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Wavefunction-based correlated ab initio calculations on crystalline solids

Alok Shukla†

Department of Physics, University of Arizona, Tucson, AZ 85721

Michael Dolg, Peter FuldeMax-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Straße 38 D-01187 Dresden, Germany

Hermann StollInstitut fur Theoretische Chemie, Universitat Stuttgart, D-70550 Stuttgart, Germany

We present a wavefunction-based approach to correlated ab

initio calculations on crystalline insulators of infinite extent.It uses the representation of the occupied and the unoccupied(virtual) single-particle states of the infinite solid in terms ofWannier functions. Electron correlation effects are evaluatedby considering virtual excitations from a small region in andaround the reference cell, keeping the electrons of the restof the infinite crystal frozen at the Hartree-Fock level. Themethod is applied to study the ground state properties of theLiH crystal, and is shown to yield rapidly convergent results.

I. INTRODUCTION

Density functional theory [1,2] (DFT) is a very pow-erful and elegant approach to many-body systems. Itsfoundations are based upon the Hohenberg-Kohn (HK)theorem [1] which states that the total energy of theground-state of a many-particle system in a given exter-nal potential can be expressed as a universal functional ofits electronic charge density. Thus, within a DFT-basedformalism, one avoids constructing the many-body wave-function of the system, and instead computes directlyground-state properties like the total energy, magnetiza-tion, lattice constant etc. from its charge density. How-ever, the exact form of the functional embodied in HKis unknown. Therefore approximations are required, themost important one being the local-density approxima-tion (LDA) developed by Kohn and Sham [2]. LDA hasproven to be extremely successful for the ground-stateproperties of weakly correlated systems. However, forstrongly correlated systems such as the ones containing felectrons, its results are far from satisfactory. There havebeen many improvements proposed to the LDA [3], buttheir applicability depends on the system involved, i.e.,the present DFT-based approaches are not amenable tosystematic improvements.

The alternative approach to the problem of electroncorrelations aims at obtaining the many-particle wave-function of the system, by approximately solving thecorresponding Schrodinger equation within a given one-particle/many-particle basis set. Since the Hamiltonian

of the system under consideration is generally knownbeforehand, there are no uncertainties of principle in-volved in this approach. One can systematically improvethe approach by enlarging the basis set or by includ-ing more terms in the expansion of the wavefunctionof the system. However, such ”wavefunction-based” ap-proaches scale quite unfavourably with the size of thesystem. Therefore, in their ab initio form, they havebeen applied mainly in quantum chemistry for small tomedium-sized molecules. In condensed-matter physics,wavefunction-based many-body approaches are generallyused in conjunction with model Hamiltonians, particu-larly when the correlations are strong. Usually variousapproximations have to be made [4] and an extension ofab initio quantum-chemistry type of methods to infinitesystems did not seem feasible for a long time.

Two recent developments may possibly bring ab ini-

tio wavefunction-based calculations for infinite systemswithin the reach of routine computations: (a) advent ofmore powerful computers (b) use of local excitations inelectronic structure calculations. The first of the afore-said developments is technological in nature, and its im-portance is self-explanatory. The second development isbased on the fact that the correlation hole of an elec-tron is a real-space phenomenon. Local or localized or-bitals can immediately tell us as to which electrons willinteract with each other strongly, and which ones weakly.Although this has been realized long ago in solid-statetheory [5] as well as quantum chemistry [6] only recentlyhave people started following it in full earnest [7–9]. Re-alizing the potential behind the localized-orbital-basedapproaches in the electronic structure theory of solids,one of us developed the so-called “incremental scheme”[10]. The incremental scheme is basically an expansionof the total corrrelation energy per unit cell of a solidwritten in terms of interactions of increasing complexityamong the electrons assigned to localized orbitals (Wan-nier functions) comprising the solid under consideration.Calculations based upon the incremental scheme havebeen performed on a variety of solids [11]. However, thereis one possible drawback associated with the computa-tions so far, namely, in all correlation calculations theinfinite crystal was modelled by a finite cluster. For theionic systems the cluster was embedded in a Madelung

1

field, while for the covalent ones the saturation of thedangling bonds with hydrogen was the substitute for theenvironment.

The purpose of the present work is to study electroncorrelation effects in a crystalline insulator within the in-cremental scheme, without compromising on the infinitenature of the system in any way. Thus by comparing theresults of the present work with calculations done by theprevious cluster-type approximation, we can critically ex-amine both approaches. Bulk LiH was chosen as a firsttest case for our method. To the best of our knowledgethis is the first calculation employing a wavefunction-based approach that does not truncate an infinite solidinto a finite cluster. Although the computational effortsinvolved is more than for the cluster-based calculations, itis not excessive. Therefore, we believe, that with furtheralgorithmic improvements, the present approach can bemade viable for solids containing many atoms in a unitcell.

The remainder of the paper is organized as follows. Insection II we briefly discuss the theoretical backgroundof the present work. In section III we present the resultsof our calculations performed on bulk LiH. Finally, insection IV we present our conclusions.

II. THEORY

The starting point of our approach is the representa-tion of the restricted Hartree-Fock (RHF) ground stateof the solid in terms of Wannier functions (WFs). It isaccomplished by solving the RHF equations of the in-finite crystal directly in the Wannier representation, asrecently described [12]. The applicability of the approachto periodic insulators of reduced dimensions was demon-strated in another work of ours [13], where we used itto study the ground states of an LiH chain and trans-polyacetylene, at the RHF level. Assuming that the sys-tem consists of N unit cells (N → ∞) each of which has2nc electrons, its RHF ground state will be described bync doubly occupied WFs per unit cell. If we use Greekindices α, β etc. to denote the occupied WFs localizedin a given unit cell, the RHF wavefunction of a crystalis a Slater determinant composed of Wannier functions{|α(Rj)〉; α = 1, nc; j = 1, N}. The latter form an or-thonormal set

〈α(Ri)|β(Rj)〉 = δαβδij , (1)

and Wannier functions localized in different unit cells aretranslated copies of each other

|α(Ri + Rj)〉 = T (Ri)|α(Rj)〉, (2)

where the operator T (Ri) represents a translation by lat-tice vector Ri. LiH has 2 occupied WFs per unit cell.Although, in our approach, the WFs are assumed to be

the linear combinations of Gaussian lobe-type functions[14], the theory presented here is quite independent ofthe choice of the basis set.

The next step is to choose a region of space — hence-forth to be called the “correlation region”(C) — in whichthe correlation effects will be computed. The infiniteregion of space lying outside the correlation region willbe called the “environment”(E). In the correlation cal-culations, we consider virtual excitations from the occu-pied WFs of the correlation region only, keeping the elec-trons of the environment “frozen” to the occupied (RHF)space. Because of the frozen character of the environmentWFs, one can immediately sum up their contribution toobtain an effective one-electron potential called the “en-vironment potential”

Uenvpq =

∑

α(Rj)∈E

(2〈pα(Rj)|1

r12|qα(Rj)〉

−〈pα(Rj)|1

r12|α(Rj)q〉) , (3)

where p and q are two arbitrary basis functions and thefactor of two in the first term is due to the spin sum-mation. Uenv

pq is evaluated using the Ewald summationtechnique [11], and one is left with an effective Hamilto-nian for the electrons located in the region C. The physi-cal interpretation of Uenv

pq is obvious — it represents theinfluence of the electrons of E on those of C.

The correlation contribution to the total energy perunit cell is computed using the aforesaid incremental ex-pansion [10]

∆E =∑

i

ǫi +1

2

∑

i6=j

∆ǫij +1

3!

∑

i6=j 6=k

∆ǫijk + . . . , (4)

where the summation over i involves Wannier functionslocated in the reference cell, while those over j and kinclude all the Wannier functions of the crystal. Theǫi (“one-body” increment) are computed by consideringvirtual excitations only from the i-th Wannier function,freezing the rest of the solid at the HF level. The “two-body” increment ∆ǫij is defined as ∆ǫij = ǫij − ǫi − ǫj

where ǫij is the correlation energy of the system ob-tained by correlating two distinct Wannier functions iand j. Thus ∆ǫij represents the correlation contributionof electrons localized on two “bodies” i and j. Simi-larly higher-order increments are defined. For ionic sys-tems such as LiH, generally one- and two-body incre-ments account for the bulk of correlation contribution —three-body increments being negligible. The incremen-tal expansion of Eq.(4) is completely independent of themethod of computation. Since LiH is a small system,for the present calculations of increments it was possibleto use the full configuration-interaction (CI) approachwhich in the physics literature is known as the exact di-agonalization method. For larger systems, however, full-CI will not be feasible. In that case one can resort to any

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of the other available size-extensive correlation schemes[4].

We finally comment upon the construction of the spaceof virtual orbitals used to compute the correlation effects.We find it absolutely essential that the virtual orbitals bealso localized. For this purpose we adopted the approachsuggested by Pulay [8]. It consists of first orthogonaliz-ing the basis functions localized in the reference cell —using projection operators — to the occupied Wannierfunctions of the reference cell, and to those located in upto third-nearest neighbor unit cells. This yields a set ofvirtual functions orthogonal to the occupied space, butnot orthogonal amongst each other. Orthogonality in thevirtual space is finally achieved by a symmetric orthog-onalization procedure which does not destroy the localcharacter of the functions. In the end, we obtain virtualWannier functions which are somewhat delocalized dueto their orthogonalization tails, however, quantitativelythe delocalization is not very significant. Since occupiedand virtual orbitals in our approach form an orthonormalset, almost the whole machinery of electron correlationtreatments developed during the last decades in ab initio

quantum chemistry for atoms and molecules can also beapplied to treat polymers or solids.

III. CALCULATIONS AND RESULTS

Among the recent studies of bulk LiH, the most note-worthy are HF level studies of Dovesi et al. [15] andof Bellaıche and Levy [16], and DFT-based studies ofBellaıche and coworkers [17,18]. For the sake of com-parison, in our study we used the Gaussian lobe repre-sentation of the polarized contracted basis sets used byDovesi et al. [15], i.e., [3s,1p] on hydrogen and [2s,1p]on lithium. The experimental fcc geometry of the sys-tem was assumed with hydrogen on the (0, 0, 0) positionand Li on (0, 0, a/2) position, a being the lattice con-stant. First all-electron Wannier HF calculations wereperformed at fifteen lattice constants in the range 4.0 —4.225 A. The results provided us with the optimized lat-tice constant and cohesive energy of LiH at the HF level.From the orbitals of these calculations the appropriateenvironment potentials were constructed (cf. Eq.(3)),which were subsequently used to perform incremental-scheme-based correlated calculations, employing the full-CI code of Knowles and Handy [19]. At the correlatedlevel we resorted to the frozen-core approximation by in-cluding the Li Wannier functions of region C also in Uenv.Since Li 1s type orbital is highly localized and core-like,this approximation is quite reasonable and is analogousin spirit to the pseudopotential calculations performedroutinely in condensed-matter physics.

In table I we present the results of incremental cal-culations performed for the infinite solid for the latticeconstant of 4.07 A. The increments reported there include

the one-body increment corresponding to the correlationenergy of the polarized hydrogen anion of the referenceunit cell and the two-body increments involving the in-teraction of the reference hydrogen anion with those sit-uated in up to third-nearest neighbor unit cells. Note therapid decrease of the two-body increments with increas-ing distance between the ions which is of van der Waalstype. The reason behind this can be intuitively under-stood as follows. When, for the purpose of the compu-tation of correlation effects, atoms (or ions) are virtuallyexcited, they develop local (virtual) electric dipole mo-ments. Therefore, the two-body increment correspondsto the interaction of two localized (virtual) dipole mo-ments, the interaction between which is nothing but thevan der Waals interaction. A noteworthy point is thatlocalized orbitals are absolutely essential for this argu-ment to hold and, within a Bloch orbital approach, sucha simplification is not possible. For the sake of compari-son, in table I we also present the results of calculationsperformed on a finite cluster modelling the bulk LiH, us-ing the same basis set and geometry. The finite clustercalculations were performed with MOLPRO program [20]and include only the most important increments namelyone-body and the nearest-neighbor two-body increments.The maximum disagreement of ≈ 1.6 × 10−3 Hartreesbetween the two calculations is very encouraging, andessentially defines the error bars of the numerics.

The equilibrium lattice constant, cohesive energy andthe bulk modulus obtained from our calculations — bothat the HF and correlated level — are presented in tableII together with results of other authors [21–27]. Be-fore accounting for the zero-point vibrations of the nu-clei, the DFT-based calculations systematically underes-timate the lattice constant. Since the hydrogen nucleusis very light, one would expect the energy of its zero-point vibrations to make non-negligible contributions tothe total energy. Indeed, Roma et al. [27] applying den-sity functional perturbation theory (DFPT) found thatthe inclusion of the zero-point energy increases the equi-librium lattice constant of LiH by 0.08 A. Bellaıche etal. [18], by adding the lattice-constant-dependent zero-point energies obtained by Roma et al. [27], to their pre-viously derived DFT energies [17] were able to obtainnear-perfect agreement with the experimental lattice con-stant in case of the Wigner expression for the correlationenergy, whereas their result is still 0.06 A too small forthe more modern Perdew-Zunger parametrization of theCeperly-Alder quantum Monte Carlo correlation energy.For the wavefunction-based approaches table II showsthat the HF approximation yields a too large lattice con-stant due to an overestimation of the ionic radii, par-ticularly for the anion. Once electron correlation effectsare taken into account, we get an agreement of betterthan 0.01 A with the experimental value of the latticeconstant, without accounting for zero-point effects.

The experimentally measured cohesive energy of LiH,

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corresponding to its dissociation into Li+ and H−, is0.346 a.u. [22]. This, when corrected by the differenceof the electron affinity of H and the ionization potentialof Li, leads to a cohesive energy of 0.176 a.u. with respectto the separation into free Li and H atoms. In the presentcase, at the correlated level, the Li core is held frozen.Thus the theoretical cohesive energies both at the HFand the correlated level are obtained by subtracting fromthe total energy per unit cell, the HF limits of the ener-gies of free Li (7.4327 a.u. [28]) and H (0.5 a.u.) atoms.From table II it is obvious that for the HF cohesive en-ergy there is very good agreement among the results ofall authors. The difference of 0.0006 a.u. between ourresults and the one of Dovesi et al. [15] is possibly dueto our use of Gaussian lobe functions instead of Carte-sian Gaussians. By subtracting our HF cohesive energy(0.1294 a.u.) from the experimental one (0.1760 a.u.),one gets ≈ 0.0466 a.u. as the “experimental” value ofthe correlation contribution to the cohesive energy perunit cell. The correlated cohesive energy that we obtainin the present work is 0.1644 a.u. and corresponds to93.4 % of the experimental value. The calculated corre-sponding correlation contribution of 0.0350 a.u. is 75.1% of the experimental value.

In order to account for a higher percentage of the cor-relation contribution we would need to apply larger basissets including additional diffuse and also higher angularmomentum functions. Unfortunately, since our programis still under development, such large-scale calculationsare not possible at present. Nevertheless, one can esti-mate the effect of the basis sets incompleteness for thetime being from finite cluster calculations using extendedbasis sets [29]. We obtained a value of -0.035486 a.u. forthe one-body contribution and -0.001544 a.u. for thenearest-neighbor two-body contributions. Since we areinterested in the values of the corresponding contribu-tions obtained with the extended basis sets for the infi-nite solid within the Wannier function approach, we scalethese numbers according to the ratios of the correspond-ing numbers in the two columns of table I and obtain theestimates -0.033652 a.u. and -0.001230 a.u., respectively.By comparing these values to the corresponding numbersobtained with the smaller basis set for the bulk case, weobtain corrections of -0.004335 a.u. and -0.000553 a.u.for the one-body and two-body term, respectively, lead-ing to a total correlation energy gain of -0.0110 a.u. Ourso-corrected estimate of the correlation contribution tothe cohesive energy per unit cell of 0.0460 a.u. (0.0350a.u. + 0.0110 a.u.) is in almost perfect agreement withthe corresponding experimental number (≈ 0.0466 a.u.).We want to point out here, that further technical im-provements in our implementation of the method willmake calculations with extended basis sets for the in-finite system feasible and therefore extrapolations fromfinite cluster calculations unnecessary. The fact that anextension of the basis sets leads to a calculated value

essentially in agreement with the experimental result un-derlines the potential of the proposed method, i.e., thepossibility of systematic improvement.

The results of our Wannier-function-based calcula-tions, using the smaller basis set of Dovesi et al. [15],show that in a sufficiently wide range of values aroundthe equilibrium lattice constant (15 points in the in-terval 4.0 A ≤ a ≤ 4.225 A) to high accuracy (cor-relation coefficient 0.999983) the correlation energy de-pends linearly on the lattice constant a (Ecorr(a.u.) =−0.034030 + 0.002935 ∗ (a(A) − 4.06)). This is in agree-ment with an assumption made in previous work on NiOby Doll et al. [11]. Such a “linear law” is important be-cause it allows one to perform correlated calculations onlyfor two values of the lattice constant near its equilibriumvalue and to deduce the other values by interpolation.Assuming also a linear behavior of the correlation contri-butions obtained with the extended basis sets mentionedabove [29], we derive an estimated equilibrium latticeconstant of 4.054 A compared to the actually calculatedvalue of 4.067 A and the experimental result of 4.060 A.

Finally, we discuss the influence of correlations on thebulk modulus of LiH. The experimentally measured val-ues reported in literature [23–25] are actually obtainedby fitting a pressure-volume (p(V)) relationship to ex-perimental data points. The values for 7LiD are onlyabout 1 % to 3 % larger than for 7LiH [23,27] and some-times a common value is reported [25]. Depending onthe chosen ansatz for the p(V) equation of state (Birch,Murnaghan, Vinet) the values reported for 300 K rangefrom 31.1 GPa to 36.2 GPa, with error bars of 1 % to6 %. In addition to this uncertainty, the values have tobe corrected to 0 K, which roughly corresponds to anincrease by 6 % [24,26], i.e., a range from 33.0 GPa to38.4 GPa. A direct comparison of our calculated values,which do not rely on any fitting of an equation of stateand are for 0 K, and the effective experimental parame-ters is to be viewed with care. From table II it is obviousthat our HF value of 33.4 GPa is in within the interval ofexperimental values, whereas our correlated (estimated)value of 36.5 (37.9) GPa appears to be slightly too large.Roma et al. [27] observed a reduction of the bulk modu-lus by 4 GPa (≈ 10 %) due to zero-point effects. Usingtheir zero-point corrections Bellaıche et al. [18] were ableto obtain close agreement with the experimental valuesof the bulk modulus in case of the Wigner correlationenergy expression (33.9 GPa), whereas for the Perdew-Zunger parametrization also a relatively large value wasobtained (36.1 GPa). We note that by adding the correc-tions calculated by Roma et al. [27] to our energy-volumedata, we end up with correlated results which are not ingood agreement with the experimental values, i.e., a lat-tice constant of 4.143 A and a bulk modulus of 30.7 GPa.Since our uncorrected values appear to be stable with re-spect to systematic improvements of the calculation andare also quite close to the experimental values, we believe

4

that the zero-point corrections obtained by Roma et al.may be slightly too large, despite the fact that previousestimates were even significantly larger [30].

IV. CONCLUSIONS AND FUTURE DIRECTIONS

In conclusion, a new wavefunction-based approach hasbeen presented, which allows the ab initio determinationof electron correlation effects in a crystalline insulator,without abandoning its infinite character. The approachmerges ideas from quantum chemistry and solid-statephysics, and has the benefit of being amenable to sys-tematic improvements of results. It has been shown to beviable by a first successful application to an ionic crys-tal, namely bulk LiH. The applicability of the presentapproach to covalent insulators and semiconductors suchas diamond, Si, GaAs etc. is essentially straightforward.The only trivial modification in those cases will be thatthe occupied (and virtual) single-particle states will con-sist of bond-centered Wannier functions, instead of theatom-centered Wannier functions encountered in case ofionic systems. As a matter of fact, the viability of thepresent local approach to electron correlations in cova-lent systems has already been demonstrated by Paulus etal. [11]—albeit within a finite-cluster approximation—intheir study of Group IV semiconductors. Its generaliza-tion within the infinite-crystal method presented in thiswork will be the subject of a future investigation.

As far as the treatment of excited states of crystalsis concerned, the quasi-particle excited states such as anelectron (hole) in the conduction (valence) band can betreated straightforwardly using the present approach, asthese excited states can be regarded as the ground statesof N +1 (N−1) electrons, where N is the total number ofelectrons in the unperturbed system. Of course, the elec-tron correlation effects in these cases will be more com-plicated as compared to the N -particle ground state, ow-ing to the polarization cloud of the extra electron (hole)in the system. Applicability of this approach to quasi-particle excitations was demonstrated by Grafenstein etal. [31,32], who computed the correlated dispersion rela-tions of holes in the valence bands of Group IV semicon-ductors. Its generalization to the infinite crystal, as wellas extension to the conduction-band dispersion relations,will also be investigated in future.

As far as the general computational viability of ourapproach is concerned, undoubtedly, at present, it ismore demanding than the DFT-based approaches. How-ever, we believe that by utilizing modern order-N typemethods of electronic-structure theory, it can be madecompetitive with such methods. Conceptually speaking,there is nothing in our approach that restricts its appli-cability to systems containing unit cells of certain shapes,sizes, or number of electrons. The frontiers in these areaswill be decided by the progress in the computer hardware,

and by the improvements in the algorithms of electronicstructure theory. However, in order to be more specificabout the shapes and sizes of the systems which can bestudied using the present approach, we believe that moreexperience with it is essential. Work along these direc-tions is presently underway in our group.

ACKNOWLEDGMENTS

The authors are grateful to Dr. G. Roma for provid-ing the actual data for the zero-point energy correctionsplotted in Ref. [27].

† e-mail address: shukla@physics.arizona.edu[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B 864

(1964).[2] W. Kohn and L.J. Sham, Phys. Rev. 140, A 1133 (1965).[3] For a recent review of density-functional-based methods

beyond the LDA see, e.g., P. Ziesche, S. Kurth, and J.P.Perdew, Comput. Mat. Sci. 11, 122 (1998).

[4] See, e.g., P. Fulde: Electron Correlations in Molecules

and Solids, 3rd edn. Springer Series in Solid-State Sci-ences, Vol. 100, (Springer, Berlin, 1995).

[5] See, e.g., J. Friedel: The Physics of Metals, ed. by J.H.Ziman (Cambridge University Press, Cambridge 1969).

[6] See, e.g., T. Gilbert in Molecular Orbitals in Chemistry,

Physics, and Biology, Ed. P.O. Lowdin and B. Pullman,Academic Press, New York, (1964).

[7] G. Stollhoff and P. Fulde, J. Chem. Phys. 73, 4548(1980).

[8] P. Pulay, Chem. Phys. Lett. 100, 151 (1983).[9] C. Hampel and H.-J. Werner, J. Chem. Phys. 104, 6286

(1996).[10] H. Stoll, Phys. Rev. B 46, 6700 (1992); Chem. Phys.

Letters 191, 548 (1992).[11] See, e.g., B. Paulus, P. Fulde and H. Stoll, Phys. Rev.

B 54, 2556 (1996); K. Doll, M. Dolg, P. Fulde, and H.Stoll, Phys. Rev. B 55, 10282 (1997).

[12] A. Shukla, M. Dolg, H.Stoll and P. Fulde, Chem. Phys.Lett. 262, 213 (1996); A. Shukla, M. Dolg, P. Fulde, andH. Stoll, Phys. Rev. B 57, 1471 (1998).

[13] A. Shukla, M. Dolg, and H. Stoll, Phys. Rev. B 58, 4325(1998).

[14] H. Preuss, Z. Naturf. A 11, 823 (1956); J.L. Whitten, J.Chem. Phys. 39, 349 (1963).

[15] R. Dovesi, C. Ermondi, E. Ferrero, C. Pisani, and C.Roetti, Phys. Rev. B 29, 3591 (1984).

[16] L. Bellaıche and B. Levy, Phys. Rev. B 54, 1575 (1996).[17] L. Bellaıche and K. Kunc, Int. J. Quantum Chem. 61,

647 (1997).[18] L. Bellaıche, J.M. Besson, K. Kunc, and B. Levy, Phys.

Rev. Lett. 80, 5576 (1998).

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[19] P.J. Knowles, N.C. Handy, Comput. Phys. Commun. 54,75 (1989).

[20] MOLPRO is a package of ab initio programs written byH.-J. Werner and P.J. Knowles, with contributions fromJ. Almlof, R.D. Amos, M.J.O. Deegan, S.T. Elbert, C.Hampel, W. Meyer, K. Peterson, R. Pitzer, A. J. Stone,and P.R. Taylor.

[21] J.L. Anderson, J. Nasise, K. Philipson, and F.E. Pretzel,J. Phys. Chem. Solids 31, 613 (1970).

[22] O.L. Anderson, J. Phys. Chem. Solids, 27, 547, (1966).[23] D.R. Stephens and E.M. Lilley, J. Appl. Phys. 39, 177

(1968).[24] J.M. Besson, G. Weill, G. Hamel, R.J. Nelmes, J.S. Love-

day, and S. Hull, Phys. Rev. B 45, 2613 (1992).[25] P. Loubeyre, R. Le Toullec, M. Hanfland, L. Ulivi, F.

Datchi, and D. Hausermann, Phys. Rev. B 57, 10403(1998).

[26] D. Gerlich and C.S. Smith, J. Phys. Chem. Solids 35,1587 (1974).

[27] G. Roma, C.M. Bertroni, and S. Baroni, Solid State Com-mun. 98, 203 (1996).

[28] E. Clementi and C. Roetti, Atomic Data and NuclearData Tables, Vol. 14 (Academic Press, New York, 1974).

[29] The basis set for hydrogen was [6s,5p,4d] and the onefor Li was [5s,4p] type. These basis sets are, respectively,referred to as AV5Z and VQZ and were taken from theMOLPRO program library [20].

[30] J.L. Martins, Phys. Rev. B 41, 7883 (1990).[31] J. Grafenstein, H. Stoll, and P. Fulde, Chem. Phys. Lett.

215, 610 (1993).[32] J. Grafenstein, H. Stoll, and P. Fulde, Phys. Rev. B 55,

13588 (1997).

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TABLE I. Various increments to the correlation en-ergy in Hartrees of bulk LiH computed by the Wan-nier-function-based approach presented in this work. Forcomparison, the results obtained using the method of Dollet al. [11] are presented, for the most important increments,under the heading of Finite Cluster. NN stands for nearestneighbors. All the calculations were performed at the latticeconstant of 4.07 A.

Correlation Increment Wannier Function Finite Cluster

one-body -0.029317 -0.030915two-body (1NN) -0.000677 -0.000850two-body (2NN) -0.000113 —two-body (3NN) -0.000023 —

TABLE II. Equilibrium values of various quantities for thebulk LiH obtained in this work as compared to those of otherauthors, and the experiment. Quantities under the columnDFT refer to slightly different exchange-correlation function-als used (see Ref. [18] for details). The theoretical resultspresented below do not include the effect of zero-point vibra-tions, unless specified otherwise. Lattice constants are in A,energies in a.u. (i.e., Hartree) and the bulk moduli are inGPa.

Quantity HFa HFb HFc Correlateda DFTd Experiment

Lattice Constant 4.106 4.102 4.110 4.067 3.91–3.98 4.060f

3.99–4.05e

Cohesive Energy 0.1294 0.1302 0.1305 0.1644 — 0.1760g

Bulk Modulus 33.4 34.1 35.0 36.5 33.9–36.1e 33.0–38.4h

31.3–36.2i

a This workb Ref. [15]c Ref. [16]d Ref. [18], only all-electron results are considered.e Includes the effect of zero-point motion.f Extrapolated T=0 value [21]g The cohesive energy reported here is the “atomic” co-hesive energy obtained by correcting the “ionic” cohesiveenergy reported in ref. [22]. See text for an explanation.h estimated for 0 K by scaling 300 K values with 1.06[24,26]i 300K [23–25].

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