Electronic structure, lattice energies and Born exponents for alkali halides from firstprinciplesC. R. Gopikrishnan, Deepthi Jose, and Ayan Datta Citation: AIP Advances 2, 012131 (2012); doi: 10.1063/1.3684608 View online: http://dx.doi.org/10.1063/1.3684608 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/2/1?ver=pdfcov Published by the AIP Publishing

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AIP ADVANCES 2, 012131 (2012)

Electronic structure, lattice energies and Born exponentsfor alkali halides from first principles

C. R. Gopikrishnan, Deepthi Jose, and Ayan Dattaa

School of Chemistry, Indian Institute of Science Education and ResearchThiruvananthapuram, CET Campus, Thiruvananthapuram, Kerala-695016, India

(Received 3 December 2011; accepted 9 January 2012; published online 1 February 2012)

First principles calculations based on DFT have been performed on crystals of halides(X = F, Cl, Br and I) of alkali metals (M = Li, Na, K, Rb and Cs). The calculated latticeenergies (U0) are in good agreement with the experimental lattice enthalpies. A newexact formalism is proposed to determine the Born exponent (n) for ionic solids. Thevalues of the Born exponent calculated through this ab-initio technique is in goodagreement with previous empirically derived results. Band Structure calculationsreveal that these compounds are wide-gap insulators that explains their optical trans-parency. Projected density of states (PDOS) calculations reveal that alkali halideswith small cations and large anions, have small band gaps due to charge transferfrom X → M. This explains the onset of covalency in ionic solids, which is popularlyknown as the Fajans Rule. Copyright 2012 Author(s). This article is distributed undera Creative Commons Attribution 3.0 Unported License. [doi:10.1063/1.3684608]

I. INTRODUCTION

Ionic compounds are ubiquitous materials and are characterized by their highly crystallinenature, high melting points and strong miscibility in polar media.1, 2 Ionic salts like NaCl and KClare essential to maintain osmotic pressure inside the cell as well in signal transmission across thecells.3, 4 Ionic salts have gained substantial importance recently due to the ability of ionic liquidsto dissolve a variety of organic substance including cellulose.5, 6 Based on structural tailoring inthe choice of the size and shape of the cations and anions, it is possible a design molecule specificsolvent.7 This has opened an interesting possibility in harnessing energy from biomass.8

From an atomistic picture, ionic crystals are probably the simplest system to understand sincethe interactions among the ions are purely electrostatic in origin. Hence, the description of theirground state energies (and thus the structure, following the variation theorem) is exact within thepoint charge model. This apparent simplicity is deceptive because Coulomb potential is a slowlydecaying function and hence contributions from higher order multipole interactions need to betaken into consideration.9, 10 Even within the point charge model, the number of ions in the crystalbeing of the order of NA (Avogadro’s number), a proper summation of all charges across the latticeis an involved task. Series summation method for the calculation of Madelung constant (M) forsimple crystals is imprecise since such series are only conditionally convergent though numericallyreasonable convergence is attainable.11, 12 Hence, this problem is generally tackled numerically incomputational chemistry through the Ewald’s summation technique that gives excellent convergencewith increase in the size/periodicity of the system.13

However, for the stability of a crystal, apart from the attractive Coulomb forces, short rangerepulsive interactions are required to prevent the collapse of the structure to a point. This repulsiveinteraction primarily arises due to electron – electron repulsion between two closed shell atoms.Unfortunately, the spacial profile (variation with respect to the interatomic/interionic distance, r)of the repulsive interaction is unknown analytically. Hence, one relies on empirical functional

aayan@iisertvm.ac.in

2158-3226/2012/2(1)/012131/8 C© Author(s) 20122, 012131-1

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012131-2 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

forms like (1/rn) or (e-r/ρ) to describe their behavior. In either case, one empirically determinesthe parameters: n and ρ. The 1/rn dependence of the repulsive interaction is well – known in theliterature as Born relationship and n is known as the Born exponent.14 Also, previous reports haveindicated that an e-r/ρ form (Born-Mayer exponential law) describes the repulsive interactions moreaccurately.15 Nevertheless, the exact values for either n or ρ remain mere parameters, somethingwhich is undesirable in molecular quantum chemistry. In principle, the Born exponents can be derivedfrom the polarizability of atoms in a particular electronic configuration15 or from the compressibilitydata.16 Thus, for ionic solids one needs to rely on the Born exponent of the cation and the anionand then average them for the compound.14 Such, simple arithmetic mean does not consider otherimportant factors like charge transfer (CT). Hence, it is important to be able to calculate the Bornexponent, n with much higher accuracy that accounts for processes like CT as well as electroncorrelations. Also, a first principles calculation is further desirable to be able to calculate n withoutnecessarily depending on experimental compressibility data, particularly for new ionic materials forwhich good experimental data might be unavailable or difficult to obtain.

Therefore, in order to develop a formalism wherein first principles calculations are possibleon ionic lattices, we perform ab-initio calculations for the simplest ionic compounds namely, alkalihalides. However, the formalism being proposed is sufficiently general to be implemented forany lattice including covalent atomic solids like diamond, transition metal (TM) oxides or evenperovskites. In the following section, we briefly discuss the theory of interionic interactions anddescribe the method of calculation of Born exponent directly from DFT calculations. Followed bythat we discuss the results of band structure calculations for these compounds and discuss CT fromthe halide ion to the cation. Finally, we conclude the manuscript with our conclusions and futureprospects.

II. THEORY AND COMPUTATION

The electrostatic theory describes the interaction experienced by a point charge, due to thepresence of N other charges through the Coulomb’s Law

Ei = e

4πε0

N∑i �= j

Z j

ri j(1)

However, electroneutrality of simple binary ionic solids ensure that there are equal number ofions of opposite charges. Hence, in an ionic compound like MZ+XZ- where Z+ and Z- represent thecharge on the cation and anion respectively, the interaction energy is given by

Ec = e2

4πε0

N∑i �= j

Z+ Z−ri j

(2)

For nearest neighbor (NN) ion-pairs i and j, the distance between the ions, r can be convenientlyrepresented in terms the unit cell vector length (a) for most lattices. However, the long range ofCoulomb interactions make it mandatory that we consider interactions from all ions [attractive(between Z+ and Z-) and repulsive (between Z+ and Z+; Z- and Z-)] in crystal. Fortunately, thesymmetry of the crystal allows one to sum up the contributions easily. Hence, the summation overall ions in equation (2) is determined numerically and is called the Madelung Constant (M) which isa constant for a crystal of a particular symmetry. Hence, equation (2) reduces to

Ec = e2

4πεo

M Z+ Z−r

(3)

Apart from this electrostatic interaction, the crystal lattice has contributions from repulsiveinteraction between the closed shell ions, which within Born’s formalism is written emperically as

ER = B

rn(4)

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012131-3 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

where B and n are parameters to be determined. Of course a crystal is composed of Avogadro’snumber, NA, of ions. Therefore, the overall lattice energy (U) is the sum of equations (3) and (4) forN ions:

U = e2

4πε0

NA M Z+ Z−r

+ NA B

rn(5)

For the equilibrium structure (r = r0), U is a minima with dU/dr = 0. Thus, B can eliminated fromequation (5) which can now be rewritten as the Born-Lande equation for the lattice energy,

U0 = NA M Z+ Z−e2

4πε0r0

(1 − 1

n

)(6)

It can be clearly seen from equation (6) that apart from n, all other quantities are known for a crystalof a particular space group.

We propose that a very accurate measure of n can be determined through first principles bycalculating the heat of formation of an ionic solid (�H). Based on the Hess’s law for the Born-HaberCycle, �H = Hatomization + HIE + HEA + HBDE + U0 where Hatomization, HIE, HEA, HBDE and U0

represent the enthalpies of atomization of the metal, ionization energy of the metal, electron affinityof the halide, bond dissociation of the X2 molecule and lattice energy respectively. Since, apart fromthe lattice enthalpy (U0), all other terms in the Hess’s law are known, once �H is known from firstprinciples, U0 is automatically determined. This immediately gives a recipe for the calculation theBorn exponent (n) from equation (6).

All the calculations reported in this manuscript are obtained using the Vienna ab initio simulationpackage, VASP which is plane wave based DFT code.17 The projected augmented wave (PAW)method is chosen to represent the ionic potentials. The calculations were performed within theGeneral-Gradient Approximation (GGA), and the Perdew-Burke-Ernzerhof (PBE) prescription forthe exchange-correlation functional.18 However, since GGA is well-known to underestimate the bandgap in materials due to lack of proper localization of holes and electrons, few test calculations havealso been performed through hybrid functionals like HSE06,19 PBE020 and B3LYP.21 The initialinput structure for all the alkali halides were obtained from the Inorganic Crystal Structure Database(ICSD).22 These structures were then relaxed using the conjugated gradient (CG) method, with forcetolerance of 0.02 eV Å-1. During the relaxation of the structure, no restrictions were imposed on thesymmetry of the cell, volume of the cell or the position of atoms inside the cell. The energy cutofffor the plane waves was set to 500 eV. The k-mesh in the Brillouin zone was set as 3×3×3 and itwas centered at the �-point. For calculation of the thermodynamic properties like the enthalpy offormation, phonon calculations were performed for the full Brillouin zone on the relaxed structures.The energies are corrected for the zero point energy (ZPE) and heat capacity and are reported for298 K. Ground State structures before and after relaxation and their energies are provided in thesupporting information.23

III. RESULTS AND DISCUSSION

For all the structures, the initial input structures were the rhombohedral primitive unit cells.However, we have verified that even if one uses the conventional unit cells (like BCC for CsCl, CsBrand CsI and FCC for rest of the alkali halides), the results are very similar. In Table I, the latticevectors for the initial (relaxed) structure, the calculated (experimental) heat of formation, calculated(experimental) lattice enthalpy for the alkali halides are reported. For all the structures the change inthe unit cell length is < 3% upon relaxation. As can be seen, the calculated enthalpy of formation arein good agreement with the known experimental values. Based on the literature values24 for bond dis-sociation energies (BDE) of 154.8 kJ/mol, 239.4 kJ/mol, 190.2 kJ/mol and 148.9 kJ/mol for F2, Cl2,Br2 and I2 respectively; Electron affinity (EA) of -328.1 kJ/mol, -348.5 kJ/mol, -324.442 kJ/mol and-295.1 kJ/mol for F, Cl, Br and I respectively; Ionization energy (IE) of 520. 2 kL/mol, 495.8 kJ/mol,418.8 kJ/mol, 402.5 kJ/mol and 375.6 kJ/mol for Li, Na, K, Rb and Cs respectively and atomizationenergy of 147.1 kJ/mol, 97.4 kJ/mol, 76.9 kJ/mol, 75.8 kJ/mol and 64.0 kJ/mol for Li, Na, K, Rb andCs respectively, the lattice enthalpies are calculated through the Born-Haber cycle. It is worthwhile

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012131-4 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

TABLE I. Structural parameters, Enthalpy of formation and Lattice enthalpies for alkali halides.

Lattice vector Enthalpy of formation, Lattice Enthalpy,(in Å) �H (in kJ/mol) at 298 K U0 (in kJ/mol)

Compound Initial Relaxed Calculated Experimental Calculated Experimental

LiF 4.02 4.0537 -558.868 -573±21 -975.397 -1034LiCl 5.13 5.1401 -339.338 -464±13 -777.746 -840.1LiBr 5.50 5.5074 -273.327 -418±21 -711.164 -781.2LiI 6.00 6.0106 -217.469 -347±13 -664.082 -718.4NaF 4.62 4.6831 -514.566 -477 -857.043 -914.2NaCl 5.64 5.6859 -340.236 -407.9 -704.591 -770.3NaBr 5.97 6.0360 -280.729 -362.8 -644.513 -728.4NaI 6.47 6.5322 -234.616 -304.2 -607.176 -680.7KF 5.35 5.4036 -514.601 -490±21 -759.534 -812.1KCl 6.29 6.3788 -366.843 -423±8 -633.654 -701.2KBr 6.60 6.7113 -316.408 -378.7±8 -582.648 -671.1KI 7.07 7.1909 -274.079 -326±13 -549.096 -632.2RbF 5.64 5.7216 -502.606 -490 -730.632 -780.3RbCl 6.58 6.7105 -365.424 -444±21 -615.328 -682.4RbBr 6.85 7.0335 -316.785 -385±21 -566.119 -654.0RbI 7.34 7.4995 -277.830 -331±13 -535.940 -616.7CsF 6.01 6.0983 -499.302 -502±42 -688.149 -743.9CsCl 4.12 4.2007 -360.234 -435±21 -571.442 -629.7CsBr 4.29 4.3921 -315.442 -416±13 -525.672 -612.5CsI 4.57 4.6821 -280.682 -335±21 -499.688 -584.5

TABLE II. Calculated Born exponent (nDFT) from DFT, Experimental Born Exponent (nκ ) from compressibilitymeasurements, Experimental Born exponent (nα) from polarizability measurements, DFT band gap (Eg

DFT), experimentalband gap (Eg

expt)29 and effective quasi particle masses for alkali halides. The empty entries correspond to experimentallyunavailable data. The scaled band gaps (Eg

scaled) are defined as: EgDFT × xav where xav = ∑

iN(Eg

expt/ EgDFT)/N = 1.7;

N=number of alkali halides with known experimental band gaps.

Born Exponent (n) Band Gap in eV Effective MassesCompound nDFT nκ nα Eg

DFT Egscaled Eg

expt electrons me* holes mh*

LiF 5.39 5.9 — 8.94 15.3 13.6 2.16 8.60LiCl 5.65 8.0 — 6.32 10.8 9.4 1.15 4.12LiBr 5.16 8.7 — 4.91 8.4 7.6 0.83 3.07LiI 5.60 — 4.83 4.24 — — 0.71 2.50NaF 5.75 — — 6.16 10.6 11.7 1.38 9.06NaCl 5.70 9.1 6.39 5.00 8.6 8.5 0.93 4.56NaBr 5.02 9.5 6.50 4.09 7.0 7.7 0.81 3.75NaI 5.45 — 6.94 3.55 — — 0.63 2.93KF 6.11 7.9 8.52 6.07 10.4 10.7 1.34 9.07KCl 5.96 9.7 7.55 5.05 8.7 8.4 1.09 6.21KBr 5.13 10.0 7.26 4.31 7.4 7.4 1.01 4.72KI 5.34 10.5 7.53 3.81 6.5 6.0 0.83 3.98RbF 6.59 — 6.60 5.51 9.5 10.4 1.16 11.53RbCl 6.00 — 8.54 4.82 8.3 8.2 1.10 5.23RbBr 4.96 10.0 7.71 4.20 7.2 7.5 0.97 5.94RbI 5.26 11.0 7.65 3.76 6.4 6.2 0.86 3.67CsF 6.73 — 7.22 5.29 9.1 10.9 1.09 10.80CsCl 5.97 — 8.40 5.00 8.6 8.3 0.88 5.50CsBr 4.93 — 8.60 4.25 7.3 7.3 0.80 4.92CsI 5.20 — 7.58 3.75 6.4 6.3 0.72 4.44

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012131-5 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

FIG. 1. Band Structure for NaCl at (A) GGA level and (B) B3LYP level of theory. The Fermi Energy (EF) is scaled to 0.0 eV.The high symmetry points are defined as �(0,0,0); F(0,1/2,0); B(1/2,0,0) and G(0,0,1/2).

to mention here that all the above quantities are of course calculable very accurately through ab-initiowavefunction based methods and one does not necessarily rely on experimental BDE, EA, IE andheat of atomization. However, since the main focus of the current work is to provide an alternativeframework for the estimation of the lattice energy (U0), we used the experimental values. The cal-culated lattice enthalpies are also in good agreement with the experimental values. It is important tonote that the present DFT calculations give a much superior estimate that for U than point chargemodels with empirical corrections.1

Nevertheless, for a comparison of the Born exponent (n) calculated from the DFT latticeenthalpies through equation (6), we tabulated the calculated n, experimentally determined n frompolarizability data15 as well as n obtained from compressibility measurements16 in Table II. Forcomparison, the calculated and experimental band gaps are also reported. Interestingly, while thecalculated Born exponents for alkali halides are in reasonably good agreement with n obtained fromboth the compressibility data and polarizability measurements, the agreement is better with thepolarizability data. This arises primarily due to the fact that compressibility of ionic solids is verysmall (∼10-12 m2N-1) and accurate measurements are difficult. However, very accurate measures forthe polarizabilities of ions are available through optical measurements. Hence, we believe that ourcalculated, nDFT is quite accurate. Nevertheless, it is important to note that the agreement betweenthe calculated Born exponents with the experimental data is rather modest for the CsCl series dueto significant covalent nature of these compounds which makes the point charge description withinthe Born-Lande equation inaccurate.25

While, our calculations do suffer from the well-known error of underestimation of the band-gapwithin the GGA in comparison to the experimental measurements,26, 27 the qualitative trends acrossthe halide series are in good agreement.28 For example, the lowest band gaps correspond to thealkali iodides. For example for the Cs ion, the experimental (calculated) band gap decreases from5.29 eV (10.9 eV) in CsF to 3.75 eV (6.3 eV) in CsI. Clearly, the band gap decreases in the seriesF > Cl > Br > I for an alkali metal ion. Hybrid calculations of course provide a better estimateof the band gaps. For example, in NaCl, the HSE06, PBE0 and B3LYP functionals calculate bandgaps of 6.48 eV, 7.08 eV and 6.73 eV respectively which are somewhat in better agreement withEg

expt = 8.5 eV. Nevertheless, since GGA is able to account for the overall trend of variation of band

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012131-6 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

FIG. 2. Density of States (DOS) for (A) LiF, (B) LiCl, (C) LiBr, (D) LiI, (E) NaF, (F) NaCl, (G) NaBr, (H) NaI. (I) KF, (J)KCl, (K) KBr, (L) KI, (M) RbF, (N) RbCl, (O) RbBr, (P) RbI, (Q) CsF, (R) CsCl, (S) CsBr, (T) CsI.

gap across the alkali halide series, all calculations are performed within GGA. The calculated bandstructure for NaCl (at the GGA and the hybrid B3LYP level of calculations) is shown in Fig. 1. Thequalitative features are similar for other alkali halides as well. Interestingly, me* << mh* suggestingthat creating excitons similar to semiconductors should be very difficult which explain the insulatingnature of these alkali halides. Also, in harmony with the variation in the band gaps, the effectiveelectron masses decrease for either F→Cl→Br→I for a fixed cation or Li→Na→K→Rb→Cs fora fixed anion.

For a detailed understanding for the origin of decreasing band gap for the alkali halides alongthe series F→Cl→Br→I, the density of states (DOS) for the alkali halides are plotted in Fig. 2. Theprojected density of states (PDOS) for the alkali ions and the halide ions are also shown.

From the DOS, it is clearly observed that the valence electrons are mostly composed of thehalide electrons while the conduction electrons are from the alkali ion. For a fixed cation, along the

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012131-7 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

FIG. 3. Electron Localization Function (ELF) for (A) LiF (isosurface value = 0.600) and (B) LiI (isosurface value = 0.168).

series F→Cl→Br→I, the valence electrons become progressively less stabilized. Similarly, for afixed anion, along the series Li→Na→K→Rb→Cs, the conduction electrons become progressivelyless destabilized. This leads shrinking of the band gap for either the case of a fixed cation withvariation along F→Cl→Br→I or fixed anion with variation along the series Li→Na→K→Rb→Cs.Therefore, as can be seen from Table II, LiF has a high band gap while CsI has a low band gap in thealkali halide series. Interestingly, the lattice energies also follow a similar pattern with the highestlattice energy for LiF and the lowest lattice energy for CsI. Clearly, LiF is most ionic while CsI isleast. This can be easily understood through a two-state Donor-Acceptor model of the perturbationtheory.28 Ionic states arise due to poor overlap between the Donor (Halide) and Acceptor (Alkaliion) states. However, strong donors like Br- and I- that have a large and diffused 4p and 5p orbitalallow interactions with the alkali metal ion. This leads to charge-transfer from the halide ion to thealkali ion. This accounts for the covalency (and associated reduced band gap) of alkali bromides andiodides. As a corollary to this effect, strong acceptors like Rb halides and Cs haildes also have poorionicity (good covalency) and hence smaller band gaps due the large and diffused 5s and 6s orbitalsrespectively. This is qualitatively observed from the plots of the electron localization function (ELF)shown in Fig. 3. Clearly, the electrons are much more localized for LiF than LiI. The electrons in LiIshow substantial delocalization and there is significant electron density in between the ions whichis representative of covalent like bonding. This rationalizes the smaller gap for alkali iodides dueto CT induced covalency. This we believe provides a microscopic explanation for the Fajans Rulewhich empirically determines the onset of covalency in ionic solids.30

IV. CONCLUSION

The heat of formation and the lattice energies for alkali halides determined through first princi-ples calculations are in good agreement with the experimental enthalpies and lattice energies. While,known formulae like the Kapustinskii equation predict lattice energies for alkali halides that arebetter in agreement with experimental results (within 5%), it still relies on parameters like the ionicradii.31 Hence, reasonable agreement of first principles calculations with experimental enthalpies isencouraging. These calculations provide an alternative formalism to calculate the inter-electronicrepulsions between ions, the Born Exponent (n). First principles determination of the Born exponentis important particularly because the exact nature of the repulsive interaction is unknown analyticallyand they have been only determined empirically till now. Also, since the pseudo-potentials used inthese calculations are sufficiently general, these calculations can be readily used for new ionic ma-terials like ionic liquids with short range ordering. Nevertheless, the computed Born exponents arein good agreement with the available empirical data. These alkali halides act as wide gap insulatorsbut the band gaps reduce considerably due to charge transfer from the halide ion to the cation. A

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012131-8 A. Datta, C. Gopikrishnan, and D. Jose AIP Advances 2, 012131 (2012)

reduction in the band gap make these solids less stable with considerable reduction in their latticeenergies. The onset of covalency in ionic solids as attributed by the Fajans rule arises from suchcharge transfer effects. We believe that the success of first principles calculations to determine em-pirical parameters is an important step towards application of such calculations in materials scienceand engineering wherein the huge complexity and variety can be better understood.

ACKNOWLEDGMENT

The authors thank DST – Fast Track scheme and CSIR for partial funding. CRG thanks DSTfor the INSPIRE fellowship. The authors thank Dr. Sujith Vijay for helpful discussions.

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