Interaction of diatomic germanium with lithium atoms

Interaction of diatomic germanium with lithium atoms: Electronic structure and stability G. Gopakumar, Peter Lievens, and Minh Tho Nguyen Citation: The Journal of Chemical Physics 124, 214312 (2006); doi: 10.1063/1.2202096 View online: http://dx.doi.org/10.1063/1.2202096 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/124/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Comparison of density functionals for energy and structural differences between the high- [ 5 T 2g :(t 2g ) 4 (e g ) 2 ] and low- [ 1 A 1g :(t 2g ) 6 (e g ) 0 ] spin states of iron(II) coordination compounds. II. More functionals and the hexaminoferrous cation, [ Fe ( NH 3 ) 6 ] 2+ J. Chem. Phys. 122, 044110 (2005); 10.1063/1.1839854 Spectroscopic properties of novel aromatic metal clusters: NaM 4 ( M = Al,Ga,In ) and their cations and anions J. Chem. Phys. 120, 10501 (2004); 10.1063/1.1738112 First principles study of the stability and electronic structure of the icosahedral La 13 , La 13 1 , and La 13 +1 clusters J. Chem. Phys. 120, 5081 (2004); 10.1063/1.1647513 Electronic structure and chemical bonding of B 5 and B 5 by photoelectron spectroscopy and ab initio calculations J. Chem. Phys. 117, 7917 (2002); 10.1063/1.1511184 Geometries and spectroscopic properties of germanium and tin hexamers ( Ge 6 ,Ge 6 + ,Ge 6 ,Sn 6 ,Sn 6 + , and Sn 6 ) J. Chem. Phys. 115, 3121 (2001); 10.1063/1.1386795

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THE JOURNAL OF CHEMICAL PHYSICS 124, 214312 共2006兲

Interaction of diatomic germanium with lithium atoms: Electronic structure and stability G. Gopakumar Department of Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium

Peter Lievens Laboratory of Solid State Physics and Magnetism, Department of Physics and Astronomy, University of Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

Minh Tho Nguyena兲 Department of Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium

共Received 11 October 2005; accepted 11 April 2006; published online 7 June 2006兲 Quantum chemical calculations were applied to investigate the electronic structure of mono-, di-, and trilithiated digermanium 共Ge2Lin兲 and their cations 共n = 0 – 3兲. Computations using a multiconfigurational quasidegenerate perturbation approach based on complete active space self-consistent-field wave functions, and density functional theory reveal that Ge2Li has a 2B1 ground state with a doublet-quartet energy gap of 33 kcal/ mol. Ge2Li2 has a singlet ground state with a 3Au- 1A1 gap of 29 kcal/ mol, and Ge2Li3 a doublet ground state with a 4B2- 2A2 separation of 22 kcal/ mol. The cation Ge2Li+ has a 3B1 ground state, being 13 kcal/ mol below the open-shell 1 B1 state. The computed electron affinities for diatomic germanium are EA共1兲 = 1.9 eV, EA共2兲 3− = −2.5 eV, and EA共3兲 = −6.0 eV, for Ge−2 , Ge2− 2 , and Ge2 , respectively, indicating that only the monoanion is stable with respect to electron detachment, in such a way that Ge2Li is composed by Ge−2 · Li+ ions. An “atoms-in-molecules” analysis shows the absence of a ring critical point in Ge2Li. An electron localization function analysis on Ge2Li supports the view that the Ge–Li bond is predominantly ionic; however, a small covalent character could be anticipated from the analysis of the Laplacian at the Ge–Li bond critical point. The ionic picture of the Ge–Li bond is further supported by a natural-bond-order analysis and the Laplacian of the electron density. The calculated Li affinity value for Ge2 is 2.08 eV, while the Li+ cation affinity value for Ge−2 is 5.7 eV. The larger Li+ cation affinity value of Ge−2 suggests a Ge−2 Li+ interaction and thus supports the ionic nature of Ge–Li bond. In GeLi4 and Ge2Li, the presence of trisynaptic basins indicates a three-center bond connecting the germanium and lithium atoms. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2202096兴 I. INTRODUCTION

There has been continuing interest in small elemental and molecular clusters1,2 as they provide a bridge toward the understanding of how matter evolves from atoms to bulk. This interest extends to the clusters of silicon and germanium not only because of their well-known technological importance as semiconductors, but also due to their possible role in surface growth processes and potential new applications in nanoelectronics. Gas-phase metal clusters generally adopt kinetically stable geometries that may not be relevant fragments of the bulk solid. They possess unusual physicochemical properties, thanks to the coordinative unsaturation and dangling bonds. The past experimental3–9 and theoretical10–17 studies on small germanium clusters focused mostly on the lowest energy electronic structures. The knowledge about the structural identity of a cluster is important since the cluster properties, specifically clusters’ relative stability and the a兲

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associated electronic structure, depend on their geometry. The first experimental studies on Gen clusters date back to 1954. Recent experimental studies included mass spectra, atomization energies, photofragmentation, photoionization, photoelectron spectroscopy, electronic gaps, ion mobility measurements, etc. Determination of geometries, dissociation energies, electronic structure, and electron affinities have abundantly been investigated in most of the theoretical studies made in the past. However, relatively little attention has been paid on the properties of metal-doped Gen clusters. While the pure germanium clusters are chemically reactive18 and is thus unsuitable as a building block of self-assembly materials, endohedral metal doping can dramatically change their structures and properties. For example, by a suitable choice of the metal dopant, it is possible to design metallic as well as semiconducting nanotubes using Ge clusters.19 On the other hand, organolithium compounds form a special family. Their unusual structures, bonding mechanism, and chemical reactions provide some of the reasons why they have been a subject of both theoretical and experimental investigations during the past years.20,21 In these compounds, the lithium atom plays an important role, and the nature of

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© 2006 American Institute of Physics

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the C–Li bond was a subject of intense debate.21 In this context, we set out to investigate the interaction of Ge2, the simplest Ge cluster, with lithium, the simplest metal atom, as a preliminary step toward the understanding of the role of lithium in stabilizing larger germanium clusters. For this purpose, we used high level ab initio molecular orbital and density functional calculations to obtain some quantitative thermochemical parameters and insights into their structure and bonding mechanism and, in particular, into the nature of the Ge–Li chemical bond. As far as we are aware, experimental studies on Ge2Lin compounds are not reported yet. II. METHODS OF CALCULATION

Density functional theory 共DFT兲 using the popular B3LYP functional and ab initio molecular orbital 共MO兲 theory calculations were carried out to investigate the structure and energies of diatomic germanium, its anion, and their mono-, di-, and trilithiated isomers 共Ge2Lin, n = 0 , 1 , 2 , 3兲. As a preliminary step, initial geometry optimizations were performed at the Hartree Fock 共HF兲 level, followed by the B3LYP functional calculations, in conjunction with dp-polarization plus diffuse-functions 6-311+ + G共d , p兲 basis set in the framework of the unrestricted formalism 共UHF and UB3LYP兲. Harmonic vibrational frequencies were computed at the same level in order to characterize the located structures. The relevant minima were then reoptimized using coupled-cluster CCSD共T兲 theory, and their electronic structure was analyzed at the multiconfigurational level, in particular, using complete active space self-consistent-field 共CASSCF兲 wave functions, that usually corrects for nondynamical or quasidegenerate correlation effects within the active space. As for a preliminary calibration of theoretical levels, the results were first compared with the available experimental and theoretical values. The active spaces selected for the computations will be discussed in subsequent sections. It is obvious that for the electronic states that are multiconfigurational in nature, evaluation of the dynamical correlation energy is indeed necessary to obtain quantitative results. One such approach is the multireference MøllerPlesset perturbation theory known as the MRMP2 method,22 in which a second-order perturbation correction scheme is applied to the multiconfigurational reference wave functions, generated for each state separately. It is known that a separate application of perturbative treatments to electronic states having very close energies sometimes leads to a reverse state ordering, and it is unphysical if they are of the same symmetry,23 or in a case where a root flipping occurs, or when the perturbation series diverges due to the existence of intruder states.24 The latter problem, which arises from the “near singularities” caused by very small or vanishing energy denominators of the corresponding perturbation expansions, often occurs because high-lying states within the complete active space frequently have zeroth-order energies that are quasidegenerate with zerothorder states in the orthogonal space. The quasidegenerate orthogonal-space states that are disrupting the perturbative convergence are called “intruder states,” and it is believed that disruption often occurs in the MRMP treatment when the

reference state is a high-lying state and a basis set with diffuse functions are used. Such kind of problems could be circumvented by the application of a quasidegenerate perturbation theory 共QDPT兲 where a small energy denominator shift value is used. In this view, the multiconfigurational quasidegenerate perturbation 共MCQDPT兲 method,25 which is based on the Van Vleck perturbation theory, is expected to give a more correct description. The effective Hamiltonian in MCQDPT contains off-diagonal corrections as well as single-state corrections to the diagonal terms, thereby providing corrected energies at second order for all states included in the model space simultaneously. The main advantage of a multistate perturbation approach is the simultaneous determination of the energies of several states of interest including degenerate or quasidegenerate states. In our MCQDPT computations, an intruder-state-free technique has been adopted by using a small energy denominator shift value. For these computations, the LANL2DZdp basis set with an effective core potential26 共ECP兲 has been employed and the structures were characterized by vibrational frequencies determined at the CASSCF level. The content of the present article is organized as follows. Initial calibration of our theoretical methodologies along with a general discussion of diatomic germanium and its anions are reported in the first section. The electronic structure of lithiated clusters will be discussed in the following sections. Finally, we carried out an “atoms-in-molecules” 共AIM兲 and “electron localization function” 共ELF兲 analysis along with the natural-bond-order 共NBO兲 charges, performed on some of the lithiated molecules considered, to obtain insights toward the interaction of lithium atoms with Ge2. All computations reported were performed with the GAUSSIAN 27 28 29 30 31 98, GAMESS, MOLPRO, AIM2000, and TOPMOD suites of programs. III. RESULTS AND DISCUSSION A. Ge2 and its anions

There have been extensive studies on small germanium clusters during the past two decades. Recently, Xu et al.32 discussed the ground electronic state properties of germanium clusters 共Gen, n = 1 – 6兲 using various DFT functionals. The Ge2 is characterized at the multireference configuration interaction 共MRCI兲 level33 as having a triplet 3⌺−g ground state, and a low-lying 3⌸u state, with an equilibrium bond distance of 2.42 Å in the ground state. Our CCSD共T兲 / 6-311+ + G共d , p兲 computations are in agreement with previous results.34 For ab initio computations, the LANL2DZdp basis set was used and the 28 electrons on each germanium atom, i.e., for 1s, 2s, 3s, 2p, 3p, and 3d electrons, have been modeled by an ECP. The active space employed thus includes the 4s and 4p orbitals on each germanium atom. In the case of Ge2, it contains eight electrons in eight orbitals, referred to hereafter as CASSCF共8,8兲. The total and relative energies are tabulated in Table I and the molecular orbitals illustrated in Fig. 1S of Electronic Supplementary Information.42 Unless otherwise stated, the energetic values mentioned hereafter refer to those obtained using MCQDPT2 computations based on CASSCF wave functions. Thus, we

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TABLE I. Calculated total and relative energies of small germanium clusters at UB3LYP/ 6-311+ + G共d , p兲 and MCQDPT2/ECP 共the effective core potentials adopted here are LANL2DZdp ECP for Ge兲 levels. Total energy 共a.u.兲 relative energy in parentheses in kcal/mol Molecule Ge2

Ge−2

Ge2− 2

Ge3− 2

State

Leading electron configuration

⌺−g

¯s␴2g , s␴2u , p␲2u , p␴2g

3

⌸u

¯s␴2g , s␴2u , p␲3u , p␴1g

2

⌸u

¯s␴2g , s␴2u , p␲3u , p␴2g

2

⌺+g

¯s␴2g , s␴2u , p␲4u , p␴1g

1

⌺+g

¯s␴2g , s␴2u , p␲4u , p␴2g , p␲0g

3

⌺+u

¯s␴2g , s␴2u , p␲3u , p␴2g , p␲1g

2

⌺+u

¯s␴2g , s␴2u , p␲4u , p␲2g , p␴1g

4

⌺+g

¯s␴2g , s␴2u , p␲4u , p␲2g , p␴1g

3

UB3LYPa

MCQDPT2

−4153.9642 共0兲 −4153.9559 共4.8兲

−7.3943 共0兲 −7.3918 共1.6兲

−4154.0353 共0兲 −4154.0302 共3.2兲

−7.4555 共0兲 −7.4472 共5.2兲

−4153.9446 共0兲 −4153.9003 共27.8兲

−7.3317 共0兲 −7.2809 共31.9兲

−4153.7240 共0兲 −4153.6547 共43.5兲

−7.0581 共0兲 −7.0220 共22.6兲

a

The B3LYP total energy values are scaled by zero point energies at the same level.

were able to derive a 3⌺−g ground state for Ge2, with a quasidegenerate 3⌸u state, in agreement with the reported MRCI results.33 The former state results from the occupancy of the unpaired electrons in the two degenerate p␲u orbitals, whereas in the latter the unpaired electrons occupy the p␲u and the bonding p␴g orbital. The energy gap between both states amounts to 1.6 kcal/ mol and is thus close to the MRCI value of 1.9 kcal/ mol and the experimental result35 of 1.98 kcal/ mol. Note that DFT computations using the UB3LYP functional overestimated this energy gap to 4.8 kcal/ mol. Because there is an obvious electron transfer from lithium atoms to Ge within the doped clusters, it is of importance to examine the relative stability of the Ge−2 anion. For this monoanion, a 2⌸u ground state was confirmed, which has the occupancy of the unpaired electron in the p␲u MO. The higher-lying 2⌺+g results from occupancy in the p␴g orbital. The change in the bond length in going from the neutral to the anion ground state is marginal, resulting in the slight shortening of the latter, due to the occupancy of the added electron in a Ge–Ge bonding MO. MCQDPT computations provide a 2⌺+g − 2⌸u energy gap of 5.2 kcal/ mol for the anion, whereas the B3LYP method gives rise to a smaller gap of 3.2 kcal/ mol. We extended the same methodology to investigate the electronic state ordering resulting from the addition of the second electron. The Ge2− 2 dianion has been found to have a 1 + ⌺g ground state with a triplet 3⌺+u excited state, the latter corresponds to a 共p␲u兲1共p␲g兲1 configuration. The energy difference between the 1⌺+g ground state and 3⌺+u triplet excited state is derived to be 31.9 kcal/ mol. The elongation of the

Ge–Ge bond in the triplet state, as compared to that of the ground state, is apparent and no doubt due to the occupancy of an unpaired electron in the antibonding p␲g molecular orbital. DFT/UB3LYP computations underestimate this energy gap to 27.8 kcal/ mol, for the same electronic state ordering. Subsequent addition of an electron leads to the formation 2 + of a Ge3− 2 trianion, which is characterized to have a ⌺u 4 + low-spin ground state, with a higher-lying quartet ⌺g state, at both MCQDPT/ECP and UB3LYP/ 6-311+ + G共d , p兲 levels 共see Table I for total and relative energies兲. In the case of the trianion, the UB3LYP method gives a large doubletquartet energy gap of 43 kcal/ mol, which is almost double of that of 22.6 kcal/ mol by MCQDPT. The electron affinities of diatomic germanium were calculated from B3LYP energies using the following expressions: EA共1兲 = E共Ge2兲 − E共Ge−2 兲,

共1兲

EA共2兲 = E共Ge−2 兲 − E共Ge2− 2 兲,

共2兲

3− EA共3兲 = E共Ge2− 2 兲 − E共Ge2 兲.

共3兲

The calculated electron affinity values are EA共1兲 = 1.9 eV, EA共2兲 = −2.5 eV, and EA共3兲 = −6.0 eV. The EA共1兲 is indeed positive, whereas both EA共2兲 and EA共3兲 are very negative. Accordingly, only the Ge−2 monoanion is stable with respect to electron detachment. To probe further the existence of the di- and trianion, total energies of their lower-lying states have been plotted as a function of the Ge–Ge bond length. Here, the single point electronic energy computations have been performed at different Ge–Ge bond lengths at the CASPT2/aug-cc-pVTZ level, using the MOLPRO suite of programs,29 and the results are in agreement with MCQDPT. However, when the electron affinity is negative, the calculations employing a very large basis set with extended diffused functions may undergo a variational collapse, i.e., the variational energy would tend towards the ground state of the neutral molecule. Carefully considering this fact, these potential energy curves will not be discussed in detail hereafter, but they are made available in the Electronic Supplementary Section for information 共see Figs. 2S and 3S of Electronic Supplementary Information42兲. We only note that in their lowest-lying states, both dianion and trianion of germanium dimer correspond to energy minima. At this stage, the aforementioned results appear to confirm the reliability of the MCQDPT method. Another interesting finding is that the DFT/UB3LYP method has been proven equally good as the perturbation approach in predicting the energetic ordering of the lower-lying states, but it tends to markedly overestimate the energy gaps between them. This is due no doubt to the multiconfigurational character of many electronic states considered. B. Interaction of Ge2 with lithium atoms 1. Ge2Li

The same methodologies as in the previous section are applied here. Geometry optimizations were initially carried

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TABLE II. Calculated total and relative energies of small germanium clusters at UB3LYP/ 6-311+ + G共d , p兲 and MCQDPT2/ECP 共the effective core potentials adopted here are LANL2DZdp ECP for Ge, the LANL2DZ basis set is implemented for lithium atoms兲 levels. Total energy 共a.u.兲 relative energy in parentheses in kcal/mol Molecule Ge2Li

Ge2Li+

Ge2Li2 共D2h兲

Ge2Li+2

Ge2Li3

Ge2Li+3

State

Leading electron configuration

B3LYPa

MCQDPT2

−4161.5321 共0兲 −4161.4885 共27.4兲

−14.9106 共0兲 −14.8569 共33.7兲

−4161.2885 共0兲 −4161.2825 共3.8兲

−14.6756 共0兲 −14.6540 共13.6兲

−4169.1068 共0兲 −4169.0728 共21.4兲

−22.4130 共0兲 −22.3664 共29.2兲

−4168.8889 共0兲 −4168.8229 共41.4兲

−22.2126 共0兲 −22.1503 共37.8兲

−4176.6395 共0兲 −4176.6035 共22.5兲

−29.8742 共0兲 −29.8379 共22.8兲

−4176.4757 共0兲 −4176.4380 共23.7兲

−29.7301 共0兲 −29.6845 共28.6兲

2

B1

¯2a21 , 1b22 , 3a21 , 1b11 , 4a21

4

A2

¯2a21 , 1b22 , 3a21 , 1b11 , 4a11 , 2b12

3

B1

¯2a21 , 1b22 , 3a21 , 1b11 , 4a11

1

B1

¯2a21 , 1b22 , 3a21 , 1b11 , 4a11

1

A1

¯2a21 , 1b21 , 2b22 , 3a21 , 4a21 , 1a02

3 A2 共 3A u兲

¯2a21 , 1b21 , 2b22 , 3a21 , 4a11 , 1a12

2

A1

¯2a21 , 1b21 , 2b22 , 3a21 , 4a11 , 1a02

4

B1

¯2a21 , 1b21 , 2b22 , 3a11 , 4a11 , 1a02 , 2b11

2

A2

¯3a21 , 1b21 , 4a21 , 2b22 , 5a21 , 1a12 , 2b01

4

B2

¯3a21 , 1b21 , 4a21 , 2b22 , 5a11 , 1a12 , 2b11

1

A1

¯3a21 , 1b21 , 2b22 , 4a21 , 5a21 , 1a02 , 2b01

3

B1

¯3a21 , 1b21 , 2b22 , 4a21 , 5a11 , 1a02 , 2b11

a

The B3LYP total energy values are scaled by zero point energies at the same level.

out at the HF level followed by B3LYP calculations in conjunction with 6-311+ + G共d , p兲 and vibrational frequency calculations. For MCQDPT2 calculations, an effective core potential was modeled employing the LANL2DZdp basis set for Ge, but for the lithium atom, the LANL2DZ basis set was adopted. From the previous section it is evident that the incorporation of the effective core potential at this level of theory does not have a considerable effect in the energy ordering of the electronic states and the relevant energy gaps. Note again that for CASSCF computations, the lithium 1s orbital was kept frozen. The active orbitals thus include the 4s and 4p orbitals on each germanium atom and the 2s orbital of the lithium, implying nine electrons in nine orbitals referred to hereafter as CASSCF共9,9兲. Geometrical parameters for doublet and quartet states, along with the lowerlying states of the Ge2Li+ cation, are given in Fig. 1. The active orbitals for the ab intio computations illustrated in Fig. 2 are labeled under C2v point group and include 4 a1, 1 b1, 1 a2, and 3 b2 orbitals. Interaction of Ge2 with one lithium atom leads to three distinct structural isomers, namely, a C2v structure with lithium being on the C2 axis, a linear structures of D⬁h symmetry in which the lithium atom connects the two germanium atoms 共Ge–Li–Ge兲, and a linear structure with C⬁v

symmetry 共Ge–Ge–Li兲. Each structural entity gives rise to two spin manifolds, the doublet and the quartet. In both doublet and quartet states, the linear form Ge–Li–Ge was found to have degenerate imaginary frequencies. The lowest-lying 2 ⌸u doublet state is characterized to have a small imaginary frequency. Following the mode associated with this small vibrational frequency, a C2v symmetric minimum was invariably obtained. The linear Ge–Ge–Li has energetically higherlying electronic states compared to the C2v symmetric structure at the B3LYP/ 6-311+ + G共d , p兲 level. The quartet state of the former has an imaginary frequency of magnitude 64 cm−1 at the same level, corresponding to the bending

FIG. 1. Selected CASSCF/ECP geometrical parameters of the Ge2Li 共9,9兲 and Ge2Li+ 共8,9兲 cations considered in some lower-lying electronic states. Bond lengths are in angstroms and bond angles in degrees.

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FIG. 2. Shape of the nine natural orbitals of the Ge2Li and its cation selected for the CASSCF computations.

motion.34 However, in the present study we have concentrated mainly on the lowest energy C2v symmetric structure. The C2v minimum structure possesses a 2B1 doublet ground state, with several low-lying excited states. The lowest-lying 4A2 quartet state is being about 33 kcal/ mol above the doublet ground state 共Table II兲. The geometrical change from the doublet ground to the quartet excited state is significant; the Ge–Li bond distance decreases by an amount of 0.04 Å, whereas the Ge–Ge bond distance increases by 0.35 Å. The ⍜ angle, denoted as the bond angle between two Li–Ge bonds, is also increased by 7.6°, upon transition from the doublet to quartet electronic state. Considering a Ge−2 Li+ interaction, the change in the Ge–Ge bond length upon addition of the Li+ cation to the Ge−2 anion is marginal, namely, 0.01 Å. Calculated configuration interaction 共CI兲 coefficients suggest a leading electronic configuration for Ge2Li: 2 B1 : ¯ 共2a1兲2共1b2兲2共3a1兲2共4a1兲2共1b1兲1. The 1b1 and 4a1 MOs are mainly from the ␲-bonding MOs of Ge2 共Fig. 2兲. The quartet state results from an occupancy of unpaired electrons in the orbitals 1b1, 4a1, and 2b2. The 2b2 orbital is formed mainly from a Ge2 ␲-antibonding orbital. It can be concluded that the geometrical change resulting in the elongation of the Ge–Ge bond is due to the occupancy of an electron in this antibonding MO. Removal of an electron from Ge2Li leads to the forma-

tion of a Ge2Li+ cation for which both MCQDPT and UB3LYP methods predict a triplet 3B1 ground state. A singlet 1 B1 state is found energetically lying 13.6 kcal/ mol above the ground 3B1 state. It is interesting to see the geometrical difference in the lower-lying states of the cation Ge2Li+ 共Fig. 1兲. The changes in the Ge–Ge and Ge–Li bond lengths and Ge–Li–Ge bond angle are not marginal, up to 0.1 Å and 1.6°. Examination of the CI coefficients suggests a predominant electronic configuration, ¯共2a1兲2共1b2兲2共3a1兲2共1b1兲1 共4a1兲1 for both states of Ge2Li+. Geometric changes upon ionization are more considerable. The Ge–Li bond length increases by an amount of 1.7 Å and the Ge–Li–Ge bond angle decreases by 4.1°. A logical explanation for the changes can again be provided from the MO’s shape. In the ground state of the neutral Ge2Li, the 4a1 MO, which is also a bonding molecular orbital with respect to the interaction between Ge2 and lithium, is doubly occupied, whereas in the cation, the occupation number is only 1 for both lower-lying states. Occupancy of two electrons in this bonding MO energetically favors the neutral ground state and shortens the Ge–Li bond length. An energy difference of 147.5 kcal/ mol is calculated between the ground states of the neutral and the cation, implying an ionization energy of 6.40 eV, which is small compared to that of 7.9 eV of Ge2, but closer to that of 5.39 eV for the lithium atom. Thus, due to its presence, attachment of lithium tends to lower the I.E. of the doped cluster, and thereby facilitates electron removal. UB3LYP computations are in agreement with the MCQDPT value suggesting an ionization energy of 6.63 eV 共152.9 kcal/ mol兲. As quantum chemical methods usually underestimate IE’s by up to 0.2 eV, it seems reasonable to predict a value IEa共Ge2Li兲 = 6.8± 0.2 eV. 2. Ge2Li2 and Ge2Li3

Progressive addition of lithium atoms to Ge2Li yields Ge2Li2 and Ge2Li3. For ab initio MO computations, the same procedure has been adopted as in the case of Ge2Li, giving rise to an active space of ten electrons in ten orbitals 关CASSCF共10,10兲兴, thus including four a1, three b1, one a2, and two b2 orbitals. Optimized geometrical parameters of the lower-lying states of the neutral molecule, and the cation, along with the shape of the active orbitals are illustrated in Figs. 3共a兲 and 4, respectively. Several possible geometries for the neutral form have been located and characterized with vibrational frequencies. It turns out that only a C2v form corresponds to the equilibrium structure. As in Ge2Li, let us first consider the geometrical changes in going from the neutral to its cation, at various lower-lying electronic states. For the neutral Ge2Li2, the closed-shell singlet 1A1 ground state is energetically 29.0 kcal/ mol below the triplet excited 3Au state 共D2h symmetry, see Table III兲. However, for the sake of comparison, the molecular orbitals are labeled under C2v symmetry and the latter state is resolved to 3A2. The UB3LYP/ 6-311+ + G共d , p兲 level gives a smaller energy gap of 21.4 kcal/ mol. CI coefficients show that these states are similar by the following leading electronic configurations: 1 A1 : ¯ 共2a1兲2共1b1兲2共2b2兲2共3a1兲2共4a1兲2共1a2兲0 and 3 A2共 3Au兲 ¯ 共2a1兲2共1b1兲2共2b2兲2共3a1兲2共4a1兲1共1a2兲1. There is a

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214312-6

J. Chem. Phys. 124, 214312 共2006兲

Gopakumar, Lievens, and Nguyen

FIG. 3. Selected CASSCF/ECP geometrical parameters of 共a兲 Ge2Li2 共10,10兲 and Ge2Li+2 共9,10兲 cations, and 共b兲 Ge2Li3 共11,11兲 and Ge2Li+3 共10,11兲 cations, considered in some lower-lying electronic states. Bond lengths are in angstroms and bond angles in degrees.

considerable geometrical change in going from the singlet ground to the triplet excited state, in such a way that the molecule acquires planarity in the high-spin manifold. The Ge–Ge bond length and Ge–Li–Ge bond angle increase by amounts of 0.1 Å and 3°, respectively. This can be again understood on the basis of the occupancy of unpaired electrons at the high-spin excited state. One unpaired electron occupies the antibonding 1a2 MO and thereby induces an elongation to the Ge–Ge bond. The planar geometric configuration further adds little bonding character to the 1a2 MO, with respect to Ge2 and Li interaction, and this can be expected to balance, to a certain extent, the antibonding character of the Ge–Ge interaction. An energetically higher-lying triplet state can be anticipated due to the occupancy of an unpaired electron in the antibonding molecular orbital, which is unoccupied at the closed-shell ground state. Regarding Ge2Li2 as a combination of the dianion Ge2− 2 and two Li+ cations, it is noted that the Ge–Ge bond length is only marginally altered upon lithiation. Removal of an electron from Ge2Li2 leads to the formation of the Ge2Li+2 cation, which retains the same symmetry in both doublet and quartet states. The corresponding doublet 2A1 ground state 共C2v兲 is found to be 37 kcal/ mol below a 4B1 quartet excited state. The lowest-lying state of the cation is expected to arise from the neutral ground state upon removal of an electron from the highest occupied molecular orbital 4a1. A marginal increase of the Ge–Ge bond length is noted, as the 4a1 MO is having the bonding Ge–Ge interaction. CI coefficients derived from CASSCF wave functions point out a dominant electronic configuration of 2 A1 : ¯ 共2a1兲2共1b1兲2共2b2兲2共3a1兲2共4a1兲1共1a2兲0共2b1兲0 and

FIG. 4. Shape of the ten natural orbitals of the Ge2Li2 and its cation selected for the CASSCF computations.

B1 : ¯ 共2a1兲2共1b1兲2共2b2兲2共3a1兲1共4a1兲1共1a2兲0共2b1兲1 for the cation. Within the quartet, manifold the molecule acquires planarity with an increase of 0.1 Å in the Ge–Ge bond length. Further lithiation on Ge2Li2 yields Ge2Li3, which is having a doublet 2A2 ground state and a lower-lying quartet 4B2 state. Geometrical parameters for the lower-lying states of the neutral and cation are illustrated in Fig. 3共b兲, and the shape of the molecular orbitals included in the active space in Fig. 4S of the Electronic Supplementary Information.42 The lithium 1s orbitals are again kept frozen; the active space thus includes the Ge 4s and 4p orbitals and the 2s lithium orbitals, comprising of 11 electrons in 11 orbitals 关CASSCF 共11,11兲兴. These consist of five a1, three b1, one a2, and two b2 MOs. In this case, both the MCQDPT and DFT computations predict a 2A2 ground state for Ge2Li3, which is energetically lying about 22.0 kcal/ mol below the high-spin 4 B2 state. The leading electronic configurations are as follows: 4

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214312-7

J. Chem. Phys. 124, 214312 共2006兲

Diatomic germanium with lithium

TABLE III. The calculated Li affinity and Li+ cation affinity for Ge2 and Ge−2 anions 共values are in eV兲 at B3LYP/ 6-311+ + G共d , p兲. Property

Ge2

Ge−2

Li+ – A Li–A

1.07 2.08

5.77 ¯

Ge2Li3: 2A2: ¯ 共3a1兲2共1b1兲2共4a1兲2共2b2兲2共5a1兲2共1a2兲1共2b1兲0 4

B2: ¯ 共3a1兲2共1b1兲2共4a1兲2共2b2兲2共5a1兲1共1a2兲1共2b1兲1 .

Geometrical changes from the doublet to the quartet state are also considerable: elongation of the two Ge–Li and Ge–Ge bonds by an amount of 0.1 Å and reduction of the Li–Li–Li bond angle 关␪ in Fig. 3共b兲兴 by an amount of 18°. In the quartet manifold, unpaired electrons occupy antibonding 1a2 and 2b1 MOs, with respect to the Ge–Ge bond, and this leads to an elongation of the Ge–Ge bond to 2.524 Å. Removal of an electron from the 1a2 SOMO of the neutral Ge2Li3 ground state gives rise to a Ge2Li+3 cation having a closed-shell singlet 1A1 ground state. Both the MCQDPT and B3LYP methods are in agreement with each other indicating a lowest-lying triplet excited state. The triplet state is located to have D3h symmetry. To facilitate the comparison, the molecular orbitals and electronic states are, however, labeled under C2v point group. The singlet-triplet 1A1- 3B1 separation is evaluated at 28 kcal/ mol. The B3LYP method again underestimates this quantity to a value of 23 kcal/ mol. It could be concluded that while the energetic ordering of the electronic states can be well predicted with the B3LYP method, this method could not be used for quantitative examination of transition energies. CI 共CASSCF兲 coefficients point out the dominant electronic configuration of Ge2Li+3 : 1A1 : ¯ 共3a1兲2共1b1兲2共2b2兲2 共4a1兲2共5a1兲2共2b1兲0 and 3B1 : ¯ 共3a1兲2共1b1兲2共2b2兲2共4a1兲2 共5a1兲1共2b1兲1. The geometrical change from singlet to the triplet excited state is similar to that of the neutral counterpart. An elongation of the Ge–Ge bond by an amount of 0.1 Å is clearly due to the occupancy of the unpaired electron in the molecular orbital 2b1 which is, mostly, unoccupied at the singlet state, and the Li–Li–Li bond angle 关␪ in Fig. 3共b兲兴 increases by an amount of 14°. Comparing the neutral and cationic lowest-lying electronic states indicates a contraction in the Ge–Ge bond length by an amount of 0.2 Å upon ionization; at the same time the Ge–Li bond lengths increase by an amount of 0.1 Å. Based on the total energies, we are in a stage to calculate the Li+ cation affinity 共denoted as Li+ . A兲 of Ge−2 and the lithium affinity 共Li.A兲 of Ge2 using the following equations; the obtained values are summarized in Table III. Li+A共Ge−2 兲 = − 关E共Ge2Li兲 − 兵E共Ge−2 兲 + E共Li+兲其兴,

共4兲

Li−A共Ge2兲 = − 关E共Ge2Li兲 − 兵E共Ge2兲 + E共Li兲其兴.

共5兲

The lithium affinity could be considered as the measure of the degree of stabilization attained by the dimer molecule upon lithiation. According to the present definition, a positive lithium affinity corresponds to a stabilization, while a

negative value indicates a destabilization. In this case, both calculated lithium affinity and Li+ cation affinity values are positive, suggesting that the doped dimer attains a certain stability upon lithiation. For Ge2, the calculated lithium affinity value is 2.08 eV, and is, indeed, smaller than the Li+ cation affinity of Ge−2 anion, which amounts to 5.7 eV. The large Li+ cation affinity of Ge−2 thus supports a Ge−2 Li+ interaction in Ge2Li. In summary, calculations on the interaction of Ge2 with lithium atoms leading to the formation of mono-, di-, and trilithiated compounds provide us with the following conclusions: the lithium atom mostly behaves as a bridging entity towards Ge2; and the linear structures has energetically higher-lying electronic states on the potential energy surface. However, it is important to notice the role of the lithium atom in stabilizing the Ge2. Careful examination leads to the following questions: 共i兲 Whether lithium really bridges both germanium atoms? 共ii兲 What is the nature of the Ge–Li bond? 共iii兲 How lithium stabilizes the molecule? In order to find some elements of answer to the above questions, we performed an atoms-in-molecules 共AIM兲 analysis on some selected systems, and this is discussed in the following section. In addition, a detailed investigation using the electron localization function 共ELF兲 and NBO analysis has also been carried out.

C. Nature of the Ge–Li bond

The AIM is a useful tool providing valuable information about the structure and bonding in molecules.37,38 AIM has thus been used to investigate the structure and bonding in traditional organolithium compounds and even supports the ionic nature of the C–Li bond.36 According to the AIM theory, a critical point 共CP兲, where the gradient of the electron density vanishes, holds chemical information and allows us to define atoms and chemical bonds within a molecule. The main questions that we considered here were as follows: 共i兲 As to whether there is a bond connecting the two germanium atoms and lithium? Otherwise stated, whether the lithium atom is really bridging the Ge2 molecule? 共ii兲 Ultimately, what is the nature of the Ge–Li bond? The wave functions used for the AIM analysis were generated at the B3LYP level in conjunction with the 6-311G** basis set using the GAMESS suite of programs. Then, the critical points were located and the bond paths were plotted using the AIM2000 suite of programs. Interestingly, for Ge2Li we were not able to locate a ring critical point, i.e., the part of the molecular graph, which bounds a ring surface. The molecular graph of Ge2Li comprises of two bond critical points and three attractors, i.e., the nuclei 共see Fig. 5 top right corner兲. The ellipticity, a quantity defined as ␧ = 共␭1/␭2 − 1兲;

␭1 艋 ␭2 艋 ␭3 ,

where ␭1, ␭2, and ␭3 are the eigenvalues of the Hessian, measure the behavior of the electron density in the plane tangential to the interatomic surface at the bond critical point. The ellipticity value ranges from zero to infinity and is widely regarded as the quantitative index of the ␲ character

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214312-8

J. Chem. Phys. 124, 214312 共2006兲

Gopakumar, Lievens, and Nguyen

FIG. 5. Energy profile, for the bifurcation mechanism, as a function of the distance between lithium atom and the Ge–Ge geometric center. The top right corner represents the resolved structure of the conflict mechanism. The dotted lines indicate the ellipticity of the Ge–Li bond critical point at different points.

of the bond. The bond critical point connecting both Ge atoms has an ellipticity value of 0.24 suggesting a certain ␲ character to the Ge–Ge bond. This is in agreement with ab initio calculations, where the leading electronic configuration suggests the occupancy of an unpaired electron in the 1b1 MO. The bond critical point between Ge and Li is lying close to the lithium atom and has an ellipticity of 0.87. Figure 5 共top right corner兲 represents a resolved form of the conflict structure. Any motion of Li close to the Ge2 unit will give rise to a bifurcation mechanism.36,37 To illustrate this point, we plotted the energy profile as a function of the distance between Li and the Ge–Ge geometric center, generating the molecular graphs at each point. At a distance of 1 Å between the Li and the Ge–Ge geometric center, there exists a ring critical point; the lithium atom is bridging the two Ge atoms at this geometry. Upon increase of this distance, the ring critical point merges with the Ge–Li bond critical points as expected in a bifurcation mechanism 共see Fig. 5兲. At a distance of 1.5 Å, the ring critical point annihilates upon uniting with the Ge–Li bond critical points 共BCPs兲 and thereby leads to a conflict structure, where two germanium atoms compete for forming a bond with the lithium atom. Such a conflict structure can be resolved to one of the component structures by an infinitesimal distortion in the molecule. However, it is interesting to note the behavior of the ellipticity of Ge–Li BCPs following the bifurcation mechanism. The relevant values are also plotted in Fig. 5 using dotted lines. At 2.3 Å being the geometric energy minimum, the elipticity becomes 0.89, which is the same, even for the resolved form of the conflict structure. Analysis of the Ge–Li bond critical point is expected to provide more information about the nature of the Ge–Li bond. With this goal in mind, we evaluated the charge density 共␳b兲, Laplacian 共Lb兲, and the ellipticity 共␧兲 at the Ge–Li bond critical points in a series of compounds and the results are summarized in Table IV. The electron density Laplacian, measured at a bond critical point and defined as Lb = ⵜ␳2BCP, usually helps us understand the nature of the bond involved. Accordingly, a value Lb ⬍ 0 indicates a closed-shell interaction, i.e., the charge is predominantly contracted towards each of the nuclei, whereas a positive Lb ⬎ 0 value suggests a shared interaction, i.e., the electronic charge is concentrated

in the internuclear region. According to the above definition, ionic bonds, bonds in van der Waals molecules, and noble gas clusters are all closed interactions. In contrast, covalent or polar bonds are shared interactions. In the former, mostly, the electron density at the bond critical point will be low, of the order of 10−2, whereas in the latter, it will be of the order of 10−1. Examination of the electron density values in compounds ranging from Ge–Li to Ge2Li3 suggests that they maintain at constant and lower values. In the case of Ge–Li and GeLi2, the ellipticity values are found to be close to zero, indicating a certain ␴-type bond between the Ge and Li atoms. In the case of Ge2Li, Ge2Li2, and Ge2Li3, the ellipticity values are 0.865, 0.663, and 0.744, respectively, indicating a certain ␲ character to the corresponding Ge–Li bond. It could be noted that in the present system, the evaluated electron density values are too low with small positive Lb values. This leads us to the conclusion that the Ge–Li bond has a very small covalent character. For additional approach, we performed ELF and NBO analyses on these molecules. The ELF is a simple measure of the electron localization in atomic and molecular systems.39 The ELF values are always in a range of 关0;1兴 and relatively large where the electrons are unpaired or formed into pairs with antiparallel spins. The zero flux surfaces of the ELF separate the electron density space into basins 共⍀i兲, thus help us define and calculate the properties of core, chemical bond, TABLE IV. The charge density 共␳b兲, Laplacian 共Lb兲, and ellipticity 共␧兲 calculated at the bond critical point between Ge and Li atoms in different molecules.

Molecule

Charge density at the bond critical point 共 ␳ b兲

Laplacian of ␳ at bond critical point 共Lb兲

Ellipticity 共␧兲 at bond critical point

GeLi GeLi2 GeLi3 Ge2Li Ge2Li2 Ge2Li3

0.021 0.022 0.026 0.023 0.022 0.022

0.014 0.014 0.021 0.011 0.015 0.019

0.0 0.0 0.080 0.865 0.663 0.744

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214312-9

Diatomic germanium with lithium

J. Chem. Phys. 124, 214312 共2006兲

FIG. 6. Cut planes of GeLin 共n = 1 , 2 , 3 , 4兲; the red balls represents the core basins of lithium atoms.

and lone pairs.40 The corresponding basins are mainly classified into two types, i.e., core and valence basins. While the former are mainly located around the nuclei—always occur when the atomic number is larger than 2—the latter are characterized by their synaptic orders, i.e., the number of the core basins that share a common boundary surface with the valance basin. Monosynaptic basins represent the lone pairs and the disynaptic basins belong to the covalent bonds. The integral of the electron density over ⍀i shows the population of the given basin. As a preliminary step, we performed an ELF analysis on the GeLin 共n = 1 , 2 , 3 , 4兲 compounds, and then extended the same methodology to Ge2Li. The calculations were performed using the TOPMOD suites of programs and the ELF isosurfaces were visualized using the gOpenMol software.41 The ELF cut planes of GeLin 共n = 1 , 2 , 3 , 4兲 are illustrated in Fig. 6, and those of Ge2Li are represented in Fig. 7. The mean electronic populations computed for the basins localized for each molecule are included in Table V. Finally, for the sake of comparison we have also performed a NBO analysis; at the UB3LYP/ 6-311+ + G共d , p兲 level, the obtained results are listed in Table VI. In GeLi, the germanium core basins have a population of 27.6 electrons 共this value being the sum of all core basins of germanium兲, and lithium core basins 2.12 electrons. The valence germanium basin V共Ge兲 has a population of 2.12 electron and the V共Ge, Li兲 basin 3.15 electrons. The basin population thus shows consistency in going from GeLi to GeLi2, except for the fact that the lone-pair-type valence germanium basin is absent in the latter 共see Fig. 6 and Table V兲. In the case of GeLi3, there exist six V共Ge, Li兲 basins around the germanium atoms, whereas there are only two in GeLi2 and one in GeLi. The V共Ge, Li兲 population is reduced to 1.22 electrons in GeLi3 from 3.15 of GeLi and GeLi2. The shape of the V共Ge, Li兲 basins in GeLi and GeLi2 shows that they are of similar shape even though it is a little bit distorted in GeLi3. The ELF cut plane of GeLi4 has a different picture as the V共Ge, Li兲 basins are absent in the molecule. There are

instead four V共Ge, Ge, Li兲 basins, each having an electronic population of 2.05 electrons. The occurrence of this trisynaptic basin indicates the origin of three-center-two-electron bonds in GeLi4. The NBO charges tabulated in Table VI indicate that the lithium atom is always donating one electron to the germanium atom, thereby acquiring a positive charge. This fact, indeed, supports the ionic nature of the Ge− – Li+ bond. From GeLi to GeLi4, the negative charge on the germanium atoms shows a regular increase along with the positive charge on the lithium atoms. In GeLi5, it is observed to decrease and in GeLi6 it has the largest negative charge on the germanium atom with the smallest positive charge on each lithium atom. The Wiberg bond indice 共Wi兲 features a lowest value in GeLi6, thus suggesting a weaker interaction between the germanium and lithium atoms. According to the NBO charges, GeLi4 could be expected to be the most stable molecule, since it has the maximal ionic interactions as indicated by NBO charges and Wiberg indices. Having investigated the monogermanium molecules, we are now in a position to examine the nature of the Ge–Li bond in Ge2Li. The ELF isosurfaces and the cut planes for Ge2, its anion, and Ge2Li are illustrated in Fig. 7. In Ge2, the V共Ge, Ge兲 basins show a certain ␲ character to the Ge–Ge bond, this is in accordance with the MO results, where the triplet state with electron occupation in the p␲u orbitals is the lowest state. The V共Ge, Ge兲 electronic population is 4.27 and the C共Ge兲 population is 27.72. The ELF isosurface of Ge−2 anion is quite different from the neutral counterpart 共see Fig. 7兲. Two V共Ge兲 basins with electronic population of 3.05 are observed in the anion. The V共Ge, Ge兲 basin population is reduced to 1.9 electrons, whereas the C共Ge兲 population remains almost unchanged as compared to the neutral molecule. The two V共Ge兲 basins can better be regarded as the lone electron pairs on each of the germanium atoms. The ELF picture of the Ge2Li is quite similar to that of the Ge−2 anion; i.e., the V共Ge兲 basins having electronic population of 2.7 and two V共Ge, Ge兲 basins of electronic population 0.85

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214312-10

J. Chem. Phys. 124, 214312 共2006兲

Gopakumar, Lievens, and Nguyen

FIG. 7. Cut planes and ELF isosurfaces of Ge2, Ge−2 , and Ge2Li 共␩ = 0.7兲.

resemble those of the anion Ge−2 . This can be interpreted as the result of an interaction between Ge−2 and Li+. There are also two trisynaptic V共Ge, Ge, Li兲 basins in Ge2Li, situated above and below the plane, having each an electronic population of 2.65 indicating a three-center bond between Ge and Li atoms. The NBO charges indicate a certain double bond character for the Ge–Ge bond in the Ge−2 anion, and the bond order is the same as in Ge2Li. Both the ELF and NBO results thus tend to support the strong ionic character of the Ge–Li bond. Similar comparisons are also performed with Ge2Li2 and Ge2Li3, where the NBO charge calculations point out an ionic picture of Ge–Li bond in both molecules. The Ge–Ge bond order of 3.09 in the dianion is preserved in Ge2Li2 共2.66兲, whereas in Ge2Li3, it shows a

considerable lowering as compared to that of the trianion. It can be expected that this will give a small preference to the former 共Ge2Li2兲 during interaction of the germanium dimer with lithium atoms. IV. CONCLUSIONS

In the present theoretical study, we analyzed the electronic structure of mono-, di-, and trilithiated Ge2 and their cations. Based on quantum chemical results, the following conclusions could be drawn: 共i兲

Ge2Li possesses a doublet 2B1 ground state, with a doublet quartet energy gap of 33 kcal/ mol 共at the MCQDPT/ECP level兲.

TABLE V. The mean electronic populations computed for basins localized in GeLin 共n = 1 , 2 , 3 , 4兲 and Ge2Li. Basins Molecule

C共Ge兲a

C共Li兲

V共Ge兲

V共Ge, Ge兲

V共Ge, Li兲

V共Ge, Ge, Li兲

GeLi GeLi2 GeLi3 GeLi4 Ge2 Ge−2 Ge2Li

27.61 27.57 27.51 27.61 27.72 27.55 27.62

2.03 2.02 2.02 2.03 ¯ ¯ 2.02

2.12 ¯ ¯ ¯ ¯ 3.05 2.70

¯ ¯ ¯ ¯ 4.27 1.90 0.85

3.15 3.17 1.22 ¯ ¯ ¯ ¯

¯ ¯ ¯ 2.05 ¯ ¯ 2.65

a

Sum of all core basins of Ge.

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214312-11

J. Chem. Phys. 124, 214312 共2006兲

Diatomic germanium with lithium

TABLE VI. Calculated Wiberg indices 共Wi兲 and NBO charges of various germanium lithium complexes at B3LYP/ 6-311+ + G共d , p兲 level. Wiberg indices in a.u. 共Wi兲

NBO charges 共a.u.兲

Molecule

Ge–Ge

Ge–Li

Ge

Li

GeLi GeLi2 GeLi3a GeLi4 GeLi5 GeLi6 Ge2 Ge−2 Ge2Li Ge2− 2 Ge2Li2 Ge3− 2 Ge2Li3

¯ ¯ ¯ ¯ ¯ ¯ 1.50 2.28 2.05 3.09 2.66 3.03 2.0

0.24 0.37 0.34 0.29 0.25, 0.5 0.15 ¯ ¯ 0.25 ¯ 0.22 ¯ 0.14, 0.23

−0.76 −1.54 −2.41 −3.37 −3.06 −3.65 0.0 −0.5 −0.35 −1.0 −0.74 −1.5 −1.2

0.76 0.77 0.80 0.84 0.66, 0.41 0.61 ¯ ¯ 0.71 ¯ 0.74 ¯ 0.84, 0.72

a

At B3LYP level this structure has two imaginary frequencies corresponding to the elongation of one of the Ge–Li bonds having magnitudes of 28.3 and 19.8 cm−1, respectively.

共ii兲

Ge2Li+ cation has a high spin 3B1 ground state. The triplet-singlet gap is estimated to be around 13 kcal/ mol. 共iii兲 The results of density functional theory 共B3LYP兲 are in qualitative agreement with the MCQDPT values. Usually the 共U兲B3LYP functional gives smaller or larger energy differences, even though the energy ordering of the electronic states could be reproduced. 共iv兲 The di- and trilithiated Ge2Lin, n = 2 – 3, molecules and their cations are having low-spin ground electronic states. The cations can better be modeled as Ge2 and n共Li+兲 cations. 共v兲 The calculated electron affinities of diatomic germanium amount to EA共1兲 = 1.9 eV, EA共2兲 = −2.5 eV, and EA共3兲 = −6.0 eV, and only the Ge−2 anion is likely to be stable with respect to the electron detachment. 共vi兲 The larger Li+ cation affinity value of Ge−2 共compared to the lithium affinity value of Ge2兲 suggests a Ge−2 Li+ interaction and thus supports the ionic nature of Ge–Li bond. 共vii兲 Investigation using the AIM approach reveals the absence of a ring critical point in the Ge2Li molecule. 共viii兲 The ELF and NBO analyses lead to a conclusion that the Ge–Li bond is predominantly ionic. In the case of GeLi4 and Ge2Li, the presence of the trisynaptic basins points out a three-center bond connecting the germanium and lithium atoms. We would anticipate that the design of alkali metal doped germanium clusters is an emerging subject for experimental research, and hope that the present computational results provide some insights into the electronic structure of larger lithium doped germanium clusters.

ACKNOWLEDGMENTS

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