Density functional studies of LiN and LiN+ (N = 2–30) clusters: Structure, binding and charge distribution

Density Functional Studies of LiN andLiN

þ (N ¼ 2–30) Clusters: Structure,Binding and Charge Distribution

NEETU GOEL,1 SEEMA GAUTAM,2 KEYA DHARAMVIR2

1Department of Chemistry, Center of Advanced Studies in Chemistry, Panjab University,Chandigarh 160014, India2Department of Physics, Center of Advanced Studies in Physics, Panjab University,Chandigarh 160014, India

Received 15 September 2010; accepted 23 November 2010Published online 8 March 2011 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.23022

ABSTRACT: Density functional calculations using B3LYP/6-311G method have beencarried out for small to medium-sized lithium clusters (LiN,N¼ 2–30). The optimizedgeometries of neutral and singly charged clusters, their binding energies, ionization potential,electron affinity, chemical potential, softness, hardness, highest occupied molecular orbitaland lowest unoccupied molecular orbital (HOMO–LUMO) gap, and static dipolepolarizability have been investigated systematically. In addition, we study the distribution ofpartial charges in detail using natural population analysis (NPA) in small-sized clusters (LiN,N¼ 2–10), both neutral and cationic, and demonstrate the correlation between symmetry andcharge. Uniform distribution of charges in cationic clusters confirms them to be energeticallymore favorable than the neutral counterparts. Whenever possible, results have beencompared with available data. An excellent agreement in every case supports new results asreliable predictions. A careful study of optimized geometries shows that Li9 is derivable frombulk Li structure, i.e., body centered cubic cell, and higher clusters have optimized shapesderived from this. Further, the turnover form two to three dimensional structure occurs atcluster sizeN ¼ 6. The quantity a1/3 (a ¼ polarizability) per atom is found to be broadlyproportional to softness (per atom) as well as inverse ionization potential (per atom). Thepresent work forms a sound basis for further study of large-sized clusters as well as otheratomic clusters. VC 2011 Wiley Periodicals, Inc. Int J Quantum Chem 112: 575–586, 2012

Key words: DFT; clusters; charge distribution; binding energy; stability; symmetry

Correspondence to: N. GOEL; e-mail: [email protected]

Contract grant sponsor: Department of Science andTechnology (DST), India.

Contract grant number: 100/IFD/6358/2007-08.Contract grant sponsor: Council of Scientific and Industrial

Research (CSIR), New Delhi.

International Journal of Quantum Chemistry, Vol 112, 575–586 (2012)VC 2011 Wiley Periodicals, Inc.

1. Introduction

C lusters made up of the lightest metal lithiumpresent a good starting point for theoretical

understanding of metal clusters as Li has onlyone delocalized electron outside of a noble gasshell. Along with its fellow alkali metals, it isexpected to closely mimic the free electron gas.This is particularly relevant while performing cal-culations within the framework of density func-tional theory (DFT) as the original approximationsto DFT were based upon analytic expressionsderived from exchange-only treatment of the freeelectron gas [1]. Though the structureless jelliummodel [2–5] accounts nicely for the relative stabil-ity of s-valence metal clusters, experiments [6, 7]and calculations [8] suggest that Li clustersexhibit a complex interplay between geometry,electronic structure, and size. This cautionsagainst the use of large approximations in the the-oretical method employed to study Li clusters.Thus, first principle atomistic calculations basedon lowest energy configuration are essential togain a deeper understanding of the size evolutionof geometric structure and electronic/opticalproperties of Li clusters. The investigation ofmetal clusters provides an interesting example,where experiment and theory have currentlyreached almost similar levels of sophistication tocompliment each other [2, 8, 9]. In the last twodecades, numerous experimental studies havebeen performed on Li clusters via spectroscopicmethods, i.e., electron spin resonance (ESR) [10],two-photon ionization [11, 12], infrared andRaman spectra [2, 13], and thermochemical (massspectrometric) measurements [14]. However, it isdifficult to generate size specific clusters and todetermine their structures experimentally. On thetheoretical side, there is an immense array ofstudies on Li clusters involving both DFT as wellas traditional quantum chemistry approaches. Inthe set of DFT studies, Zhao and coworkers [15]have studied medium-sized LiN clusters (N ¼ 20,30, 40, 50); Fournier et al. [16] have employedKohn-Sham theory with local spin density (LSD)and gradient corrected functional (for N ¼ 5–20)and used a Tabu search algorithm for structureoptimization; Gardet et al. [17] have proposed astructural growth pattern based on the pentagonalbipyramid and have studied the electronic prop-erties of clusters with up to 20 atoms. Jose andGadre [18] recently reported a molecular electro-

static potential (MESP) guided method for build-ing LiN clusters (N ¼ 4–58) and optimizing smallclusters by a DFT based method and larger onesvia molecular tailoring approach. Ghosh and co-workers [19, 20] have made DFT based studies ofelectronic properties of Li clusters with focus onionization potential and polarizability. Jones et al.[21] have reported DFT study of structure andbonding in Li clusters (up to 10 atoms) and theiroxides. Wu and Jones [22] have presented jointexperimental/DFT study of neutral and hydro-genated clusters. There have also been a numberof other studies such as those using Hartree-Focktheory [6, 23] with or without electron correlationcontributions, Moller Plesset (MP) level of theory,and Configuration interaction (CI) methods [24,25]. Except for a few [15, 18, 23(a)], the previoustheoretical studies are limited to cluster size N �20. It is thus desirable to investigate medium-sized Li clusters as they represent the bridgebetween molecular regime and the macroscopicscale. Note that most of the cited articles [except25, 26] are focused on the study of geometry ofneutral clusters and their properties. The objectiveof the present work is to carry out a systematicanalysis of small- to medium-sized (N ¼ 2–30)neutral as well as singly charged Li clusterswithin the DFT formalism. We endeavor to lookat many different aspects of neutral and cationicLi clusters including their stability, distribution ofcharge, electronic and optical properties and ana-lyze their size dependence.

2. Computational Details

All calculations were performed using theGaussian 03 program package [27]. Equilibriumgeometric structures of the neutral and the corre-sponding cationic clusters were fully optimizedby using the B3LYP method; a gradient-correctedfunctional that involves both the value of electronspin densities and their gradients. It is a DFTmethod using three parameter Becke’s gradientcorrected exchange functional (B3) [28] with thegradient corrected correlation functional of Lee,Yang, and Parr [29]. This is a popular gradient-cor-rected exchange functional proposed by Becke in1988; a widely used gradient-corrected correlationfunctional is LYP functional of Lee, Yang, andParr (LYP). This exchange-correlation functional isreported to have been employed for DFT-based

GOEL, GAUTAM, AND DHARAMVIR

576 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 112, NO. 2

theoretical investigation of alkali metal clusters [30]and Li clusters in particular [15, 18–20]. Optimiza-tions were performed with the 6-311G basis set, asplit valence basis set with polarization functions.The present theoretical approach is able to com-pute the structural, electronic, and optical proper-ties of small lithium cluster in a rather satisfactorymanner. While performing geometry optimization,the stable structures of LiN clusters up to N ¼ 8,considered in the present work are the same as theones reported in literature [18, 25]. For larger clus-ters (8 � N � 30), the geometries were obtained bycapping the most stable isomer of LiN�1 with aLi-atom at the appropriate position. The presenceof four distinct convergence criteria namely forces,root mean square of the forces, displacement,and the root mean square of the displacementprevent a premature identification of the mini-mum [31]. For the cluster sizes N � 18, thedefault self consistent field convergence of 10�8

is used. For the larger clusters, 10�6 convergencelimit is employed. The ground state of each ofthe lithium clusters has the lowest possible mul-tiplicity that can be obtained for a given numberof unpaired electrons. The electronic and opticalproperties presented here include highest occu-pied molecular orbital and lowest unoccupiedmolecular orbital (HOMO–LUMO) gap, adiabaticand vertical ionization potentials (IP), electron af-finity, chemical hardness, softness, chemicalpotential, static dipole polarizability, and chargedistribution in terms of natural population analy-sis (NPA) [32, 33]. To the best of our knowledge,the study of charge distribution in Li cluster hasnot been reported earlier. It is of interest toreveal how the electronic charge is distributed inthe neutral cluster and compare it with the cati-onic cluster and hence correlate symmetry andcharge distribution. After geometry optimiza-tions, partial charges were obtained from NPA.The atomic charges calculated by NPA are reli-able and usually independent of basis sets. Fre-quency analysis was also performed at the finaltheoretical level to confirm that the optimizedstructures are the genuine minima on the poten-tial energy surfaces of corresponding clusters.

The binding energy (BE) per atom of optimizedneutral clusters is computed as

Eb=N ¼ ðNE1 � EðLiÞNÞ=N (1)

where E1 is the energy of single lithium atom,and E (LiN) is the total energy of LiN cluster.

For cationic clusters, the formula used for BEper atom is

Eb=N ¼ fðN � 1ÞE1 þ EðLiþÞ � EðLiþNÞg=N (2)

where E (Liþ) is the energy of single Liþ cation,and E (LiN

þ) is the total energy of LiNþ.

2.1. GEOMETRIES AND SIZE DEPENDENCEOF ENERGY

The lowest-energy structure of neutral as wellas cationic lithium clusters LiN (N � 30) atB3LYP/6-311G level are displayed in Figures 1and 2. The optimized energies of LiN and LiN

þ

are listed in Table I, and in all cases, we findexcellent agreement with the values quoted in lit-erature [18]. The optimized bond length of Li2 is2.71 A, close to the experimental bond length [34]of 2.67 A. Li2

þ shows an increase in bond lengthto 3.15 A. This is in accordance with the molecu-lar orbital binding model according to which theremoval of a binding electron reduces the electrondensity between nuclei, causing an increase inbond length. For Li3, the isosceles triangularstructure is the lowest energy isomer withC2v symmetry having bond lengths 2.76 A and3.32 A [see Fig. 1(a)], the linear structure being0.186 eV higher in energy. This may be due to sp-hybridization. Li3

þ has equilateral triangularstructure with somewhat increased bond length,2.97 A. This is because of extra charges (see TableII) which are expected to introduce extra Cou-lomb repulsion. The current work finds that thelowest energy isomer of Li4 belongs to the D2h

point group, with Li–Li distances of 2.59 A and3.02 A [see Fig. 1(a)]. Boustoni et al. [25] havereported three local minima for Li4 with D2h, C2v,D4h symmetry obtained by CI method with themost stable one belonging to D2h point groupwith Li–Li distances of 2.69 A and 3.16 A. Thepresent stable structure of Li4 is also in agreementwith the one reported by Gardet et al. [17]. TheLi4

þ cation also belongs to D2h point group withbond lengths 2.75 A and 3.12 A (see Fig. 2). Theshort diagonal having been extended because ofCoulomb repulsion (see Table II) between the di-agonal atoms (1, 4). The most stable structure ofLi5 cluster is ‘‘W-shaped planar’’ with C2v symme-try, with bond lengths 2.99 A, 2.89 A, and 3.09 A[see Fig. 1(a)]. Boustoni et al. [25] also report thesame structure as the most stable isomer, withbond lengths 3.09 A, 3.08 A, and 3.08 A. Li5

þ

DENSITY FUNCTIONAL STUDIES OF LiN AND LiNþ

VOL. 112, NO. 2 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 577

cation has a different shape, a more symmetricone—a centered rectangle or two nearly equilateraltriangles placed tip to tip with bond lengths 2.83 Aand 3.10 A (see Fig. 2). This also agrees well withthe reported structure [25]. Gardet et al. [17] andBoustoni et al. [25] have reported three local min-ima for Li6 belonging to C5v, C2v, and C3v pointgroup at DFT and CI level of theory, respectively,and suggest that the most stable isomer of Li6belongs to C5v symmetry. In the present work too,the lowest energy isomer is found to belong to C5v

point group. In contradiction, Temelso and Sherrill[26] suggest that D4h is the global minimum among

Li6 isomers, being energetically favored over thelocal minimum C5v structure. The Li6 clusterpresents the turn over from 2D geometry to 3D ge-ometry [shown in Fig. 1(a)] in agreement with liter-ature [18]. The Li6

þ cation has a shapeless symmet-ric than the neutral cluster—obtained by shiftingone atom of the pentagon of Li6 cluster to the op-posite ‘‘capping’’ position. The lowest energy struc-ture of Li7 is, in consistence with the literature [17,25], a pentagonal bipyramid belonging to the D5h

point group. Li7þ also has similar structure. When

an atom is introduced at the central position of thisbipyramid, the lowest energy isomer of Li8 is

FIGURE 1. Optimized geometries of neutral Li cluster. The label below each cluster indicates its point symmetrygroup. (a) Li2 to Li15 and (b) Li20. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]



obtained. In the current work, the distance of cen-tral atom from both the tip atoms is 2.44 A in con-trast to the reported distances [18] of 3.00 A and2.89 A. The growth pattern of Li8 is Li6 [C5v] ! Li7[D5h] ! Li8 [D5h]. Similar structures have beenreported by Jones et al. [21] using DFT (MD) simu-lation. In Li8

þ, the pentagon gets extended withbond lengths 3.13 A (3.02 A in Li8), while the cap-

ping atoms (7, 8) get squeezed toward the planarpentagon.

Interestingly, the Li9 cluster (C4v symmetry)has a shape which can be derived from the body-centered-cubic (BCC) unit cell by rotating one ofthe cube faces in its own plane by 45�. This cell[see Fig. 1(a)] has two square faces with bondlengths 3.07 A (for the face consisting of atoms 2,

FIGURE 2. Optimized geometries of cationic Li cluster. The label below each cluster indicates its point symmetrygroup. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



3, 8, 9) and 3.45 A (for the square face consistingof atoms [4–7]). Neighboring atoms in the twoplanes are separated by 3.11 A. In Li9

þ, both thesquares are of equal bond length, i.e., 3.16 A (seeFig. 2). The Li10 cluster can be derived from Li9 byplacing an apex atom on the bigger square of clus-ter at a distance of 3.25 A from the plane, while

the atom at the body-centre moves slightly towardthe apex atom. Clusters larger than Li10 can beformed by adding one atom to the most stable iso-mer at a suitable ‘‘capping’’ position. For stablecationic clusters, optimizations were performedby giving charge þ1 to respective neutral geome-tries followed by optimization. Likewise, Li11 and

TABLE IThe energies of optimized neutral (LiN) and cationic (LiN

1 ) clusters at B3LYP/6–311G level of theory.

Cluster size (N) LiN (a.u.) LiNþ (a.u.)

1 �7.4909 �7.28492 �15.0139 (�15.0147) �14.8207 (�14.8218)3 �22.5227 �22.36964 �30.0529 (�30.0547) �29.8869 (�29.8884)5 �37.5724 �37.41946 �45.1031 (�45.1101) �44.9523 (�44.9539)7 �52.6414 �52.49748 �60.1697 (�60.1757) �60.0086 (�60.0109)9 �67.6939 �67.5650

10 �75.2076 (�75.2095) �75.0826 (�75.0732)11 �82.7495 �82.615412 �90.2850 (�90.2689) �90.1435 (�90.1499)13 �97.8152 �97.675114 �105.3513 �105.220515 �112.8891 �112.757016 �120.4259 �120.288317 �127.9514 �127.813618 �135.4237 (�135.4863) �135.3408 (�135.3467)19 �143.0326 �142.873520 �150.5671 (�150.5506) �150.4236 (�150.4163)22 �165.6162 (�165.6202) �165.4799 (�165.4891)24 �180.6679 (�180.6767) �180.5389 (�180.5489)26 �195.7307 �195.602228 �210.7745 �210.655330 �225.8235 �225.7023

For comparison, values from Ref. [18] are given in parentheses.

TABLE IICharge distribution in neutral (LiN) and cationic (LiN

1 ) clusters in terms of NPA.

Size (N) Charges in LiN Charges in LiNþ

2 (1,2) 0.0 (1,2) 0.53 (1) �0.145, (2) �0.005, (3) 0.15 (1,2,3) 0.3334 (2,3) �0.25, (1,4) 0.25 (1,4) 0.098, (2,3) 0.405 (1,3) 0.011, (2) 0.35, (4,5) �0.187 (1,2,4,5) 0.354, (3) �0.4186 (1–5) 0.130, (6) �0.656 (1,2,3,5,) 0.289, (4,6) �0.3967 (2–6) 0.178, (1,7) �0.195 (2–6) 0.499, (1,7) �0.7508 (2–6) 0.515, (7,8) 0.591, (1) �3.759 (2–6) 0.302, (7,8) 0.377, (1) �1.2629 (4–7) 0.303, (2,3,8,9) 0.270, (1) �1.791 (2–9) 0.249, (1) �0.993

10 (4–7) 0.402, (2,3,8,9) 0.305, (10) 0.539, (1) �3.364 (2–9) 0.320, (1) �1.925, (10) 0.298

The numbers in parentheses indicate respective position of atoms in the cluster (as shown in Figs. 1 and 2).



Li12 also have structures derivable from the Li9 ge-ometry. However, clusters larger than this do notshow perceivable symmetry or systematic changein geometry. Notably, the optimized structure forthe Li13 cluster is not the icosahedron, which is thecase with most of the metallic clusters.

A close look at the clusters LiN with N ¼ 13,14, and 15 show a clear propensity for 8-coordina-tion rather than 12. Thus, we may conclude thatthose metals (notably the alkali metals), whichhave BCC structure in the bulk phase, preferthose structures in the clusters which have a max-imum number of 8-coordinated atoms.

2.2. CHARGE DISTRIBUTION

Table II presents a comparison between chargedistributions in neutral and cationic clusters interms of NPA. Li2

þ has equal positive charge(0.5C) on both atoms (1, 2). Hence, an increase inbond length from 2.67 A in Li2 to 3.15 A in Li2

þ

is observed on account of Coulomb repulsion. It isclear from Table II that Li3 has non-uniform chargedistribution; however in Li3

þ, all three atoms (1, 2,3) have equal charge (0.33C). In tetramer Li4, pairdiagonal atoms (1,4) and (2,3) acquire equal but op-posite charges (0.25C); however upon ionization,redistribution of electron density puts positivecharges on all four atoms resulting in an increase inbond length from 3.02 A in Li4 to 3.12 A in Li4

þ.As shown in Table II, charge distribution is not uni-form in Li5; whereas in Li5

þ, all the corner atoms(1, 2, 4, 5) have equal positive charge (0.354C),while the central atom (3) has a negative charge(�0.418C). Li6 has equal charge (0.13C) on five ofthe atoms (1–5) constituting the planar pentagon[see Fig. 1(a)], while the capping atom (6) has anegative charge (�0.656C). In Li6

þ, the two cappingatoms (4, 6) have equal negative charges (�0.396C),while the rest of the positive charge is sharedequally (0.289C) by all the remaining atoms (1, 2, 3,5). In Li7, all the atoms (2–6) on the planar penta-gon [see Fig. 1(a)] have equal positive charge(0.178C), and capping atoms (1, 7) have equal nega-tive charge (�0.195C). In Li7

þ, redistribution of elec-tron density gives larger positive charges (0.499C)on the atoms (2–6) of the planar pentagon andlarger negative charges (�0.75C) on the cappingatoms (1, 7). As a result, the caps are squeezed to-ward the pentagon which has expanded slightly byincreasing the bond length up to 3.21 A (see Fig. 2).In Li8, all atoms (2–8) have average positive charge(0.5C) except the central atom (1) which has a nega-

tive charge (�3.759C). The situation remains sameupon ionization; in Li8

þ, these charges become0.3C and �1.262C, respectively. The Li9 clusterbeing a twisted BCC cell with a central atom (1)[see Fig. 1(a)] has equal charges over each of thesquare faces but different for the two faces; oneface consisting of atoms (4–7) have equal positivecharge (0.303C), while atoms belonging to the othersquare face (2, 3, 8, 9) have charge 0.27C. The cen-tral atom (1) acquires a negative charge (�1.791C).However, ionization redistributes the electron den-sity equally (0.249C) among all the corners atoms(2–9) of the twisted BCC cell and again the centralatom (1) acquires a negative charge but withdecreased magnitude (�0.993C). In Li10 which hasa cap atom (10) over the twisted BCC unit cell ofLi9 [see Fig. 1(a)], again the sense of charge distri-bution is same as in the case of Li9 except that theapex atom (10) has (0.539C) charge. On ionization,charges are uniformly distributed over the corneratoms (2–9) of twisted BCC unit with the apexatom (10) acquiring a charge 0.298C. Further downthe table, similar trends are seen to persist.

The neutral cluster has unevenly distributedcharge—the central atom has a high negativecharge, while positive charges are distributed overthe surface. In cationic clusters, the distribution ofcharge remains the same, but the surface is nowmore evenly charged. Till the 10-atom cluster, thecentral atom acquires a lower (less negative)charge when ionized, particularly at size N ¼ 9, 10(see Table II). For N > 10, the charge acquired bythe central atom when ionized is not systematic.On the basis of these observations, we can corre-late symmetry and charge distribution for neutraland cationic clusters in the following way:

1. Charge distribution in cationic clusters ismore uniform than in neutral ones. It sup-ports the observation that cationic clustersare energetically more favorable than neutralones and hence should have higher BE.

2. Positive charges reside on the surface of thecluster and negative charges are trappedinside. Thus, when ionization occurs, electronsare removed first from peripheral atoms.

2.3. BINDING ENERGY OF NEUTRALAND CATIONIC CLUSTERS

The BEs of neutral and cationic Li clusters aredepicted in Figure 3. One can notice a pronounced



general trend of increasing cluster stability withincreasing size for both neutral and cationic clus-ters. However, increase in stability is not smoothand exhibits even–odd oscillations. Cahuzac andcoworkers [35] have studied unimolecular dissocia-tion path way and BEs of lithium clusters fromevaporation experiments. For neutral atomic clus-ters, the agreement with experimental counterpart[35] is fairly good as shown in Figure 3(a). Magicnumbers have been observed in mass spectra ofclusters; metal clusters of size N ¼ 2, 8, 20, and 40are found to be more abundant [36]. Knight et al.[37] have explained the occurrence of these magicnumbers in terms of an electronic shell model. The

extra stability at the magic number is not visible inthe BE plot, but sharp peaks corresponding tomagic numbers are clearly shown in the plot ofHOMO–LUMO gap (Fig. 4). The pronouncedmaxima at N ¼ 7, 9, 11, 17 [see Fig. 3(b)] in caseof cationic clusters show their enhanced stabilitywhen the cluster has odd number of atoms. Thechange of spin multiplicity of lower energy stateis reflected in the oscillatory behavior of BE (seeFig. 3). The inverse stability of cationic clusteralso derives from the fact that it is difficult toremove an electron from a doubly occupiedHOMO of the closed shell system. The cationicclusters have higher BE than the correspondingneutral cluster. This is in agreement with the con-clusion obtained from NPA analysis.

2.4. THE HOMO–LUMO GAP

The HOMO–LUMO gap is considered to be animportant parameter to study the electronic stabil-ity of small clusters. When HOMO–LUMO gap isplotted against cluster size (Fig. 4), we find sharppeaks for N ¼ 2, 6, 8, 16, and 20. These clustershave large energy gaps and are chemically morestable relative to the neighboring odd-numberedclusters. This can be explained through the electronpairing effect. The odd (even)-sized clusters havean odd (even) total number of electrons and theHOMO is singly (doubly) occupied. The electronsin doubly occupied HOMO have stronger effectivecore potentials as screening is weaker for electronsin the same orbital than for inner shell electrons.This explains the sharp peaks corresponding to

FIGURE 3. Binding energy per atom versus clustersize for: (a) neutral cluster and (b) cationic cluster.

FIGURE 4. Gap energy (HOMO–LUMO) versus clustersize for neutral Li clusters.



even-sized clusters in Figure 4. This led us tostudy only even-sized clusters for N > 20 in thecurrent work. Thus even (odd) clusters with adoubly (singly) occupied HOMO are expected tobe more (less) chemically stable, consistent withmagic number clusters, of size N ¼ 2, 8, 20, and40 that have been observed in mass spectra [36].

2.5. IONIZATION POTENTIALS, CHEMICALPOTENTIALS, HARDNESS, SOFTNESS

Table III presents the values of adiabatic ioni-zation potential (AIP), vertical ionization potential(VIP), vertical electron affinity (VEA), chemicalpotential (l), hardness (g), and softness (S)obtained at B3LYP/6-311G level of theory. AIP isobtained as the difference between the energies ofthe appropriate neutral form and the cationic oneat their optimized geometries. VIP is obtained asthe difference between the energies of the appro-

priate neutral form and cationic one at optimizedneutral geometry. The comparison of AIP withits experimental counterpart [38] as a functionof cluster size is presented in Figure 5. A niceagreement of calculated AIP with the AIP deter-mined through photoionization experiments [38]is seen. Like BE, AIP values also exhibit even–oddoscillations with maxima at N ¼ 4, 8, 12, 20.Even-numbered clusters have higher AIP thantheir neighboring odd-sized systems. As dis-cussed earlier, electron removal is difficult fromthe doubly occupied HOMO of a closed shell sys-tem than singly occupied HOMO of an open-shell. Chemical potential signifies the direction ofelectron transfer from one system to another [39].Chemical hardness has been used as an electronicproperty to characterize the relative stability ofmolecules. Based on a finite differences approxi-mation and Koopman’s theorem, these quantitiesare defined as:

TABLE IIIVIP, AIP, VEA, l, g, and S of LiN calculated at B3LYP/6–311G level of theory.

Size (N) AIP (eV) VIP (eV) VEA (eV) l (eV) g (eV) S (eV)

1 5.60 (5.36)a 5.60 0.136 �2.86 (�3.02)b 2.73 (2.53)b 0.18 (0.19)b

2 5.25 (5.14)a 5.32 0.046 �2.68 (�2.77)b 2.63 (2.47)b 0.18 (0.20)b

3 4.17 (4.08)a 4.28 0.471 �2.37 (�2.28)b 1.90 (1.90)b 0.26 (0.26)b

4 4.52 (4.31)a 4.53 0.422 �2.47 (�2.47)b 2.05 (2.01)b 0.24 (0.24)b

5 4.16 (4.02)a 4.36 0.590 �2.47 (�2.50)b 1.88 (1.76)b 0.26 (0.28)b

6 4.10 (4.20)a 4.63 0.340 �2.48 (�2.51)b 2.14 (2.12)b 0.23 (0.23)b

7 3.92 (3.94)a 4.05 0.612 �2.33 (�2.31)b 1.71 (1.68)b 0.29 (0.29)b

8 4.38 (4.16)a 4.44 0.193 �2.31 (�2.61)b 2.12 (1.98)b 0.23 (0.25)b

9 3.51 (3.29)a 3.63 0.324 �1.97 (�1.98)b 1.65 (1.55)b 0.30 (0.32)b

10 3.40 (3.95)a 3.47 0.430 �1.95 (�2.31)b 1.52 (1.79)b 0.32 (0.27)b

11 3.65 (3.47)a 3.76 0.585 �2.17 1.58 0.3112 3.76 (3.60)a 3.99 0.598 �2.29 1.69 0.2913 3.74 (3.29)a 3.92 0.814 �2.36 1.55 0.3214 3.56 (3.54)a 3.56 �2.2515 3.59 (3.30)a 3.59 0.985 �2.28 1.30 0.3816 3.74 (3.44)a 4.07 0.808 �2.43 1.63 0.3017 3.75 (3.22)a 4.23 0.614 �2.42 1.81 0.2818 2.25(3.31)a 3.70 1.090 �2.39 1.30 0.3819 4.33 3.78 1.040 �2.41 1.37 0.3620 3.91(3.31)a 4.50 0.636 �2.56 1.93 0.2522 3.71(3.39)a 3.89 0.985 �2.43 1.45 0.3424 3.51(3.24)a 3.73 1.148 �2.44 1.29 0.3926 3.49 4.27 1.079 �2.67 1.59 0.3128 3.24 4.11 1.268 �2.68 1.42 0.3530 3.39 3.94 0.642 �2.29 1.65 0.30

VIP, vertical ionization potential; AIP, adiabatic ionization potential; VEA, vertical electron affinity; l, chemical potential; g, hard-ness; S, softness; LiN, neutral lithium clusters.a For the sake of comparison, experimental values from Ref. [37] of AIP are listed in parentheses.b For the sake of comparison, literature values from Ref. [20] of l, g, S are listed in parentheses.



m ¼ ðVIP� EAÞ=2 (3)

l ¼ �ðVIPþ EAÞ=2 (4)

S ¼ 1=ðVIP� EAÞ (5)

Hardness is interpreted as the resistance towardchange in number of electrons. According to theprinciple of maximum hardness [40], the hardnessof a system becomes maximum at equilibrium geo-metries [20], and the stability is directly related tothe higher values of hardness [41]. The chemicalpotentials and hardness values of the clusters alsoexhibit the even–odd oscillations, very similar tothe size evolution of IP and EA values. It can beseen in Table III that even numbered clusters, i.e.,N ¼ 2, 4, 6, 8, 12, 16, 20 have higher values of gcompared to odd numbered ones. It further con-firms the stability of even numbered clusters.

2.6. STATIC DIPOLE POLARIZABILITY (a)

This quantity provides a measure of the extentof distortion of electron density under the effectof an external electric field. The mean polarizabil-ity is calculated from diagonal elements of thepolarizability tensor as:

a ¼ ðaxx þ ayy þ azzÞ=3 (6)

The comparison among calculated and experi-mental values is displayed in Table IV. The calcu-lated values at B3LYP/6-311G level of theory arein fairly good agreement with experimental val-

ues [42]. However, this method fails to depict theexperimentally observed minimum at cluster sizeN ¼ 6 [42], where it shows a continuous mono-tonic increase. On the lines of Chandrakumar et al.[20], our results show good correlation betweenthe cube root of polarizability and inverse of ioni-zation potentials and a linear relationship betweenaverage softness of lithium clusters and a1/3/N.These results are shown in the linear plots pre-sented in Figure 6. Only the Li20 cluster has allthree diagonal elements of the polarizability tensorequal (axx ¼ 1159.038, ayy ¼ 1171.054, azz ¼1187.068). The corresponding structure has beenpresented in Figure 1(b) and is evidently a remark-ably symmetric structure.

3. Summary and Conclusions

We have systematically investigated the size de-pendent electronic properties of small- to medium-sized (N ¼ 2–30) neutral as well as singly charged

FIGURE 5. Adiabatic ionization potential versus clus-ter size for neutral Li clusters. Broken Line correspondsto experimental results [37].

TABLE IVStatic dipole polarizability of neutral lithium clusters(LiN) calculated using B3LYP/6–311G method andthe experimental results.

Size (N)Calculated

valueExperimental values

from Ref. 41

2 184.90 221.333 296.56 232.804 336.84 326.605 442.92 428.496 476.25 360.347 502.23 538.488 506.42 561.429 579.89 601.23

10 666.57 701.7811 700.23 801.6512 743.89 955.4913 833.57 929.8514 936.09 925.8015 938.20 991.1916 922.48 939.3017 1058.08 1170.0818 1179.45 1166.0319 1125.21 1435.9420 1172.39 1201.1222 1362.98 1336.0824 1505.9826 1604.3728 1749.3430 1220.43



Li clusters within the DFT frame work usingB3LYP functional with 6-311G basis set. The turnover from 2D to 3D for Li clusters occurs at thesize N ¼ 6. It is pertinent to mention here that itsfellow alkali members show this turnover at suc-cessively larger clusters [30]. For N � 9, the moststable structure is derivable from the BCC unit celland can be termed a ‘‘twisted BCC cell.’’ All thesestructures are optimized in such a way that the sur-face of each cluster consists of triangles. The trendof charge distribution over the clusters is very inter-esting. In each cluster, neutral as well as cationic,the surface atoms have positive charges, whilethose in the interior have negative charges. The sur-face of a cationic cluster however has a more even

distribution of charges as compared to the corre-sponding neutral one. These distributions are rele-vant for the study of reactivity as well as doping.

The energies of neutral clusters match verywell with experimental values for N � 10. Theenergy of a cationic cluster is lower than its neu-tral counterpart throughout the series, showingthat the cationic system is relatively stable. Simi-lar to the BE, the HOMO–LUMO gap also showseven–odd oscillations. For N ¼ 2, 6, 8, 16, and 20,the clusters have larger HOMO–LUMO gap andhave relatively high chemical stability comparedto neighboring odd-numbered clusters. Similarobservations hold for hardness (g). We have alsocalculated softness (S) and polarizability (a). Inter-estingly, a1/3 is proportional to softness S as wellas inverse of ionization potential. Strictly speak-ing, these relations hold good in atomic systems,the concerned quantities being related to size.Our empirical observations show that argumentssimilar to the atomic system would work for thesesmall- and intermediate-sized clusters.

The current work establishes that the 8-coordina-tion is visible in small-sized Li cluster, i.e., Li9. Thisis an indication that the Li clusters are likely toattain the bulk property at relatively small size. Thecharge distribution reported here is crucial whiledesigning application-based Li cluster with desiredelectronic properties by doping or by substitution.

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Documents

Density functional studies of LiN and LiN+ (N = 2–30) clusters: Structure, binding and charge distribution