 101

The aromaticity concept is used in chemistry to distinguish some special types of molecules, which are particularly stable and present similar chemical reactivity. The name is derived from early known examples such as benzene and its derivatives that present pleasant aroma. Aromaticity is associated with high stability, uniform bond lengths intermediate between single and double bonds, and a ring current that ­produce a chemical shift in the NMR spectra. However, like many other chemical concepts, its origin is empirical and qualitative. Like many important concepts in chemistry, aromaticity has been generalized. It was used at the beginning only for planar π-conjugated hydrocarbons, but very soon it was also used for hetero atoms, and for noncompletely planar molecules, such as biphenyl. Later it was also generalized for nonplanar molecules, especially for caged molecules like fullerenes [1,2]. Over the last decade, a new very interesting generalization was proposed. The aromaticity concept in all-metal clusters and the existence of a σ aromaticity [3], which open the possibility of conflicting aromaticity, that is, σ aromatic and π antiaromatic or vice versa [4]. Huckel proved that planar molecules with 4n + 2 π electrons are aromatic, whereas molecules with 4n π electrons are antiaromatic [5]. The rule changes for open shell molecules [6]. Over the past 50 years there has been continuous work to bring out a sharp definition of the term and a clear way of quantifying it [7]. However, one has to bear in mind that it is an empirical chemical concept, and as such, it is neither directly measurable in the lab nor the expectation value of some quantum mechanic operator. Hence, trying to theoretically model aromaticity necessarily needs abandoning the formal theory and making an empirical interpretation of some derived formula at some point. 95

96

Aromaticity and Metal Clusters

There are various geometric and magnetic criteria to measure the degree of a­ romaticity of a molecule [8]. In the past, some criteria based on electron density has also been put forward, like the one to be presented here [9]. The Electron Localization Function (ELF) has been proposed by Becke and Edgecombe [10] and first applied to chemical bonding by Savin et al. [11] and Silvi and Savin [12]. It gives us information about the region of the space where it is more probable to find an electron or a localized electron pair. It depends only on the ­density, and it can be in fact extracted from experimental measurements [13]. Some thorough reviews of its significance and applications can be found in the literature [9,11,14,15]. Now, a short rederivation of the ELF for different parts of the total density in complete analogy with the original derivation will be presented. Usually, the aromaticity concept is associated with delocalization of the electrons occupying orbitals of π symmetry. Therefore, it seems natural to start with a separation of the density according to this type of symmetry:

   ρ(r ) = ρσ (r ) + ρπ (r ).

(5.1)

Of course, it is straightforward to generalize to cases where other types of symmetries are present.

    ρ(r ) = ρσ (r ) + ρπ (r ) + ρ δ (r ) + ….

Every part of the density is constructed as the sum of the square of all the orbitals having this symmetry:

 ρσ (r ) =

∑φ

i ,σ

 2 (r ) .

(5.2)



In the same manner one can define the second-order density matrix of a given symmetry and also the conditional pair density. Hence, using the Taylor series expansion of the spherical average conditional pair density of a given symmetry is as follows:



 1 1 (∇ρΩ )2 2 … PΩ (r , s ) =  τ Ω − s + , 2 4 ρΩ 

(5.3)

where Ω represents the symmetry (σ, π, δ, …) and the term τΩ is the kinetic energy density associated to the orbitals with symmetry Ω

τΩ =

∑ (∇φ

i ,Ω

)2 .



(5.4)

The conditional pair density gives us the probability of finding an electron at a  distance s of another electron localized at the point r . After Becke and Edgecombe

Using the Electron Localization Function to Measure Aromaticity

97

[10] this Taylor series expansion contains all the information about the electron localization. The smaller the possibility of finding a second electron at a distance s  of the electron localized at the point r , the more localized is the electron. Of course, they did the argumentation for the total density but it is perfectly applicable to the part of the density with symmetry σ or π. Hence, the main element of localization is the two first terms of the expansion: DΩ =

1 1 (∇ρΩ )2  τ − 2  Ω 8 ρΩ 

(5.5)

which can be used for another explanation of the ELF due to Savin et al. [11]. The first term on the right side of the last equation represents the kinetic energy density of the noninteracting system, and the second one is the kinetic energy density because of the von Weiszaecker term. It is well known that the von Weiszaecker term is exact for the hydrogen atom, which has the most localized electron of the periodic table of the elements. It is also exact for the helium atom in an independent electron model (i.e., Hartree–Fock) which is the most localized pair of electrons in the periodic table of elements. Finally, it is also exact for any boson particles which are in the ground state always completely localized. Therefore, it is to expect that the region of the space, where the von Weiszaecker kinetic energy density is a good approximation to the kinetic energy density of the real system, corresponds to the region of the space where there is a good probability of finding an electron or a pair of electrons localized. Note again that the argumentation does not need to talk about the total density. It is the same for any partition of the density. Remember that the kinetic energy is completely additive. Since the function D Ω presents strong oscillations it is easier for the analysis to do first a mapping to a function varying between zero and one:

ELF(ρΩ ) = (1 + χΩ2 )−1 ,



(5.6)

where



χΩ =

DΩ DΩ0

(5.7)



with

DΩ0 = cFρΩ5/ 3



(5.8)

the kinetic energy density of the homogeneous electron gas. Note that the way of writing the ELF in Equation 5.6 makes it clear that it is a function which indicates some density, not necessarily the total density. It may be also applicable to any distribution of charges obeying the Pauli Exclusion Principle. Of course, the evaluation of the ELF with the total electron density of a molecule is not

98

Aromaticity and Metal Clusters

the addition of the ELF’s evaluated value with the partition of the density. In our ­special case, in a loose language, we are just looking at the probability of finding two π electrons or two σ electrons in some regions of the space. We are completely ­missing the probability of finding a π electron and a σ electron in this region of the space.

5.1 Topological Analysis of ELF to Determine Aromaticity Using the traditional derivation and interpretation of the ELF, it is possible to ­analyze with this function any convenient partition of the density. In particular, in the cases where it is possible in a semirigorous way to separate the density in its σ, π, δ … . Symmetries, the ELF can also be partitioned to do a separate analysis. As it was already mentioned a high value of the ELF, near to 1, is interpreted as a region of the space where it is probable to find a localized electron or electron pair. Hence, the maxima of the function, called the attractors, play a significant role. A basin is the volume enclosed by all the gradient lines ending up at the same attractor. Hence, the attractors are enclosed by the minima of the function. Therefore, at low values of the ELF an isosurface could contain more than one attractor, and eventually at a lower value one has an isosurface containing all the attractors. When the isosurface goes to higher values of the ELF the basins will begin to split. It will occur exactly at the minima of the function, and they are called bifurcation points. The higher the bifurcation point, the more fluctuating is the charge between the two connected basins. Inversely, a bifurcation point close to zero implies that there is almost no interaction among the electrons at each basin. As an example, a pictorial ELF description and a bifurcation scheme of acetaldehyde are shown in Figure 5.1. In the ELF topological analysis, 12 attractors were recognized, three core attractors corresponding to C1, C2, and O, which are named as C(C1), C(C2), and C(O), respectively. Seven valence attractors, corresponding to four C–H bonds, V(C1,Hi = 1–3) and V(C2,H4), another one to C–C single bond, V(C1,C2), and the last two valence basins corresponding to C–O double bond, Vi = 1,2(C2,O). Finally, it is possible to appreciate two valence basins corresponding to oxygen lone pairs, Vi = 1,2(O). The basin reduction (separation) is obtained increasing the ELF isosurface value; the corresponding bifurcation diagram and some selected ELF isosurfaces of each reduction domain are shown in Figure 5.1. Therefore, it is expected that the analysis of the bifurcation points gives us information about the delocalization and/or aromaticity of the molecule. In the following paragraphs, a proposed aromaticity scale based on the ELF will be briefly discussed and two applications presented. Following those ideas, it has been proposed to do the topological analysis on an ELF formed only by the π orbitals and the other one formed only by the σ orbitals [16,17]. This scheme has shown that the bifurcation analysis of the ELFπ and ELFσ characterizes in a reasonable way the π and σ aromaticity in a variety of systems going from organic molecules to atomic clusters [18,19]. The average bifurcation value of ELFσ and ELFπ was used to construct an aromaticity scale for chemical systems [17]. The proposed scale was validated with a series

Using the Electron Localization Function to Measure Aromaticity

99

V(C2,H4) O

H1 H2

H3

C1

V(C1,H1) V(C1,H2) V(C1,H3)

C2 H4

V(C1,C2)

V(C2,O) V(C2,O)

V1(O) V2(O)

Core basins

Figure 5.1  ELF bifurcation diagram and some selected ELF isosurfaces of acetaldehyde.

of well-known molecules and then used to evaluate global aromaticity on aluminumbased clusters, which present σ aromaticity and π antiaromaticity. The proposed scale correctly predicted the aromaticity of [Al4]2− and the overall antiaromatic character of the controversial [Al4]4− cluster. The same scheme of separated σ and π functions was used to study the electronic structure and properties of a silabenzene series and of the new aromatic planar species, Si6Li6 [19]. Figure 5.2 shows ELF isosurfaces of total, sigma, and pi functions of planar D6h isomer of Si6Li6. Although this planar structure does not correspond to the more stable isomer [20], it possesses similar electronic properties to those observed in the benzene aromatic structure [19,21]. The electronic delocalization and  stability of planar Si6Li6 were evaluated by π aromatic–cation interaction in [Si6Li8]2+ and [Si6Li6Na2]2+. Both systems were characterized as minima conserving the D6h symmetry. The cyclic structure does not suffer appreciated changes in its

Total function

σ function

π function

Figure 5.2  Total, σ, and π ELF isosurfaces of Si6Li6 (D6h).

100

Aromaticity and Metal Clusters

geometrical parameters and the bifurcation analysis was very similar to the one of the isolated Si6Li6. There was a slight decrease in the π delocalization which was attributed to π–cation interaction but the π cyclic aromatic system was unchanged. In the next paragraph, the aromatic π–cation interaction in host–guest complex in C32 fullerenes will be analyzed. It is well known that in π–cation interactions there are strong forces between the cation and the π-face of an aromatic structure. Electrostatic forces, polarization, and the charge distribution of aromatic species appear to play a dominant role in this kind of interactions [22,23]. Fullerenes smaller than C60 contain adjacent pentagons and the curvature of the carbon surface is much higher than one of their large homologues. As a result the π-conjugation in the small fullerenes is curved and weakened, especially in the pentagon fusion sites. The unusual stability of C32 has been ascribed to its spherical aromaticity arising from its completely filled 32 π-electron valence shell [2]. The HOMO–LUMO gaps obtained for the analyzed complexes (Table 5.1) are higher than other results published before, and comparable to those obtained in very stable homologue fullerenes [23]. The H–L gaps in all endohedral cation–C32 inclusion complexes is higher than the one of C32. The inclusion energy (Table 5.1) indicates that encapsulation of Li+, Mg2+, Ca2+, and Al3+ is favorable whereas inclusion of Na+ and K+ is energetically unfavorable which is in agreement with the cation size. Some isosurfaces of the ELF topological analysis of C32 and the endohedral ­complexes can be seen in the second column of Figure 5.3. C32 is characterized by two fragments formed by the fusion of three six-membered rings, which interact through two fusion apices formed by three pentagons. The population in the three bonds of central fused six-membered rings is higher in the system (2.8e−) and lower bond population is located in the apices of the fused five-membered rings (2.25e−). There is also a no-bonding basin over carbon atoms of apices (light gray color) with population of one-third electron. A similar ELF profile was obtained for endohedral complexes, the population varies slightly and the main changes occur over no-bonding basins where the population decreases to 0.18e−. The ELF bifurcation analysis of the separated σ and π components shows that C32 has an aromaticity (electronic delocalization) similar to naphthalene according Table 5.1 ELF Bifurcation Values, HOMO–LUMO Gap (H–L) and Inclusion Energies of the Analyzed Endohedral Cation–Fullerene Complexes System C32 C32…Li+ C32…Na+ C32…K+ C32…Mg2+ C32…Ca2+ C32…Al3+

ELFσ Function

ELFπ Function

Total ELF Function

H–L Gap (eV)

Inclusion Energy (eV)

0.64 0.69 0.64 0.67 0.63 0.68 0.62

0.63 0.63 0.64 0.59 0.64 0.64 0.63

0.69 0.68 0.69 0.68 0.69 0.68 0.68

2.66 2.76 2.75 2.65 2.84 2.74 2.86

— −0.66 1.41 5.74 −4.95 −0.82 −19.3

Using the Electron Localization Function to Measure Aromaticity C32

C32…Li+

C32…K+

101

C32…Ca+2

π-function

σ-function

Total function

Figure 5.3  ​Some selected ELF isosurfaces of the analyzed systems.

to the aromaticity scale mentioned before. The delocalization is highly concentrated over the fragments formed by the fusion of three six-membered rings. A similar bifurcation analysis was found in the cation–C32 inclusion complexes. The bifurcation values are close to 0.62, 0.65, and 0.68 for π, σ, and total functions in the studied systems, respectively (see Table 5.1). These values indicate that the electronic structure in the complex remains almost undisturbed. The π-electron delocalization is conserved which could be interpreted as an indicative of π aromatic– cation interaction. Summarizing, an efficient partition of the ELF allows us to quantify the aromaticity character of different types of molecules ranging from standard organic aromatic molecules such as benzene to the relative new aromatic atomic clusters and ­fullerenes derivatives.

Acknowledgments We would like to acknowledge the financial support from FONDECYT through grants 1080184, 3080033, and 11060197. JCS also acknowledges the Universidad Andres Bello for the support received through the project DI-47-08/R.

REFERENCES

1. Chen, Z.; King, R. B. Spherical aromaticity: Recent work on fullerenes, polyhedral boranes, and related structures. Chem. Rev. 2005, 105, 3613. 2. Lu, X.; Chen, Z. Curved pi-conjugation, aromaticity, and the related chemistry of small fullerenes ( χC, a transfer of electronic charge is expected from C to B. Using the chemical potentials, µ B = (∂EB /∂nB )V = µ 0B + 2 ηB ∆n , where the partial derivative is taken at constant external potential VB, and a corresponding equation for C, combined with the equilibrium condition μB = μC and with the relation μ = −χ, the charge transfer becomes B

∆n =

χ B − χC . 2( ηB + ηC )



(7.4)

The difference of electronegativities drives the electron transfer, and the sum of the hardness parameters acts as a drag, or resistance. Further corrections [2] modify

121

Reactivity of Metal Clusters

only the numerator of Equation 7.4. An estimation of the energy change in the ­process is then



∆E BC = −

1 ( χ B − χ C )2 . 4 ηB + ηC

(7.5)

7.2.2 Application to Clusters The hardness varies as the size of the cluster increases. Using a spherical droplet model for metallic clusters leads to the result η = e2/2R, where e is the electron charge and R is the cluster radius [3,4]. This reflects the fact that, as R increases, I and A approach the work function of the metal at the rate 1/R. For small and medium size clusters, I ≠ A, which is a consequence of the energy gaps existing between the discrete electronic levels. For clusters of semiconducting or insulating materials, the gap at the Fermi level remains in the transition to the bulk material. For clusters, the main interest resides on the deviations of η from the averaged curve as the cluster size grows. Electronic shell effects impart structure to the functions I(X N), A(X N), and η(X N). The value of I(X N) shows a sizable drop between clusters of sizes Nc and Nc + 1, where Nc indicates the size of a cluster with closed electronic shells. Those cluster sizes are known as magic numbers. On the other hand, A(X N) drops between Nc − 1 and Nc. A cluster of size Nc has a large I and a small A, and consequently a large value of η is expected. The expectation has been verified by calculations for Na clusters using the spherical jellium model [4]. In this model, electrons fill shells in the order 1s, 1p, 1d, 2s, 1f, 2p, … and clusters with closed shells occur for a number of electrons equal to 2, 8, 18, 20, 34, 40, 58, …. Since Na is monovalent, local maxima of the hardness are obtained [5] for cluster sizes N = 2, 8, 18, 20, 34, 40, 58, … . Transcending the spherical approximation, a similar argument predicts the appearance of even–odd oscillations in the reactivity, clusters with even number of electrons being less reactive than their odd neighbors. An exact treatment in the framework of the Density Functional Theory (DFT) [2] leads to I = −ϵHOMO (X N), where ϵHOMO (X N) is the energy of the highest occupied molecular orbital of the neutral species. Similarly, A = − HOMO ( X N− ) where ϵHOMO (X N− ) is the energy of the HOMO of the anionic cluster X N− . For approximate exchangecorrelation energy functionals these relations for I and A are approximate. If the change of the single-particle energies due to charging the cluster is neglected, we may approximate HOMO ( X N− ) =  LUMO ( X N ), where now ϵLUMO(X N) is the energy of the lowest unoccupied molecular orbital of the neutral cluster. Then, one obtains

η( X N ) =

1 ( ( X ) − HOMO ( X N )). 2 LUMO N

(7.6)

Clusters with a large HOMO–LUMO gap are therefore chemically hard. Clusters having closed electronic shells belong to this class; their resistance to either donating or accepting electronic charge makes these clusters less reactive than the clusters with a small gap.

122

Aromaticity and Metal Clusters –1 –2 –3

EF(–3.49)

–4

* A26

KS eigenvalues (eV)

–5 –6

EF(–6.62)

–7

EF(–7.05)

–8

* A11

–9

* A10

–10 –11

* A5

–12 –13

* A2

–14 –15 –16

AI13

AI13H

*

A1

AI– 13

− Figure 7.1  Energies of the electronic levels of Al13, Al13 , and Al13H. For Al13 the energies for spin-up and spin-down electrons are shown in different columns. The HOMO state is indicated as EF. The labeled molecular levels of Al13H have a sizable H–1s contribution.

We now apply these concepts to the understanding of the chemical properties of − the Al13 cluster. Figure 7.1 gives the energies of the electronic levels of Al13, Al13 , and Al13H obtained from DFT calculations [6]. The structure of Al13 is an icosahedron with an atom at the center. This icosahedron has a small oblate deformation: the six atoms in two opposite triangular faces are closer to the center than the six equatorial atoms. Figure 7.2a shows the cluster in an orientation allowing to see those two faces well, which are formed by Al atoms (1, 2, 3) and (4, 5, 6). Al13 has 39 valence electrons, an odd number, and the energies of the electronic orbitals with spin up and spin down are slightly different. In Figure 7.1, the HOMO is indicated as EF and the HOMO and LUMO states of Al13 have opposite spin orientations. The HOMO– LUMO gap is small, 0.18 eV, and the cluster is chemically reactive. In fact, the electronic configuration of Al13 in the spherical jellium model is 1s2 1p6 1d10 2s2 1f 14 2p5, short of just one electron to fill the 2p shell. Reduction of the spherical symmetry to the Ih symmetry gives the following electronic configuration for − Al13 : 1ag2 1tu6 1hg10 2 ag2 1t26u 1gu8 2t16u . The main photoelectron spectroscopy (PES) features of Al3− correspond to electron detachment from the 1t2u, 1gu, and 2t1u orbitals [7]. The neutral cluster reacts with a hydrogen atom to give Al13H and the reaction is  exothermic. The H atom sits on the surface of Al13. Calculations indicate that the most stable positions are the hollow sites on the triangular faces and the bridge positions between two neighbor Al atoms [6,8–10], with the binding energies being

123

Reactivity of Metal Clusters (a)

(b) 3

7

1

8

4

5

10

12

11

6

7

13

9

13

4

10

2

8

12 11

3

2 1

6

5

9

Figure 7.2  Two views, (a) and (b) of the structure of Al13H. The H atom is represented by the small sphere. The Al atoms are the large spheres numbered 1–13.

similar. The most favorable faces are those two closer to the cluster center (see ­Figure 7.2a). The electronic structure of Al13H fits into the picture of a closed-shell cluster: a large HOMO–LUMO gap can be seen in Figure 7.1. The figure also shows − the electronic levels of Al13 . These levels present large degeneracies because the − icosahedron is not distorted, but the HOMO–LUMO gaps of Al13 and Al13H are very − similar, 1.81 and 1.77 eV, respectively. The upward shift of the energies in Al13 is due to its excess charge. Al13H has been explored as a candidate to form cluster-assembled solids [11]. By assembling together Al13H clusters in a cubic superlattice, the internal structure of the clusters appears to be preserved to a large extent. Al13H further reacts with hydrogen. However, the reactivity of Al13H is lower than the reactivity of Al13. This can be appreciated by comparing the binding energies of the last H atom in Al13H and Al13H2; these are 2.92 and 2.38 eV, respectively [12]. The difference of 0.54 eV arises from the fact that Al13H is a closed-shell cluster. Both binding energies are still large because H is very reactive, and additional H atoms can be added. The values of η(Al13Hx) with x = 1 – 13, obtained from Equation 7.3, are [12]: 2.39, 1.92, 2.21, 2.01, 2.37, 1.89, 2.41, 2.00, 2.45, 1.88, 2.45, 2.02, and 2.99 eV. These values show an oscillating pattern, with larger values for clusters with even number of valence electrons (n = 39 + x), that is, with odd x, and lower values for clusters with even x. In the even-x clusters, the HOMO is double-occupied, while in the odd-x clusters this level is split and the single-occupied HOMO and LUMO states have opposite spin orientations. A correlation with the reaction energies of H atoms successively added to Al13H is expected, and the reaction energies, 2.38, 2.75, 2.48, 2.88, 2.21, 2.84, 2.46, 3.06, 2.12, 2.93, 3.00, 3.09 eV, show the expected oscillating trend. Similar conclusions are reached by analyzing the reactivity of Al13 with lithium [13]. The calculated binding energy of the first Li atom is 2.43 eV, and the binding energy of a second Li is 1.80 eV. The ionization potentials of Al13Li and Al13Li2 are 6.6 and 3.14 eV, respectively. The drop is substantial and Al13Li2 can be viewed as an

124

Aromaticity and Metal Clusters

alkali superatom. The interaction of Al13 with H and Li is, however, very different. The charge density in Al13H shows that the state of H can be described as a negatively charged impurity screened by the surrounding electron gas [6]. The stabili­ zation of extra charge around H is interpreted by noticing that the electronegativity of H is higher than that of Al13. The labeled electronic levels of Al13H in Figure 7.1 have a sizable H–1s contribution. In contrast, the Li atom in Al13Li loses part of its electronic charge, which is transferred to the Al13 cluster, and the binding is partially ionic. The ionicity increases if a heavier alkali atom is attached to the Al cluster. In Al13K, Al13Rb, and Al13Cs, the Al cluster behaves as a halide and an ionic descrip− − − K +, Al13 Cs+, is appropriate. Alkali halides form crystals, and Rb +, and Al13 tion, Al13 Khanna and Jena [14] have proposed forming ionic crystals with these supermole− K + solid [15] assuming a CsCl-like structure showed cules. Simulations of an Al13 that bonds are formed between the different Al13 units, indicating that the solid is less ionic than expected, and that the assembled solid resembles crystalline fcc Al. An improved proposal started with BAl12Cs. In BAl12, the B atom occupies the central position of the icosahedron. The choice of this cluster was motivated by the fact that it is isoelectronic with Al13 and more stable. On the other hand, the radius of Cs+ is larger than the radius of K+, and the (BAl12)− anions are separated more efficiently in the assembled solid. The calculated total energy of a solid with the CsCl structure built by approaching (BAl12)− and Cs+ units shows a minimum as the lattice constant is decreased. Further reduction of the lattice constant induces a change of the structure of (BAl12)− from icosahedral to cubo-octahedral and a new minimum appears. The two minima are separated by a barrier of 0.3 eV, and Ashman et al. [16] suggested that it might be possible to assemble this ionic solid. As examples of the experimental confirmation of the ideas relating the reactivity of clusters to their electronic structure, we mention the work of Castleman and coworkers [9,17]. These authors have studied the reaction of positive and negatively charged Al clusters with oxygen. An etching reaction occurs, and negligible reactiv− − ity of Al 7+, Al13 , and Al 23 was observed. These three clusters are not only unreactive; they are produced by reactions of larger clusters. In the spherical jellium model, these three species are closed-shell clusters with 20, 40, and 70 electrons, respectively, and consequently they have sizable HOMO–LUMO gaps. Other feature observed was the odd–even alternation in the values of the reaction rate constants for the etching reactions as the size of the clusters increases. For cationic clusters this + alternation begins with Al18 , clusters with even number of atoms showing higher reactivity. The reason is that a cluster Al +N has n = 3N − 1 valence electrons, and n is an odd number for N even, and an even number for N odd. Electrons are paired in clusters with even n but clusters with odd n have one unpaired electron and are more reactive. For Al −N , a related trend is observed. The super-halogen character of Al13 is a subject of debate. Bergeron and coworkers [9,18] have concluded that the super-halogen character is confirmed in their − study of the reaction of Al13 with HI. They propose that the experimental reaction − outcomes are dominated by the pseudo-halide nature of Al13 : a stable Al13I − species − forms in which the Al13 framework is preserved even in the reactive environment of an I atom. Experiments [7] and calculations [19] indicate that the electron affinity − of neutral Al13 is 3.4–3.6 eV. The I atom has a lower affinity, 3.06 eV, so Al13 is

Reactivity of Metal Clusters

125

expected to keep the extra electron when interacting with iodine [18]. Extensive experiments [9] of halogenation of Al −N indicate that Al13I − is very stable (other very stable clusters in the Al NI− series were found). The high stability is revealed by the resistance to O2 attack. However, Al13Cl− is not a magic cluster in the Al N Cl− series. First, the electron affinity of the Cl atom (3.62 eV) is similar to that of Al13. In addi− tion, while it costs 4.65 eV to remove an Al atom from Al13 , the binding of the AlCl − molecule is 5.23 eV. This means that Al13Cl would be unstable with respect to frag− mentation into Al12 and AlCl. Furthermore, it costs 3.87 eV to remove an Al atom from neutral Al13, and fragmentation or etching of Al13Cl− would be even easier if the Al13 moiety is closer to the neutral state than to the anionic one. That is, the key to the stability of Al13X−, X being a halogen atom, is the ability of Al13 to remain − charged: Al13X− is only stable when Al13 is bound to a halogen atom of affinity lower than that of neutral Al13. Halogenation experiments [9] have also produced very stable clusters in the series Al N I 2−, and Al13 I 2− is one of them. The stability of this − cluster suggests that Al13 can maintain its integrity even when interacting with more than one reactive atom. DFT calculations [20] indicate that attachment of an I atom − on top of an Al atom of Al13 gives rise to substantial localization of the HOMO on the opposite Al vertex. This is an active site, and a second I atom easily binds to that site to form Al13 I 2− . This prediction was tested in experimental work by the same group [20]. By analyzing the reaction of Al −N first with I2 vapor and then etched by O2, they found that the Al N I −x clusters formed exhibit high stability when x is an even number. The stability is obtained because complementary pairs of I atoms are attached to opposing Al atoms. The work of Han and Jung [21] provides additional insight on the super-halogen nature of Al13. They analyzed the electronic density in neutral Al13X with X = F, Cl, Br, I, and electronic charge transfer from Al13 to the halogen atom was found in all cases. The effective positive charges of Al13 in Al13X are q = 0.70, 0.47, 0.38, and 0.26, res­ pectively, in units of the electron charge. For Al13 in Al13X2, the corresponding charges in are q = 1.40, 0.91, 0.71, and 0.41, nearly doubling the above values. The electronegativity of Al13, calculated from Equation 7.3, is 4.66 eV, and the electronegativities of F, Cl, Br, and I are 10.66, 8.32, 7.60, and 6.87 eV, respectively. The charge transfers correlate perfectly with the difference χ(X) − χ(Al13), as expected from established chemical ideas. Therefore, a first impression is that in its interaction with halogen atoms Al13 does not show a typical halogen behavior. A closer look reveals that charge transfers of the magnitude given above also occur in the interaction between two different halogen atoms. For instance, the charges transferred toward F in FI and FBr are 0.48 and 0.39e−, respectively [21]. Second, why is there charge localization of the HOMO orbital δ+ δ− of Al13I− on the aluminum moiety? The chemical picture in neutral Al13I is Al13 I with δ = 0.26. When an electron is added, it becomes predominantly localized on the Al cluster [18,20,21]. This is mainly an effect of electrostatics [22]: the attraction by the δ+ positive Al13 and the repulsion from the negative Iδ−; or more precisely, the interaction of the electron with the (δ+, δ−) dipole. The charge localization compensates for the charge lost by Al13 in neutral Al13I, and the excess charge of Al13I− is shared by the two α − (1− α ) − moieties. The final chemical picture is Al13 , where α can be estimated as 0.65, I and one can conclude that the metal cluster behaves as a halogen in this special case. One may ask why α is not zero, that is, why the I atom does not achieve the full anionic

126

Aromaticity and Metal Clusters

configuration I−. In our opinion this is because a closed shell I− anion attached to Al13 would result in a lower binding energy, compared to the state when the electron is shared. This is also reflecting a fundamental difference in charging atoms or clusters. Halogen anions, X −, are isoelectronic with noble gases and then unreactive: from the point of view of this work, their HOMO–LUMO gap and hardness are large. In contrast, the gaps in anionic clusters are smaller and those clusters are still reactive; in fact, clusters can accommodate more than one excess electron because the electron–electron repulsion is lowered, especially as the cluster size increases (see Chapter 5 of Ref. [4]). The chemical picture in Al13 I 2− is similar to that for Al13I −: the extra electron is mainly localized on the metal cluster, to compensate for the charge lost in the neutral state, and the net negative charge of the Al moiety is α = 0.38. In Al13 I3− this negative charge decreases to 0.10e−. Replacing I by a more electronegative halogen, Br, Cl, or F, the positive charges of Al13 in neutral Al13X or Al13X2 are larger than for the case of X = I, and in spite of the localization effect of the extra electron, the net charge of the metal moiety is expected to be closer to zero, that is, a chemical picture close to (Al13)-X− will be appropriate. In these cases, Al13 does not behave as a superhalogen. In summary, concerning its interaction with halogen atoms, only in charged Al13I− does Al13 behave as a halogen species. Of course, no doubt it behaves as a halogen in its interaction with electropositive species as the alkali atoms. Experiments and calculations show that the series Al14 I −x exhibits high stability for odd x [20]. This series has been described [20] as consisting of an Al14 I3− core upon which new I atoms occupy positions on-top of opposite Al atoms. It was proposed that, with 42 electrons, Al14 behaves as an alkaline-earth superatom, and in the 2+ Al14 I −x it achieves the character of an Al14 di-cation. Analysis of the charge distributions indicates that the directionality of the I–Al bonds switches from polar toward the Al atoms in clusters based on Al13, to polar toward the I atoms in clusters based on Al14. The gain in energy as successive I atoms are added was calculated as

∆E x = E (Al N I −x −1 ) + E (I) − E (Al N I −x )

(7.7)

in terms of the DFT total energies, for N = 13 and 14. For N = 13, ΔEx shows an even–odd alternation in which the values for odd x are 0.9 eV lower than those of even x, indicating the higher stability of clusters with even x. For N = 14, ΔEx shows the even–odd alternation after x = 3, with larger values for odd x. The work of Han and Jung [21] offers a different interpretation. These authors have noticed that, in neutral Al13I and Al14I clusters, the effective charges on the Al moiety are very similar, +0.26 and +0.25, respectively, and the same occurs for Al13I2 and Al14I2, with charges +0.41 and +0.52. This is a result of the similar electronegativities of Al13 and Al14. These small charges are in conflict with an alkaline-earth behavior. The results discussed above are consistent with the principle of Maximum Hardness, proposed by Pearson [23], which states that molecular systems at equilibrium present the highest values of hardness. Therefore, it is reasonable that the clusters with the highest hardness are those with closed electronic shells. Also consistent with this principle is the odd–even oscillation in the reactivity. DFT calculations also display an odd–even oscillation of the hardness of sodium and copper clusters [24,25].

127

Reactivity of Metal Clusters

As another example, it is useful to recall the interaction between alkali atoms and carbon nanostructures [26,27]. C6H6, C24H12, C20H10, and C21H9 form a series of ­carbon clusters with the border atoms saturated by hydrogen. Benzene and C24H12 are planar. In the other two, the C20 and C21 parts are curved fragments of the C60 fullerene. The HOMO–LUMO gaps of these four hydrogenated clusters are 5.0, 2.8, 2.2, and 0.50 eV, respectively [26]. Benzene is then expected to be the least reactive and C21H9 the most reactive one. In agreement with this expectation, the calculated interaction energies of atomic Cs with these clusters increase as the gap of the clusters decreases. The idea also works if we compare the interaction of alkali atoms with graphene and carbon clusters [27]. The interaction of Li with C24H12 is very weak, the binding energy obtained in a DFT calculation is only 0.025 eV. In contrast, an extended graphene layer is a zero gap semimetallic system, and the binding energy of Li to graphene is about 1.10 eV.

7.3 Local Reactivity Indexes In addition to the global indexes, local indexes provide detailed information on the reactivity [28]. A way of introducing these local indexes [29] is to begin with the  energy of a system as a functional of the electron density ρ(r ) in DFT [2], that is,

∫

   E[ρ] = V (r )ρ(r ) dr + F[ρ].





(7.8)

 The first term gives the interaction of the electrons with an external potential V(r) (due to the atomic nuclei), and F[ρ] is a universal functional of the electron density, consisting of kinetic, classical electrostatic and exchange-correlation energies. Since  the functional E[ρ] for a fixed V(r ) is stationary for the true density, one has the Euler equation µ=



 ∂E[ρ] ∂F[ρ]  = V (r ) +  , ∂ρ(r ) ∂ρ(r )

(7.9)

where μ is the electronic chemical potential. Actually, μ and the electronegativity χ are  the same quantities with opposite signs [30]. Evaluating the energy change of Equation 7.1 considering explicitly the changes of the electron density, one obtains [29]



∆E = µ∆n +

1 (∆n)2 2

∫∫

  ∂ 2 F[ρ] ∂ρ(r ) ∂ρ(r ′ )     dr dr ′ . ∂n ∂ρ(r )∂ρ(r ′ ) ∂n

(7.10)

By comparison with Equations 7.1 and 7.2 we notice that the integral in Equation 7.10 is the hardness η



η=

∫∫

  ∂ 2 F[ρ] ∂ρ(r ) ∂ρ(r ′ )     dr dr ′. ∂n ∂ρ(r )∂ρ(r ′ ) ∂n

(7.11)

128

Aromaticity and Metal Clusters

    The response function K (r , r ′ ) = (∂ 2 F[ρ]) / (∂ρ(r )∂ρ(r ′ )) is known as the hardness kernel [31,32]. It tells us how that part of the energy which is exclusively electronic (E[ρ] minus the interaction with the external potential) responds to two-point changes     δρ(r )δρ(r ′) of the electron density. Then, K(r , r ′) is weighted in the integral by the   product f (r ) f (r ′), where



   ∂ρ(r )  f (r ) =   .  ∂n  V

(7.12)

This function measures the local changes of the electron density due to a small  global variation in the number of electrons, with a fixed external potential. f(r ) distinguishes between regions where the changes of the electron density are large from   regions where changes are small. Multiplied by the response kernel, K(r , r ′), to those density changes and integrated over the volume of the system one obtains the global hardness η.  The function f(r ) can be positive or negative, and it appears that regions where its magnitude is large are the regions of the cluster accepting or releasing charge in chemical reactions. If the chemical reaction is easy or not, depends on the integrated  response to those local changes. Since the changes δρ(r ) are produced by the effect of an approaching reacting species, it looks reasonable to expect that regions where   the magnitude of f(r ) is large are more favorable to host reactant species. f(r ) can be expressed in an alternative, but equivalent form [2,33–35]



  ∂µ   . f (r ) =   ∂V (r )  n

(7.13)

  Although μ is a global quantity, f(r ) is a function of r , known as the Fukui func tion [2,33–36]. Written in this way, f(r ) is interpreted as giving a measure of the  change of the chemical potential μ when a local change δV(r ) is induced in the  external potential V(r ), for instance by the presence of reagent species. This reinforces the view that the regions of the cluster surface with large absolute values of  f(r ) are the most reactive, and that reactant atoms or radicals, will preferentially attach at those regions.  It is useful to consider the Fukui functions from above, f +(r ), and from below,  f −(r ), corresponding to changes in the number of electrons from n0 to n0 + Δn or to n 0 − Δn, respectively, in Equation 7.12 [2,37,38]. Regions of the cluster where f −(r) is large, easily give up electronic charge and are reactive toward electron-poor reactants (electrophiles). Regions where f +(r) is large stabilize an uptake of charge and are reactive toward electron-rich reactants (nucleophiles). Using a finite difference approximation, the Fukui functions can be written as [37,38]

   f − (r ) = ρ V ,n (r ) − ρ V , n −1(r ),



   f + (r ) = ρ V, n +1(r ) − ρ V, n (r )

0

0

0

0



(7.14) (7.15)

129

Reactivity of Metal Clusters

as a difference of the densities of the system with no electrons (neutral), n0 − 1 ­(cationic) and n 0 +1 electrons (anionic system) in the external potential V.

7.4 Hydrogen as a Probe of the Local Reactivity of Clusters 7.4.1 Interaction of H with Al13 We consider again the reactivity of H with the cluster Al13. The H atom will attach to those sites for which its approach to Al13 results in a large response (change) of the chemical potential μ. As discussed above, the H atom takes up charge from the cluster, so the relevant reactivity function in this case is f −(r). Figure 7.3a shows a plot of f −(r) in a plane cut through the cluster. This plane contains the central Al atom and the atoms labeled 3, 9, 5, and 10 in Figure 7.2b. The highest values of f −(r) occur in regions corresponding to the two faces of the icosahedron closer to the cluster center. Consequently, concerning the regions most susceptible to H attack, there is agreement between the predictions provided by the index f −(r) and the results of the total energy DFT calculations. To make this point clear, Figure 7.3b shows the electron density of Al13H in the same plane. The position of the H atom corresponds to the region of highest f −(r) in the Al13 cluster.

7.4.2 Reactivity of Gold Clusters Gold has a noble, chemically unreactive character in its bulk form. However, Au clusters and nanoparticles are active catalysts in low-temperature CO oxidation, (a)

(b)

20

14 12 10

Distance (a.u.)

Distance (a.u.)

15

10 5

8 6 4 2

0

0

5

15 10 Distance (a.u.)

20

0

0

2

4

6 8 10 Distance (a.u.)

12

14

Figure 7.3  (a) Contour plot of the Fukui function f −(r) for the cluster Al13 calculated as the difference of electron densities in Equation 7.14. The plane of the plot contains atoms 3, 9, 5, 10 (see Figure 7.2b) and the central atom of the cluster. Continuous lines stand for positive ­values, dashed lines for negative values, and dashed–dotted lines are nodal lines. (b) Contour plot of the electron density in Al13H. (Reproduced from Mañanes, A. et al. J. Chem. Phys. 2003, 119, 5128. With permission from the American Institute of Physics.)

130

Aromaticity and Metal Clusters

p­ ropene epoxidation, water-gas-shift reaction, and other interesting chemical ­reactions. For this reason it is important to know the influence of the size, shape, and chemical state of gold nanoparticles on their reactivity, as well as the effect of the surface support. One possible application for the gold catalysts is as active elements in fuel cells, catalyzing the reaction between hydrogen and oxygen to produce electric power and water as residual elements. Two favorable features of these catalysts, namely their high selectivity and the ability of running reactions at relatively low temperatures, place them as possible candidates for this task. Therefore, it is interesting to understand well the interaction of gold clusters and nanoparticles with hydrogen. In more general terms, and making connection with one of the main purposes of this paper, hydrogen can be used as a probe of the local reactivity of gold clusters, in a similar way to the study presented above for the interaction between H and Al13; the interaction between H and gold clusters can be used to test the local reactivity parameters. Gold clusters are chemically active, and it is easier to control fundamental parameters as their size or the charge state, compared to gold nanoparticles. A striking property of small AuN clusters is that their structures are planar. Experiments have shown planar structures up to N = 12 for anions [39] and up to N = 7 for cations [40]. These observations are supported by theoretical calculations. DFT calculations of Fernández et al. [41] predict planar structures for Au +N , AuN, and Au −N clusters up to N = 7, 11, and 12, respectively. The preference for planar structures is due to relativistic effects that change the chemistry of gold clusters in comparison to copper or silver: the electronic 6s and 5d states are unusually close in energy in gold, and their hybridization enhances the tendency for planarity in the small clusters [42]. The binding energies of atomic hydrogen adsorbed on planar gold clusters, obtained from DFT calculations [43] are shown in Figure 7.4 for clusters with N = 4−10. The panel corresponding to each cluster contains information on the adsorption energies on different cluster sites. The largest binding energies, highlighted with circles, occur at sites on the perimeter of the cluster with the H atom in the same plane. In most cases, the H atom is in a bridge position between two Au atoms. In contrast, adsorption on central sites above the cluster plane is weaker. The only special case is Au8, which has a large hole at the cluster center where the H atom fits well. The H binding energy at this site is only slightly larger than at the perimeter. All these results reveal a strong directionality of the H-cluster bonding. The preference for Au–H–Au configurations with the H atom in a bridging position is attributed to the optimized overlap between the H 1s electronic orbital and the Au 5d orbitals. The H atom then serves as a probe of the local site reactivity of gold clusters. Such localized reactivity is an intrinsic property of the small gold clusters, and can be generalized to other adsorbates and to the interaction of the cluster with other nanostructures. Small AuN ­clusters interacting with oxide surfaces (MgO and TiO2) prefer upright conformations with the cluster plane perpendicular to the surface [43]. Returning to the Fukui functions f −(r) and f +(r) of Equations 7.15 and 7.14, these two have been plotted in Figure 7.5 for the AuN clusters. Both functions are distributed over the rim of the clusters. A connection is therefore established between the spatial distribution of the Fukui functions and the preferred H binding sites, on the perimeter of the cluster. Another interesting feature is that f −(r) and f +(r) are nearly

131

Reactivity of Metal Clusters 1.69

3.17

2.07

2.32

1.65

2.05

1.94

1.71

1.29

1.56

2.72

2.57

3.03

1.42

2.04

2.22

2.39

2.05

2.72

1.12

2.48

1.04

1.62

1.68

1.50

1.80

0.94

2.37 1.70 2.02

1.71

2.13

2.93 1.31

2.44

2.40 2.90

1.90

1.35

2.39

0.82

1.83

2.30

Figure 7.4  Interaction of H with small planar AuN clusters, N = 4–10. For each gold cluster, the information on the local equilibrium structures and binding energies (in eV) of a single H atom at different sites of the cluster is assembled together. Circles highlight the most stable adsorption sites.

equal in Au5, Au7, and Au9, the clusters with odd number of electrons; in contrast, f −(r) and f +(r) show some differences in Au4, Au6, Au8, and Au10, the clusters with even number of electrons. The different behavior of clusters with odd and even number of electrons is easily explained. Assuming that the shape of the electronic orbitals changes little upon charging the cluster, f +(r) and f −(r) can be further approximated by the electron densities of the frontier orbitals:

  f + (r ) = ρLUMO(r ),

(7.16)



  f − (r ) = ρHOMO(r ).

(7.17)

Actual comparison with f +(r) and f −(r) obtained from Equations 7.14 and 7.15   shows that the approximations by ρLUMO (r ) and ρHOMO (r ) is very reasonable [43]. In the even-n clusters Au4, Au6, Au8, and Au10, the HOMO state is doubly occupied with one spin up electron and one spin down electron, and an energy gap exists between

132

Aromaticity and Metal Clusters f+ 4

5

f–

f+

f– 8

9

6 10 7

Figure 7.5  Fukui functions f +(r) and f −(r) of AuN clusters, N = 4–10, from Equations 7.15 and 7.14. Values of 0.003 e/A3 have been chosen for the surface contours.

the HOMO and the LUMO levels. The spatial distributions of the HOMO and LUMO are rather different, and then f +(r) and f −(r) show some differences. But in the odd-n clusters Au5, Au7, and Au9, the HOMO and LUMO states are nearly degenerate in energy and the corresponding orbitals have nearly the same shape, because the only differences arise from the spin polarization of the exchange-correlation contribution to the effective potential, small in Au clusters. The main message from Figure 7.5 is that the most reactive part of these clusters is the perimeter, and that reactive species will attach at those regions of the cluster. This holds for species donating electronic charge to the cluster as well as for those accepting electronic charge. Equations 7.16 and 7.17 state that, in chemical reactions of a cluster, charge is added to the LUMO and is subtracted from the HOMO. But it should be noticed from Equation 7.12 that this works when the external potential V is not changed. In practice, the presence of the donor or the acceptor species modifies V, and consequently the charge transfer is not necessarily canonical, that is, in accordance with the prediction from the Fukui functions. An example is provided in the work of Martínez et  al. [26], who have studied the interaction of Cs with carbon clusters. Here, we focus on the interaction of Cs with C20H10 and C21H9. These two clusters are fragments of C60 with the peripheral C atoms saturated by H. In both cases, Cs transfers electronic charge toward the carbon cluster. The Fukui functions f +(r) of both C20H10 and C21H9 are localized on the convex side, that is, on the outer surface of the fullerene fragment. The calculations show that the charge transfer is canonical when the Cs atom is absorbed on the outer surface of the clusters: the distribution of the charge transferred follows the Fukui function prediction. On the contrary, if Cs is adsorbed on the concave, inner surface of the cluster, the charge transfer does not follow that prediction. It becomes distributed over the inner and outer surfaces of the carbon cluster, to avoid, it seems, an excessive distance between the charge transferred and the positively charged atom left behind.

Reactivity of Metal Clusters

133

7.5 Summary To conclude, we have shown that, despite the complexity of their structural and electronic properties, the reactivity of small metallic clusters can in many cases be understood in simple terms with the use of both global reactivity indexes (ionization potentials, electron affinities, and chemical hardness, which provide a measure of the global cluster reactivity) and local Fukui functions (which allow the identification of special reactive sites within a cluster). All of them can be easily evaluated within the framework of Density Functional Theory.

Acknowledgments This work was supported by MEC of Spain (Grants MAT2008-06483-C03-01 and -03) and Junta de Castilla y León (Grants VA017A08 and GR23). L. M. Molina ack­nowledges a Ramon y Cajal fellowship. J. A. Alonso acknowledges an Ikerbasque Fellowship of the Basque Foundation of Science and the hospitality shown by members of the Donostia International Physics Center.

REFERENCES

1. Parr, R. G.; Pearson, R. G. Absolute hardness: Companion parameter to absolute electronegativity. J. Am. Chem. Soc. 1983, 105, 7512. 2. Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules. Oxford University Press: Oxford, 1989. 3. Mañanes, A.; Membrado, M.; Pacheco, A. F.; Sañudo, J.; Balbás, L. C. A mass formula for the energy of metal clusters. Int. J. QuantumChem. 1994, 52, 767. 4. Alonso, J. A. Structure and Properties of Atomic Nanoclusters. Imperial College Press: London, 2005. See Chapter 5. 5. Alonso, J. A.; Balbás, L. C. Hardness of metallic clusters. Struct. Bonding 1993, 80, 229. 6. Mañanes, A.; Duque, F.; Méndez, F.; López, M. J.; Alonso, J. A. Analysis of the bonding and reactivity of H and the Al13 cluster using density functional concepts. J. Chem. Phys. 2003, 119, 5128. 7. Li, X.; Wang, L. S. Experimental search and characterization of icosahedral clusters: Al12X− (X = C, Ge, Sn, Pb). Phys. Rev. B 2002, 65, 153404. 8. Kawamura, K.; Kumar, V.; Sun, Q.; Kawazoe, Y. Magic behavior and bonding nature in hydrogenated aluminum clusters. Phys. Rev. B 2001, 65, 045406. 9. Bergeron, D. E.; Castleman, A. W.; Morisato, T.; Khanna, S. N. Formation and properties of halogenated aluminum clusters. J. Chem. Phys. 2004, 121, 10456. 10. Han, Y. K.; Jung, J.; Kim, K. H. Structure and stability of Al13H clusters. J. Chem. Phys. 2005, 122, 124319. 11. Duque, F.; Mañanes, A.; Molina, L. M.; López, M. J.; Alonso, J. A. Computer simulation of cluster assembling. Int. J. Quantum Chem. 2002, 86, 226. 12. Jung, J.; Han, Y. K. Structure and stability of Al13Hn (n = 1–13) clusters: Exceptional stability of Al13H13. J. Chem. Phys. 2006, 125, 064306. 13. Martínez, J. I.; Castro, A.; Alonso, J. A. Density functional study of the structural and electronic properties of aluminium-lithium clusters. J. Comput. Methods Sci. Eng. 2008, 7, 355. 14. Khanna, S. N.; Jena, P. Designing ionic solids from metallic clusters. Chem. Phys. Lett. 1994, 219, 479.

134

Aromaticity and Metal Clusters

15. Liu, F.; Mostoller, M.; Kaplan, T.; Khanna, S. N.; Jena, P. Evidence for a new class of solids. First-principles study of K(Al13). Chem. Phys. Lett. 1996, 248, 213. 16. Ashman, C.; Khanna, S. N.; Liu, F.; Jena, P.; Kaplan, T.; Mostoller, M. (BAl12)Cs: A cluster-assembled solid. Phys. Rev. B 1997, 55, 15868. 17. Leuchner, R. E.; Harms, A. C.; Castleman, A. W. Thermal metal cluster anion reactions: Behavior of aluminum clusters with oxygen. J. Chem. Phys. 1989, 91, 2753; Aluminum cluster reactions. 1991, 94, 1093. 18. Bergeron, D. E.; Castleman, A. W.; Morisato, T.; Khanna, S. N. Formation of Al13I–: Evidence for the Superhalogen Character of Al13. Science 2004, 304, 84. 19. Ashman, C.; Khanna, S. N.; Kortus, J.; Pederson, M. R. Isomers in Al13-N InN (N = 0–4) clusters, in Cluster and Nanostructure Interfaces, Eds. P. Jena, S. N. Khanna, B. K. Rao, World Scientific: London, 2000, 383. 20. Bergeron, D.E.; Roach, P. J.; Castleman, A. W.; Jones, N. O.; Khanna, S. N. Al cluster superatoms as halogens in polyhalides and as alkaline earths in iodide salts. Science 2005, 307, 231. 21. Han, Y. K.; Jung, J. Does the “superatom” exist in halogenated aluminum clusters? J. Am. Phys. Soc. 2008, 130, 2. 22. Leung, H.; Naumkin, F. Y. Induced super-halogen behavior of metal moieties in halogen-doped clusters: LinI(-) and AlnI(-), n = 13, 1, 2, 3 J. Phys. Chem. A 2006, 110, 13514. 23. Pearson, R. G. Chemical Hardness: Applications from Molecules to Solids, Wiley-VCH: Weinheim, 1997. 24. Mineva, T.; Russo, N.; Toscano, M. Odd-even alternation of global hardnesses in the Nan (n = 2–9) clusters. Int. J. Quantum Chem. 2000, 80, 105. 25. Jaque, P.; Toro-Labbé, A. Characterization of copper clusters through the use of density functional theory reactivity descriptors. J. Chem. Phys. 2002, 117, 3208. 26. Martínez, J. I.; López, M. J.; Alonso, J. A. Theoretical study of the reactivity of cesium with benzene and graphitic CxHy clusters. J. Chem. Phys. 2005, 123, 074303. 27. Martínez, J. I.; Cabria, I.; López, M. J.; Alonso, J. A. Adsorption of lithium on finite graphitic clusters. J. Phys. Chem. C 2009, 113, 939. 28. Geerlings, P.; De Proft, F.; Langenaeker, W. Conceptual density functional theory. Chem. Rev. 2003, 103, 1793. 29. Ghosh, S. K. Energy derivatives in density-functional theory. Chem. Phys. Lett. 1990, 172, 77. 30. Parr, R. G.; Donelly, R. A.; Levy, M.; Palke, E. W. Electronegativity: The density functional viewpoint. J. Chem. Phys. 1978, 68, 3801. 31. Berkowitz, M.; Parr, R. G. Molecular hardness and softness, local hardness and softness, hardness and softness kernels, and relations among these quantities. J. Chem. Phys. 1988, 88, 2554. 32. Nalewajski, R. F. General relations between molecular sensitivities and their physical content. Z. Naturforsch. A 1988, 43, 65. 33. Parr, R. G.; Yang, W. Density functional approach to the frontier-electron theory of chemical reactivity. J. Am. Chem. Soc. 1984, 106, 4049. 34. Yang, W.; Parr, R. G. Hardness, softness, and the fukui function in the electronic theory of metals and catalysis. Proc. Natl. Acad. Sci. USA 1985, 82, 6723. 35. Mortier, W.; Ghosh, S. K.; Shankar, S. Electronegativity-equalization method for the calculation of atomic charges in molecules. J. Am. Chem. Soc. 1986, 108, 4315. 36. Fukui, K. Role of Frontier orbitals in chemical reactions. Science 1982, 218, 747. 37. Ayers, P.; Parr, R. G. Variational principles for describing chemical reactions: The Fukui function and chemical hardness revisited. J. Am. Chem. Soc. 2000, 122, 2010. 38. Ayers, P.; Morrison, R. C.; Roy, R. K. Variational principles for describing chemical reactions: Condensed reactivity indices. J. Chem. Phys. 2002, 116, 8731.

Reactivity of Metal Clusters

135

39. Furche, F.; Ahlrichs, P.; Weis, P.; Jacob, C.; Gilb, S.; Bierweiler, T.; Kappes, M. M. The structures of small gold cluster anions as determined by a combination of ion mobility measurements and density functional calculations. J. Chem. Phys. 2002, 117, 6982. 40. Gilb, S.; Weis, P.; Furche, F.; Ahlrichs, Kappes, R. Structures of small gold cluster cations (Aun+, n  K0, we have τsp ≈ τ0  A

∑∑ A

Z AZB Ψ|Ψ RAB B> A

∑∑

occ

∑ i

∑ µ ,ν

ψi | ψi

(20.22)     

 Pµν Sµν  .  N 

(20.23)

(20.24)

392

Aromaticity and Metal Clusters

The separation of the density matrix elements in Equation 20.24 gives σ ENN =





π ENN =

Nσ N

∑∑

Nπ N

∑∑

A

A

Z AZB , RAB B> A



Z AZB . RAB B> A



(20.25)

(20.26)

The expressions of the σ and π energies, within the context of this new separation scheme, are

σ σ E σ = ESCF + ENN

and

π π E π = ESCF + ENN ,

respectively. This new separation scheme was applied to different six-, five-, and four-membered systems [18], demonstrating that it can be used to study a large variety of neutral and charged cyclic systems formed with different atoms. In the next sections, this new separation scheme will be validated considering different six- and five-membered systems, and it is then applied to a series of dicationic chalcogenid ring systems such as S24 + , Se 24 + , and Te 24 +. For all the studied cyclic systems, the distortions were performed following the scheme as the one we proposed in previous studies [9–11].

20.2.2 A New Aromaticity Index (AI) Based on the Relationship between σ − π Energy Separation and Aromaticity The σ and π energy curves calculated starting from the distortions of cyclic systems can be properly described using the harmonic potential approximation. If the σ and π energy curves are harmonic potentials, the sum of both energy contributions is also a harmonic potential. Based on these arguments, the difference of the total potential energy between the equilibrium structure R0 and its closest distorted structure can be calculated. We introduce this energy value as a new aromaticity index (AI). Within the harmonic potential approximation, the potential energy can be expanded as a Taylor series up to the second order as E ( R0 + ∆r ) = E ( R0 ) + g | ∆r +

R0

1 k ∆r 2 , 2 R 0

(20.27)

where Δr = r − R0 represents the increment of the distortion applied to the equilibrium structure R0 until the distorted structure R1 is found, r is the resonance coordinate, k is the curvature of the harmonic potential, and g is the gradient calculated at R0.

393

Are Dicationic Chalcogenid Ring Systems Aromatic?

The distortions of all the systems studied were performed as in our previous works, where the problem of distortions was extensively analyzed. Therefore, for details related to this topic, we would like to address the reader to Refs. [9–11]. In the equilibrium structure R0, the k term can be expressed as the second derivative of the total potential energy E with respect to the resonance coordinate r as  d2 E  k= 2 .  dr  r = R



0

(20.28)

The new AI introduced here was applied to several five and six aromatic ring molecules. In Table 20.1, the AI values we obtained are compared with the values reported in the literature employing other kinds of AIs. At the bottom of Table 20.1 the definition of each reported method is given.

Table 20.1 Aromaticity Indices (AI) of Five- and Six-Membered Rings Calculated Employing Different Methods Molecule

rcva

χ πzz /nπb

DREc

HSREd

χMe

REf

AIg

Benzene Pyridine 1,3,5-Triazine Pyrimidine 1,2,3,5-Tetraazine Pyrazole Pentaazine Pyrrole Imidazole 1,2-Oxazole Furan Hexaazine 1,3-Oxazole

1.437 1.422 1.404 1.411

10.05 9.67 8.88 9.28

22.6 23.1

0.390 0.348

20.2

0.297

54.8 49.2 37.9 43.1

45.8h 43.3h 44.9h 40.6h

1.096 1.407 1.124 1.075 1.032 1.081 1.426 1.042

7.43 8.18 7.48 7.17 6.98 7.25 7.86 7.03

0.330

42.6

40.4i

5.3 15.4

0.233 0.251

4.3

0.044

47.6 44.3 37.5 43.1

40.5j 40.0i 34.3i 27.2i

39.2

26.2i

10.18 9.51 9.31 9.13 7.01 5.46 4.60 4.57 4.06 2.92 2.36 1.87 1.83

a b

c d e f g h i j

Bond order criterion [19] based on the valence bond method [20]. Diamagnetic susceptibility of the π system orthogonal to the ring plane. The χzz component was calculated with the SINDO1 method [21,22]. The nπ term is the π electron number. Values are given in −106(cm3 mol−1). Dewar resonance energy [20,23]. Values are given in kcal mol−1. Hess and Shaad resonance energy per electron [24]. Values are given in terms of β. Molar diamagnetic susceptibility [25]. Values are given in −106(cm3 mol−1). Resonance energy [26–29]. Values are given in kcal mol−1. AI values obtained in this work. Values are given in kcal mol−1. Thermodynamic date [30]. Thermodynamic date [31]. Thermodynamic date [32].

394

Aromaticity and Metal Clusters

As Table 20.1 shows, the RCv, χ πzz / nπ, and χM methods assign to the benzene molecule the largest aromaticity values and to the 1,2-oxazole molecule the smallest aromaticity values. In contrast, both the Dewar resonence energy (DRE) and the Hess and Shaad resonence energy per electron (HSRE) methods identify furan as the least aromatic molecule. At the same time, the DRE method establishes the pyridine molecule as the most aromatic system, whereas the benzene molecule results the most aromatic molecule if the HSRE method is applied (see Table 20.1). Moreover, as Table 20.1 shows, our method and the RE method agree in the assignments of the benzene molecule as the most aromatic system and of the 1,3-oxazole molecule as the least aromatic cyclic system. Considering the results reported in Table 20.1, the decreasing order of the aromaticity for the cyclic systems studied with the different employed methods results as following: RCv benzene > hexaazine > pyridine > pyrimidine > pentaazine > 1,3,5-triazine >  pyrrole > pyrazole > furan > imidazole > 1,3-oxazole > 1,2-oxazole χ zzπ /nπ benzene > pyridine > pyrimidine > 1,3,5-triazine > pentaazine > hexaazine >  pyrrole > pyrazole > furan > imidazole > 1,3-oxazole > 1,2-oxazole DRE pyridine > benzene > pyrimidine > imidazole > pyrrole > furan HSRE benzene > pyridine > pyrazole > pyrimidine > imidazole > pyrrole > furan χM benzene > pyridine > pyrrole > imidazole > furan = pyrimidine > pyrazole >  1,3-oxazole > 1,3,5-triazine > 1,2-oxazole RE benzene > 1,3,5-triazine > pyridine > pyrimidine ≈ pyrrole ≈ pyrazole ≈ imidazole > 1,2-oxazole > furan > 1,3-oxazole AI in this work benzene > pyridine ≈ 1,3,5-triazine > pyrimidine > 1,2,3,5-tetraazine > pyrazole >  pentaazine ≈ pyrrole > imidazole > 1,2-oxazole > furan > hexaazine > 1,3-oxazole We notice that the new AI introduced in this work gives an aromaticity trend very similar to the one produced by the RE method.

20.2.3 Validation of the new Aromaticity Index to Six- and ­Five-Membered Rings This section presents the results of the validation of our newly developed AI considering different six- and five-membered rings employing the previously discussed σ and π harmonic potential approximations.

395

Are Dicationic Chalcogenid Ring Systems Aromatic? Benzene 50 50 45 45 σ 40 40 π 35 35 kπ = –894.4 30 30 25 25 20 20 Δk = 346.4 15 15 10 10 kσ = 1240.8 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

Potential curves

Figure 20.1  Harmonic potentials of the σ and π energies of benzene. The force constants of the σ and π curvature are given in kcal mol−1 Å−2.

From Figures 20.1 and 20.2, it can be observed that for the six-membered ring systems studied, the harmonic potential curves (dotted curves) adjust very well to the curvature of the σ and π energies (given as points and triangle, respectively). Using the σ and π harmonic potentials, we can obtain the energy difference between the equilibrium R0 structure and the distorted structure, R1, by employing the Equation 20.27. As an example, for the benzene molecule, the obtained harmonic potential adjusted to the π curve is

Eπ = 26.2154 − 447.22r 2

(20.29)

Eσ = 0.17143 + 620.422r 2

(20.30)

and for the σ curve,

is obtained. As Equations 20.29 and 20.30 show, for the benzene molecule, the gradients of the π and σ energies vanish. Similar results were obtained for all the other sixmembered systems investigated. From the second derivatives of Equations 20.29 and 20.30, with respect to the resonance coordinate r, the force constants π and σ, in kcal mol−1 Å−2, are obtained. The calculated values of kπ (−894.4 kcal mol−1 Å−2) and kσ (1240.8 kcal mol−1 Å−2) for the benzene are shown in Figure 20.1. Considering these data and Equation 20.27, for benzene an AI value equal to 10.18 kcal mol−1, is obtained, with



Eπ ( R0 + ∆r ) − Eπ ( R0 ) = ∆Eπ =

1 k 2 π

∆r 2 ,

(20.31)

R0

representing the change of the π potential energy, and



Eσ ( R0 + ∆r ) − Eσ ( R0 ) = ∆Eσ =

1 k 2 σ

∆r 2 , R0

(20.32)

396

Aromaticity and Metal Clusters

Pi energy (kcal/mol)

1, 2, 3, 5–Tetraazine Potential curves 50 50 45 45 σ 40 40 π 35 35 k = –935.9 30 30 π 25 25 20 20 Δk = 238.8 15 15 10 10 kσ = 1174.7 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Sigma energy (kcal/mol)

Pyrimidine Potential curves 50 50 45 45 σ 40 40 π 35 35 kπ = –926.1 30 30 25 25 20 20 Δk = 310.1 15 15 10 10 kσ = 1236.2 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Hexaazine Potential curves

50 50 45 45 σ 40 40 π 35 35 kπ = –981.9 30 30 25 25 20 20 Δk = 156.3 15 15 10 10 kσ = 1138.2 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Pi energy (kcal/mol)

Pentaazine Potential curves

Pi energy (kcal/mol)

Pi energy (kcal/mol)

50 50 45 45 σ 40 40 π 35 35 kπ = –969.7 30 30 25 25 20 20 Δk = 317.6 15 15 10 10 kσ = 1287.3 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

50 50 45 45 σ 40 40 π 35 35 kπ = –1033.6 30 30 25 25 20 20 Δk = 63.5 15 15 10 10 kσ = 1097.1 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Pi energy (kcal/mol)

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

1, 2, 3,–Triazine Potential curves

Sigma energy (kcal/mol)

Sigma energy (kcal/mol)

Sigma energy (kcal/mol)

Sigma energy (kcal/mol)

Pyridine Potential curves

50 50 45 45 σ 40 40 π 35 35 kπ = –911.6 30 30 25 25 20 20 Δk = 331.4 15 15 10 10 kσ = 1243.0 5 5 0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Figure 20.2  Harmonic potentials of the σ and π energies of pyridine, 1,3,5-triazine, pyrimidine, 1,2,3,5-tetraazine, pentaazine, and hexaazine. The force constants of the σ and π curvatures are given in kcal mol−1 Å−2.

representing the change of the σ potential energy, respectively. The algebraic sum of Equations 20.31 and 20.32 is the total potential energy or the AI,



AI = ∆E = ∆Eπ + ∆Eσ =

1 k 2 π

∆r 2 + R0

1 k 2 σ

∆r 2 R0

(20.33)

introduced here. Unless other criteria are specified, the Δr applied to all the studied molecules corresponds to the last distorted structure. In benzene, the Δr has a value of 0.242486 Å (Figure 20.1). Equation 20.33 can also be expressed as

AI = ∆E =

1 ∆k ∆r 2 . 2

(20.34)

397

Are Dicationic Chalcogenid Ring Systems Aromatic?

The term Δk is the algebraic sum of the σ and π force constants. Δk obtained for the benzene molecule is shown in Figure 20.1. Following the same procedure, the values of kσ, kπ, Δk, and the AI for all the other studied six-membered rings were obtained as well. These values are graphically displayed in Figure 20.2. Table 20.2 shows the values of Δr used as final points of the distortions of the R0 structures for the six-membered rings. The corresponding values of kπ, kσ, and Δk are also reported. The fact that the σ and π energy curves have a quadratic behavior can be very helpful for the prediction of the aromaticity of other types of ring systems. It is easy to understand that in order to describe the approximate behavior of the σ and π harmonic curvatures and to calculate the corresponding AI, one needs just to know the σ and π energies of the nondistorted optimized structure and of one distorted structure of a ring system. Table 20.2 gives the AI values calculated for all the here studied six-membered rings. Table 20.2 shows the calculated aromaticity decreases as follows: benzene > pyridine > 1,3,5-triazine > pyrimidine > 1,2,3,5-tetraazine > pentaazine > hexaazine. We notice that, if in the above sequence the AI value corresponding to 1,2,3,5-tetraazine is neglected, the aromaticity trend obtained is almost coincident to the trend reported by employing the χ πzz / nπ method. The only exception is that the order between 1,3,5-triazine and pyrimidine are inverted. However, the similarity observed comparing the aromaticity trend obtained in this work with the one given by the χ πzz / nπ method will change as soon as five-membered rings are included in the series of the considered systems. In fact, as it was already previously pointed out, in this case, the global aromaticity trend obtained in the present work is very similar to the one reported with the RE method. In order to obtain the AI values of the studied five-membered ring systems, the same procedure previously described for the six-membered rings was applied. However, in this case, the gradients of the σ and π harmonic potentials evaluated in R0 were also taken into account (apart the force constants). As an example, let us Table 20.2 Resonance Coordinate of the Last Distorted R1 Structure, kπ, kσ, and Δk of the Six-Membered Rings Molecule Benzene Pyridine 1,3,5-Triazine Pyrimidine 1,2,3,5-Tetraazine Pentaazine Hexaazine

Δr

kπ

kσ

Δk

0.242486 0.239564 0.242156 0.242722 0.242337 0.242722 0.242487

−894.4 −911.6 −969.7 −926.1 −935.9 −981.9 −1033.6

1240.8 1243.0 1287.3 1236.2 1174.7 1138.2 1097.1

346.4 331.4 317.6 310.1 238.8 156.3 63.5

Note: The r values are given in Å and the k values are given in kcal mol−1 Å−2.

398

Aromaticity and Metal Clusters

consider the case of the pyrrole molecule. For pyrrole, the harmonic potential adjusted to the π energy curve is

Eπ = 35.4369 + 126.954r − 210.152r 2

(20.35)

and the harmonic potential adjusted to the σ energy curve is

Eσ = 9.08436 − 124.647r − 430.012r 2 .

(20.36)

The first derivative of the π and σ harmonic potentials (Equations 20.35 and 20.36) with respect to the resonance coordinate r, evaluated in r = R0, gives the gπ and gσ gradients with magnitudes of 126.9 and −124.6 kcal mol−1 Å−1, respectively, whereas the second derivative gives kπ and kσ equal to −420.3 and 860.0 kcal mol−1 Å−2, respectively (see Figure 20.3). The Δk value for pyrrole is 439.7 kcal mol−1 and is shown in Figure 20.3. With these results, an AI value of 4.57 kcal mol−1 for the pyrrole molecule is obtained as



AI = ∆E = gπ ∆r +

1 1 k ∆r 2 + gσ ∆r + kσ ∆r 2 , 2 π 2

(20.37)

where the terms gπ and gσ are the magnitudes of the gradients of the π and σ harmonic potentials in R0, respectively. For convention, the Δr (Equation 20.37) applied to the five-membered systems corresponds to the distortion of the R0 structure to the resonance structure R1. For pyrrole, the value of Δr is equal to 0.208497 Å. For all the other studied five-membered rings, the magnitudes of the gradients, the force constants, and the AI values were obtained by applying the same procedure. The results are presented in Figure 20.4. In order to scale the AI values obtained for the five-membered rings with respect to the ones obtained for six-membered rings, the AI value obtained for each fivemembered ring molecule is multiplied by the dimensionless factor 2/3. This ratio is derived considering that in the formation of the five-membered rings, two double bonds and three single bonds are, respectively, involved.

σ π

100

80

80

60

60

40 20

40

kπ = –420.3

Δk = 439.7 kσ = 860.0

0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 Resonance coordinate

20 0.30

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

100

Pyrrole Potential curves

0 0.40

Figure 20.3  Harmonic potentials of the σ and π energies for pyrrole. The force constants of the σ and π are given in kcal mol−1 Å−2.

399

Are Dicationic Chalcogenid Ring Systems Aromatic?

Sigma energy (kcal/mol)

σ π

100

80

80

60

60

40

kπ = –448.1

40

20

Δk = 429.7 kσ = 877.8

20

Pi energy (kcal/mol)

Pyrazole Potential curves

100

0 0 –0.40–0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

40

40

20

kπ = –280.6 Δk = 445.2 kσ = 725.8

20

0 0 –0.40–0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

80

Furan Potential curves

σ π

60 40 20

60 kπ = –661.5 Δk = 444.2 kσ = 1105.7

40 20

80 60

40

kπ = –646.6

40

20

Δk = 422.3 kσ = 1068.9

20

100

100 80

60

100

0 0 –0.40–0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

100

80

σ π

Pi energy (kcal/mol)

60

1, 2–oxazole Potential curves

0 0 –0.40–0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

1, 2–oxazole Potential curves

80

σ π

60 40 20

100 80 60

kπ = –534.0 Δk = 428.3 kσ = 962.3

40 20

Pi energy (kcal/mol)

80

60

100 Sigma energy (kcal/mol)

80

σ π

100

Sigma energy (kcal/mol)

Imidazole Potential curves

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

100

0 0 –0.40–0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Figure 20.4  Harmonic potentials of the σ and π energies of pyrazole, pyrrole, imidazole, 1,2-oxazole, furan, and 1,3-oxazole. The force constants of the σ and π curvatures are given in kcal mol−1 Å−2.

Table 20.1 lists the AI values calculated for the studied five-membered rings. As Table 20.1 shows, the aromaticity of the five-membered ring molecules decreases as follows: pyrazole > pyrrole > imidazole > 1,2-oxazole > furan > 1,3-oxazole. We notice that the aromaticity trend of the five-membered ring systems studied here is very similar to the aromaticity trend predicted by employing the RE method. However, our predicted AI values show a better differentiation among the different five-membered rings (see Table 20.1).

20.3 RESULTS AND DISCUSSION In this section, the new separation σ − π energy scheme presented in Section 20.2 is applied to three chalcogenid ring systems such as S24 + , Se 24 + , and Te 24 +. For these

400

Aromaticity and Metal Clusters

Table 20.3 Aromaticity Index (AI) of s24+ , se24+ , and te24+ Molecule S2+ 4

AIa 7.15

NICS(0)b −10.6

NICS(1)c −7.9

Se 2+ 4

5.88

−9.8

−7.6

Te 2+ 4

4.26

2+ Note: For S2+ 4 and Se 4 , the values obtained by NICS, for comparison, are also reported. a AI calculated in this work. The values are given in kcal mol−1. b NICS data (in ppm) evaluated in the center of the ring [33]. c NICS data (in ppm) evaluated at 1 Å above the center of the ring [33].

three clusters, the corresponding AI value was also calculated employing our new AI implementation. The obtained AI values are reported in Table 20.3. For comparison, 2+ for the S2+ 4 and Se 4 systems, the AI values reported in the literature are also given [33]. To the best of our knowledge, for the Te 2+ 4 cluster no aromaticity values were previously reported. The three dicationic structures were optimized by employing the Vosko, Wilk, and Nusair (VWN) functional [34] in combination with the double zeta valence plus polarization basis set (DZVP) basis sets optimized for local functionals [35]. During the optimization the molecular plane was kept constant in all cases. These three dicationic cyclic systems, S24 + , Se 24 + , and Te 24 +, containing each 22 valence electrons, are examples of isolated planar homo atomics four-membered species with D4h symmetry [36–42]. The reported experimental bond distances for 2+ 2+ S2+ 4 [43,44] and for Se 4 [45] are of 1.98 and 2.28 Å, respectively. For the Te 4 system, differences in the bond distances can be found in the literature depending on the corresponding salt. For example, in the Te4(Al2Cl7)2 compound, the bond distance is of 2.660 Å [46], in Te4(AlCl4)2 of 2.669 Å [46], and in the Te4(SbF6)2 of 2.673 Å, respec2+ tively [47]. The bond distances calculated in the present work for the S2+ 4 , Se 4 , and 2+ Te 4 ring systems are 2.056, 2.326, and 2.730 Å, respectively. We notice that these values are just a bit longer than the reported experimental values. In Figure 20.5, the σ and π energy curves obtained for these three chalcogenid systems are illustrated. The σ energy curves are indicated by circles and the π energy curves are indicated by triangles, respectively. In all cases, at the equilibrium structure R0, the σ energy curves have a positive σ curvature, whereas the π energy curves have a negative curvature. Moreover, as we can see from Figure 20.5, in these ­systems the π curvature is smaller than the σ curvature. As a consequence, the ­directional force of the σ electrons will keep the cyclic structures with equal bonds. 2+ 2+ Figure 20.5 shows that, as we move from the S2+ 4 to Te 4 , passing through Se 4 , the π energy curve increases, whereas the σ energy curve decreases. The AI values ­calculated in this work for S24 + , Se 24 + , and Te 24 + are 7.15, 5.88, and 4.26 kcal mol−1, respectively (see Table 20.3). Over the last years, the concept of aromaticity from the magnetic point of view has been often been used for the study of metallic cyclic systems and of other

401

Are Dicationic Chalcogenid Ring Systems Aromatic? S42+

Potential curves

20 15 10 5

kπ = –295.8

25 σ π

20 15 10

Δk = 243.3

5

kσ = 539.1

Pi energy kcal/mol)

Sigma energy (kcal/mol)

25

0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate Se42+

25

20

20

15 10 5

kπ = –238.3

σ π

15 10

Δk = 199.9

5

kσ = 438.2

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

25

Potential curves

0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate Te42+

25

20 15 10 5

20 kπ = –268.2 Δk = 144.8 kσ = 413.0

σ π

15 10 5

Pi energy (kcal/mol)

Sigma energy (kcal/mol)

25

Potential curves

0 0 –0.40 –0.30 –0.20 –0.10 0.00 0.10 0.20 0.30 0.40 Resonance coordinate

Figure 20.5  Harmonic potential curves of the σ and π energies of S24+ , Se 24+ , and Te 24+ .

­ olecules formed with elements of the group 16 of the periodic table. In this context, m the magnetic aromaticity has been understood as a potential property of the compounds to be the place where electron circulation occurs upon the effect of a magnetic field. The aromatic compounds generate a diamagnetic (diatropic) current opposites to the applied magnetic field (B0) whereas the antiaromatic compounds generate a paramagnetic current (paratropic), which enforces the effect of the applied magnetic field [48,50–54]. The diatropic and paratropic terms refer to their effects in the chemical shift of a trial nucleus [33]. Often, the ring current is measured via the NICS [48,49] at the center of the ring or close to it. De Proft and co-workers [33] determined the aromaticity of a series of inorganic 2+ monocyclic compounds, including the S2+ 4 and Se 4 planar clusters. These authors obtained maps of the induced current density with the CTOCD-DZ/6-311G**

402

Aromaticity and Metal Clusters

(Continue Transformation of the Origin of the Zero Diamagnetic Current Density) 2+ method [33]. For both systems, S2+ 4 and Se 4 , a similar behavior in the total current density (σ + π) was found. In both cases, the current is dominated by a peripheric diatropic circulation of electrons (π electrons), which incorporates a central paratropic current (σ electrons). The π contribution to the ring current in these dicationic systems comes from the four HOMO electrons. Considering these results, S2+ 4 and Se 2+ resulted in being aromatic [33]. The calculated NICS(0) values in the ring center 4 2+ Se are −10.6 ppm and −9.8 ppm, for S2+ and , respectively (see Table 20.3). On the 4 4 contrary, the corresponding NICS(1), calculated at 1 Å above the center of the ring, are −7.9 ppm and −7.6 ppm, respectively [33]. 2+ Table 20.3 shows that the AI values calculated for S2+ 4 and Se 4 employing our new method reproduce the same trend as the one given by the NICS values reported in Ref. 2+ [33]. Therefore, the dicationic S2+ 4 is more aromatic than the dicationic Se 4 . Considering 2+ these results, for the dicationic Te 4 , for which ours is the only aromaticity value reported so far, one could expect an NICS value smaller than the one reported for Se 2+ 4 , in agreement with the aromaticity trend we have calculated for these systems. We analyzed their molecular orbitals (MOs) with the aim to gain more insight into the stability of these three chalcogenid systems. In Figures 20.6 through 20.8, the most relevant MOs for the dicationic ring systems, S24 + , Se 24 + , and Te 24 +, are respectively depicted. Figures 20.6 through 20.8 show that, for these systems, the HOMO is double

HOMO

HOMO – 1

HOMO – 2 dxz

dyz

dxy

dz2 dx2–y2

HOMO – 3

HOMO – 4 px

pz

py

HOMO – 5

z y

x

HOMO – 6

Figure 20.6  Correlation diagram of the p and d atomic orbitals with the most relevant molecular orbitals of S2+ 4 .

403

Are Dicationic Chalcogenid Ring Systems Aromatic?

HOMO

HOMO – 1

HOMO – 2 dxz

dyz

dxy

dz2 dx2–y2

HOMO – 3 HOMO – 4

px

pz

py

HOMO – 5 z HOMO – 6

x

y

Figure 20.7  Correlation diagram of the p and d atomic orbitals with the most relevant molecular orbitals of Se 2+ 4 . HOMO

HOMO – 1 HOMO – 2 dxz

dyz

dxy

dz2 dx2–y2 HOMO – 3 HOMO – 4

px

pz

py

HOMO – 5 z y

x

HOMO – 6

Figure 20.8  Correlation diagram of the p and d atomic orbitals with the most relevant molecular orbitals of Te 2+ 4 .

404

Aromaticity and Metal Clusters

LUMO + 1

LUMO

HOMO

HOMO – 1

Figure 20.9  Diagram of the most relevant molecular orbitals of the Cu8 cluster.

degenerate and in general the three systems show the same MO order. The only ­difference is in the energy separation of the MOs, of course. Interestingly, we notice that these MOs are very similar to the most relevant MOs of a metallic cluster, Cu8 [55], which are depicted in Figure 20.9. However, the Cu8 cluster is a system that is neither planar nor aromatic. The MOs of the Cu8 cluster are occupied following the well-known jellium model. The jellium model assumes that the valence electrons of the atoms of the clusters move freely in a uniform jellium of positive charges and gives a shell structure for the electronic states of the cluster (the shell model), very similar to the atomic shell structure. In this way, the valence electrons of the atoms of the clusters occupy shells of s, p, and d characters on the whole system. Moreover, these MOs have nodal properties, which show the s, p, and d characters of the cluster as a whole. As shown in Figure 20.9, these cluster orbitals can be observed as the HOMO, the LUMO, and the adjacent MOs. The HOMO of the Cu8 cluster is a double degenerated fully occupied MO, whereas the HOMO − 1 is a nondegenerated MO (Figure 20.9). These two MOs have the shape of an atomic p orbital, whereas the LUMO and LUMO + 1 have the shape of d-type atomic orbitals. These cluster MOs can also be appreciated in the dicationic chalcogenid systems (see Figures 20.6 through 20.8). Figures 20.6 through 20.8 show that, in the dicationic chalcogenid systems, the HOMO is a double degenerated MO with a shape of d-type atomic orbital. Moreover, the nondegenerated HOMO − 2 and the double-degenerated HOMO − 6 have the shape of p-type atomic orbitals whereas the single occupied HOMO − 3, HOMO − 4, and HOMO − 5 have the shape of d-type atomic orbitals. For the Cu8 cluster, the energy gap between the cluster MOs and the localized MOs is 2.77 kcal mol−1. The dicationic chalcogenid systems present also an energy gap between the cluster MOs and the localized MOs. Following the shell model, the S24 + , Se 24 + , and Te 24 + systems have d shells, which are completely occupied. This situation confers a large stability to these dicationic chalcogenid systems. Therefore, we can conclude that, based on the

Are Dicationic Chalcogenid Ring Systems Aromatic?

405

MOs analysis, the aromaticity trend of the dicationic chalcogenid S24 + , Se 24 + , and Te 24 + systems is unclear. However, on the basis of the new AI calculated here, the order of the aromaticity of these systems decreases as follows:

S24 + > Se 24 + > Te 24 + .

20.4 CONCLUSIONS In this work, a new all-electron DFT implementation of the separation of σ and π energies based on the Born–Oppenheimer approximation was presented. This new separation scheme is independent of the assignment of the atomic nuclear charges. A new AI taking into account the σ and π energy separation was also introduced for the first time in the literature. The introduction of this new AI is based on the fact the σ and π energy curves can be described with good accuracy in terms of a harmonic potential. This is a very important result since the σ and π energy curves of cyclic systems can be obtained directly from the information contained in the minimum structure of the studied system. It was shown that the σ and π energy curves calculated for different six- and five-membered systems can be very well described using the harmonic potential approximation. It was demonstrated that for six- and five-membered rings, the aromaticity trend obtained with the new AI is much similar to the aromaticity trend given by the resonance energy. Based on the proposed methodology, the benzene molecule is the most aromatic system among the different six-membered rings studied, whereas the hexaazine molecule is the least aromatic. For the five-membered rings, the pyrazole is the most aromatic and the 1,3-oxazole the least aromatic system. The two proposed new implementations were 2+ successfully applied to three dicationic chalcogenid systems such as S2+ 4 , Se 4 , and 2+ Te 4 . For these systems, the σ and π energy curves and the AI were calculated. For 2+ the S2+ 4 and Se 4 systems, the aromaticity trend obtained is similar to the one reported in the literature using the NICS method. For the dication Te 2+ 4 planar cluster, the aromaticity value was calculated for the time. This cluster results to be the least aromatic among the three studied dicationic chalcogenid systems. With the aim to gain more insight into the stability of these chalcogenid ring systems, a detailed study based on the analysis of their most relevant MOs was also performed. Following the shell model, the shape of the MOs of S24 + , Se 24 + , and Te 24 + is similar to the cluster MOs of a metallic cluster, Cu8, which is neither planar nor aromatic. On the basis of the analysis of the MOs, it is not possible to give a clear conclusion about the aromaticity of these three dicationic chalcogenid systems. Nevertheless, on the basis of the AI calculated with the presented new implementations, one can 2+ 2+ conclude that the S2+ 4 ring system is more aromatic than the Se 4 and Te 4 chalgogenid ring systems.

ACKNOWLEDGMENTS This work was financially supported by the CONACYT projects 48775-U and ­60117-U.

406

Aromaticity and Metal Clusters

REFERENCES

1. K. Jug, K., P.C. Hiberty, S.S. Shaik, σ-π energy separation in modern electronic theory for ground states of conjugated systems, Chem. Rev. 101, 1477, 2001. 2. S.S. Shaik, A. Shurki, D. Danovich, P.C. Hiberty, A different story of π-delocalization—The distortivity of π-electrons and its chemical manifestations, Chem. Rev. 101, 1501, 2001. 3. E. Hiickel, Quantum theory of double linkings, Z. Phys. 60, 423, 1930. 4. R.B. Woodward, R. Hoffmann, Stereochemistry of electrocyclic reactions, J. Am. Chem. Soc. 87, 395, 1965. 5. L. Salem, Intermolecular orbital theory of interaction between conjugated systems. 1. General theory, J. Am. Chem. Soc. 90, 543, 1968. 6. R.S. Mulliken, W.B. Person, Molecular Complexes, Wiley, New York, 1969. 7. S.S. Shaik, P.C. Hiberty, J.M. Lefour, G. Ohanessian, Is delocalization a driving force in  chemistry—benzene, allyl radical, cyclobutadiene, and their isoelectronic species, J. Am. Chem. Soc. 109, 363, 1987. 8. P.C. Hiberty, D. Danovich, A. Shurki, S.S. Shaik, Why benzene possess a D6h symmetry— a quasi-classical state approach for probing π-bonding and delocalization ­energies, J. Am. Chem. Soc. 117, 7760, 1995. 9. K. Jug, A.M. Köster, Influence of σ-electron and π-electron on aromaticity, J. Am. Chem. Soc. 112, 6772, 1990. 10. A.M. Köster, P. Calaminici, G. Geudtner, Z. Gómez-Sandoval, separation of σ and π energies, J. Phys. Chem. A 109, 1257, 2005. 11. G. Geudtner, Z. Gómez-Sandoval, F. Janetzko, P. Calaminici, First principle σ-π energy separation, Theor. Chem. Acc. 114, 137, 2005. 12. W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133, 1965. 13. B.I. Dunlap, J.W.D. Connolly, J.R. Sabin, 1st-row diatomic-molecules and local density models, J. Chem. Phys. 71, 4993, 1979. 14. J.W. Mintmire, B.I. Dunlap, Fitting the Coulomb potential variationally in linear-­ combination-of-atomic-orbitals density functional calculations, Phys. Rev. A 25, 88, 1982. 15. J.W. Mintmire, J.R. Sabin, S.B. Trickey, Local density functional methods in 2-dimensionally periodic systems—hydrogen and beryllium monolayers, Phys. Rev. B 26, 1743, 1982. 16. R.G. Parr, W. Yang, Density functional theory of atoms and molecules, Oxford University Press, New York 1989. 17. A.M. Köster, P. Calaminici, R. Flores-Moreno, G. Geudtner, A. Goursot, T. Heine, F. Janetzko, S. Patchkovskii, J.U. Reveles, A. Vela and D.R. Salahub, deMon2k, The deMon developers, 2004. 18. Z. Gómez-Sandoval, Ph.D. Thesis, CINVESTAV, Mexico, 2004. 19. K. Jug, A bond order approach to ring current and aromaticity, J. Org. Chem. 48, 1344, 1983. 20. K. Jug, A.M. Köster, Aromaticity as a multidimensional phenomenon, J. Phys. Org. Chem. 4, 163, 1991. 21. K. Jug, D.N. Nanda, SINDOl. 2. Application to ground-states of molecules containing carbon, nitrogen and oxygen-atom. Theor. Chim. Acta. 57, 95, 107, 131, 1980. 22. K. Jug, R. Iffert, J. Schulz, Int. J. Quantum Chem. 32, 265, 1987. 23. A.T. Balaban, D.C. Oniciu, A.R. Katritzky, Aromaticity as a cornerstone of hetero­ cyclic chemistry, Chem. Rev. 104, 2777, 2004. 24. B.A. Hess Jr., L.J. Schaad, Hückel molecular orbital π resonance energies—New approach, J. Am. Chem. Soc. 93, 305, 1971. 25. T.D. Gierke, H.L. Tigelaar, W.H. Flygare, Calculation of molecular electric dipole and quadrupole-moments, J. Am. Chem. Soc. 94, 330, 1972.

Are Dicationic Chalcogenid Ring Systems Aromatic?

407

26. C.W. Bird, Heteroaromaticity. 5. A unified aromaticity index, Tetrahedron 48, 335, 1992. 27. C.W. Bird, Heteroaromaticity. 10. The direct calculation of resonance energies of azines and azoles from molecular dimensions, Tetrahedron 53, 13111, 1997. 28. C.W. Bird, Absolute hardness as a convenient criterion of heteroaromaticity, Tetra­ hedron 53, 3319, 1997. 29. C.W. Bird, The relationship of classical and magnetic criteria of aromaticity, Tetra­ hedron 52, 9945, 1996. 30. K. Bystrom, The stabilization of 1,3,5-triazine derived from measurements of the enthalpies of combustion and sublimation, J. Chem. Thermodynamics 14, 865, 1982. 31. P. Jimenez, M.V. Roux, C. Turrion, F. Gornis, Thermochemical properties of N-heterocyclic compounds. 1. Enthalpies of combustion, vapor-pressures and enthalpies of sublimation, and enthalpies of formation of pyrazole, imidazole, indazole and benzimidazole, J. Chem. Thermodynamics 19, 985, 1987. 32. J.B. Pedley, R.D. Naylor, S.P. Kirby, Thermochemical Data of Organic Compounds, 2nd ed., Chapman & Hall, London, 1986. 33. F. De Proft, P.W. Fowler, R.W.A. Havenith, P.v.P. Schleyer, G.v. Lier, P. Geerlings, Ring currents as probes of the aromaticity of inorganic monocycles: P5-AS5-S2N2, S3N 3− , S4N 3+, S4N 24+ , S5N 3+ S24+S, and Se 24+, Chem. Eur. J. 10, 940, 2004. 34. S.H. Vosko, L. Wilk, M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin-density calculations—a critical analysis, Can. J. Phys. 58, 1200, 1980. 35. N. Godbout, D.R. Salahub, J. Andzelm, E. Wimmer, Optimization of gaussian-type basis-sets for local spin-density functional calculations. 1. Boron through neon, optimization technique and validation, Can. J. Phys. 70, 560, 1992. 36. R.J. Gillespie, J. Passmore, Adv. Inorg. Chem. Radiochem. 17, 49, 1975. 37. A. Cisar, J.D. Corbett, Synthesis and crystal-structure of a salt containing tritelluride(2−) anion, Inorg. Chem. 17, 49, 1975. 38. I.D. Brown, D.B. Crump, R.J. Gillespie, Crystal structure of tetrase-lenium bis(hydrogen disulfate), Inorg. Chem. 16, 632, 1977. 39. J. Passmore, G.W. Sutherland, P.S. White, Preparation and crystal-structures of (S7I)4S44(ASF6)6 and S4(ASF6)2.0.6SO2—a convenient synthesis of hexafluoroarsenate salts of chalcogen homoatomic cations, J. Chem. Soc. Chem. Commun. 330, 1980. 40. J. Passmore, G.W. Sutherland, P.S. White, Preparation and x-ray crystal-structure of μ-iodo-bis(4-iodo-cyclo-heptasulfur) tris(hexafluoroantimonate)-bis(arsenic trifluoride), [(S7I)2I](SBF6)3.2ASF3, and (S7I)4S4(ASF6)6 containing the iodo-cyclo-heptasulfur(1+) and the tetrasulfur(2+) cations, Inorg. Chem. 21, 2717, 1982. 41. J. Passmore, G.W. Sutherland, T.K. Widden, P.S. White, C.H. Wong, The preparation of S7brasf6, S7brsbf6, and the preparation and crystal-structure of (S7BR)4S4(ASF6)6 containing the bromo-cycloheptasulfur(1+) cation, Can. J. Chem. 63, 12097, 1985. 42. R. Faggiani, R.J. Gillespie, J.F. Sawyer, J.E. Vekris, Structures of nonadecasulfur(2+) bis[hexafluoroantimonate(1−)] S19(SBF6)2, octasulfur(2+) di-μ-fluoro-dodecafluoro­tri­ antimonate(1−) hexafluoroantimonate(1−) S8(SB3F14)(SBF6), and tetrasulfur(2+) di-μ-fluorodifluorodiantimony(2+) μ-fluoro-tetrafluorodiantimony(1+) pentakis[hexafluoroantimonat e(1−)] S4(SB2F4)(SB2F5)(SBF6)5, Acta. Crystallogr. C45, 1847, 1989. 43. P.W. Fowler, E. Steiner, Ring currents and aromaticity of monocyclic π-electron systems C6H6, B3N3H6, B3O3H3, C3N3H3, C5H 5− , C 7H 7+, C3N3F3, C6H3F3, and C6F6, J. Phys. Chem. A 101, 1409, 1997. 44. P.W. Fowler, R.W.A. Havenith, E. Steiner, Ring current and electron delocalisation in an all-metal cluster, Al4-4(2−), Chem. Phys. Lett. 359, 530, 2002. 45. I.D. Brown, D.B. Crump, R.J. Gillespie, D.P. Santry, Structure of Se 2+ 4 ion, Can. Chemi. Commun. (London) 15, 853, 1968.

408

Aromaticity and Metal Clusters

46. T.W. Couch, D.A. Lokken, J.D. Corbett, Crystal-structures of tetratellurium(2+) tetrachloraluminate and heptachlorodialuminate, Te 24+ (AlCl −4 ) and Te 24+ (Al 2 Cl 7− )2, Inorg. Chem. 11, 357, 1972. 47. G. Cardinal, R.J. Gillespie, J.F. Sawyer, J.E. Vekris, Charge-transfer interactions in the square-planar chalcogen cations, M 2+ 4 —preparation and crystal-structures of the compounds (Se 24+ )(Sb 2 F42+ )(Sb 2 F52+ )(SbF6− )5 , (Se 24+ ) (AlCl 4− )2 , and (Te 24+ ) (SbF6− )2 , J.  Chem. Soc., Dalton Trans. 765, 1982. 48. P.v.P. Schleyer, C. Maerker, A. Dransfeld, H. Jiao, N.J.R. van Eikema Homines, Nucleusindependent chemical shifts: a simple and efficient aromaticity probe, J. Am. Chem. Soc. 118, 6317, 1996. 49. P.v.P. Schleyer, H. Jiao, N.J.R. van Eikema Hommes, V.G. Malkin, O.L. Malkina, An evaluation of the aromaticity of inorganic rings: refined evidence from magnetic properties, J. Am. Chem. Soc. 119, 12669, 1997. 50. P.v.P. Schleyer, H. Jiao, What is aromaticity? Pure Appl. Chem. 68, 209, 1996. 51. L. Pauling, The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys. 4, 673, 1936. 52. F. London, The quantic theory of inter-atomic currents in aromatic combinations, J. Phys. Radium 8, 397, 1937. 53. J.A. Pople, Proton magnetic resonance of hydrocarbons, J. Chem. Phys. 24, 1111, 1956. 54. J.A. Elvidge, L.M. Jackman, Studies of aromaticity by nuclear magnetic resonance spectroscopy. 1. 2-pyridones and related systems, J. Chem. Soc. 859, 1961. 55. K. Jug, B. Zimmermann, A.M. Köster, Growth pattern and bonding of copper clusters, Int. J. Quantum Chem. 90, 594, 2002.

of Aromaticity 21 Study in Phosphazenes Prakash Chandra Jha, Y. Anusooya Pati, and S. Ramasesha Contents 21.1 Introduction...................................................................................................409 21.2 Methodology.................................................................................................. 412 21.3 Results and Discussion.................................................................................. 414 21.4 Summary....................................................................................................... 420 References............................................................................................................... 420

21.1 Introduction The definition of aromaticity and aromatic character of conjugated organic and inorganic systems has been of long-standing interest to chemists. Widely different properties such as molecular geometry, stabilization energy, and magnetic properties have been employed in the literature for characterizing aromaticity [1]. Yet there is no single measure for quantifying aromaticity. The general consensus is that aromatic molecules are planar and have equal bond lengths; they have 4n + 2 π electrons and large resonance energies. It is difficult to quantify aromaticity using any of these criteria. However, cyclic delocalization of mobile π electrons results in ring currents that are responsible for anomalous magnetic properties such as magnetic anisotropies, exalted magnetic susceptibilities, and large chemical shifts [2]. These properties appear to afford both a theoretical and an experimental way of quantifying aromaticity. The magnetic exaltation E is the difference in the average magnetic susceptibility (averaged over all orientations of the molecule with respect to the applied magnetic field) and empirically estimated susceptibility χM [3]. The latter is calculated as the sum of the atomic susceptibilities and an increment due to the presence of double bonds [4,5] χM = ∑AχA + nλC=C, where n is the number of double bonds). Systems with a positive E are classified as aromatic and a zero E as nonaromatic [6]. Flygare [7] has suggested that the exaltation values depend upon the nonlocal effects and are more qualitative than quantitative and, in many cases, such as for noncyclically delocalized molecules it depends on estimates and not direct measurements. Flygare considered the anisotropy in magnetic susceptibility Δχ = χzz − (1/2)(χxx + χyy), the difference in the out-of-plane component and the average of in-plane components of magnetic susceptibilities, as a criterion for aromaticity. 409

410

Aromaticity and Metal Clusters

In the early 1960s, nuclear magnetic resonance (NMR) studies showed that the protons in benzene and other aromatic compounds came into resonance at lower magnetic fields than those in the corresponding unsaturated carbon systems. Pople [8] attributed this to the ring current due to mobile π-electrons in aromatic systems set in motion when a magnetic field is applied perpendicular to the molecular plane. The direction of the current is such that a proton in the plane of the molecule experiences paramagnetic field, leading to a downfield shift, and the ones perpendicular to the plane experience a diamagnetic field (upfield shift) due to this ring current. Thus, the NMR chemical shifts can be related to the aromatic ring currents and have often been used to characterize aromaticity. Gomes et  al. have computed the current–density maps to interpret the diatropic and paratropic ring currents in organic molecules [9]. Schleyer et  al. proposed nucleus-independent chemical shift (NICS) values as the ­criteria for aromaticity [10]. They calculated the magnetic shielding for a test magnetic dipole situated at the ring center. Using this method, one can calculate the contribution from each ring in a polycyclic molecule unlike the case of magnetic exaltation or anisotropy, which is the measure of the overall effect of applied magnetic field. Soos et al. first obtained the ring currents as a second-order correction to energy, E2, in the presence of the magnetic field. E2 is proportional to the induced magnetic ­susceptibility [11]. They calculated the ring currents within interacting models that include the electron–electron interactions explicitly. The ring currents calculated were exact and served as a benchmark for comparisons. Applying the technique developed by Soos et al. [11], the ring currents for many polycondensed aromatic systems have been calculated by Ramasesha et al. [12] and the aromatic character of different rings in polycondensed conjugated systems have been analyzed. Aromaticity as a concept is not limited to only carbon systems. Indeed, according to the Hückel theory, any cyclic planar system with 4n + 2 electrons delocalized in the ring should exhibit aromaticity. It is possible to consider many inorganic systems that meet this criterion. Borazine is an example of such a system; this system is even isoelectronic with benzene. However, the electronegativity differences as well as participation of lone pairs in delocalization could imply weaker aromaticity in the inorganic systems. Recently, there is resurgence of interest in the study of aromaticity in inorganic molecules. Alexandrova et al. have studied the aromaticity of a small boron cluster with six atoms and its ionic counterparts [13], and from the current density maps, they observe a paramagnetic ring current. Fowler et al. have computed the ring current for inorganic analogs of benzene, borazine, boroxine, and so on [14]. They have obtained the current density maps and found that the diamagnetic current circulated around N in borazine and O in boroxine, whereas in the case of benzene it is through all the carbon atoms. They have obtained the aromaticity scale for these molecules by calculating the magnetic anisotropy with respect to benzene. On this scale, they find s-triazine slightly less aromatic than benzene but more aromatic than borazine. Boroxine is the least aromatic. Using the ring current index, Jug [15] concluded s-triazine to be as aromatic as benzene and borazine and boroxine to be moderately aromatic. Schleyer et al. have calculated the NICS values for a series of inorganic analogs of benzene. Their results are comparable with the results of Jug. They find that the aromaticity decreases with the ring size (C6H6 > Si6H6 ≈ Ge6H6). They also observed that borazine is not aromatic [2].

Study of Aromaticity in Phosphazenes

411

In this context, polyphosphazene -(NPX2)n -, an inorganic polymer, which was discovered more than 100 years ago [16], is a system of considerable interest. This polymer is very similar to conjugated carbon polymers, but contains N–P backbone instead of the C–C backbone found in the latter. Allcock et al. have synthesized short chains of polyphosphazene and found from x-ray diffraction studies that these systems show bond-length alternation (Peierls instability) similar to that observed in the conjugated system, polyacetylene [17]. The bond length alternation as measured from these studies is 0.07 Å. Ferris et al. have carried out ab initio calculations, and the calculated bond alternation is found to be 0.05 Å [18]. Sun has shown from the density functional theory (DFT) calculations, including the electron–electron correlation that the extent of bond alternation varies from 0.03 Å to 0.07 Å depending upon the substituents [19]. These results clearly show the similarities between phosphazenes and conjugated carbon systems. Thus, we should also expect to observe diamagnetic ring currents in cyclic phosphazenes in much the same way as in annulenes. Experimentally, small cyclic phosphazenes have been known for a long time, and they are found to have equal N–P bond lengths [20], just as the annulenes. The trimer cyclophosphazene has a structure similar to its carbon counterpart, benzene, with all the ring bond lengths equal to 1.61 Å and the bond angles between N–P–N and P–N–P being approximately 120°. According to the Craig’s model, the molecule has six delocalized π-electrons, and hence it should be aromatic [21]. However, Dewar has proposed his island model in which the electrons are localized on three atoms (N–P=N) ↔ (N=P–N) forming an island [22]. This implies that the system is not aromatic. However, the calculated resonance energy is quite large. On the contrary, unlike the organic conjugated systems, cyclic polyphosphazenes do not show absorptions in the near-ultraviolet region. Moreover, the absorption maximum remains nearly the same irrespective of the length of the polymer. Hence, there is no supporting evidence for the aromaticity of these systems. There have been several semiempirical and ab initio studies on polyphosphazenes. These have mainly focused on their geometry, electronic and molecular ­structure, and the possible conformations. The structure and stability of di- and tricyclophosphazenes have been studied by Trinquier by ab initio calculations [23]. Ferris et al. have also studied the electronic structure of (F2PN)3, (F2PN)4, and OP(F2) NP(F2)NPF3 using ab initio molecular orbital calculations [18]. The geometries of both the linear chains and cyclic systems have been obtained. The effect of ligand electronegativity on the geometry and electronic structure of these systems has been studied by replacing the halide groups by hydrogen. It is observed that the bond length alternation persists even in the presence of hydrogen. It is also found that the chemical bond and charge distribution is highly polarized through π-bond; nitrogen is negatively charged compared with phosphorus. The molecular conformations of the polyhalophosphazenes have been studied by Tanaka et al. [24]. Meyer first proposed a helical structure for crystalline compounds [25]. Allcock et al. have reported the two polymorphs with cis–trans planar structures with the variation of temperature for both chloro- and fluoro- polymers from x-ray diffraction studies [26]. Tanaka et al. have carried out a semiempirical calculation within the tight-binding self consistent field (SCF) method [24]. They showed that the cis–trans planar structure is the most stable ­conformation. Sun included the electron–electron interaction and

412

Aromaticity and Metal Clusters

studied the ­conformations of these polymers using the DFT calculations [19]. He observed that the twisted helical structure is energetically more stable than the cis– trans planar structure because of the intramolecular electrostatic interaction. The bond length alternation for the unsubstituted phosphazene is half of that for polyacetylene. They found that the π-bond is due to the negative hyperconjugation from the lone pair of electrons on N to the σ bond of P–X, where X is any halogen atom. Besides the ground-state properties, linear and nonlinear optical properties of these systems have been studied by Ferris et  al. using the semiempirical valence electron molecular orbital method [18]. An earlier paper by Jha et al. reports ­nonlinear optical coefficients of linear polyphosphazenes as a function of the system size [27]. However, there seems to be no systematic study of the magnetic properties of cyclic polyphosphazene so far. However, it is worth mentioning that there is some heuristic study of such systems by Craig et al., who have examined the effect of variation of the electronegativity on the magnetic susceptibility within the Hückel approximations for the cyclic compounds formed by second and third row elements [21]. They find a smooth exponential decay of magnetic susceptibility with increase in the difference between the electronegativity of two elements. It should be emphasized at this point that inclusion of electron–electron interaction is essential to explain even qualitatively optical as well as magnetic properties of conjugated systems [11,28]. In this chapter, we report the study of optical and magnetic properties of cyclic polyphosphazene within an interacting model, namely Pariser–Parr–Pople (PPP) model. We study the effect of ligands on these properties. In the next section, we give a brief account of the computational method; this will be followed by a section dealing with our results and discussion.

21.2 Methodology Computation of magnetic susceptibilities requires gauge invariant orbitals [29]. Pople had done necessary approximations to calculate the matrix elements of the molecular Hamiltonian in the presence of magnetic field using gauge invariant atomic orbitals [8]. The Hamiltonian in the presence of a magnetic field is given by 2



HB =

1  eA  P0 + + V (r ),  2m  c 

(21.1)

where V(r) is the potential energy operator. A is the vector potential for a given magnetic field B, and depends on the choice of gauge. Hence the molecular orbital coefficients become gauge dependent. To overcome this problem, Pople used gauge invariant orbitals by transforming the atomic orbital φj as



e   χ j = φj exp  − i A j .r   c  

(21.2)

where Aj is the vector potential at site j. This transformation leads to the modification of transfer integral terms in the Hamiltonian. Pople introduced some approximations to calculate the matrix elements of the Hamiltonian. Under these approximations,

413

Study of Aromaticity in Phosphazenes

the hamiltonain for a noninteracting model can be written in the second quantized notation as H 0 ( B) =

∑∑t < ij >

exp(if )ai+σ a jσ + exp(−if )a +jσ aiσ  ,  

0

σ

(21.3)

where ai+σ (aiσ ) creates (annihilates) an electron with spin σ in the orbital at site i. t0 is the transfer integral or the Hückel resonance integral between the bonded sites i and j. f is the London’s phase factor for a magnetic field B applied perpendicular to the molecular plane and is given by f = (eSB/cN), where S is the area enclosed by the aromatic ring of size N. Magnetic properties are calculated as the second-order correction to energy. Expanding the above equation till second order in f, one obtains H 0 ( B) = H 0 + ifv− −



1 2 f v+ , 2

(21.4)

where H0 is the Hückel Hamiltonian in the absence of magnetic field, and v± is given by v± = t 0



∑ (a

+ iσ i +1σ

a

± ai++1σ aiσ ).

iσ

(21.5)

The effect of electron–electron interactions on electronic as well as magnetic properties is well known. Inclusion of electron–electron interaction gives the correct picture of energy levels in the aromatic molecules [28]. Soos et al. showed that the ring current of charged annulene is larger than the neutral annulenes by taking an interacting model [11]. They have obtained the ring currents in an interacting model given by H = H 0 ( B) + H int ,





H int =

Ui

∑ 2 n (n i

i

i

− 1) +

∑ V (n ij

i

(21.6)

− zi )(n j − z j ).

ij

(21.7)

Ui is the onsite correlation energy of the orbital at site i. The interorbital or intersite electron–electron repulsion parameter Vij is given by Ohno parameterization [30]:



 28.974  2  2  Vij = 14.397  r + ij  Ui + U j    

−1 / 2

,

(21.8)

where the distance rijs are in Å, the energies Ui and Vij are in eV. zi is the local chemical potential of the orbital given by the number of electrons contributed by the orbital i to the conjugation backbone. Although all these parameters are well para­ meterized for carbon and nitrogen [31,32], for other atoms they have not been determined. In an earlier paper by Jha et al., the values for all these quantities for N and P have been parameterized by comparing with optical spectra of phosphazenes

414

Aromaticity and Metal Clusters

from experiments [27]. We have used the U values obtained from that study, that is, U = 11.64 and 14.12 eV for N and P, respectively. The site energy was taken to be 0.0 and −5.8 eV for P and N, respectively. To study the effect of varying the ligand, we have changed the site energy of N from 5.8 to 0.0 eV, in steps of 1 eV, keeping the site energy of P at 0.0 eV. We have employed the exact diagonalization method with diagrammatic valence bond basis [31]. It is a full configuration interaction (CI) calculation. Besides obtaining ground-state and excited-state energies and properties such as charge and spin density, charge–charge correlation, we have obtained the ring current values (E2). E2 is a second-order correction to the energy in the presence of the magnetic field applied perpendicular to the plane of the molecule. E2 is calculated as the sum of two terms involving v+ and v− operators Equation 21.5, and is given by



E 2 = Ψ 0 v+ Ψ 0 + 2

∑ k ≠0

2

Ψ k v− Ψ0

(21.9)

,

( E k − E0 )



where ψ0 and ψk are the ground- and excited-state wavefunctions, and E0 and Ek are the ground- and excited-state energies of the unperturbed Hamiltonian, respectively. The first term contributes to the diamagnetic susceptibility and the second term is the van Vleck paramagnetic susceptibility. While calculating the second term, we have used the correction vector (CV) method to avoid summing over all the excited states [33]. In the CV method, we define a correction vector |φ(1)〉by the equation

( H − E0 I ) ϕ (1) = v− G ,

(21.10)



where |φ(1)〉 is formally given by ϕ (1) =

∑ k

Ψ k v− Ψ0

Ψk

( E k − E0 )

(21.11)

and the second term in Equation 21.9 can be evaluated as

2 ϕ (1) v− Ψ 0 .



(21.12)

Since the Hamiltonian matrix H is known in the chosen basis (Valene bond or Slater determinant), |φ(1) 〉 can be obtained by solving Equation 21.10 in the same basis. Since |ψ0 〉 is also known, evaluation of Equation 21.12 is straightforward [33,34].

21.3 Results and Discussion We have taken a planar geometry with equal bond lengths of 1.63 Å between the nearest neighbor atoms P and N. We have considered only valence orbitals at each site, and electron contribution is taken in terms of site energy. Earlier studies showed that the effect of the bonding is negligible [35]. We have studied the cyclic

415

Study of Aromaticity in Phosphazenes

p­ hosphazene and their substituted compounds up to a system size of 7 units. The optical gaps for the phosphazenes are reported in the earlier paper for 6 units. Here we will focus on the ground- and excited-state properties as a function of variation of electronegativity of the ligand group and also discuss the aromaticity and antiaromaticity of these compounds. The criteria for aromaticity are defined in several ways. One of them is the planarity of the molecule. The others are bond alternation, stability, and the Hückel’s 4n + 2 rules. However, these rules cannot be applied for inorganic molecules. Schleyer et al. have used magnetic susceptibility computed at the ring center (i.e., NICS) as the criteria for aromaticity/nonaromaticity of inorganic molecules [10]. We have used the second-order correction to energy, E2, which is proportional to the induced magnetic susceptibility as the aromaticity index. A positive E2 corresponds to a diamagnetic susceptibility (the system is aromatic) and a negative value to a paramagnetic susceptibility (the system is nonaromatic or antiaromatic). In Figure 21.1 we show E2 values as a function of Δε, the difference in site energy between N and P sites, for a noninteracting model. Site energy at P is taken as zero and at N it is varied. Hence, the difference in site energy between P and N is alternatively written also as site energy at N. All the 4n systems show paramagnetic behavior, whereas 4n + 2 systems are diamagnetic. The ring current decreases exponentially with increase in the difference in electronegativity between N and P. This is due to the reduction in delocalization of electrons between the orbitals. Craig et al. found a similar behavior within the Hückel model for six electron systems [21]. As the system size increases, the E2 value becomes nearly zero even for a small Δε. When we introduce the electron–electron interactions, the nature of the curve is different. When Δε is small, the 4n systems have finite E2 values. As difference in electronegativity increases, the E2 values increase and when it is approximately 3.5 eV, the E2 value diverges (Figure 21.2). This divergence decreases with increase 15 6 8 10 12 14

E2(eV)

10

5

0

−5

−10 −6

−5

−4

−3

−2

−1

Site energy difference

Figure 21.1  E2 vs. difference in site energy at N and P, for the Huckel model as a function of system size.

416

Aromaticity and Metal Clusters 2 –1

6 10 14

(a)

–4 –7

E2

–10 –13

8 12

(b)

550 350 150 –50 –6

–5

–4

–3

–2

–1

Site energy difference

Figure 21.2  E2 vs. difference in site energy at N and P, for PPP model as a function of system size; (a) 4n + 2 rings and (b) 4n rings.

in the system size. On further increase in Δε, E2 value reduces to zero. Even for 4n + 2 systems, the E2 values initially decrease with system size and reaches a minimum around the same point and then increases again. The reduction in electron delocalization with increase in Δε reduces the ring current. To understand the reason behind this unusual behavior, we have calculated the optical gap and singlet–triplet gaps. We have obtained the charge and spin density, spin–spin correlations, and charge–charge correlations. We have used reflection sym­ metry about the plane perpendicular to the molecular plane to obtain the optical gap. The optical gap is the energy difference between the two lowest states in the A1 and A2 subspaces. Figure 21.3 shows the optical gap as a function of electronegativity for different system sizes. For 4n systems, at low values, the ground state is in A1 subspace. As the difference in electronegativity increases, the optical gap decreases initially and around 3.5 eV it is zero, that is, the states belonging to two different symmetry subspaces become degenerate. After this critical value of Δε, the ground state switches symmetry, and is in the A2 subspace. Upon increasing the Δε further, the optical gap increases quite sharply. In case of the 4n + 2 systems, the optical gap increases slowly in the beginning, and then increases quite sharply. The crossover is smooth unlike in 4n systems. The ionization potentials of the cyclic tetramer phosphonitrilic chloride (9.80 eV) are lower than the trimer (10.26 eV) and pentamer (9.81 eV). The trend is the same for the fluoride series. The optical gap shows a similar behavior with respect to the size of the system for a chosen value of Δε, that is, the optical gap for a tetramer is lower than that for the hexamer and trimer. As the

417

Study of Aromaticity in Phosphazenes 4.5

6 8 10 12 14

S−S gap

3.5

2.5

1.5

0.5

−0.5 −6

−5

−4 −3 −2 Site energy difference

−1

Figure 21.3  Optical gap vs. difference in site energy at N and P, for different ring sizes of cyclic phosphazenes.

electronegativity increases, the ionization potential remains the same irrespective of the size of the system. From Figure 21.3 we can see that at large values, the optical gap is constant except for the hexamer. We have given the singlet–triplet gap in Figure 21.4 as a function of Δε. As Δε is increased, the singlet–triplet gap remains constant (0.5 eV), till the transition point. After the transition point, the gap increases almost linearly with increase in the electronegativity. As in the case of optical gap, spin gap also shows a sharp change for 4n systems and it varies smoothly for the 4n + 2 system. At large Δε, the gap seems to converge to a single point irrespective of the system size. We have characterized the ground state by calculating the charge density as a function of Δε. We have plotted the ratio of charge density on P to the charge density on N in Figure 21.5 for different ring sizes. In the unsubstituted phosphazene molecules (corresponding to Δε of 5.8 eV), the N has almost two electrons (1.8), whereas P has nearly 0 (≈0.2) electrons. This was observed from earlier quantum chemical calculations [18]. Substituting the hydrogen by more electronegative halide groups or other alkyl or aryl groups corresponds to reduction in the difference between the electronegativity of N and P. Therefore, as we move along the curve to the right, we see that the N increases slowly, indicating the charge transfer from N to P. As the transition point is reached, the extent of charge transfer increases sharply. There is complete transfer of an electron from N to P; in a range of 0.5 eV, both the electrons on N will be transferred to P. After the transition point, the change in charge density ratios is very small. All the curves meet at a single point corresponding to a Δε of 3.51 eV. This implies that if one could replace the H group by another group with the electronegativity corresponding to 3.50 eV, irrespective of the system size, both the N and P will have equal charge density. This point also corresponds to the peak in the ring current plot, in Figure 21.2b.

418

Aromaticity and Metal Clusters 7 6 5 4 3

S−T gap

2 1

Huckel model 6 8 10 12 14

3 2 1 0 −6

PPP model −5

−4 −3 −2 Site energy difference

−1

Figure 21.4  Singlet–triplet gap vs. difference in site energy at N and P, for different ring sizes of cyclic phosphazenes.

1

6 8 10 12 14

ρP /ρN

0.8

0.6

0.4

0.2 −6

−5

−4 −3 Site energy difference

−2

Figure 21.5  Ratios of charge density, ρP to ρN vs. difference in site energy at N and P.

419

Study of Aromaticity in Phosphazenes (a)

(b) 3 1.8 2.8 3.8 4.8 5.8

1

1.8 2.8 3.8 4.8 5.8

n_i·n_j

n_i·n_j

2

0.5

1

0

0

2

4 |i-j|

0 0

8

6

(b) 2

4 |i-j|

8

6

Figure 21.6  Charge–charge correlation for 14 site ring: (a) for P with other atoms and (b) for N with other atoms, for different site energy at N.

In Figure 21.6 we plot charge–charge correlations for the largest system we have studied. Since the system is cyclic, there are only eight distinct correlations between N and P with other atoms. Figures 21.6a and b give correlations between P and N with other atoms, respectively. The correlations between the atoms decay slowly with the distance. As the Δε is increased from 1.8 to 5.8 eV, N–N correlations increase. This is in conformity with the fact that with the increase in the electronegativity difference, there is an increased charge transfer from P to N. Charge–charge correlations as a function of system size are given in Figure 21.7a and b. Here we give only correlation between the same sites versus Δε as a function of system size. Figure 21.7a gives the correlations between N and N and 21.7b gives (b)

n_P·n_P

2.5

6 8 10 12 14

2

1

6 8 10 12 14

n_N·n_N

(a) 3

0.8

1.5 0.6 1 –5

–4

–3 Site energy

–2

–5

–4

–3 Site energy

–2

Figure 21.7  Nearest-neighbor-site charge–charge correlation for (a) P with P and (b) N with N for different system sizes.

420

Aromaticity and Metal Clusters

that for P and P. The magnitude of correlation increases for N with N and decreases for P with P as Δε is increased from 1.8 to 5.8. The curves for all the system sizes meet at a single point like in the case of ratios of charge density plot. This indicates the transition point to be at 3.5 eV.

21.4 Summary We have studied larger phosphazene molecules within exact diagonalization method using the diagrammatic valence bond basis. Both optical and magnetic properties are studied as a function of variation of substituents. We observe symmetry crossover in the ground state of the system, as a function of Δε. The ring current values also show a divergence at the same point. The charge density, charge–charge correlations, and spin–spin correlations calculations confirm the transition to be at 3.5 eV.

REFERENCES



1. Schleyer, P. v. R.; Jiao, H. What is aromaticity? Pure and Appl. Chem. 1996, 68, 209– 218; Cyranksi, M. K.; Krygowski, T. M.; Katrizky, A. R.; Schleyer, P. v. R. To what extent can aromaticity be defined? J. Org. Chem. 2002, 67, 1333–1338; Havenith, R. W. A.; Jenneskens L. W.; Fowler, P. W. Ring currents that survive bond alternation in ­constrained 8-pi and 6-pi monocycles, Chem. Phys. Lett. 2003, 367, 468–474; Fowler, P. W.; Steiner E.; Jenneskens, L. W. Ring current aromaticity in triplet states of 4n pi ­electron monocycles, Chem. Phys. Lett. 2003, 371, 719–723. 2. Schleyer, P. v. R.; Jiao, H.; Hommes, N. J. R. van E.; Malkin, V. G.; Malkina, O. L. An  evaluation of the aromaticity of inorganic rings: refined evidence from magnetic properties, J. Am. Chem. Soc. 1997, 119, 12669–12670. 3. Pacault, A.; Hoarau, J.; Marchand, A. Aspects recents du diamagnetisme, Adv. Chem. Phys. 1961, 3, 171–238. 4. Gomes, J. A. N. F.; Mallion, R. B. Aromaticity and ring currents, Chem. Rev. 2001, 101, 1349–1384. 5. Lazzeretti, P. Ring currents, Progr. Nucl. Magn. Resonance Spectrosc. 2000, 36, 1–88. 6. Dauben, H. J.; Wilson, J. D.; Laity, J. L. Diamagnetic susceptibility exaltation in ­hydrocarbons, J. Am. Chem. Soc. 1969, 91, 1991–1998. 7. Flygare, W. H. Magnetic interactions in molecules and an analysis of molecular ­electronic charge distribution from magnetic parameters, Chem. Rev. 1974, 74, 653–687. 8. Pople, J. A. Molecular orbital theory of aromatic ring currents, Mol. Phys. 1958, 1, 175–180. 9. Gomes, J. A. N. F. Topological elements of the magnetically induced orbital current densities, J. Chem. Phys. 1983, 78, 4585–4591. 10. Schleyer, P. v. R.; Maerker, C.; Dransfeld, A.; Jiao, H.; Hommes, N. J. R. van E. Nucleus independent chemical shifts: a simple and efficient aromaticity probe, J. Am. Chem. Soc. 1996, 118, 6317–6318. 11. Kuwajima, S.; Soos, Z. G. Aromaticity and electron correlation, J. Am. Chem. Soc. 1987, 109, 107–113. 12. Anusooya, Y.; Chakrabarti, A.; Pati S. K.; Ramasesha, S. Ring currents in condensed ring systems, Int. J. Quant. Chem. 1998, 70, 503–513. 13. Alexandrova, A. N.; Boldyrev, A. I.; Zhai, H-J.; Wang, L.-S.; Steiner, E.; Fowler, P. W. Structure and bonding in B6− and B6: planarity and antiaromaticity, J. Phys. Chem., A 2003, 107, 1359–1369.

Study of Aromaticity in Phosphazenes

421

14. Fowler, P. W.; Steiner, E. J. Ring currents and aromaticity of monocyclic pi-electron systems C6H6, B3N3H6, B3O3H3, C3N3H3, C5H 5− , C7H 7+, C3N3F3, C6H3F3, C6F6, Phys. Chem. A 1997, 101, 1409–1413. 15. Jug, K. A bond order approach to ring current and aromaticity, J. Org. Chem. 1983, 48, 1344–1348. 16. Rose, H. Ann. Chem. 1834, 11, 131–138. 17. Allcock, H. R.; Tollefson, N. M.; Arcus, R. A.; Whittle, R. R. Conformation, bonding and flexibility in short-chain linear phosphazenes, J. Am. Chem. Soc. 1985, 107, 5166–5177. 18. Ferris, K. F.; Risser, S. M. Bond alternation in phosphonitrilic compounds, Chem. Phys. Lett. 1990, 174, 333–337. 19. Sun, H. Molecular structures and conformations of polyphosphazenes, a study based on density functional calculations of oligomers, J. Am. Chem. Soc. 1997, 119, 3611–3618. 20. Allcock, H. R.; Kugel, R. L.; Stroth, E. G. Phosphonitrilic compounds, XIII. structure and properties of poly(difluorophosphazene), Inorg. Chem. 1972, 11, 1120–1123. 21. Craig, D. P.; Heffernan, M. I.; Mason, R.; Paddock, N. L. Delocalization and magnetic properties of the phosphonitrilic halides, J. Chem. Soc. 1961, 108, 1376–1382. 22. Dewar, M. J. S.; Lucken, E. A. C.; Whitehead, M. A. The structure of the phosphonitrilic halides, J. Chem. Soc. 1960, 2423–2439. 23. Trinquier, G. J. Structure, stability and bonding in cyclodiphosphazene and cyclotriphosphazene, Am. Chem. Soc. 1986, 108, 568–577. 24. Tanaka, K.; Yamashita, S.; Yamabe, T. Electronic structure of poly(dichlorophos­ phazene), Macromolecules 1986, 19, 2062–2064. 25. Meyer, K. H.; Lotmar, W.; Pankow, G. W. Helv. Chim. Acta 1936, 19, 930–948. 26. Allcock, H. R.; Kugel, R. L.; Stroth, E. G. Phosphonitrilic compounds XIII. Structure and properties of poly(difluorophosphazene), Inorg. Chem. 1972, 11, 1120–1123. 27. Jha, P. C.; Krishnan, A.; Das, P. K.; Ramasesha, S. Nonlinear optical properties of linear chain phosphazenes (PN)x, J. Chem. Phys. 2002, 117, 2873–2881. 28. Soos, Z. G.; Ramasesha, S.; Galvao, D. S. Band to correlated crossover in alternating Hubbard and Pariser Parr Pople chains: nature of the lowest singlet excitation of conjugated polymers, Phys. Rev. Lett. 1985, 71, 1609–1612; Soos, Z. G.; Ramasesha, S. Valence bond theory of linear Hubbard and Pariser-Parr-Pople models, Phys. Rev. 1984, B29, 5410–5422. 29. London, F. J. Phys. Radium 1937, 8, 397–409. 30. Ohno, K. Some remarks on Pariser-Parr-Pople method, Theor. Chim. Acta 1964, 2, 219–227. 31. Soos, Z. G.; Ramasesha, S. Diagrammatic valence bond theory, in Valence Bond Theory and Chemical Structure, Ed. D. J. Klein and N. Trinajstic, Elsevier, Amsterdam, 1990, 81–109. 32. Albert, I. D. L. A Theoretical Study of the Linear and Nonlinear Optical Properties of Conjugated Systems. Ph.D. thesis, 1991. 33. Soos, Z. G.; Ramasesha, S. Valence bond approch to exact nonlinear optical properties of conjugated systems, J. Chem. Phys. 1989, 90, 1067–1076. 34. Albert, I. D. L.; Das, P. K.; Ramasesha, S. A model exact calculation of SHG coefficients in polysubstituted push-pull polyenes, Chem. Phys. Lett. 1990, 168, 454–460. 35. Ferris, K. F.; Samuels, W. D.; Exarhos, G. J. Electrical and Optical Polymer Systems, Eds., D. L. Wise, G. E. Wnek, D. J. Trantolo, T. M. Cooper and J. D. Gresser, Marcel Dekker, New York, NY, 1998, 663–680.

Index A Ab initio molecular dynamics, 173, 206. See also Jellium model; Ultimate jellium model (UJM) orbital theory, 7 ACID. See Anisotropy of the current induced density (ACID) Adaptive natural density partitioning (AdNDP) method, 58 bonding elements recovery, 59, 61 Ga3− cluster and cyclogallene, 62–63 Hg6− 4 cluster analysis, 58–60 Sn 6− 5 cluster analysis, 60–62 ADEs. See Antiaromatic destabilization energies (ADEs) Adiabatic ionization potential, 210 AdNDP method. See Adaptive Natural Density Partitioning (AdNDP) method AFM metals. See Antiferromagnetic metals (AFM metals) AIM. See Atom in the molecule (AIM) Al 2− 4 compounds aromaticity, 248 CMO-NICS, 265 eigenvectors, 258, 259 HOMO, 258 magnetic response, 255 MAl4−clusters structure, 57 MCI contributions, 257 MCI over 4 Al Atoms, 256 molecular orbital energy levels, 23 NICS, 11, 259 ring, 328 ring current maps, 260, 261, 262, 263, 264, 265 structure of MAl4, 328 Alkali atoms, 143 Al13, 126 and carbon nanostructures, 127 magnetic moment variation, 154 magnetic response in, 255 stability, 156 z-component contour lines, 197 Alkali metal clusters, 153 alkali addition energy variation, 156–157 aromaticity, 194, 248 ionization potential, 143 magnetic moment variation, 153–154 magnetic superatom, 157

mass spectra, 138 models to explore, 206 one-electron energy levels, 155 spd overlaps, 154, 155–156 stability, 273, 274 synthesis and structure of aromatic, 399 All-metal cluster, 56 aromatic hydrogen bonds, 303 aromaticity, 56, 95 [Be6Zn2]2−, 371 gas-phase, 309 NICS values, 303, 304 planar, 57 Anisotropy, 410 ACID mappings, 301 consequence, 148 of the current-induced density (ACID), 3, 45–46, 345 magnetic, 41, 410 magnetic shielding, 410 magnetic susceptibility, 41, 305, 409 MCA, 148 total polarizabilities, 114 Annulenes, 272–273 (4n + 2) π electron rule, 192 Hückel, 38, 39 Möbius, 39 topological resonance energy, 39 Antiaromatic destabilization energies (ADEs), 299 Antiaromaticity, 351 estimation, 74 heterocycle uranium ring core, 351 magnetotropicity, 356 metal atom role, 309 NICS value, 6, 80 paratropic magnetic response, 358 paratropic NICS, 352 π–aromaticity, 99, 302 radial, 57 σ-aromaticity, 99, 302, 359 topological resonance energy, 39 Antiferromagnetic metals (AFM metals), 145 coupled state, 157 AO. See Atomic orbital (AO) ARCS. See Aromatic ring current shielding (ARCS) Aromatic compounds, 2, 297 Ab initio calculations, 196 all-metal, 247, 323–324

423

424 Aromatic compounds (Continued) analysis methods, 70 benzene, 278 bonds, 281 characterization, 3 diamagnetic current, 401 diatropic ring currents, 287 electron count rules, 272 electron delocalization, 246 energy levels comparison, 23 hydrogenation, 20–22 magnetic properties, 281 metal-containing, 339, 346 polyaromatic aromaticity, 44 stabilty, 3, 279, 300 substitution reactions, 32 Aromatic-fluctuation-index-(FLU), 3 Aromaticity, 1, 104, 187, 278, 297–298, 323, 410, 415. See also Benzene ACID, 45–46 all-metal aromaticity, 371 aromatic sextet theory, 2, 31–32 ASE, 3 atoms in molecules theory, 43–44 bond lengths and orders, 281 borazine, 200 categories, 69, 70, 47 chemical bond interpretation, 56 conflicting, 58, 302 descriptors, 302, 345 Diels–Alder reaction, 42 different aromatic molecules, 4 different indexes, 298, 308 d orbital, 350 energetic measures, 37–41 estimation, 74 features, 81 hardness, 42, 43 index (AI), 35–36 inorganic chemistry, 350 inorganic monocyclic compounds, 401–402 isodesmic reaction, 279–280 magnetic measures, 41–42 magnetic properties, 281 measuring categories, 3, 6 metal atom role, 309 multidimensional, 32 multifold, 5 NICS value, 6, 80 NMR chemical shifts, 410 obstructions, 46–47 π–aromaticity, 99, 350, 371 π electrons, 280 Polansky index, 44–45 polarizability, 45–46 PSM, 278

Index radial, 57 reactivity, 42 related terms, 32 resonance energy, 42 scale, 410 σ-aromaticity, 99, 282, 323, 350 σ–π energy separation, 392–394 stability, 279 structural measures, 35–37 tetramer clusters, 5–6 theoretical methods, 7 three-dimensional, 32, 283 trend, 397 1,3,5-tridehydrobenzene cation, 282 uses, 95, 194 Aromaticity Index (AI), 298, 304, 392–399, 400 BAC index, 299 benzene σ and π energies, 395 classification, 298 energetic criteria, 299 HOMA index, 299 magnetic criteria, 300 potential curves, 396 potential energy, 392–394 rationalization, 308 reactivity criteria, 300 structural criteria, 298 validation, 394 Aromatic molecules, 409 Aromatic ring-current shielding (ARCS), 3, 70, 188, 248, 300 Aromatic sextet theory, 2 Aromatic stabilization energies (ASEs), 3, 70, 40–41, 299 ASEs. See Aromatic stabilization energies (ASEs) Atomic charges, 375–377 Atomic orbital (AO), 57 correlation diagram, 402–404 Atom in the molecule (AIM), 252. See also Hirshfeld-I Atom obey Hund’s rules, 145 Attractors, 98, 252 Au20 cluster, 55

B B3LYP. See Becke three-parameter Lee–Yang– Paar (B3LYP) BAC. See Bond alternation coefficient (BAC) Bader method, 252 Basin, 98 hopping method, 164 no-bonding, 100 BCP. See Bond critical points (BCP) BE. See Binding energies (BE)

425

Index Be6Zn2 clusters bond stretch isomers, 382 electronegativity, 380 electrophilicity, 381 hardness, 381 optimized geometries, 378–379 properties, 374, 375–377 Becke three-parameter Lee–Yang–Paar (B3LYP), 7, 208 Benzene, 1–2, 20–21, 31, 33, 272, 281. See also Aromaticity aromaticity index, 393 aromatic stabilization energy, 40 cyclobutadiene, 31, 189–192 diatropic net ring current, 193 external magnetic field, 193 Hückel-Breslow rules, 33 induced magnetic field, 194 isolation, 31, 278 magnetic field effect, 189 metal interaction, 21 molecular formula, 1 nodal ring effect, 189 physico-chemical evidences, 33 π-electrons, 33 potential curves, 395 puckering extent, 82–86 quantum mechanics, 192–193 resonance energy, 41 shielding isosurface, 189, 191 structure and stability, 2–3, 278, 280 Benzenoid hydrocarbons, 4, 339 Benzin. See Benzene Bernal structure, 172 Bicarburet of hydrogen. See Benzene Bifurcation points, 98 Bimetallic clusters, 150 alkali metal clusters, 157 anionic clusters, 5 aromaticity, 302, 306 noble metal clusters, 151–153 Binding energies (BE), 8 atomic hydrogen on gold clusters, 131 cluster size, 235 Biot–Savart’s law, 188 BLA. See Bond-length alteration (BLA) Block-localized wave function (BLW), 299 BLW. See Block-localized wave function (BLW) Bond alternation coefficient (BAC), 36, 299 Bond critical points (BCP), 207, 209 Bond-length alteration (BLA), 71 energy fragmentation analysis, 77 σ–π analysis, 75–77 Borazine, 200, 400 analogous, 272 cyclic planar system, 410 induced magnetic field, 200

isolines, 201 orbital overlap, 74 paramagnetic contributions, 200 paratropic regions, 200 π orbital contribution, 201 Boroxine, 410 Buckyball structure, 55

C C60 cluster, 55 Cambridge Crystallographic Database (CCSD), 71 Canonical molecular orbital (CMO), 199, 303, 57 CASSCF. See Complete active space selfconsistent field (CASSCF) Cationic clusters, 9. See also Dianionic clusters; Neutral clusters electron, 326 HOMO energy eigenvalues, 13 HOMO spatial orientations, 10, 12 lowest-energy structure, 9 multifold aromaticity presence, 10 NICS values, 11 odd–even alternation, 124 π electrons, 326 synthesis, 11 CCSD. See Cambridge Crystallographic Database (CCSD) CD. See Current density (CD) n-center–two electron (nc–2e) bonds, 58 Charge–charge correlations, 416, 419 Charge density, 417 versus site energy difference, 418 Charge–dipole interaction model, 114–115 Charge transfer (CT), 71 charge localization, 79 extent of, 237 HOMO–LUMO gap, 239 split charge formalism, 113 Chemical hardness, 106 Chemical potential, 106, 111, 120, 127 change in, 109 Chemical reactivity criterion, 3 local, 413 CI. See Configuration interaction (CI) cis conformation, 21 Clemenger–Nilsson model (CNM), 142, 276–277 optimized energies, 277 second-order energy difference, 143 Clusters, 55 assembled materials, 104 cationic, 9–11 dianionic, 7–9 experimentally isolated, 55–56

426 Clusters (Continued) involving metal atoms, 301 neutral, 6, 11 parents, 166 positively charged, 6 singly charged, 6 standard basis sets, 7 CMO. See Canonical molecular orbital (CMO) CNM. See Clemenger–Nilsson model (CNM) Coarse-grained variables, 105 Complete active space self-consistent field (CASSCF), 351 Configuration interaction (CI), 163, 206, 414 Contenders, 333 Correction vector (CV), 414 Coulomb contribution, 107 electrons repulsion, 389 CP. See Critical point (CP) Critical point (CP), 208 CT. See Charge transfer (CT) Current density (CD), 3, 193 HOMO-(1), 260 induced, 45, 188 ipsocentric mappings, 301, 303 ring currents indication, 45 CV. See Correction vector (CV) Cyclobutadiene, 32, 33 antiaromatic molecule, 34 Cyclophosphazene, 411. See also Benzene optical gap versus site energy difference, 417 shielding isosurface, 189, 191 singlet–triplet gap versus site energy difference, 418

D DAFH. See Domain-averaged Fermi-hole (DAFH) DDA. See Discrete dipole approximation (DDA) Deformation parameter, 209, 212, 224 magnitude different, 213 variation, 219 Delocalization, 40, 101, 410. See also Aromaticity cyclic, 41 electrons responsibilities, 56 π bonding, 350 π-electrons, 101, 411 Densities of states (DOSs), 283, 289, 290 Density expansion coefficients, 389 Density functional perturbation theory, 108 atom–atom hardness approximation, 112–113 atomic charge, 110 atomic dipole, 110

Index chemical potential, 109, 111–112 cluster dipole moment, 109 coarse graining, 109 energy change, 108–109, 111 hardness kernel, 110 Taylor-series expansions, 110 Density functional theory (DFT), 7, 70, 105–106, 121, 143, 164, 206, 305, 411 correlation functional test, 354 energy characterization, 105–107 GIAO, 352 quantum mechanical description, 104–105 scalar relativistic, 352 σ–π energy separation, 388 Density-functional tight-binding (DFTB) method, 163, 165–166, 228. See also Density functional theory (DFT) function comparison, 179–180 HOMO–LUMO gap, 176 Kohn–Sham potential, 228 magic Na N cluster, 174 Na cluster binding energy, 168 Na cluster radial distance, 178–179 Na cluster stability function, 172, 174 total-energy minima structures, 169 Dewar resonence energy (DRE), 38, 394 DFT. See Density functional theory (DFT) DFTB method. See Density-functional tight-binding (DFTB) method Diamagnetic, 254 contribution extraction, 300 ring current, 84, 401, 410, 411 susceptibility, 41, 301, 414 Dianionic clusters, 7. See also Cationic clusters; Neutral clusters electronic properties, 8, 9 geometrical parameters, 8, 9 HOMO spatial orientation, 10 lowest-energy structure, 7 resonance energy, 8 Diatropic cyrrent. See Diamagnetic—ring current Dicationic chalcogenid ring systems, 387 Diffusion function, 7 Discrete dipole approximation (DDA), 108 Domain-averaged Fermi-hole (DAFH), 249 analysis, 250 correlation functional, 249 eigenvectors, 259 DOSs. See Densities of states (DOSs) Double zeta valence plus polarization basis set (DZVP), 400 DRE. See Dewar resonence energy (DRE) DZVP. See Double zeta valence plus polarization basis set (DZVP)

427

Index E EA. See Electron affinity (EA) EAM. See Embedded-atom method (EAM) Effective mass approximation (EMA), 228 18-electron rule, 152–153 Electron affinity (EA), 120, 137, 373. See also Electrophilicity Al13, 124, 145 [Be6Zn2]2−, 378 Cl atom, 125 Cu N clusters, 144–145 photoelectron spectroscopy, 144 size variation indication, 145 Electron configuration, 271. See also Phenomenological shell model (PSM) central Be atom, 361 central C atom, 361 18-Electron configuration, 153 Hf 3 ring, 333 metal clusters, 272 natural, 362 U ring atoms, 355 Electron count rules, 272 Electron delocalization, 249 aromaticity and, 330 Al 2− 4 , 196 computational methods, 255 direct measurement, 246 Fermi-Hole analysis, 249–250 Hirshfeld-I, 252 Multicenter indices, 250 π-electron delocalization, 101 ring current maps, 253 Electron density function, full, 104–105 Electronegative dopant, 274, 276 Electronegativity, 120, 372, 374. See also Electron affinity (EA) Al13, 125 [Be6Zn2]2−, 374, 380, 382 differences, 120 E2 values and, 415 optical gap and, 416 Pauling, 289 χ parameter, 106 Electronic polarization, 104 Electron localizability indicator (ELI), 284 Electron localization function (ELF), 3, 5, 44, 70, 96, 248, 284 aromaticity measuement, 301 bifurcation analysis, 99, 100, 101 C32 characterization, 100 density construction, 96 isosurfaces, 99, 101 kinetic energy density, 97 localization element, 97 plot for trimeric aggregate, 90

Taylor series expansion, 96, 97 topological analysis, 98 Electron repulsion integrals (ERIs), 389, 390 Electrophilicity, 372, 374 [Be6Zn2]2−, 381, 382 Electrostatic interaction model, 107 aggregate expression, 108 dipole field tensor expression, 108 induced dipole, 107 ELF. See Electron localization function (ELF) ELI. See Electron localizability indicator (ELI) EMA. See Effective mass approximation (EMA) Embedded-atom method (EAM), 163–167 larger magic Na N cluster, 174 Na clusters binding energy, 168 Na clusters radial distances, 178–179 Na clusters stability function, 172 oblate clusters, 178 prolate clusters, 178 structural comparison, 179–180 total-energy minima structures, 169 EN, 36, 37 Energetic criterion, 3, 70 EOS. See Equation of state (EOS) Equation of state (EOS), 167 ERIs. See Electron repulsion integrals (ERIs) Evolutionary algorithms. See Genetic algorithm (GA) Evolutionary algorithms, 166 Excess property, 3 Exchange-correlation energy functional, 389 External potential, 105, 106

F Ferromagnetic (FM) metals, 145 FF. See Fukui function (FF) 5f orbitals nodality, 349 role, 351 FLU. See-Aromatic-fluctuation-index-(FLU) FM metals. See Ferromagnetic (FM) metals (4n + 2) rule, 3, 272, 330. See also Hückel model 4nπ rule, 2–3. See also (4n + 2) rule Frank–Kasper tetrahedron, 289 Free-electron theory, 138 Fukui function (FF), 106, 128–129. See also Reactivity descriptor Au N clusters, 132 [Be6Zn2]2−, 375–377 condensation, 373

G GA. See Genetic algorithm (GA) Gauge-including atomic orbitals (GIAO), 7, 352 values for tiara complexes, 80

428 Gauge-including magnetically induced current (GIMIC), 248, 310 Generalized gradient approximation (GGA), 207, 352 Genetic algorithm (GA), 232 GEO, 36, 37 GGA. See Generalized gradient approximation (GGA) GIAO. See Gauge-including atomic orbitals (GIAO) GIMIC. See Gauge-including magnetically induced current (GIMIC) Global hardness, 12, 13 AB3 clusters, 18 Q2r2 clusters, 15 tetramer clusters, 18 X3Y clusters, 20 Gold clusters, 129, 283 binding energies, 130 features, 130 Fukui functions, 130, 131, 132 HOMO and LUMO states, 131, 132 hydrogen interaction, 131 properties, 130 Gradient-based approximation. See Generalized gradient approximation (GGA)

H Halogens, 143 Hamiltonian, 139 Hückel Hamiltonian, 413 in magnetic field, 412 matrix, 389 model, 140 motion equation, 141 noninteracting model, 413 spheroidal harmonic oscillators, 142 TB scheme, 228 transfer integral terms, 412 valence electron movement, 140, 141 Hardness, 120, 215, 216, 217, 372, 374 [Be6Zn2]2−, 374, 381, 282 chemical, 106 global, 12, 13 kernel, 106, 110, 128 local, 106 Maximum Hardness, 126, 216 molar refractivity index, 43 mutual atom–atom, 112–113 parameters sum, 120 variation, 218 Hard sphere model, 139–140 Schrödinger equation, 138 Harmonic oscillator model of aromaticity (HOMA), 3, 345 EN and GEO, 36

Index formula, 37 index, 36, 246, 299 modified index, 37 Pauling bond order, 37 Harmonic potential curves, 401 benzene, 395 furan, 399 hexaazine, 396 imidazole, 399 1,2-oxazole, 399 1,3-oxazole, 399 pentaazine, 396 pyrazole, 399 pyridine, 396 pyrimidine, 396 pyrrole, 398, 399 1,2,3,5-tetraazine, 396 1,3,5-triazine, 396 Hartree–Fock (HF), 97, 143 exchange, 114 methods, 206 theory, 7, 43–44 Hess and Shaad resonence energy per electron (HSRE), 394 Heterocyclic systems, 77 four-membered, 78 Hexaazine, 405 harmonic potential curve, 396 Hexaferrocenylbenzene (HFB), 82 average puckering angle, 82 benzene ring retrieval, 83, 84 distortions modes, 82, 84 HOMO for benzene, 85 hydrogenation heat, 82–83 NICS values, 84 HF. See Hartree–Fock (HF) HFB. See Hexaferrocenylbenzene (HFB) Highest-occupied molecular orbital (HOMO), 7, 57, 71, 73, 175, 402–405 Al 2− 4 , 258 benzene, 85 energy eigenvalue, 8, 13 HOMO–LUMO gap, 100, 176, 237, 240 representative shapes, 10, 11 Highest spin (H.S.), 75 Hirshfeld-I, 252 AIM density function, 252 atom condensed overlap matrices, 253 weight functions, 253 HMO. See Hückel molecular orbital (HMO) HOMA. See Harmonic oscillator model of aromaticity (HOMA) HOMO. See Highest-occupied molecular orbital (HOMO) HRE. See Hückel Resonance Energy (HRE) H.S. See Highest spin (H.S.)

429

Index HSRE. See Hess and Shaad resonence energy per electron (HSRE) Hückel (4n + 2) counting rule. See (4n + 2) rule Hückel delocalization, 39, 40 electron counting rule, 330 graph, 38 Hückel annulenes closed analytical form, 38 TRE, 39 Hückel model annulenes, 280 application, 272 E2 versus Site energy difference, 415 Hückel molecular orbital (HMO), 39 Hückel Resonance Energy (HRE), 38 Hydrocarbons, 272 Hydrogenation heat, 82–83 Hydrogen molecule comparison, 21 interaction energies, 21

J Jahn–Teller (JT) distortions, 32, 71, 286 Jellium model, 56, 104, 206, 273, 404 Al13 electronic configuration in, 122 outcome, 163 spherical, 121, 124 175 ultimate, 164 JT distortions. See Jahn–Teller (JT) distortions

K K.E. See Kinetic energy (K.E.) Kinetic energy (K.E.), 77 density, 96, 97 Kohn–Sham energy, 389 equations, 107 orbitals, 44 potential, 229

L I ICSS. See Iso-chemical shielding surfaces (ICSS) Inclusion energy, 100 Induced magnetic field, 188, 190 4− analysis with Al 2− 4 and Al 4 , 195–196 aromatic and antiaromatic rings influence, 189 borazine, 200, 201 contour lines, 191–192, 197, 199 external magnetic field effect, 193 long-range influence, 194 nonconjugated systems, 189 paratropic region, 195 π orbital contribution, 194 quantum mechanics, 192–193 shielding isosurface in benzene, 189 shielding tensor computation, 188 Inorganic compounds, 70 Ionization potential (IP), 120, 137, 373 adiabatic, 210, 220 Al13Li and Al13Li2, 123 average local, 43 Li N and K N clusters measurements, 143, 144 photoionization spectroscopy, 143 size variation indication, 145 IP. See Ionization potential (IP) ISE. See Isomerization stabilization energy (ISE) Iso-chemical shielding surfaces (ICSS), 191 Isomerization stabilization energy (ISE), 299

Lagrange multiplier. See Chemical potential Langevin function, 149 LCAO. See Linear combination of atomic orbitals (LCAO) LCGTO. See linear combination of Gaussiantype orbitals (LCGTO) LDA. See Local density approximation (LDA) Linear combination of atomic orbitals (LCAO), 7 Linear combination of Gaussian-type orbitals (LCGTO), 389 Local density approximation (LDA), 143, 206, 207 Local reactivity indexes, 127 electronic chemical potential, 127 Fukui function, 128–129 hardness kernel, 128 system energy, 127 Lowest unoccupied molecular orbital (LUMO), 175. See also Highest-occupied molecular orbital (HOMO) tetramercury cluster, 292 LUMO. See Lowest unoccupied molecular orbital (LUMO)

M Magic clusters, 141, 151, 174, 224. See also Magic numbers Magic numbers, 121, 161, 163, 168, 273, 276 Au20 and Cu20, 275 + bare cluster, 324 B13 Na clusters, 224 Magnetic criteria, 3, 6, 41, 42, 70, 300

430 Magnetic exaltation, 409. See also Anisotropy Magnetic field external, 193 Hamiltonian, 412, 413 induced, 188–189 paramagnetic and diamagnetic terms, 254 ring current, 41, 188 Magnetic shielding, 300 isotropic, 193 NICS, 42 NMR parameters, 6 tensors, 300, 352 Magnetic superatom, 157 Magnetic susceptibility anisotropy, 305 of compounds, 41 computation, 412 diamagnetic susceptibility, 414 paramagnetic susceptibility, 414 smooth exponential decay, 412 Magneto-crystalline anisotropy (MCA), 148 Mass spectrometry Na clusters spectrum, 138 photofragmentation, 151 MCA. See Magneto-crystalline anisotropy (MCA); Multiply charged anions (MCA) MCI. See Multicenter bond indices (MCI); Multicenter indices (MCI) MED. See Molecular electron density (MED) MESP-guided method. See Molecular electrostatics potential guided method (MESP-guided method) Metal chalcogenide clusters, 227 applications, 228 Metal clusters, 283, 301 abundance spectrum, 138, 139 AdNDP analysis, 58–60 Al2Si2 valence MOs, 306, 307 alkaline, 273 aromaticity, 272 chemical bonding, 56–58 Clemenger–Nilsson model, 142, 143 closed electronic structure, 280–281 conflicting aromaticity, 302 energy states, 139 free-electron theory, 138 Hamiltonian model, 140, 141 jellium model, 273 magic clusters, 141 magnetic susceptibility anisotropy, 305–306 Mn-doped cluster properties, 286 NICS scanning curves, 311, 312, 313 NICS values, 308, 309 orbital analysis, 304, 305, 310 π contribution, 303 planar clusters, 283

Index PSM, 272, 273 Schrödinger equation, 138, 139, 140 second-order energy difference, 141 size observations, 137–138 SJM, 143 spherical approximation, 141–142 stability, 279 3D clusters, 289 3D harmonic oscillator potential, 140 unique feature, 302 Zn clusters, 308 Metal clusters, transition, 146 FM and AFM, 146, 146 magnetic moments measurements, 146–147, 149 structure variation, 147–148 superparamagnetism, 148–150 Metallic clusters Al13 cluster analysis, 122, 123 Al13H structure, 123 Al13 reactivity, 123–124 DFT, 121 even–odd alternation, 126 HOMO–LUMO gap, 121, 127 magic numbers, 121 spherical droplet model, 121 super-halogen character, 124–126 Metalloaromatic, 324 Metallobenzenes, 302 Osmabenzene, 339 Metallocycles, 324 Metalloid clusters, 5 Metalloligands, 340 geometries, 340 K6 unit formation, 342, 343 Lewis base sites, 340 Na 4+ 4 ring formation, 345, 346 Na ions1D chain formation, 342 ORTEP view, 341, 343, 345 solid-state structure, 341 2D K6 cluster sheet, 344 Mn-doped cluster, 286 electronic configurations, 287 MO diagram, 286 MO. See Molecular orbital (MO) Möbius annulenes, 2, 39 Molecular electron density (MED), 207 analysis, 214, 215 contours, 222, 223 Laplacian, 209 Molecular electrostatics potential guided method (MESP-guided method), 207, 219, 220, 221 application, 208 cluster growth path tracing, 210, 211, 212 MESP minima, 219 Na m clusters, 210–211

431

Index Molecular orbital (MO), 2, 71, 274, 402 Al 2− 4 aromaticity, 329 Al 2− 4 ring, 328 Al 2− 4 valence molecular orbitals, 330 analysis, 57 correlation diagram, 402–404 electron counting rule, 330 energetic ordering, 327–328 MAl −4 , 328 molecular occupation rules, 329 multiple-fold aromaticity, 330 ring axes system, 327 valence molecular orbitals, 329 Molecular system aromaticity, 70 cyclic electron delocalization, 187 electric response, 105 hardness, 126 homocyclic, 71 stable state of, 372 Molecular tailoring approach (MTA), 207 cluster optimizations, 208 geometry optimization, 224 Molecules theory, 43 bond order, 43 n-center delocalization index, 44 ELF, 44 ellipticity, 43 Hartree–Fock theory, 44 Moller–Plesset perturbation theory (MP2), 7 Monte Carlo method, 206–207 MO. See Molecular orbital (MO) MP2. See Moller–Plesset perturbation theory (MP2) MPA. See Mulliken population analysis (MPA) MTA. See Molecular tailoring approach (MTA) Mulliken population analysis (MPA), 346, 373, 375–377 Multicenter bond indices (MCI), 246 251, 345 aromaticity, 251, 345 Kohn–Sham theory, 255 orbital contributions, 257 ring current, 265 Multicenter delocalization indices, 3, 255 Multicenter indices, 250–252. See also Shared electron distribution index (SEDI) Multifold aromaticity, 5, 330 establishment, 5 metal clusters, 302 NICS values, 24 Multiply charged anions (MCA), 60

N Nanoparticles, 129 geometry optimization semiconductor, 227

size dependence, 228 Natural bond orbital (NBO), 351 NBO. See Natural bond orbital (NBO) Neutral clusters, 6, 11, 122, 306. See also Cationic clusters; Dianionic clusters analog, 6 bond lengths, 20 cis and trans configurations, 16 electronic properties, 15, 18, 20 equilibrium geometry, 12, 13 frequencies and intensities, 14 geometrical parameters, 15, 18 geometry optimization, 16, 17, 19 global hardness, 12, 13, 16 HOMO orientations, 17, 19 inverse correlation, 114 NICS. See Nuclear independent chemical shift (NICS) NMR. See Nuclear magnetic resonance (NMR) Noble metal clusters, TM-doped, 138, 150, 151 DFT calculations, 152 18-electron rule, 152–153 magic clusters, 151, 152 magnetic properties, 153 photofragmentation spectra, 151, 152 Nodal ring, 189 NP-hard problem, 162 Nuclear independent chemical shift (NICS), 41, 42, 70, 345, 375 calculation uses, 41 Nuclear magnetic resonance (NMR), 3 Nuclei, 162 parameters, 6 proton signals, 35 Nucleus-independent chemical shift (NICS), 3, 5 187, 246, 372 all-metal cluster, 303, 304 aromatic behavior, 220 benzene ring, for, 84 calculation, 6 diatropic, 352 index, 6 magnetic shielding tensors, 300, 410 metallocenes values, 304 NICS values, 23, 308, 309 paratropic, 352 ring current computation, 281, 332 scanning curves, 311, 312, 313

O Oblate shape, 178, 276 Octathio[8]circulene, 86 cooperative interactions, 89 H2 binding modes, 86, 87 HOMO plots, 86, 87 hydrogen stabilization, 89, 90

432 Octathio[8]circulene (Continued) molecular structures, 86, 87 organic analogues, 86, 87 PES for H2 binding, 87–89 stacked aggregate structure, 90 Odd–even staggering (OES), 151, 152 OES. See Odd–even staggering (OES) Optical gap, 416–417 Osmabenzene, 339

P Packing effects, 162, 173 Paramagnetic ring current, 410 Pariser–Parr–Pople model (PPP model), 412 E2 versus Site energy difference, 416 partial density of states (pDOS), 283, 284 Pauling bond order, 37 pDOS. See partial density of states (pDOS) PE. See Per π electron (PE) Pentaazine, 396 pentacoordinate U (ppU), 351 Per ring bond (PRB), 38 Per π electron (PE), 38 PES. See Photoelectron spectroscopy (PES); Potential energy surface (PES) Phenomenological shell model (PSM), 272, 273, 290 advantages, 278 Clemenger–Nilsson model (CNM), 276–277 itinerant electrons number, 275 spherical clusters, 274 superatom concept, 275 Phosphazenes, 409 bond length alternation, 412 optical gaps, 415, 417 polyphosphazene, 411 singlet–triplet gap, 418 small cyclic, 411 Photoelectron spectroscopy (PES), 122, 144 d-orbital aromaticity, 307 electronic shell structures, 164 Photoionization spectroscopy, 143 π-bond delocalization, 101, 302, 310 distortive nature, 33, 35 5c–2e p-AO-based, 62, 63 4c–2e p-AO-based, 59 occupation number variation, 62 π-electrons, 33 σ and π components, 34 substantial resonance stabilization, 33 valence bond curve-crossing diagram, 34 Planar hypercoordinate atoms, 197 boron clusters utilization, 198 planar boron wheels, 198 Ptc investigation, 197–198, 199

Index Planarity + cluster, 324 B13 Boustani structure, 325 Kohn–Sham π-orbitals, 325, 326 molecular backbone, 281 octathio[8]circulene, 86 Ricca structure, 325, 326 planar tetracoordinate carbon (ptC), 197 planar tetracoordinate element E (ptE), 351 Polansky index, generalized, 44–45 Polarizability, 45–46, 164, 221–222, 223 atomic-level response, 114 charge–dipole interaction model, 108, 114–115 comparison along clusters, 222 density functional perturbation theory, 108 electron correlation effect, 113–114 electrostatic interaction model, 107–108 Hartree–Fock exchange, 114 Na clusters, 210 Polyphosphazene, 411 helical structure, 412 Pople’s model, 194 Position-sensitive time-of-flight (PSTOF) mass spectrometer, 146 Potential energy surface (PES), 83, 208, 372 PPP model. See Pariser–Parr–Pople model (PPP model) ppU. See pentacoordinate U (ppU) PRB. See Per ring bond (PRB) Prolate shape, 178, 277 PSM. See Phenomenological shell model (PSM) PSTOF mass spectrometer. See Positionsensitive time-of-flight (PSTOF) mass spectrometer ptC. See planar tetracoordinate carbon (ptC) pte. See planar tetracoordinate element E (ptE) Pyridine, 4, 396 Pyrimidine, 396 Pyrrole harmonic potential, 398

Q QTAIM. See Quantum—theory of atoms in molecules (QTAIM) Quantum chemical descriptors, 78 mechanical basis, 32 theory of atoms in molecules (QTAIM), 44

R Rare-earth (RE), 145 RE. See Rare-earth (RE); Resonance energy (RE)

Index Reactivity, 42, 47, 119 contour plots, 129 electronegativity, 120 energy change estimation, 121 gold clusters, 129, 130–132 H2 with Al13, 129 hardness, 43, 120 HOMO–LUMO gaps, 42 local indexes, 127–129 Reactivity descriptors, 114, 372 global, 374 local, 372, 373 Relative Energy, 374 [Be6Zn2]2−, 379 Resonance energy (RE), 5, 38, 40, 42, 248, 207 Al 2− 4 system, 8 aromaticity estimation, 37–38 benzene, 41 topological, 38, 39, 326 Resonance hybrid, 2, 37 Ring current maps, 253, 255, 260 in Al 2− 4 plane, 261–262 diamagnetic terms, 254 orbital resolved, 263–264 paramagnetic terms, 254

S Scandium doped derivatives (Cu NSc), 283. See also Metal clusters and benzene, 285 CMO-NICS value, 288, 289 electronic structure, 284 geometry, 283, 284, 290 ground state, 290 isomers, 292 Jahn–Teller distortion, 286 NICS distribution in, 285 pDOS, 284, 290, 291 PSM, 290 shell orbitals, 288 SCF. See Self consistent field (SCF) Schrödinger equation, 138, 140, 141, 152, 273 SEDI. See Shared electron distribution index (SEDI) Self consistent field (SCF), 352, 389, 411 SG apparatus. See Stern–Gerlach (SG) apparatus Shared electron distribution index (SEDI), 251 Shell, 137 algebraic sum, 397 aromaticity, 392–394 filled electronic, 141, 143 harmonic potential curves, 396, 398–399 orbital combination, 286 orbitals, 288, 291 separation, 388–392

433 Single-molecule magnetic material (SMM material), 82 Singlet–triplet gaps, 417, 418 Site energy difference, 415 Six-membered rings, 75, 397 aromaticity indices, 393 benzene, 397 hexaazine, 397 pentaazine, 397 pyridine, 397 pyrimidine, 397 1,2,3,5-tetraazine, 397 1,3,5-triazine, 397 SJM. See Spherical jellium model (SJM) SMM material. See Single-molecule magnetic material (SMM material) Sodium (Na) clusters, 162, 163 abundance spectrum, 138 adiabatic ionization potential, 210, 216 Bernal structure, 172 binding energy, 167–168 critical points, 208 deformation parameter, 209, 212, 213, 219 DFTB, 165–166, 178 EAM, 166–167, 178 electronic properties, 219, 220 electronic shell structures, 164 energy gradients, 209 fragment division, 208 geometric deformation, 209 hardness, 215, 216, 217 HOMO–LUMO gap, 176 jellium model, 163 larger magic structures, 174 mass abundance spectroscopy, 207 MED, 209, 214, 215 MESP-guided method, 208, 210–211, 219 MESP minima values, 220–221 normalized eigenvalues, 177 orbital energies comparison, 175–176 overall shape analysis, 177–178 packing effects, 162 pentagonal growth, 173 polarizability, 164, 210 radial distance, 178–179 similarity function comparison, 179–181 stability function, 172 stabilization energies, 212, 213, 214 structure optimizations, 163–164 structures, 215, 216, 217, 218 symmetry classifications, 170–171, 172–173 total energies, 212, 213, 214 total-energy minima structures, 169 variation in hardness, 218 Softness, 374, 382 Spectroscopic properties, 187 Spherical aromaticity, 289, 324

434

Index

Spherical jellium model (SJM), 121, 143 Al13 electronic configuration, 122 closed-shell clusters, 124 Stern–Gerlach (SG) apparatus, 146 S-triazine, 410 Structural criterion, 3, 47, 70 aromaticity index development, 298 HOMA index, 299 Sulflower. See Octathio[8]circulene Superatoms, 145, 275 magnetic, 157, 275 Superhalogen, 145, 275 Al13 character, 124–125 Superparamagnetism, 148 energy scales measurement, 148–149 magnetic moments variations, 149–150 time scales measurement, 148

aromatic stabilization, 40–41 Hückel annulenes, 38, 39 Möbius annulenes, 39 PE, 38 PRB, 38 Transition metal (TM), 138 aromaticity in, 324 Transition metal rings contenders, 333 δ-type molecular orbitals, 331 Hf 3 ring, 333 transition metal clusters, 330 valence molecular orbitals, 331, 332, 334 TRE. See Topological resonance energy (TRE) 1,3,5-tridehydrobenzene (TDB), 282, 288 Triple aromaticity, 307

T

U

Taylor series expansion, 96, 97, 110 TB. See Tight-binding (TB) TDB. See 1,3,5-tridehydrobenzene (TDB) Tetramer binary clusters. See Neutral clusters Tetramer clusters, 6, 17 estimated values, 18 isoelectronic heteroatomic, 9 Tetramercury cluster Hg2+ 4 geometry, 292 Hg2+ 4 HOMOs and LUMOs, 292 Hg6− 4 AdNDP analysis, 58–60 Hg6− 4 compact basis sets, 60 Hg6− 4 MCA, 60 Thermodynamic stabilization, 39 aromatic stabilization, 39, 40 delocalization energy, 39 energy-level, 140 Hückel delocalization, 40 iso-and homodesmic equations, 40 Schrödinger equation, 140 3D. See Three-dimension (3D) 3D harmonic oscillator potential, 140 variation of energy levels, 142 Three-dimension (3D), 140 Tiara nickel thiolates, 78 aromatic nature, 81 crystal structures, 78 distance and angle variations, 78, 79 ground-state geometries, 79 NICS magnitude, 78, 79, 80 orbital contributions, 81 planar Nin backbone, 78 symmetric structures, 81, 82 Tight-binding (TB), 228 TM. See Transition metal (TM) Topological resonance energy (TRE), 38, 39, 326

UJM. See Ultimate jellium model (UJM) Ultimate jellium model (UJM), 164 Uranium clusters antiaromatic zones, 358 aromaticity, 359 aromatic zones, 358, 359 DFT performance, 352 equilibrium geometries, 360 excitation energies, 359, 364 magnetotropicity, 363–365 NICS Values, 358, 364 NICSzz-scan curves, 357, 363 orbital computation, 351–352 properties, 353, 354–356, 360–361 3Dvalence MOs, 356, 357, 362

V Valence bond (VB), 5, 248 curve-crossing diagram, 344 theory, 2 Van Vleck paramagnetic susceptibility, 414 VB. See Valence bond (VB) VDE. See Vertical detachment energies (VDE) VEA. See Vertical electron affinity (VEA) Vertical detachment energies (VDE), 56 Vertical electron affinity (VEA), 13 AB3 clusters, 18 Q2R2 clusters, 15 tetramer clusters, 18 X3Y clusters, 20 Vertical ionization potentials (VIP), 13 AB3 clusters, 18 Q2R2 clusters, 15 tetramer clusters, 18 X3Y clusters, 20

435

Index VIP. See Vertical ionization potentials (VIP) Von Weiszaecker term, 97 Vosko, Wilk, and Nusair functional (VWN functional), 400 VWN functional. See Vosko, Wilk, and Nusair functional (VWN functional)

W WBI. See WIBERG Bond Indices (WBI) WIBERG Bond Indices (WBI), 352

Wurtzite Bulk ZnS band structure, 231 properties, 230

Z Zeeman energy, 149 Zero point energy (ZPE), 352 Zinc-blende bulk ZnS band structure, 231 properties, 230 Zintl phases, 60–61 ZPE. See Zero point energy (ZPE)