The Accuracy of the Pseudopotential Approximation within Density-Functional Theory

Transcription

1 D. Porezag et a.: The Accuracy of the Pseudopotentia Approximation 219 phys. stat. so. (b) 217, 219 (2000) Subject cassification: Hx; Mb The Accuracy of the Pseudopotentia Approximation within Density-Functiona Theory D. Porezag (a, b), M. R. Pederson (a), and A. Y. Liu (b) (a) Center for Computationa Materias Science, Nava Research Laboratory, Washington, DC 20375, USA (b) Department of Physics, Georgetown University, Washington, DC 20057, USA (Received August 10, 1999) We have investigated the accuracy of pseudopotentia (PSP) density-functiona cacuations with respect to the corresponding a-eectron (AE) resuts for a variety of atoms and sma moecues. It is found that most of the deviations between ab-initio PSP and AE cacuations are due to the inearization of the exchange-correation functiona within the PSP approach. This probem can be eiminated by appying noninear core corrections (NLCC). We find that a correct description of spin-poarized states requires the NLCC, even for first-row atoms. This is essentia for simuations of magnetic systems and reaction processes which invove radicas. The NLCC is aso essentia for a reaistic description of eements with more ong-range core states such as akai atoms. A further improvement of pseudopotentia accuracy may be achieved by expicity incuding semi-core states in the cacuation. 1. Introduction Recent impementations of eectronic structure methodoogies can be divided into two groups. On one side, there are a-eectron (AE) approaches which consider both core and vaence eectrons expicity in the cacuation. Since core eectrons are strongy ocaized, these methods need to use ocaized basis functions such as augmented pane waves (APW) [1], inearized muffin-tin orbitas (LMTO) [2], Sater-type orbitas (STOs), or gaussian orbitas [3 to 5]. Aternativey, one can empoy pseudopotentias (PSPs) which effectivey project out the core states from the probem whie retaining the physica properties of the vaence region. Extensive discussions and reviews of this approach have appeared in Refs. [6 to 10]. Many pseudopotentia appications use a pane-wave basis set [11 to 13] but a variety of oca-orbita based impementations exist as we [14 to 21]. Currenty, a ot of work is aso aimed at mesh-based approaches [22 to 27]. Nevertheess, many PSP appications use pane-waves. The size of the basis needed for a cacuation of this type depends strongy on the shape (hardness) of the atomic pseudopotentias. Severa approaches have been proposed to construct soft pseudopotentias which minimize the numerica costs whie retaining the accuracy of the approach [28 to 31]. In this work, we use a modification of the scheme proposed by Trouier and Martins [31]. It wi be discussed in Section 2. Athough the aspect of PSP softness is vita for practica appications, the more important probem is pseudopotentia transferabiity, i.e. the abiity of the PSP to dea with eectronic configurations that differ from the one used for its construction. By definition, a PSPs that are created from first principes must reproduce the a-eectron

2 220 D. Porezag et a. eigenvaues of the eectronic reference state. In order to avoid an expicit spin dependence of the PSP, the spin-averaged ground state of the corresponding atom is usuay chosen. However, the actua ground states of most atoms are spin-poarized and the PSP must be transferabe in order to describe these states accuratey. This may not be the case for a PSP with a poor transferabiity. Simiar probems arise when atoms are paced in a moecuar environment and donate or accept eectrons. It is therefore important to systematicay study the resuts of PSP cacuations and compare them with the corresponding a-eectron data. This way, systematic PSP deficiencies can be found, understood, and corrected. Our paper is organized as foows. In Section 2, we wi describe the different eves of pseudopotentia approximations that may be used in practica cacuations and how to impement them. Section 3 focuses on discussing the accuracy of these approximations for cacuations on atoms and sma moecues. The resuts are summarized in Section Method If the density-functiona formaism is used in the usua Kohn-Sham approach [32, 33], the tota energy may be written as: where E ˆ T 0 r r Š dr r r V ext r E H r r Š E xc r r Š ; E H r r Š ˆ 1 dr r r V H r ; 2 r r ˆ Pocc jy i r j 2 ; T 0 r r Š ˆ Pocc i V H r ˆ i dr 0 r r 0 jr r 0 j ; 1 i r2 dr Y r 2 Y i r : 2 Note that atomic units for energy (1 a.u. ˆ 27:211 ev) and ength (1 a B = A) have been used in the above equations. V ext r is the externa potentia, for an atomic cacuation V ext r ˆ Z=jrj. T 0 r r Š is the kinetic energy of an interaction-free eectron gas which may be cacuated easiy using the Kohn-Sham orbitas Y i r. E xc r r Š is the exchange-correation energy which is a noninear functiona of the density r r. The exact expression for E xc is unknown but there are reasonabe approximations to it. In this paper, we have used both oca-density and the Perdew-Burke-Ernzerhof generaized-gradient approximations [34]. A generaized tota energy expression for spin-poarized systems may be obtained by aowing different Kohn-Sham orbitas for up and down eectrons and summing over both spin systems. Minimization of the expression (1) eads to the Kohn-Sham equations r2 2 V ext r V H r r Š V xc r r Š Y i r ˆ e i Y i r ; V xc r r Š ˆ de xc r r Š ; V eff r ˆ V ext r V H r r Š V xc r r Š : 3 dr r In an a-eectron cacuation, both core and vaence states are contained in the set of Kohn-Sham orbitas Y i used in the cacuation. Since core orbitas usuay change very itte if the atom is put in a different environment, it is a good approximation to assume

3 The Accuracy of the Pseudopotentia Approximation 221 that they are fixed for a certain atom type and that their effect may be simuated by a modified potentia (pseudopotentia). Experience has shown that a reiabe pseudopotentia shoud fufi the foowing conditions: 1. The (node-ess) owest-energy eigenfunction (pseudo-wavefunction) of the pseudopotentia has the same eigenvaue as the a-eectron vaence wave function. 2. The pseudo-wavefunction is identica to the a-eectron wave function outside a chosen cut-off radius r cut (norm conservation). There are ways to reax the second condition, for a discussion of these methods, see e.g. Ref. [30]. Since the a-eectron vaence functions Y for different anguar momenta (s, p,...) are different, the corresponding pseudopotentias V ps wi be -dependent. Therefore, they are caed semioca. There is a number of schemes eading to pseudopotentias which obey the two conditions stated above. Here, we wi focus on the approach suggested by Trouier and Martins [31]. Within this method, the pseudo-wavefunction Y ps is written as 8 >< Y ps r ˆ exp m 2 P c n r 2n for r r cut ; nˆ0 4 >: Y r for r r cut : The c n are determined from the foowing conditions: d n dr n Y ps r cut ˆ dn dr n Y r cut 8n ˆ 0; m match first m derivatives ; r cut 0 r 2 Y ps r 2 Y r 2 ˆ 0 norm conservation ; d 2 dr 2 Vps=scr 0 ˆ 0 smoothness condition ; 5 where V ps=scr r is the screened pseudopotentia repacing the effective a-eectron potentia in Eq. (3) and reated to Y ps r by V ps=scr r ˆ e r2 Y ps r 2Y ps r (inverted Kohn-Sham equation). In the origina Trouier-Martins approach, ony the first four derivatives of Y ps r were matched at r ˆ r cut. In order to improve the PSP smoothness in this area, we match the first six derivatives. As a resut, the eigenvaue of the a-eectron vaence wave function is usuay reproduced by the pseudo-wavefunction within 1 mev. As an exampe, Fig. 1 shows pseudo and a-eectron vaence wave functions for the suphur atom. As has been noted earier, the screened PSP V ps=scr r in a pseudopotentia cacuation is equivaent to the effective potentia V eff r in an a-eectron cacuation, r2 2 Vps=scr r Y ps r ˆ e Y ps r : 7 However, what is needed for appications of the formaism to arbitrary systems is the unscreened PSP V ps=unscr r which is equivaent to the externa potentia in an a-eectron cacuation. In the traditiona PSP approach, the pseudo-density r ps r takes the 6

4 222 D. Porezag et a. Fig. 1. Pseudo and a-eectron vaence wave functions for suphur pace of the tota density r r in an a-eectron cacuation. Therefore, the PSP Kohn- Sham equation reads r2 2 Vps=unscr r V H r ps r Š V xc r ps r Š Y ps r ˆ e Y ps r : 8 and hence V ps=unscr r ˆ V ps=scr r V H r ps r Š V xc r ps r Š : 9 However, this approach negects that in contrast to the Couomb potentia V H which is inear in the charge density, E xc and V xc are noninear functionas. Consequenty, core and vaence contributions to the exchange-correation energy cannot be separated in a simpe manner. For this reason, Louie et a. [35] introduced a scheme which takes the noninear nature of the E xc functiona into account by expicity adding a core charge density, i.e. repacing r ps r by r ps r r c r. Hence, the unscreening procedure is modified V ps=unscr r ˆ V ps=scr r V H r ps r Š V xc r ps r r c r Š : 10 This correction to the standard PSP approach is commony known as noninear core correction (NLCC). If r c is chosen as r c r ˆ r r r ps r ˆ r c fu r (fu NLCC), the corresponding PSP shoud give the same resuts as an a-eectron cacuation, provided that core reaxation effects are negigibe. However, such an approach is not feasibe for pane-wave cacuations because of the strongy ocaized nature of the true core charge density. For this reason, Louie et a. [35] suggested a partia NLCC characterized by a pseudo core density that is equivaent to the true core charge density for a radius r r core and smooth for smaer radii. Unfortunatey, the origina ansatz for r c r of Ref. [35] does not ead to a smooth transition at r ˆ r core. Therefore, we have deveoped a different procedure to determine this quantity. We set 8 < r c r ˆ exp m 1 P d n r 2n for r r core ; 11 : nˆ0 r c fu r for r r core ;

5 The Accuracy of the Pseudopotentia Approximation 223 and the d n are determined based on the foowing conditions: d n dr n rc r core ˆ dn dr n rc fu r core 8n ˆ 0; m match first m derivatives ; r core! 2 d 2 dr 2 rc r! min smoothness condition ; 12 0 Figure 2 iustrates the PSP unscreening without the NLCC and with a partia NLCC for the suphur atom. Note that the unscreened potentias depend on the magnitude of the NLCC. After the unscreened PSP has been defined, it can be used in eectronic structure cacuations. A resuts presented here have been obtained using the gaussian-orbita based density functiona code NRLMOL deveoped by Pederson, Porezag, and Jackson [4, 5, 20]. This method is based on an accurate numerica integration scheme. Besides the expansion of the wave functions into gaussian orbitas, no additiona approximations are made. The program aows for mixed a-eectron and pseudopotentia descriptions within the same cacuation, i.e. some atoms may be treated within an a-eectron Fig. 2. Upper part: vaence density and core densities with fu and partia NLCC for suphur. Lower part: semioca PSPs without and with partia NLCC for suphur (a.u. atomic units)

6 224 D. Porezag et a. scheme whie others may be described with PSPs. We have empoyed very arge basis sets and checked that a resuts are converged with respect to basis set size. The generaized-gradient (GGA-PBE) and oca-density (LDA) approximations [34] have been used to describe exchange-correation effects. 3. Resuts and Discussion Before presenting our resuts for moecues, it is worthwhie to ook at the cacuated properties of free atoms. In fact, whie the PSP construction is usuay performed based on spin-unpoarized eectronic states, the actua atomic ground state is often characterized by a non-vanishing tota spin. It is therefore important to compare the reative energies of different eectronic configurations and compare them to the corresponding a-eectron atoms. One of the most important figures is the energy difference between the spin-averaged eectronic state used in the PSP construction and the (frequenty spin-poarized) atomic ground state. This quantity wi be referred to as the atomic spin poarization energy (SPE) in the foowing. Tabe 1 summarizes the cut-off and NLCC core radii used to construct the PSPs for the eements discussed here. The spin-unpoarized eectronic ground state of the corresponding atom was used as the PSP construction reference state. 3.1 Atoms Figure 3 shows the cacuated differences between PSP and a-eectron SPEs with and without the NLCC for the first-row atoms B through Ne. These atoms are characterized by a singe, strongy ocaized 1s core she. Therefore, it is usuay assumed that they can be treated without the NLCC. However, since the vaence states in these systems are aso fairy compact, there is sti a significant overap between vaence and core charge densities. As a resut, the PSP based spin poarization energies show significant errors as compared to the corresponding a-eectron vaues if noninear core corrections are ignored. The deviations are argest for N which has a fuy poarized 2p she and a tota SPE of 3.02 ev within LDA and 3.12 ev within GGA-PBE. If the NLCC is not appied, the SPE errors amount to about 10% of the tota SPE. The deviations can be as arge as 0.3 ev (for N and O within GGA-PBE). Errors of this magnitude wi ead to discrepancies of approximatey 0.6 ev if binding energies for homo-nucear dimers are cacuated. For exampe, the AE GGA-PBE dissociation energy for O 2 is 6.2 ev whereas it is determined to be 5.7 ev in the PSP approach without the NLCC. Ta b e 1 Cut-off and core radii [in a B ] used for the pseudopotentia construction. Mn c corresponds to a conservative Mn pseudopotentia which incudes the 3s and 3p semi-core states in the vaence basis H Li B C N O F S Mn Mn c s p d NLCC

7 The Accuracy of the Pseudopotentia Approximation 225 Fig. 3. Errors of the atomic spin-poarization energies with respect to a-eectron vaues as cacuated with fu NLCC and without NLCC Incuding the NLCC greaty improves the performance of the PSP approach, eading to a maximum deviation of ony 0.02 ev (for the SPE of N). Hence, it can be concuded that SPEs of first-row atoms are very we described if a fu NLCC is appied. The question which remains to be answered is how the SPE error depends on the choice of the core radius if ony a partia NLCC is appied. Tabe 2 shows resuts for a GGA-PBE cacuation on oxygen (the behavior is very simiar for the other first-row atoms as we). For core radii of about 0.4a B or smaer, the spin poarization energies determined with a partia NLCC do not differ from the fu NLCC resuts. Ta b e 2 Atomic GGA-PBE spin poarization energies [in ev] for oxygen as cacuated with aeectron and PSP methods using NLCC and different core radii r core (first ine, given in a B ). r core ˆ 1 corresponds to a negect of NLCC AE SPE error

8 226 D. Porezag et a. Ta b e 3 Atomic LDA spin poarization energies [in ev] for Mn as cacuated with a-eectron and PSP methods using different core radii r core (first ine, given in a B ). r core ˆ 1 corresponds to a negect of NLCC AE The deficiencies of the PSP approach without the NLCC become even more apparent for transition meta atoms. These systems have strongy ocaized d eectrons. In fact, the d charge density in the 3d transition metas overaps substantiay with the 3s and 3p core density. In the foowing we consider the Mn atom which has a fuy spinpoarized 3d she. Tabe 3 shows resuts for the LDA spin poarization energies as cacuated within the a-eectron and PSP approaches using different core radii for the NLCC. It is apparent that the PSP method without the NLCC gives competey unreasonabe resuts. Further, the core radius must be chosen fairy sma (0.43a B ) in order to achieve converged resuts. However, if the NLCC is appied correcty, it eads to a very good agreement with the a-eectron cacuation. Another interesting and important property is the s±d transfer energy (i.e. the tota energy differences between the atomic s 2 d n and s 1 d n 1 eectronic configurations). We have cacuated this quantity and give the resuts in Tabe 4. The a-eectron and converged PSP + NLCC s±d transfer energies are found at 1.04 and 1.20 ev, respectivey. Note that the error increases significanty if r core is chosen too arge. Further, athough the NLCC eads to a much improved description of the s±d eectron transfer, there is sti a difference of 0.16 ev between PSP and a-eectron resuts which needs further discussion. The cacuations presented so far were based on PSPs which assigned the 3s and 3p states to the core. However, when we examine the spatia range of these states, we find that they strongy overap with the spatia range of the the 3d eectrons. Transferring an eectron from the rather diffuse 4s state to the ocaized 3d state wi increase the Couomb repusion in this area and hence ead to a reaxation of the 3s and 3p states. Assigning these states to the core makes such a reaxation impossibe. For this reason, we have constructed a `conservative' pseudopotentia for Mn which incudes the 3s and 3p states in the vaence basis. As can be seen from Tabe 4, the conservative approach eads to resuts which are in exceent agreement with the a-eectron data. Therefore, it can be concuded that the PSP approach can describe transition meta atoms with a simiar accuracy as other atom types if both noninear core±vaence interactions and semi-core reaxation effects are incuded in the method. Ta b e 4 Atomic LDA s±d transfer energies [in ev] for Mn as cacuated with a-eectron and PSP methods using different core radii r core (first ine, given in a B ). r core ˆ 1 corresponds to a negect of NLCC. The Ar core ine refers to a cacuation where the 3sp eectrons are treated as core eectrons AE Ar core Ne core

9 The Accuracy of the Pseudopotentia Approximation 227 Fig. 4. LDA cohesive energy for the LiF moecue as a function of internucear distance. The PSP+NLCC resuts for r core ˆ 1.62a B are identica to the fu NLCC 3.2 LiF diatomic moecue The preceding section was aimed at discussing energy differences induced by simpy changing the occupation numbers of the atomic one-eectron orbitas. However, in many practica appications, the most important change in the atomic eectron density occurs when the atom is incorporated into a soid, custer, or moecue. The argest reative changes wi occur for atoms with a very diffuse vaence density if they are paired with atoms characterized by compact ocaized states. Prominent exampes for such systems are akai-haogenide compounds such as LiF. In order to demonstrate the probems which occur for this combination of eements, we focus on the LiF diatomic moecue. Figure 4 shows the cohesive energy of this system as a function of interatomic distance as cacuated within LDA. The PSP cacuation without the NLCC overestimates the binding energy by about 1 ev and underestimates the bond ength by more than 0.5a B. Appying a fu NLCC corrects this deficiency and eads to a good agreement with the a-eectron resut. Partia and fu NLCC resuts agree for r core 1:62a B. The remaining difference between PSP + NLCC and AE data can be attributed to the negect of the 1s core reaxation. Note that in order to rue out additiona sources of error, the F atom has aso been treated in an a-eectron fashion here. 3.3 Structura and vibrationa properties of sma moecues The preceding sections have estabished that it is often necessary to appy the NLCC in order to obtain reiabe resuts. Whie there are aso many cases where the NLCC has negigiby sma effects, we advocate to aways use this correction since it usuay does not increase the numerica effort of a cacuation very much, especiay not in cacuations with gaussian-orbita basis sets. Consequenty, the resuts presented in the foow-

10 228 D. Porezag et a. Ta b e 5 Bond engths [in a B ] and anges for some moecues as cacuated with the a-eectron (AE) and PSP NLCC approaches. The GGA-PBE exchange-correatioin functiona has been used AE PSP AE PSP H 2 r(hh) LiH r(lih) Li 2 r(lili) LiF r(lif) N 2 r(nn) H 2 O r(oh) O 2 r(oo) (HOH) F 2 r(ff) CH 4 r(ch) CO r(co) SO 2 r(so) HF r(hf) (OSO) ing are based on cacuations with pseudopotentias that empoy the NLCC and use the cut-off and core radii summarized in Tabe 1. For a more detaied discussion on the proper choice of the NLCC core radii pease refer to Ref. [36]. Tabe 5 ists the cacuated bond engths and anges for tweve moecues as cacuated with the a-eectron and pseudopotentia approaches and the GGA-PBE exchange-correation functiona. PSP and AE resuts are in exceent agreement with a root-mean-square (RMS) error of ony 0.003a B for the bond engths. The argest errors occur for LiH and LiF which can be attributed to the negect of the Li core reaxation effects. Tabe 6 contains information about cacuated atomization energies. Again, AE and PSP data are in very good agreement, the deviations do not exceed 0.04 ev per atom. The current PSP resuts show much smaer errors than the ones pubished in an earier paper [20] which were based on BHS pseudopotentias without noninear core corrections. This is especiay true for the moecues containing Li where the previous PSP cacuations gave binding energies with errors of severa ev as compared to the corresponding a-eectron vaues. This iustrates that the NLCC is especiay important for akai atoms. Note that a atomization energies are consistenty underestimated. This is most ikey due to the negect of core reaxation effects for the moecuar species. Tabe 7 summarizes the vibrationa properties of the moecues investigated here. Frequencies, IR intensities, and Raman scattering activities have been determined with a method pubished in Ref. [37]. The PSP based vibrationa frequencies are in exceent agreement with the AE cacuations. The two methodoogies usuay differ by ony a few wavenumbers, or at most 1%. However, the behavior of the IR intensities and Ra- Ta b e 6 Atomization energies [in ev] for some moecues as cacuated with the a-eectron (AE) and PSP NLCC approaches. The GGA-PBE exchange-correation functiona has been used. Zero-point motion has been incuded in the vaues presented here AE PSP D AE PSP D H HF Li LiH N LiF O H 2 O F CH CO SO

11 The Accuracy of the Pseudopotentia Approximation 229 Ta b e 7 Harmonic vibrationa frequencies [in cm 1 ], IR intensities [in (D/A) 2 amu 1 ] and Raman activities [in A 4 amu 1 ] for some moecues as cacuated with the a-eectron (AE) and PSP + NLCC approaches. The GGA-PBE exchange-correation functiona has been used AE PSP freq. IR Ram. freq. IR Ram. H 2 S g Li 2 S g N 2 S g O 2 S g F 2 S g CO S HF S LiH S LiF S H 2 O A B CH 4 T E A SO 2 A B man activities (which depends strongy on subte changes of the moecuar charge distribution as a function of the nucear coordinates) is different. Whie many IR and Raman strengths agree very we (within 5%) there are aso cases with significanty arger deviations. For exampe, the spectra intensities of a moecues containing Li show errors of about 10%, indicating that the effect of the Li core eectrons cannot be competey negected. However, even the argest errors do not exceed the typica differences between theory and experiment which were found to be 10 to 30% in Ref. [37]. 4. Summary and Concusions We have investigated the accuracy of the pseudopotentia approximation as compared to the a-eectron approach for a variety of sma moecues using norm-conserving Trouier-Martins PSPs and a gaussian-orbita based numerica scheme. It is found that noninear core corrections are often necessary to obtain reiabe resuts, therefore, we suggest to aways use them in appications of the method to arger systems. We have shown that the PSP approach with NLCC predicts equiibrium structures, energies, and vibrationa frequencies with a very high accuracy. Resuts for IR and Raman intensities show arger errors but are sti sufficienty accurate for most practica appications. Remaining deviations between the PSP and AE methods can be expained by core reaxation effects. Consequenty, the performance of the PSP approximation can be substantiay improved by using conservative pseudopotentias, i.e. by incuding semi-core states expicity in the cacuation. In summary, it can be concuded that the combined PSP + NLCC approach is very we suited for the cass of systems investigated here.

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