theoretical studies on the spectroscopy of some

THEORETICAL STUDIES ON THE SPECTROSCOPY OF SOME INTRAGROUP IVA HETERONUCLEAR DIATOMIC MOLECULES AND THEIR IONS

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (SCIENCE) OF JADAVPUR UNIVERSITY

BY ANUP PRAMANIK, M.Sc.

DEPARTMENT OF CHEMISTRY PHYSICAL CHEMISTRY SECTION JADAVPUR UNIVERSITY KOLKATA – 700032 INDIA

CERTIFICATE FROM THE SUPERVISOR(S)

This is to certify that the thesis entitled “THEORETICAL STUDIES ON THE SPECTROSCOPY OF SOME INTRAGROUP IVA HETERONUCLEAR DIATOMIC MOLECULES AND THEIR IONS” submitted by Sri / Smt. ANUP PRAMANIK, who got his/her name registered on 25.06.2007 for the award of Ph.D. (Science) degree of Jadavpur University, is based upon his own work under the supervision of PROF. DR. KALYAN KUMAR DAS and that neither this thesis nor any part of it has been submitted for either any degree / diploma or any other academic award anywhere before.

(Signature of the Supervisor(s), date with official seal Prof. Dr. Kalyan Kumar Das, Department of Chemistry, Physical Chemistry Section, Jadavpur University, Kolkata – 700 032, India.

TO

MY FRIEND PHILOSOPHER AND GUIDE

“KOUSIK-UNCLE”

Acknowledgements The research work presented in the thesis has been performed in the Department of Chemistry, Physical Chemistry Section, Jadavpur University since January, 2006. I would like to take the opportunity to convey my thanks to the people whose constant help and encouragement have finally laid me to complete the thesis. I express my warmest gratitude to my supervisor, Prof. Dr. Kalyan Kumar Das for his kind cooperation and thoughtful advices. What he has done for me is really beyond my expectation. In each step I have learnt from him how to utilize the valuable times of our life properly. Great scientific attitude as well as nice behavior of him is truly rememberable. The financial support provided by CSIR, Govt. of India is gratefully acknowledged. Without this it was impossible to carry out my research work, whatever I have done. I am indebted to Prof. Dilip Kumar Bhattacharyya and Dr. Biplab Bhattacharjee for their moral supports and valuable discussions. I am also thankful to the Head, other teaching and nonteaching staffs of the department of Chemistry, Jadavpur University. Library and laboratory facilities of this university are also gratefully acknowledged. A lot of thanks to my lab-mates, Mr. Amartya Banerjee, Ms. Susmita Chakrabarti for their ever helping hands and cooperation. The former guy deserves a speciality for his philosophical sense and critical analysis, which helped me a lot during my research period. My heartiest love and respect to my parents and other family members. Specially, my sincere thanks to my mother, and my wife, Mitali. They have provided me continuous supports and all kinds of facilities. I can’t make him dishonored by expressing only my thanks to Kousik-uncle who induced the philosophy of science in my mind. It brings a great pleasure to me to dedicate the thesis to him.

Date: Department of chemistry, Physical Chemistry Section, Jadavpur University, Kolkata – 700032, India

ANUP PRAMANIK

Contents

Introduction

1

Plan of the thesis

4

1. A brief review of the electronic structure theory of atoms and molecules 1.1 Introduction

7

1.2 The Schrdinger equation

8

1.3 The variational principle

9

1.4 The Hartree-Fock model

10

1.5 Basis sets

12

1.6 Relativistic effects

13

1.7 Electron correlation energy and post Hartree-Fock treatments

15

1.8 References

20

2. Brief review of the computational methodology: details of the Configuration Interaction method 2.1 Introduction

22

2.2 Relativistic corrections 2.2.1 The Dirac equation

24

2.2.2 Effective core potential

25

2.2.3 Spin-orbit coupling

29

2.3 Computational methodology 2.3.1 Configuration selection technique

31

2.3.2 Role of unselected configurations

32

2.3.3 Spin-orbit interaction

34

2.3.4 Calculation of spectroscopic constants

36

2.3.5 Estimation of radiative lifetime

37

2.4 References

38

i

3. Electronic structure and spectroscopic properties of the SiC radical 3.1 Introduction

42

3.2 Computational details 3.2.1 RECPs and basis sets

44

3.2.2 SCF MOs and CI

44


46

3.3 Results and discussion 3.3.1 Spectroscopic constants and potential energy curves of Λ-S states

46

3.3.2 Spectroscopic constants and potential energy curves of Ω states

56

3.3.3 Dipole moments and transition properties

59

3.4 Summary

62

3.5 References

64

4. Electronic structure and spectroscopic properties of SiC+ and SiC− 4.1 Introduction

66


67


67


69

4.3 Results and discussion 4.3.1 Spectroscopic constants and potential energy curves of Λ-S states A. SiC+

69

−

75

B. SiC

4.3.2 Spectroscopic constants and potential energy curves of Ω states

81

4.3.3 Transition properties A. SiC+

83

B. SiC−

85

4.3.4 Dipole moments, ionization potentials, and electron affinities

86

4.4 Summary

89

4.5 References

91

5. Electronic structure and spectroscopic properties of SnC and SnC+ 5.1 Introduction

93 ii


95


95


96

5.3 Results and discussion 5.3.1 Spectroscopic constants and potential energy curves of Λ-S states A. SnC

97

B. SnC+

105

5.3.2 Spectroscopic constants and potential energy curves of Ω states A. SnC

110

B. SnC+

113

5.3.3 Transition properties A. SnC

117

B. SnC+

120

5.3.4 Dipole moments and ionization energies

122

5.4 Summary

125

5.5 References

127

6. Electronic structure and spectroscopic properties of PbC and PbC+ 6.1 Introduction

129


130


130


132

6.3 Results and discussion 6.3.1 Spectroscopic constants and potential energy curves of Λ-S states A. PbC

132

B. PbC+

138

6.3.2 Spectroscopic constants and potential energy curves of Ω states A. PbC

142

B. PbC+

147

6.3.3 Transition properties A. PbC

151

B. PbC+

154 iii

6.3.4 Dipole moments and ionization energies

157

6.3.5 Comparison of some spectroscopic properties of MC and MC+ (M = Si, Sn, Pb)

159

6.4 Summary

163

6.5 References

164

Conclusion

166

List of publications

169

iv

INTRODUCTION

The interpretation and understanding of every experimental finding requires the knowledge of theoretical background. A large number of experimental results can be brought into together by theoretical interpretation and suitable formulation. So, necessity of theoretical research is urged by its own demand. A chemical problem can be solved theoretically by proper use of physical laws and mathematical methods, often by the use of computer memory. Large number of computational methods have been developed over the years for the complete solution of chemical problems. Quantum mechanics is one such tool, which has been developed enormously with the advancement of computer hardware and softwares. Modern electronic structure theory, which is based on quantum mechanics, is capable of providing reliable predictions of quantities of chemical interest. It is not surprising that, the variational methods could be applied to systems as large as XeF6 , azulene, and guanine-cytosine base pair. Now a days, with the help of computation, a large number of organic molecules, such as protein, DNA, RNA etc. are designed theoretically. Such attempts are very much helpful to the experimentalists to reach the goal of real synthesis with prior experiences of chemical hazards. Moreover, where we are bound to our experimental limit, theoretical investigation is the only tool to interpret the natural observations. Thus structural chemistry, which is based on spectroscopic measurement, is equally balanced by experimental results as well as theoretical predictions. The space trajectory and other dynamical properties of macroscopic objects can be well studied by classical mechanics.1 However, classical mechanics fails in the domain of submicroscopic world of atom and its constituents. In the beginning of the nineteenth century, Planck’s idea of quantization was brought into a new field of mechanics mainly by Heisenberg and Schr¨odinger. The new mechanics, revealed as quantum mechanics2−10 , is the the successful treatment to describe the structural and dynamic properties of subatomic particles. Now, if the velocity of the object is comparable to that of light, one must use the relativistic mechanics of Einstein which takes into consideration of variation of mass with velocity. So, subatomic particles of low mass and having very high velocity, comparable to that of light, need to use of relativistic quantum mechanics, derived by Dirac. This uses the modified Hamiltonian, containing various relativistic correction terms including mass-velocity, spinorbit, Darwin correction and Breit interaction. Thus, depending upon the mass and velocity, the dynamics of a particle is governed by suitable mechanics and consequently it requires a proper mathematical treatment. The behavior of electrons in atoms or molecules are described by quantum mechanics. 1

Their space trajectories are described on the basis of probabilistic interpretation, according to which the stationary sates of them are fitted with time independent Schr¨odinger equations. On solving these quantum mechanical equations, which are second order differential in nature, one may get the electronic structure of atoms and molecules. By using Born-Oppenheimer approximation, in which electronic motions are treated separately from nuclear motion, the electronic Hamiltonian can be resolved and consequently the solution of it gives the structural aspects and spectroscopic information of a molecule11−16 . The task is no longer a simple one, specially for molecules having heavier atoms, there involves a large scale relativistic effects in hamiltonian and hence proper treatment is essential for that.17−19 A number of algorithms have been developed for this purpose over the past few decades with the improvement of enormous computing facilities. Many of them give the results with reasonably good accuracy. Large efforts are required for a bit of improvement of computed result. Moreover, an enormous volume of computation may be necessary for this purpose. So, there are always limitations in the accuracy. Parallel efforts are also being given for the development of computing facilities. Thus, the real challenge is to exploit these developments and carry out theoretical research to reach the stage more close to reality.

2

References 1

H. Goldstein, Classical Mechanics, Addition-Wesley, Reading, Mass., 1950.

2

L. Pauling, E.B. Wilson, Introduction to Quantum Mechanics, McGraw-Hill, 1935.

3

H. Eyring, J. Walter, G.E. Kimball, Quantum Chemistry, Wiley, New York, 1944.

4

L.I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1968.

5

J.P. Lowe, Quantum Chemistry, Academic Press, New York, 1978.

6

D.A. McQuarrie, Quantum Chemistry, University Science, Mill Valley, Calif. 1983.

7

P.W. Atkins, Molecular Quantum Mechanics, Oxford University Press, New York, 1983.

8

A. Hinchliffe, Computational Quantum Chemistry, Wiley, New York, 1988.

9

F.L. Pillar, Elementary Quantum Chemistry, McGraw-Hill, New York, 1990.

10

I.N. Levine, Quantum Chemistry, Printice-Hall, N.J., 1991.

11

R.G. Parr, Quantum theory of Molecular Electronic Structure, Benjamin, New York, 1963.

12

J.A. Pople, D.L. Beveridge, Approximate Molecular Orbital Theory, McGraw-Hill, New York, 1970.

13

J.N. Murrell, A.J. Harget, Semiempirical Self-Consistent-Field Molecular Orbital Theories of Molecules, Wiley-Interscience, New York, 1971.

14

R.S. Mulliken, W.C. Ermler, Diatomic Molecules, Academic Press, New York, 1977.

15

R.S. Mulliken, W.C. Ermler, Polyatomic Molecules, Academic Press, New York, 1981.

16

W.J. Hehre, L. Radom, P.V.R. Schleyer, and J.A. Pople, Ab Initio Molecular Orbital Theory, Wiley-Interscience, New York, 1986.

17

P. Pyykk¨ o, Relativistic Theory of Atoms and Molecules, Springer-Verlag, Berlin and New York, 1986.

18

K. Balasubramanian, Relativistic Effects in Chemistry Part A. Theory and Techniques, Wiley-Interscience, New York, 1997.

19

K. Balasubramanian, Relativistic Effects in Chemistry Part B. Applications to molecules and Clusters, Wiley-Interscience, New York, 1997.

3

PLAN OF THESIS

The aim of Quantum Chemistry is to have some information about chemical bonds. Energetic information of chemical bonds involving permutation of all elements in the entire periodic table have been collected over the years by many experimental scientists. Besides their applications, the simple diatomic molecules draw special interest in contributing the information about their bond length, bond energy etc. Intragroup IVA heteronuclear diatomic molecules have generated a special interest in recent years because of their possible applications in catalysis, sensor films and mostly, they are the building blocks of cluster materials which have interesting solid state properties. Inspite of the facilities of many modern sophisticated instruments like high resolution spectrophotometer, laser vaporization, supersonic jet expansion, matrix isolation etc. this type of molecules are rarely studied in experiment because of the difficulty of their isolation in gas phase. Even for the simplest of them (SiC), ab initio calculations were performed before the experimental detection. Very recently seven of the intragroup IVA diatomics have been energetically characterized by Knudesen effusion mass spectroscopic (KEMS) technique. To verify the available experimental data and to predict the spectroscopic characteristics it is common practice to use quantum mechanical techniques like configuration interaction (CI), complete active space self consistent field (CASSCF), couple-cluster, many-body perturbation theories, density functional theories (DFT) etc. The present thesis aims to study the electronic structure and spectroscopic properties of intragroup IVA heteronuclear diatomic molecules, specially the carbides of Si, Sn, Pb, and some of their ions. The multireference singles and doubles configuration interaction (MRDCI) calculations have been performed in the present thesis using relativistic effective core potentials and suitable Gaussian basis functions for the participating atoms. Potential energy curves of some low-lying Λ-S as well as Ω states of the molecules and ions are constructed from the estimated full-CI energies. Many avoided crossing interactions have been properly studied by analyzing the CI state functions. Spectroscopic constants like re , ωe , and Te values are calculated by fitting the potential energy curves. The variation of dipole moment functions of some low-lying states and transition moment functions involving ground and some of the excited states are followed against bond distances and subsequently radiative lifetimes of few low-lying states are computed for neutral as well as the ionic species. Vertical ionization energies (VIE) of the neutral species are reported. Electron affinity of SiC have also been verified from the MRDCI studies of SiC and its anion. Chapter 1 gives an overview of the basic quantum mechanics like time dependent 4

Schr¨ odinger equation, variational principle, Hartree-Fock model, basis sets, relativistic effect, electron correlation and post Hartree-Fock methods like MCSCF, CI etc. Chapter 2 describes the computational methodologies which are used in the calculations throughout the thesis. For many electron atoms, it is not possible to carry out all electron CI calculations. So the effective core potential method is used. The details of MRDCI method are discussed in this chapter. The configuration selection technique is also discussed in this chapter along with the corrections due to the unselected configurations. The method of including the spin-orbit coupling at the CI level is mentioned. The methods of estimation of the spectroscopic constants and radiative lifetimes are also described here. Chapter 3 deals with the results obtained from the calculations of SiC. Potential energy curves and spectroscopic constants of a large number of Λ-S states of singlet, triplet, and quintet spin multiplicities are reported and compared with the existing data. The groundstate dissociation energy of the species is computed and verified with the experimental results. E3 Π is found to be an important one which have not been studied before. The important transitions like A–X, B–X, C–X, D–X, E–X etc. are studied, at the same time radiative lifetimes of some excited states are also reported. Dipole moments of some lowlying states are computed. Finally, the effects of spin-orbit coupling on the spectroscopic properties of SiC are discussed in the chapter. Chapter 4 describes the effects of removal and addition of an electron to the neutral silicon carbide using MRDCI methodologies. Ionization energies and electron affinities of SiC are reported in this chapter. Spectroscopic aspects of the SiC+ and SiC− ions are studied in detail. The ground states of SiC+ and SiC− are 4 Σ− and 2 Σ+ , respectively. Thus, the quartet-quartet transitions for SiC+ and the doublet-doublet transitions for SiC− are of special interest. No experimental data are known, but a very few theoretical results are available for comparison. The spectroscopic constants of low-lying states of both the species up to an energy level of 6 eV are reported. Chapter 5 contains the results of the electronic structure and spectroscopic properties of SnC and SnC+ . Spectroscopic constants and some other properties of these species, at both Λ–S and Ω levels, are reported. Because of the heavier mass of Sn, the Ω states have become more important as compared to those of SiC. Hence, the spin-forbidden transitions are given a special attention. Radiative lifetimes of some low-lying states of these species have been reported in this chapter.

5

In Chapter 6, we have discussed electronic structure and spectroscopic properties of PbC and PbC+ . Lead is the heaviest element of group IVA and consequently the spin-orbit coupling has been found to be the most prominent. Hence, spectroscopic properties of the Ω states are thoroughly studied. This chapter includes dipole and transition dipole moment functions of some low-lying states. Many spin-forbidden transitions are computed, and a comparison of the spectroscopic properties of all three carbides and their monopositive ions has also been made in the last part of this chapter.

6

CHAPTER – 1

A BRIEF REVIEW OF THE ELECTRONIC STRUCTURE THEORY OF ATOMS AND MOLECULES

1.1. Introduction As we have mentioned earlier, behaviors of electrons in an atom or molecule are described by stationary state wave functions as given in time independent Schr¨ odinger equations. To have solutions of such equations is always a difficult task. If we neglect all the relativistic effects and consider the electrons to be moving in a fixed nuclear framework as in the BornOppenheimer approximation, the problem becomes much easier to deal with. But still it is a formidable task to solve these equations because of the involvement of a large number of inter-electronic interaction terms in the Hamoltonian. Approximate methods have been developed over the years to solve the non-relativistic Schr¨ odinger equation for determining the electronic structure of the molecular systems as accurately as possible. As an approximation, firstly these two body terms are converted into separate one-electron potentials. This transforms the many body problem into the effective one-body problems which is popularly known as independent particle model. The model gives the best possible solution in which the wave functions are represented by the antisymmetrized product of one-electron functions, commonly called orbitals. Next, for the construction of the single configuration state, it has to satisfy two criteria; the wave function would have minimum energy in its neighborhood, and the orbitals must have maximum overlap. The minimum energy criterion is fulfilled by Hartree-Fock model, based on the variational principle. On the other hand, the maximum overlap criterion with the exact wave function leads to the Brueckner approximation.1 The last one is not practicable to implement as it requires the knowledge of exact wave functions, while it is easier to implement the Hartree-Fock theory in practice. The minimum energy wave function of a given class can be obtained from the variational principle which can be applied in quantum chemistry. There are several post-Hartree-Fock methodologies which may then be applied for estimating the electron correlation missing in the HartreeFock approximation. However, it requires rigorous mathematical calculations and numerical methods. The basic principles and techniques, which are used in the electronic structure theory of atoms and molecules, are briefly discussed in the following sections.

7

1.2. The Schr¨ odinger equation In quantum mechanics, dynamics of a system is described by the time dependent Schr¨ odinger equation ˆ Hψ= i¯ h(∂ψ/∂t),

(1.1)

ˆ is the Hamiltonian operator consisting of the kinetic and potential energy operators where H of the system and ψ is called state function which is the function of space coordinate (r) and time (t). Now, the state of a many-body system (say, consisting n number of electrons) is given by the wave function ψ=ψ(r1 , r2 , r3 .....rn ; t) and the probability density is written as P (r1 , r2 , r3 .....rn ; t)= | ψ(r1 , r2 , r3 , .....rn ; t) |2 . The equation (1.1) shows how the wave function evolves in time. Now, the time-independent Hamiltonian operator of the n-electron system in the absence of any external field but only with the Coulomb interactions among the electrons can be written as follows h2 ¯ ˆ H=− Σ (∂ 2 /∂x2i +∂ 2 /∂yi2 +∂ 2 /∂zi2 ) +Σij 2me i

Qi Qj

| ri − rj |

.

(1.2)

ˆ does not contain time explicitly, one can apply the method of separation of variables. Since H Time dependent and time independent part of the wave function can be separated for the stationary state problem, ψ(r1 , r2 ....., rn ; t)=Ψ(r1 , r2 ....., rn ; t)e−iEt/¯h .

(1.3)

This gives rise to time-independent Schr¨ odinger equation, HΨ=EΨ.

(1.4)

There are 3n coordinates in the wave function Ψ for n electron system. In addition to the spatial coordinates, if we consider spin coordinates into account the total number of coordinates become 4n, where the spin is restricted to the value ± 12 . In the relativistic 8

treatment, the spin of the electrons appears naturally, and it is sometimes considered as an intrinsic angular momentum of the particle. The symmetry restriction is to be imposed on to the wave functions. The only acceptable solutions of the equation (1.4) are those with appropriate symmetry on the application of two particle permutation operator. For electronic systems, the wave functions must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. Time independent Schr¨ odinger equation will provide many solutions for stationary states. The lowest energy state is obviously the ground state. Schr¨ odinger equations are simplified for stationary state problems by using Born-Oppenheimer approximation,2 in which the nuclear coordinates are kept frozen and the electronic part is solved at a fixed nuclear geometry. ˆ el (r; R)Ψel (r; R)=Eel (R)Ψel (r; R) H

(1.5)

1.3. The variational principle The Schr¨ odinger equation cannot be solved exactly for many electron systems because the variables are not separable. Many approximate methods are employed for getting the solutions. The variational principle3 provides one such approximate technique to solve the time-independent Schr¨ odinger equation HΨk =EΨk , say for the k-th stationary state. In the f ) is expanded as a linear combination linear variational principle, a trial wave function (Ψ k

of basis functions {χi } f =Σ c χ , Ψ k i i i

(1.6)

where ci denotes the expansion coefficients. If the functions {χi } form a complete set, one obtains the true wave function of the system. However, truncation is needed for practical purpose. The energy functional is given as f =hΨ f |H c|Ψ f i/ hΨ f |Ψ fi E k k k k k

={Σij ci cj Hij }/{Σij ci cj hχi |χj i}.

(1.7)

The linear variational principle ensures that the trial energy calculated above is always higher

9

than the true energy of the system. In other words, there is an upper bound to the electronic f is minimized with respect to all c parameters. One gets energy. The energy functional, E k i

the exact energy if the trial function is the exact solution to the Schr¨ odinger equation i.e. Ψk . The above mentioned first-order variational conditions give the secular equations in the matrix form as Hc=Ec.

(1.8)

From the computational view point, one must construct the Hamiltonian matrix element for a given basis set used to expand the wave function in the form of the equation (1.6). The diagonalization of the the H-matrix has to be done next to get eigenvalues and eigenfunctions. The matrix elements may be computed by different semiemperical or abinitio methods.

1.4. The Hartree-Fock model Electronic motions in atoms or molecules are correlated mainly because of the Coulombic interactions among the electrons. Equation (1.2) suggests that the molecular Hamiltonian is independent of three or higher body interaction terms. However, the two-electron interaction terms are the most important and difficult to compute in the electronic structure theory. Three or higher body interactions are approximated to zero. In the independent particle model, one may write the wave function in terms of product of orbitals, which are functions of both space and spin-coordinates of electrons. Ψ(r1 , r2 , r3 ....) =Φ1 (r1 )Φ2 (r2 )Φ3 (r3 ).....

(1.9)

Here the electron-electron repulsion term is taken into consideration in an indirect manner. Each electron has been considered to be moving in the mean-field of the remaining (n-1) electrons. This gives the following set of one-electron equations hi Φi (ri )=i Φi (ri ),

i=1,2,3...n

(1.10)

where hi is an effective one-electron operator for the i-th electron which includes the meanfield interaction with other electrons. The sum of all orbital energies (i ) differs from the total energy of the system. A self-consistent field (SCF) method is employed to solve these one-electron equations (1.10) iteratively.5,6 10

One must employ the antisymmetry requirement for the many electron wave function in the next step to ensure the incorporation of Pauli exclusion principle. The antisymmetry requirement is fulfilled if the product of one-electron functions is written in the form of a Slater determinant. Ψ(r1 , r2 , r3 ....)=| Φ 1 (r1 )Φ2 (r2 )Φ3 (r3 )....Φn (rn )|

(1.11)

The use of complete set of orbitals gives rise to a complete set of determinants those span the full space of the antisymmetric many-electron wave functions. Such an independent particle model is known as the Hartree-Fock (HF) model. Two rows of the determinant will be identical if two electrons possess the same coordinates and hence the probability of such event is zero. So, in the Hartree-Fock model, electrons must have different coordinates. The determinantal form of the wave function leads to a certain correlation between their positions and movements (Fermi Correlation) for electrons with the same spin. Moreover, the orbitals in the determinant must be linearly independent and orthonormal. Instead of solving one 3n-dimensional equation, one has to solve n 3dimensional differential equations7 in the Hartree-Fock approximation. The one-electron Fock operator (hi ) has the following form: hi Φi =Ti + VN + VC + VX , where 2

∂ Ti =− 21 h ¯ 2 ( ∂x 2 +

VN =−Σα ( R

Zα

| r −rα |

VC =Σj dr2 ( R

∂2 ∂y 2

+

∂2 )Φi , ∂z 2

)Φi ,

Φ∗j (r2 )Φj (r2 )Φi (r1 ) ), r12

VX =−Σj dr2 (

φ∗j (r2 )φj (r1 )Φi (r2 ) ). r12

(1.12)

Ti is the kinetic energy, VN corresponds to all electron-nuclei attraction, VC is the Coulomb interaction and VX is the exchange interaction. This last term VX is the outcome of the antisymmetry requirement and has no classical analogue. The integration of the spin-components must be considered in addition to the spatial coordinates as each electron is associated with spin. The spin-orthogonality has a major role in deciding the zero or nonzero value of the integrals. The Coulombic interaction between 11

electrons will always occur, but the exchange interaction has non-zero values only between electrons of the same spin. Although the Hartree-Fock equations are one electron equations, the Fock operator (hi ) itself is a function of all other orbitals in the system. Iterative methods are employed to solve the equations. At first, one has to guess the trial orbitals which are used to compute the Fock operator, and one-electron equations are then solved. A new set of orbitals is constructed from the resulting orbitals and the iteration is continued. A convergence problem may be encountered if the initial guess of trial orbitals is not good enough. Different numerical techniques such as damping, scaling etc. are employed to achieve the convergence in those situations. The SCF method has become a very important technique for modeling a variety of many electron systems as it generates a number of symmetry adapted molecular orbitals which are used as basis functions. At the lowest level, the closed-shell Hartree-Fock theory gives good result for the ground state of molecules in the close vicinity at the equilibrium configuration.

1.5. Basis sets In the molecular orbital (MO) theory, the probability density for the electron in a molecule is described by a set of MOs {φi } which are constructed from the set of atomic orbitals (AO) of the constituent atoms in the molecule. The individual molecular orbital (φi ) can be expressed as linear combinations of a set of one-electron basis functions {χj } centered on each atom φi =

Pn

j=1

Cji χj ,

where Cji terms denote the expansion coefficients. To represent the MOs exactly, the basis functions χj should form a complete set, hence an infinite number of basis functions is required which is not possible in practice. So a finite number of basis functions is chosen and their choice is important for the satisfactory representation of the molecular orbitals. One may use Slater type orbitals (STO)8 χST O =Yl,m (θ, Φ)e−αr

(1.13)

as basis functions. However, these STOs are not often suitable for the numerical work.

12

Boys9 proposed another type of functions, namely, the Gaussian-type functions 2

glmn =N(x−xo )l (y−yo )m (z−zo )n e−α(r−ro ) ,

(1.14)

where N is the normalization constant, l, m, n are positive integers and α is orbital exponent which is also positive. The function glmn denotes s, p, d-type of Gaussian depending upon the value of l+m+n=0, 1, 2, respectively. The evaluation of various two-electron integrals in the Hamiltonian matrix elements is the most difficult part in the MO calculation. Furthermore, the number of these integrals increases rapidly with the number of basis functions. Wherever possible the symmetry of the molecule may be used to reduce the number of integrals to a large extent. Instead of using a single Gaussian function one can use a linear combination of a small number of Gaussians, χj =

P

i

dij gi ,

where gi s are Gaussians centered on the same atom and having the same l, m, n values as one another with different α values. χj is called a contracted Gaussian function and gi s are called primitive Gaussians, and dij terms are the suitable coefficients. The use of contracted Gaussians instead of primitive Gaussians reduces the number of variational coefficients to be determined. However, at the Hartree-Fock level, the number of MOs generated does not pose much problem. But in the large scale post Hartree-Fock calculations such as MCSCF and CI, the number of MOs can not be kept too large as it generates enormous number of configurations for a given electronic state.

1.6. Relativistic effects The velocity of light in classical mechanics is considered as infinite compared to that of the object and the light does not interact with the object of measurement. Assuming the velocity of light to be infinite, if the light is allowed to interact with the matter, one gets the non-relativistic quantum mechanics through Heisenberg’s uncertainty principle . If both the assumptions are relaxed, i.e., the velocity of light is finite relative to that of the object and there exists an interaction with the object, one should include relativistic corrections. In other words, if a particle moves at a velocity which is comparable with that of light, the non-relativistic quantum mechanics is no longer accurate. Actually, the mass of the system

13

determines the extent of the relativistic correction necessary in the calculations. The energy of the one-electron atom in the ground state with atomic number Z is -Z2 /2 in atomic units. The average velocity of the electron is of the order of Z which can be easily shown from virial theorem. The velocity of light is about 137 in atomic unit which indicates that relativistic effects can not be neglected for atoms of heavier masses. Non-relativistic quantum mechanical methods are quite satisfactory for most of the molecules having lighter atoms in the first and second row of the periodic table. But the situation is not similar for molecules containing atoms of higher rows in the periodic table where the relativistic effect comes into play. The inner electrons of heavy atoms attain faster speed due to large nuclear charges, and the speed is comparable with that of light. As for example the 1s electron of the Au atom acquires about 60% of the speed of light. As the core electrons are subjected to larger nuclear charges, the relativistic effects10−14 are significantly large for them. These core electrons in turn affect the valence space which is significant for the chemical bonding. Therefore, the chemical bonding and spectroscopic properties of these molecules are expected to change to a large extent due to the heavy nuclear masses. Different types of relativistic corrections are made for heavy atoms and molecules. These are mass-velocity correction, Darwin correction, spin-orbit correction, spin-spin interaction, Breit interaction etc. The dominant part of the relativistic correction is the mass-velocity correction which arises due to the variation in mass of the electron with its speed as it compares the speed of light. The relativistic mass is written as m= qm0 2 . 1− v2 c

The basic equation in the relativistic quantum mechanics is the Dirac equation (−i¯ h∂/∂t−V +cα · π+βmc2 )Ψ=0,

(1.15)

where π =−i¯h(∂/∂x, ∂/∂y, ∂/∂z), 

0 0 0 1

 0  α= 0 

 

0

0

0

      

0

0

−i

0

−i

0

−i

0

0

0 1 0  , 1 0 0 

1 0 0 0

−i

0

0

1

   0   0 ,   0   1

0

0 −1   , 0 0  

 

0

0 −1 0

0

14

0

 

0



1 0

 0  β= 0 

1

0

0

0  

0 −1

0 0 

and

0

0 Ψ1

 

, 0  

−1



   Ψ2   . Ψ=  Ψ  3  

Ψ4 For stationary states, time-independent one-electron Dirac equation takes the following form (−V +cα · π+βmc2 )Ψ=EΨ.

(1.16)

Results of the solution of the Dirac equation for one-electron systems are in excellent agreement with the experimental data but for many-electron systems, the applications of the Dirac equation are not so simple. Many approximate schemes based on variation as well as perturbation have been developed. In the next chapter of the thesis, we have reviewed the method of computation for many electron atoms and molecules with the consideration of relativistic effects.

1.7. Electron correlation energy and post Hartree-Fock treatments Hartree-Fock wave functions are written as Slater determinants of one electron functions under the independent particle model. The best possible determinant is chosen by using variational methods. Therefore, it means that each electron experiences an effective mean electrostatic field of all other electrons, but the motion and instantaneous positions of these electrons are not explicitly correlated. The approximation is a crude one, though it works well in some cases, especially for the ground state of the molecule. To obtain more accurate results for studying the electronic structure and properties in the low-lying excited states of the molecule, one must go beyond the Hartree-Fock approximation . Therefore, the approximation by the mean effective field is not sufficient15,16 as the electron-correlation in the post Hartree-Fock calculations becomes very important. The Hartree-Fock model does not produce accurate results because of the inadequacy 15

of including correlations between the motions of electrons. The wave function written in the single determinant form does not take into account of the electron correlation between electrons of opposite spin. The correlation of the motions of electrons having the same spin is partially, but not completely accounted for by virtue of the determinantal form of the wave function. There are many qualitative deficiencies in the description of the electronic structure of many electron system due to the omission of the correlation between electrons of opposite spin. The closed-shell Hartree-Fock calculations do not describe the dissociation of molecules correctly. The difference between the true non relativistic energy and the HartreeFock energy is the measure of the correlation energy. Eexact -EHF =Ecorrelation Therefore, one has to achieve this amount of the correlation energy by some other means for getting better results. The post Hartree-Fock methods are thus employed in quantum chemistry so that the electron correlation17−19 which has been left out in the HF treatment can be obtained. The methods like configuration interaction (CI), Many body perturbation theory (MBPT), Coupled cluster (CC), density functional theory (DFT) are among the post-Hartree Fock methods employed in quantum chemistry. The CI method is one of the most useful methods which extend beyond the Hartree-Fock model. In this method, the main concern is the choice of important configurations and elimination of others at the optimum level so that the volume of the computations does not increase very rapidly with the molecular size. Another requirement is the size consistency i.e. the method must provide additive results when applied to an assembly of isolated molecules. It is advantageous if the variational method can be applied as it ensures the upper boundness of the total energy. In order to incorporate the electron correlation within the variational principle, the wave function is expressed as a linear combination of several Slater determinants, each of which represents an individual electronic configuration. The variational method determines the best possible combination. Multiconfiguration-SCF (MCSCF) method exploits this technique in which both expansion coefficients and orbitals forming the determinants are optimized variationally while in the CI methodology, only CI coefficients are optimized. Variants of MCSCF and CI methods are available and used to solve the actual problem depending upon the computational capability and the desired accuracy. In recent years, many efficient computational techniques are available in literature to tackle the problem in carrying out 16

MCSCF and CI calculations. The lower energy molecular orbitals are generally occupied while the higher vacant ones are virtual orbitals. One can generate antisymmetric many electron functions which have different orbital occupancies. Each such many electron antisymmetrized function is a Slater determinant or a linear combination of such determinants. As a result spin-adapted configuration state functions (CSF) can be formed. The ground-state configuration is that distribution of electrons among the MOs which possess the lowest energy. For even number of electrons, say, 2n, the single configuration wave function is written as, 1

(0)

ΦG =A |φ1 α(1)φ1 β(2)........φn α(2n − 1)φn β(2n)|; S=0, Ms =0.

For odd number of electrons (2n+1), the single configuration functions are, 2

(0)

ΦG =A | φ 1 α(1)φ1 β(2)........φn α(2n − 1)φn β(2n)φp α(2n + 1)|; S=1/2, Ms =1/2

and 2

(0)

ΦG =A |φ1 α(1)φ1 β(2)........φn α(2n − 1)φn β(2n)φp β(2n + 1)|; S=1/2, Ms =-1/2.

The singly excited configurations are those distributions in which an electron has been promoted from an occupied MO say, φk to a vacant MO φs . For 2n number of electrons corresponding to singlet single-configuration wave function may be written as: 1

(1)

Φk→s = √12 [A |φ1 α(1)φ1 β(2).......φk α(2k − 1)φs β(2k)........φn α(2n − 1)φn β(2n)|

– A |φ1 α(1)φ1 β(2).......φk β(2k − 1)φs α(2k)........φn α(2n − 1)φn β(2n)|]; S=0, Ms =0. Triplet state wave functions are, 3

(1)

Φk→s = √12 [A |φ1 α(1)φ1 β(2).......φk α(2k − 1)φs β(2k)........φn α(2n − 1)φn β(2n)|

+ A |φ1 α(1)φ1 β(2).......φk β(2k − 1)φs α(2k)........φn α(2n − 1)φn β(2n)|]; S=1, Ms =0 and (1)

3

Φk→s =A |φ1 α(1).......φk α(2k − 1)φs α(2k)........φn β(2n)|; S=1, Ms =1

3

Φk→s =A |φ1 α(1).......φk β(2k − 1)φs β(2k)........φn β(2n)|; S=1, Ms =-1.

(1)

The doubly excited configurations are those distributions which are obtained by promoting 17

(0)

2 electrons from an occupied MO of 1 ΨG to one vacant MO. The single-configuration wave function is, 1

(2)

Φk→s =A |φ1 α(1)φ1 β(2).......φs α(2k − 1)φs β(2k)........φn α(2n − 1)φn β(2n)|; S=0, Ms =0.

A linear combination of these CSF gives the CI wave function. P

Ψ=

i

Ci Φi

(1.17)

The variation of coefficients Ci to minimize the energy functional leads to the determinantal equation, det(Hij -ESij )=0.

(1.18)

It is important that only those CSF will contribute in the linear combination which have the same angular momentum eigenvalues as that of the state Ψ or the CSF will have the same symmetry properties (symmetry eigenvalue) as that of the state Ψ. The number of configurations increases with the number of electrons and number of basis functions. For n electrons and p number of basis functions, the number of CSF is roughly proportional to pn . A CI calculation that includes all possible CSF with proper symmetry is a full CI calculation. Due to large number of CSF, full CI calculations are not possible to carry out except for very small molecules with small basis set. There exist variants of CI calculations to choose the proper configurations which will contribute largely to Ψ. It is, thus possible to perform limited or truncated CI calculations.20 The simplest way of limiting the CI expansion is to truncate the series in a given level of excitation. The truncated wave function can be written as Ψ=Φo +Σs>0 Cs Φs

(1.19)

where Φ0 denotes the single determinant Hartree-Fock wave function while other determinants are denoted by Φs . Ψ becomes Φ0 , if no excitation is allowed, and one gets the HF energy. The inclusion of all single excitations gives the CI wave function, ΨCIS as, virt ΨCIS =C0 Φ0 + Σocc Cia Φai , i Σa

(1.20)

where the excitation is indicated by i→a. If only single excitations are included it does not improve the wave function or energy much. In the next step, the CI is limited with double 18

excitation only and the wave function may be written as, ab ab virt ΨCID =C0 Φ0 + ΣΣocc i