I. INTRODUCTION. Periodic boundary conditions in ab initio calculations

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1 PHYSCAL REVEW 8 VOLUME 51, NUMBER 7 15 FEBRUARY Periodic boundary conditions in ab initio cacuations G. Makov and M. C. Payne Cavendish Laboratory, Madingey Road, Cambridge CB3 OHE, United Kingdom (Received 19 Juy 1994) The convergence of the eectrostatic energy in cacuations using periodic boundary conditions is considered in the context of periodic soids and ocaized aperiodic systems in the gas and condensed phases. Conditions for the absoute convergence of the tota energy in periodic boundary conditions are obtained, and their impications for cacuations of the properties of poarized soids under the zero-fied assumption are discussed. For aperiodic systems the exact eectrostatic energy functiona in periodic boundary conditions is obtained. The convergence in such systems is considered in the imit of arge supers, where, in the gas phase, the computationa effort is proportiona to the voume. t is shown that for neutra ocaized aperiodic systems in either the gas or condensed phases, the energy can aways be made to converge as O(L ) where L is the inear dimension of the super. For charged systems, convergence at this rate can be achieved after adding correction terms to the energy to account for spurious interactions induced by the periodic boundary conditions. These terms are derived exacty for the gas phase and heuristicay for the condensed phase.. NTRODUCTON Periodic boundary conditions (PBCs) have been appied extensivey in theoretica modeing of crystaine soids. ntroduction of these boundary conditions is equivaent to considering an infinite Bravais attice. The shape of the to which the PBCs are appied determines the type of attice, and the contents of the determine the attice basis. n particuar, quantum ab initio cacuations have combined periodic boundary conditions with pane-wave expansions to create a powerfu cacuationa scheme, which has been appied extensivey in the study of properties of crystaine soids. There is some interest in extending this method to aperiodic systems such as moecues, defects in soids, and disordered or iquid-condensed matter. This interest arises from the foowing considerations: (i) PBCs are a simpe way to impose the boundary conditions in cacuations of condensed matter. (ii) PBC are compatibe with pane-wave expansions, which in turn aow for reativey simpe cacuations of forces in moecuar-dynamics simuations. (iii) Unified numerica schemes can be set up to consider both periodic and aperiodic systems. There is, however, a major difference between (static) cacuations on periodic and on aperiodic systems when PBCs are used. For periodic systems, a cacuation on one unit can yied a the information that may be obtained. n an aperiodic system, there is no periodic unit. nstead the cacuation is performed on a portion of the system of interest contained in a super, which is then periodicay repeated in space for cacuationa convenience. Ony in the imit of an infinitey arge super do the cacuated resuts converge to the properties of the aperiodic system. f the aperiodicity is oca (e.g., a ocaized defect in a soid, a moecue in a gas), then the difference between resuts obtained using any finite-sized super and those obtained in an infinitey arge super arises from the spurious interactions of the system with its images in neighboring supers. f the aperiodicity exists at a ength scaes, as in the case of a random soid, then, in addition to the above, the energy wi Auctuate statisticay about its expected vaue. n this paper we consider ony the first of these two possibiities. Since the resuts required in a cacuation of an aperiodic system are obtained ony in the imit L +, where L is the inear dimension of the super, it is necessary to consider their convergence. This can depend on the quantity being cacuated. n practice, it is often sufricient to consider the convergence of the energy as a the other properties are determined through the variationa principe. The convergence of the energy, as the size of the super is increased, is determined by the ongest-ranged forces, which in genera are the eectrostatic forces. n soids, the eastic forces can aso be ong ranged and in principe can be treated simiar to the eectrostatic forces by the methods described beow. The rate of convergence can be very sow, as in the case of a charged species for which the energy converges as L The importance of the rate of convergence ies in the fact that the computationa effort is proportiona to the voume of the super. Typicay, in a Kohn-Sham-type cacuation using a pane-wave expansion and PBC, the computationa effort increases as V, nv, for an isoated moecuar species, where V, is the super voume (i.e., V, -L ). f the exact asymptotic dependence on the inear dimension of the super is known, then the energy cacuated at finite L can be corrected to provide a better, and more rapidy converging, estimate of the energy in an infinitey arge super. A step in this direction was taken by Lesie and Gian who considered the case of a charged defect in a soid and showed heuristicay that the eading term in the convergence of the defect formation energy was yl, where y is a cacuabe constant invoving the dieectric constant, for which they used an experimenta vaue /95/51(7)/4014(9)/$ The American Physica Society

2 from "+, 51 PERODC BOUNDARY CONDTONS N AB NTO CALCULATONS 4015 To obtain the convergence properties of the eectrostatic energy in the imit of infinitey arge supers, it is necessary to have an expression for the eectrostatic energy in PBCs. Such expressions were first considered, to our knowedge, eary this century in the context of the cohesive energy of ionic soids. However, despite the many years that have passed there is continuing controversy over the exact properties of these expressions. The source of much of this controversy is the fact that eectrostatic sums on an infinite attice are not aways absoutey convergent, but for some attice bases can be ony conditionay convergent or even divergent. This can introduce a certain ambiguity into the resuts, commony in the form of a constant and unknown potentia, but in some cases in the form of a constant and unknown eectric fied (see Sec. beow). The probem is sometimes compounded by the use of mathematicay idefined operations invoving divergent terms and imits of vectors. An important exception to this, was the work of de Leeuw, Perram, and Smith who considered in detai the eectrostatic energy of a neutra assemby of cassica point charges in a repeated cubic super. n what foows, we present a systematic derivation of the eectrostatic energy of an assemby of point charges together with a continuous charge distribution paced in a periodicay repeated of arbitrary geometry. This derivation is a generaization of the work of de Leeuw, Perram, and Smith. n the course of making this derivation, some conditions for modeing a crysta as an infinite soid by the introduction of PBC s are obtained, in particuar with respect to poarized crystas. Then, in Sec., the asymptotic convergence of the eectrostatic energy of neutra and charged aperiodic systems in PBC is considered. First, we show that for isoated moecuar species the rate of convergence can aways be reduced to 0 (L ). This is iustrated with severa numerica exampes. Then, for the case of aperiodic systems in condensed phases, we show that the same rate of convergence can be achieved if the aperiodic system is neutra. For a charged aperiodic system, the convergence can be reduced to 0 (L ) by introduction of the dieectric constant, which may be obtained separatey or by fitting the data. A short discussion and a summary concude this paper.. CALCULATON OF THE ELECTROSTATC ENERGY N PBC Consider a finite sampe of N periodicay repeated s, each of which contains a charge density p(r) comprised of point charges and a continuous charge distribution n (r), p(r)= gz, 5(r r;)+n(r). This charge density is periodic with the periodicity of the attice vectors, p(r+1)=p(r). The eectrostatic potentia at an arbitrary point r in the sampe is p(r)= f X sampe d r =y f dr een X X s d Es 1 r r r+1 radia moment of the charge distribution). P are the Legendre functions with coso=(r r) /r r1. For odd vaues of n, the inversion symmetry of the Bravais attice ensures that the asymptotic contributions from this mutipoe to the sum are identicay zero. f the owest nonzero mutipoe is n =0, then there is a net charge in the unit and the sum of the asymptotic terms in Eq. (3) diverges. f it is n =3, or greater, then the sum in Eq. (3) is seen to converge absoutey. f the owest nonzero mutipoe is n =2, then the sum in Eq. (3) is conditionay convergent by Dirichets test. Note that these concusions require the attice sum in Eq. (2) to be three-dimensiona. The same approach can be appied to derivatives of the potentia: One finds that, e.g., the equivaent sum for the eectric fied is absoutey convergent if the owest nonzero mutipoe is n =2 or greater. This sum is conditionay convergent if n =1 and diverges if n =0. What is the extent of the indeterminacy introduced by the conwhere the sum is over a the N s. The in which the point r is ocated, is taken as the origin of the attice. To obtain resut (2), it is necessary to assume the existence of zero-potentia Dirichet boundary conditions at infinity. f the sampe is macroscopic, then N is very arge and we woud ike to simpify the mathematics by extending the finite sum over the attice vectors in Eq. (2) to an infinite sum. This extension is equivaent to introducing periodic boundary conditions and may be performed ony if the sum does not diverge. There are two possibe cases to be considered. (i) The sum is absoutey convergent, in which case the potentia wi have converged for arge N and extending the sum to infinity wi not affect the resuts, or in physica terminoogy the surface terms make a negigibe contribution to the potentia. (ii) The sum is conditionay convergent, in which case the resut depends on the order of summation, or in physica terminoogy on the contribution of the surface terms. What are the convergence properties of the sum in Eq. (2) after it has been extended to infinity? These can be obtained by transforming the sum over the attice vectors to a sum over spherica shes s of attice sites at distances the attice origin, The convergence wi be determined by the contributions of distant shes (arge ) to the sum. n this imit, the asymptotic form of the summand in Eq. (2) is we known to be q P (cose} (4) where n is determined by q, which is the owest nonzero mutipoe of the charge distribution p (defined as the nth (3)

3 4016 G. MAKOV AND M. C. PAYNE 51 ditiona convergence? Consider first the potentia: t is conditionay convergent if and ony if the owest nonzero mutipoe is n =2 (quadrupoe). This is the owest mutipoe if the unit is neutra and does not have a dipoe. n this case, the sum for the eectric fied is absoutey convergent, so the potentia is undetermined to an additive constant. This constant is determined by the contributions of the charges at the surface of the sampe and can be identified as the eectrostatic surface barrier. Therefore, this constant cannot be determined by this cacuation. Not knowing the vaue of the constant potentia in the attice does not affect the cacuation of the energy of the unit because it is charge neutra. t aso does not affect the forces on the ions and eectrons in the as it is independent of position. The exact summation of the potentia in an infinite attice has been obtained in the Appendix and, in this case, it is equa to P(r)= J d rp(r) g(r, r) r 3V, tj(r, r)= g erfc(gr r+ ) c g+p g g /4g ig (r r) e (5) where g are the reciproca attice vectors and V, is the voume of the unit. Now consider a unit with a nonvanishing dipoe, noting that the vaue of the dipoe can be i defined when periodic boundary conditions are used. This is iustrated in Fig. 1 for a cassica point-charge distribution. As can be seen, the choice of super (a) or (b) yieds very different dipoe moments, whie the choice of super (c) yieds the same dipoe moment as (a) (a simiar argument can be extended for higher mutipoes). n an infinite attice of unit s with nonvanishing dipoes, we have shown above that the sum for the eectric fied is conditionay convergent and that the sum for the potentia is divergent. From simiar arguments to those used before for the potentia, one sees that the derivatives of the eectric fied with respect to space are absoutey convergent, so the extent of the indeterminacy is an unknown constant eectric fied. (See aso the discussion in Ref. 11. ) This means that the energy per unit of the infinite sampe and the forces on the eectrons and ions in the unit are indeterminate unti the surface contributions have been defined. n the Appendix, we have cacuated the potentia in a attice with a dipoar basis under the assumption of spherica boundary conditions and found P(r)= J d rp(r) P(r, r) r 3V, 4 + r d rrp r 3 V, where g is defined in Eq. (5). Comparison with the potentia in an infinite attice without a dipoe, (5), shows that ony the ast term in (6) originates in the dipoar contribution. ndeed it has the form of a constant eectric fied, the magnitude of which is determined by the shape of the boundary conditions and the definition of the unit. n a periodic soid, a choices of super geometry reative to the charge distribution shoud yied equivaent potentias (and energies). This is not the case with Eq. (6); however, this merey reaects the fact that the potentia in an infinite attice with a dipoar basis is not uniquey defined but depends on the contributions of the surface terms, as discussed above. Some cacuations on poarized soids have used periodic boundary conditions with an additiona assumption of zero eectric fied. This means that an externa process has caused the potentia at different surfaces to be equa. This is often the case in macroscopic sampes of poarizabe soids where impurities tend to adsorb on the surface and equiibrate the potentia. Therefore, one may appy the zero-fied condition to cacuate static properties of such a poarized crysta using an infinite attice mode. (The resuts may not necessariy appy to a pure crystaine sampe. ) The eectrostatic energy functiona that shoud be used is that of Eq. (5), which is obtained from the dipoar case, (6), by removing the term inear in the eectric fied. Cacuations of dynamic processes are ikey to invove a change in the surface configuration and, therefore, the creation of potentia differences and the associated eectric fieds (i.e., piezoeectric effects). The zero-fied approximation can sti be appied in such cacuations, but this impies the existence of an externa process, which rapidy equaizes the potentia on the different surfaces of the crysta. The existence of such an externa process, and its reevance to the dynamics, must be considered separatey for each case.. APERODC SYSTEMS N PBC FCx. 1. A section of a periodic cassica charge distribution, denoted by the + and signs, with three possibe choices for the super. When cacuating the energy of an aperiodic system using periodic boundary conditions, one is interested ony in the energy, Eo in the imit L, where L is the inear dimension of the super. The energy cacuated for a finite super E (L) differs from Eo, because of the spurious interactions of the aperiodic charge density with its images in neighboring s. Furthermore, these in-

4 . to 51 PERODC BOUNDARY CONDTONS N AB NTO CALCULATONS 4017 teractions induce changes in the aperiodic charge density itsef, which depend on L. To estimate Eo from the cacuated E(L), we need to know the asymptotic dependence of the energy on L. For simpicity, we concentrate first on the case of an isoated moecuar species in a arge super, such that the entire moecuar charge density is contained in the super. n practice, the eectron density decays exponentiay away from the moecuar and the super need ony be arge enough for the density at the surface of the to be in this regime. f there was no induced charge density, then the mutipoe moments woud be exponentiay convergent with L, whie we wish to study the convergence of the eectrostatic energy, which wi have the form of a power aw in L. The eading term in the induced charge density wi be the induced dipoe, the magnitude of which is determined by the eectric fied. f the moecue has a permanent dipoe then the induced dipoe wi be 0(L ) and its eading contribution to the eectrostatic energy 0 (L ), reaecting the dipoe-induced dipoe interactions. f the moecue has no dipoe moment, then the induced density is 0(L ) at east. [This resut is aso obtained for a moecue with a permanent dipoe moment, if the super is chosen to be cubic see beow Eq. (9). ] As we confine our discussion beow to terms in the energy that converge as 0 (L ), we can ignore the effects of induced charge distributions on the convergence of the energy of an isoated moecuar species. A. Neutra moecuar species Consider first the case of an isoated neutra moecue that has no dipoe moment. The energy is absoutey convergent and is given by Eq. (A14) in the Appendix or equivaent by direct summation of Eq. (2) and integration. The exact detais need not concern us here as it is sufficient to note that the asymptotic behavior is determined by the quadrupoe-quadrupoe interaction, which has the functiona dependence of For arge supers, the quadrupoes wi be independent of L and the energy wi be E( L)=E 0+0(L ). Consider next an isoated neutra moecue that has a dipoe moment. n this case, we first have to define what energy is being cacuated, because the energy of an infinite attice with a basis that has a dipoe moment is not we defined, as discussed above in Sec.. However, in a cacuation of an aperiodic system the infinite attice is merey a device and is not meant to correspond to any physica reaity. Therefore, the order of summation of the eectrostatic sum (i.e., the surface) can be chosen for convenience. Choice of summation over spherica shes is considered in the Appendix and the eectrostatic energy functiona is found to be E =, f d r p(r) f d r p(r)g(r, r) + f 2 d rrp(r) (&) where if there are point charges in the charge distribution, the modified functiona in Eq. (A14) of the Appendix shoud be used. Two comments shoud be made about Eq. (8). (i) t differs from commony used energy functiona (e.g., that in Ref. 1) in having an additiona dipoedependent term. n the case of an infinite periodic soid, the same arguments appy as those used above in discussing the potentia. For an aperiodic system, the absence of this term wi not change the vaue of the energy in the imit Loo, Eo, as the additiona term is 0(L ). However, if the dipoe term in Eq. (8) is not incuded then the energy wi converge to Eo with an additiona 0 (L ) term. Furthermore, since this additiona term in the energy functiona contains the position coordinate expicity, it wi aso mean that the forces wi converge more sowy to their imiting vaue. (ii) t has been noted that the dipoe moment is i defined in PBC s. n a periodic soid, a choices of super geometry reative to the charge distribution shoud yied equivaent energies. This is not the case with Eq. (8) as different choices of super wi yied different vaues of the dipoe moment and, therefore, different energies. However, this is not reay a probem as it merey reaects the fact that an infinite periodic soid with a dipoar basis does not have a we-defined energy. However, in a cacuation of an aperiodic system, different choices of super need not be equivaent. The reevant choice of super geometry reative to the charge distribution is determined by the system to be considered in the imit L. The super chosen must incude the entire aperiodic system in the same configuration as it woud be in the buk imit. f this condition is obeyed, then the dipoe moment is invariant to the choice of super as in choices (a) and (c) in Fig. 1. Choosing super (b) woud impy choosing to study a different aperiodic system in the imit L +. What is the asymptotic convergence of the energy with increasing super size in this case? From the same considerations as above, the energy is dominated by the dipoe-dipoe interactions, which have the functiona dependence 1 and, therefore, in genera, woud ead to an 0(L ) convergence. However, by choosing the super to be a cube we can make use of the specia symmetry of the Legendre functions on a cube, first pointed out in this context by Nijboer and dewette. " The contribution of the dipoes on a spherica she of radius 1 the eectric fied at the origin is 3 2 P- J =X,J, Z 1E where is the cartesian component of in direction j. This resut impies that dipoes on a cubic attice do not interact and, therefore, the energy in such a cacuation wi converge 0(L ). f it is necessary to use a super with a shape other than a cube, then the contribution to the energy arising from the dipoe interactions may be cacuated by performing the reevant attice sum, which depends on the specific attice geometry. As an exampe, we have cacuated the potentia energy of stretching a NaC1 moecue by 0.3 A from equiibrium. The cacuations were based on the Kohn-Sham

5 4018 G. MAKOV AND M. C. PAYNE 51 OQ o F L (Angstrorns) FG. 2. The potentia energy of stretching a NaC moecue by 0.3 A from equiibrium cacuated for cubic supers of side L. The trianges refer to the eectrostatic energy functiona in Eq. {8), whie the fied circes refer to the same functiona without the dipoe term. method, using pseudopotentias to represent the core eectrons and the nucei, periodic boundary conditions, and a pane-wave representation of the eectron wave functions. The super was chosen to have a cubic geometry and the moecue was ocated at the center of the cube. The cacuations were performed using the program castep, the principes of which have been described esewhere. The cacuation was performed both with the dipoar term in the eectrostatic energy functiona and without. Therefore, we expect the resuts to converge as 0(L ) and 0(L ), respectivey. The resuts are shown in Fig. 2, where one can ceary see that the energy cacuated without the dipoe term in Eq. (8) converges more sowy than the energy cacuated with this term. Aso, note that both cacuations converge to the same resut within numerica accuracy. (The numerica error increases with size due to an instabiity associated with the representation of the eectrostatic potentia in reciproca space. ) B. Nonneutra moecuar species. Cacuation of the energy of a charged system is of interest for ions or charged impurities in crystaine soids. However, the energy of a periodicay repeated eectricay charged system diverges. For an aperiodic system, practica interest is restricted to the imit of an infinitey arge super, which contains the charged system. This imit is identica to that obtained from a simiar system, which consists of the origina charged system immersed in a jeium background which fis the super and neutraizes the charge, so that the net charge is zero. For this new system, the cacuation can proceed as discussed in the previous section. The energy of the unit in this cacuation wi converge sowy, reaecting the decreasing interaction between the charge species and the jeium background as the super is taken to be arger and arger. This convergence wi have the form of a power aw in L, and in this section we obtain the asymptotic dependence on super size of the eectrostatic energy of a charge species to 0 (L ). The charge density of the immersed system consists of the density of the charged species, p, (r) and the jeium density, n0; and the point charge in p, and that arising from the interaction of p2 and no. The interaction of the point charge with p2 can be considered in the imit of the conp(r)=p, (r)+no. Assuming a tota charge q on the charged species, then n0= q/v, to ensure charge neutraity. The density may be spit into two contributions by adding and subtracting a point charge q at r0, p(r) = [q5(r ro)+ [p, (r) q5(r ro) j. The density in the first brackets on the right-hand side of Eq. (11) wi be denoted as p& and the density in the second brackets as p2. r0 can be chosen to be any point in the super. A usefu choice, which we make use of beow, is to choose r& so that p2 has no dipoe, and the origin of the coordinates is chosen to be at the center of the unit. We wish to obtain the asymptotic contributions to the energy in the imit of arge supers. The eectrostatic energy can now be considered to be the sum of three energies: p on a attice interacting with itsef, E11,pz on a attice interacting with itsef, E22, and p, and p2 on a attice interacting ony with each other, E12. E11 is we known to be the Madeung energy of a attice of point charges immersed in neutraizing jeium E 2L (12) where n is the attice-dependent Madeung constant. E22 is the interaction energy of a neutra charge density on a attice. Since the p2 has no dipoe moment (due to the choice of ro), E22 wi converge as 0(L ) as discussed above. Last, consider E12. This is the interaction energy between two di6erent neutra charge densities, which at first might be expected to aso have an 0 (L ) asymptotic form. However, this resut was obtained under the assumption that the charge densities are ocaized and independent of super size. Whie for p2 this assumption hods, it does not hod for p where n0 is obviousy dependent on the super voume. The expicit form of E12, by simpe generaization of the resuts of the Appendix, is E,&=im f d r p, (r) s 0 X f d rp2(r)t(r r, s), (13) where g is defined in Eq. (A9), and pj refers to j density of Eq. (11). p2 does not have a dipoe moment, because of the choice of ro, which impies that the attice sum in g is absoutey convergent up to an irreevant constant potentia. This aows us to separate the energy in Eq. (13) into two parts; that arising from the interaction between p2

6 51 PERODC BOUNDARY CONDTONS N AB NTO CALCULATONS 4019 vergence parameter g +0, i.e., in the imit of a attice sum over the attice vectors. The eading term in this interaction is between the point charge q and the quadrupoe (second radia) moment of p2, Q. f the attice is chosen to be simpe cubic, then this interaction vanishes for the same symmetry reasons as discussed for the dipoe-dipoe interaction at Eq. (10) above. The interaction of the jeium with pz can be considered in the opposite imit of q, which means a attice sum over the reciproca vectors g. The jeium has no spatia structure and, therefore, ony the g=o contribution to the potentia (see Appendix) need be considered. This eads to the L dependent contribution E2= d rp2rr +0 3 V, (14) where V, =L for a cube. The asymptotic resut for the tota eectrostatic energy of a charged species on a cubic attice is (b) L (Angstroms) L (Angstrorns) FG. 3. (a) The ionization potentia of a Mg atom cacuated in cubic supers of side L (fied trianges). The same after appication of the Madeung correction Eq. (12) (fied squares). The same after appication of the present correction Eq. (15) (fied circes). (b) Expanded section of (a). 2 E =E 2L +O(L ), (15) where Q is the quadrupoe moment given by the integra in Eq. (14) and Eo is the desired eectrostatic energy of the isoated species. An exampe iustrating the size dependence of the energy on size in Eq. (15) is shown in Fig. 3. The ionization energy of an Mg atom has been cacuated by the methods described above using the eectrostatic energy of Eq. (8). Aso shown are the convergence after the appication of the Madeung correction of Eq. (12) and after the more extended correction of Eq. (15). t is cear that as the corrections are appied, convergence increases rapidy. C. Aperiodic system in condensed matter Consider first a ocaized (point) defect in a crystaine soid. n this case, the charge density can be considered to be the sum of two contributions the periodic charge density of the underying crystaine soid p (r), and the charge density of the aperiodic defect p,z(r); p(r) =p(r)+ p,p(r), p (r+)=p(r). (16) The aperiodic density p, (r) wi be a ocaized charge density simiar to a moecuar density. The eectrostatic energy can then be considered to be the sum of the interactions between (i) the periodic charge density and itsef, which is independent of L; (ii) the periodic and aperiodic charge densities, which is aso independent of L (and equa to the interaction of a singe isoated aperiodic density with an infinite periodic density); and (iii) the aperiodic densities ocated in different supers, which is L dependent. Note that the periodic part of the density must fufi the conditions of Sec., namey, it shoud not contribute a net charge in the unit or a net dipoe. t is reasonabe to assume that this mode wi be vaid for modeing any species in a condensed phase that is homogeneous beyond some ength scae (e.g., an ion in a soution). What is the asymptotic dependence of the aperiodic density and its mutipoes on the super dimensions? The aperiodic density depends on L through two mechanisms. One mechanism, as in the case of the isoated moecue, is changes in the charge distribution induced by interactions of the aperiodic charge with its images. The convergence with L of this case has been discussed above, and was found to converge faster than O(L ). The other source of L dependence in p, is the dieectric response of the periodic medium to the aperiodic density. The asymptotic term arising from this response cannot be obtained by the methods used above for the isoated moecue as it invoves the induced charge density, which impies noneectrostatic contributions to the energy. nstead, the phenomenoogica approach of Lesie and Cxian in which the potentia is reduced by the dieectric constant c can be appied. This correction is exact in the 1imit of arge L, and we appy it to the resuts of Sec.

7 G. MAKOV AND M. C. PAYNE B to obtain, for the case of a charged aperiodic system in a cubic super, E =Eo- q e 2.c 3L, c. +0(L ). (7) n this case, Q is the second radia moment ony of that part of the aperiodic density that does not arise from dieectric response or from the jeium, i.e., is asymptoticay independent of L. The two parameters e and Q are properties of the periodic density and the aperiodic density, respectivey. They may either be cacuated expicity (e can be obtained separated by considering the response of the periodic density to a point test charge, q «e, where e is the unit charge and for which Q =0), or by fitting the data to expression (17). For neutra aperiodic systems, the size dependence wi be unaffected by the introduction of the dieectric constant and wi remain 0(L ), assuming a square super for dipoar aperiodic densities. V. SUMMARY AND DSCUSSON n this paper, we have derived exacty the eectrostatic energy functiona for an infinite periodic attice with a basis, by generaizing and extending the work of de Leeuw, Perram, and Smith. n doing so we obtained conditions for using an infinite soid as a mode for a rea soid. These conditions are that the unit shoud be neutra and have no dipoe moment. f these conditions are not fufied, then the energy per unit of the soid diverges or is indeterminate uness further boundary conditions at the surface are imposed. However, static properties of a poarized soid can sti be cacuated under the assumption of zero eectric fied. This reaects the existence of externa processes that can cause the potentia at a the surfaces to be equa. Cacuations of dynamica processes under the zero-fied assumption, impy the existence of a faster externa mechanism, which equiibrates the potentia over the crysta surfaces. The convergence of the energy of a ocaized aperiodic system with respect to super size was aso considered in detai. The convergence is dominated by the eectrostatic interactions of the species under consideration and its images. We have shown that by suitabe choice of super geometry, as we as the empoyment of the exact eectrostatic energy functionas, the energy can be cacuated so that it wi converge to its imiting vaue as 0(L ). For neutra systems without a dipoe, this is aways the case. For systems with a dipoe moment, this is the case ony if a cubic super is used and if the eectrostatic energy functiona is derived correcty. n particuar, the commony used eectrostatic energy functiona was found to be missing a dipoar term, the addition of which increases the rate of convergence from 0 (L ) to 0(L ). For nonneutra isoated moecuar systems, we found the expicit asymptotic dependence on L to order 0 (L ) for a cubic super. Correcting the cacuated energy by adding these terms we found it converged rapidy, as expected. For an aperiodic system in condensed matter, we argued that the gas-phase terms may be generaized by introducing an empirica dieectric constant. These resuts for cacuations on aperiodic systems make the use of PBC highy competitive compared to the isoated system on a grid approach. The main benefit of this approach was that spurious interactions with the attice images were avoided. We have shown that these interactions can be converged rapidy with super size. This aows e%cient use of PBC in the study of aperiodic systems in the condensed phase. t is aso our beief that considerations simiar to those empoyed in this work can be appied to the convergence of ong-ranged eastic forces in microscopicay deformed soids, which are simiar in their form to eectrostatic forces. ACKNOWLEDGMENTS The authors acknowedge usefu discussions with Professor D. Vanderbit and with Dr. S. Crampin, Dr. A. de Vita, and Dr. G. Rajagopa. One of us (G.M. ) acknowedges the FCO-Core Foundation for financia support. APPENDX n this appendix the eectrostatic energy per of a charge distribution in a periodicay repeated super, is cacuated by the method of de Leeuw, Perram, and Smith who studied the reated probem of point charges in a cubic attice. The expression for the eectrostatic energy is E= d rpr r =, f d r p(r) g f d rp(r),, (A) cd& and the is charge neutra f d r p(r)= f d r n(r)+ gz;=0, 1 (AZ) where n (r) is the continuous part of the density and there are aso point charges, with the ith point charge having a charge z; and ocated at r;. As discussed in the main body of this work, the attice sum for the potentia in (A 1) can be conditionay convergent. To make the sum absoutey convergent, we intro- duce the convergence factor e s1. n the absoutey convergent sum we are aowed to exchange the sum and the integras; therefore, we consider the sum g(x, s) = g e s 1 x+ (A3) This sum can be transformed into a doube sum as in Eq. (3) in the main body of the paper. Then the sum over 1 is a one-dimensiona conditionay convergent series, for which the introduction of a convergence factor induces convergence to a definite imit. The choice of how to transform the attice sum to a one-dimensiona sum is arbitrary and rejects the surface geometry. The present choice corresponds to a sum over spherica shes. ntroducing the integra expression for the Gamma function (n), which after simpe rearrangement is

8 x+ y 2a 1 1 (a) o t(x, s) can be rewritten g( ) PERODC BOUNDARY CONDTONS N AB NTO CALCULATONS 4021 te t as )/2 y f (/2 t(x+ ) s 0 erfc(r x+ ),) X+ (A4) (A5) The integra in (A5) is singuar when s =0, and this singuarity is at the t =0 imit of the integra. f the integration range is spit arbitrary into two ranges [0,2 ] and [2, oc ], then the second integra is immediate and g(x, s) becomes equa to,, ( ) (A6) Erfc(z) is the compementary error function, which for 1/2 1 arge z has the asymptotic form, z e z. This ensures that the second attice sum in (A6) is absoutey convergent for a vaues of s, or in other words, that the singuarity at s =0 is in the first attice sum as expected. Appying the 0 transformation, vaid for a Bravais attices 3/2 t)x+ g /4t ig x C g (A7) to the first attice sum, we obtain a new attice sum over the reciproca attice vectors g and the first term on the right-hand side of (A6) becomes a2 y f dt(t+ ) V, o /2 y f ) (/2 t(x+) s 0 "t [g /4(s+t) j ig xt/(t+s) [st/(t+s) jx (AS) For arge g, the sum in (AS) is absoutey convergent independent of s. However, the g=o term is singuar for s =0; therefore, we separate it from the sum and change variabes from t to u =t(s+t) Then w. e take the imit s 0 for the remaining, absoutey convergent sum over g&0 and the absoutey convergent sum in (A6) over, to obtain )(x, s we obtain c go g g /4g ig (r r) e e (A12) For the specia case of a neutra charge distribution, we obtain for the energy E =, erfc(21 ) 4 1 g2/4 2 X+ 1 go g2 +, V,s 0 sux due 1/2 u Note that the entire singuarity in s is contained in the third term on the right-hand side of (A9). Expanding the third term for the case of sma s, we obtain x +O(s). (A10) V, g 3V, P(r)= f d rp(r) P(r, r) (r 2r r), (A) 3V, f d r p(r) f rp(r)g(r, r) 2 + d rrpr (A13) where use of the charge neutraity condition (A2) was aso made. This resut for the energy depends on the fact that we chose a spherica surface for the performance of the sum. This choice was impicit in our choice of convergence factor as mentioned above. t shoud be noted sr+ that the choice of the convergence factor e + eads to the same resut ony without the unusua dipoe term in (A13). However, in this case the convergence factor does not appy to a one-dimensiona sum and is not known to have a definite imit as s0. Thus, the resut shoud be considered as a coincidence rather than one of mathematica significance. f the charge distribution contains discrete point charges, additiona care needs to be taken in appying equation (A12) as it contains unphysica, and diverging sef-interaction terms when 1=0. For this case the energy is found to be E=, f d r n(r) f d r n(r)+2+z, 5(r r,. ) erfc( 2 r r+ r, r = r r+ X P(r, r) d rrpr 3 V, g g z;z p(r,, r )+ JW (A9) with g defined as =im. r r erfc(gr r+ r r+ ) ) gz,2, + 4 y 1 g /4s) ig (r r) Using (A10) and (A9) in (A) and taking the imit s +0, + erfc(2 ) g2/4 2 2g wo i geo g (A14)

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