first-principles responses of solids to atomic...

First-principles responses of solids to atomic displacements and homogeneous electric fields:Implementation of a conjugate-gradient algorithm

Xavier GonzeUnite de Physico-Chimie et de Physique des Materiaux, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

!Received 7 November 1996"

The changes in density, wave functions, and self-consistent potentials of solids, in response to small atomicdisplacements or infinitesimal homogeneous electric fields, are considered in the framework of the density-functional theory. A variational principle for second-order derivatives of the energy provides a basis forefficient algorithmic approaches to these linear responses, such as the state-by-state conjugate-gradient algo-rithm presented here in detail. The phase of incommensurate perturbations of periodic systems, that are, likephonons, characterized by some wave vector, can be factorized: the incommensurate problem is mapped on anequivalent one presenting the periodicity of the unperturbed ground state. The singularity of the potentialchange associated with an homogeneous field is treated by the long-wave method. The efficient implementa-tion of these theoretical ideas using plane waves, separable pseudopotentials, and a nonlinear exchange-correlation core correction is described in detail, as well as other technical issues. #S0163-1829!97"05016-9$

I. INTRODUCTION

The accurate prediction of material properties is one ofthe pivotal goals of computational condensed matter physics.Current efforts aim at increasing the accuracy of the predic-tions, the complexity of the systems studied, and the numberof properties predicted, altogether for a decreasing computa-tional cost. I will focus on the efficient prediction of re-sponses of periodic systems to different perturbations, usingthe local-density approximation !LDA" to density-functionaltheory1 !DFT" as a basic underlying tool. With this tech-nique, changes in total energy due to adiabatic perturbationsare obtained within a few percent of experimental data.2 Ex-ceptions to this gratifying picture are well characterized anddiscussed in the literature.2,3The perturbations that will be considered belong to the

following two classes: !a" collective displacements of atomscharacterized by a wave vector, either commensurate or in-commensurate with the underlying lattice, that altogethergenerate a basis for the description of phonons, and !b" ho-mogeneous static electric fields. The present paper describesthe computation of the first-order derivatives of the wavefunctions, density, and self-consistent potential with respectto these perturbations. The subsequent computation of vari-ous second-order derivatives of the total energy is describedin an accompanying paper !P2".4The responses of crystalline solids to external perturba-

tions, like electric fields or atomic displacements, have beencalculated within the DFT using various methods.5–12 Thesimplest is a direct approach5 in which one freezes a finite-amplitude perturbation into the system and compares the per-turbed system with the corresponding unperturbed one !e.g.,the frozen-phonon technique". However, in this approach, itis was impossible to handle perturbations incommensuratewith the periodic lattice, or potentials linear in space !corre-sponding to homogeneous electric fields", while commensu-rate perturbations were handled through the use of super-cells, sometimes with a considerable increase of computing

time. Still, its simplicity has attracted many researchgroups.13 The recent appearance of O!N" algorithms forphonons14 as well as Wannier function approach to the di-electric constant15 could reduce the above-mentioned disad-vantages.There is also a dielectric-matrix approach in which one

calculates dielectric matrices from the unperturbed ground-state wave functions.6–8 Incommensurability is not a problemin this technique. However, the whole spectrum of thevalence- and conduction-band wave functions is required,which can be computationally demanding, and responses toatomic displacements cannot be obtained when the electron-ion interaction is represented by a nonlocal pseudopotential.Baroni, Giannozzi, and Testa10,16 !BGT" demonstrated the

power of a perturbative approach, appearing also in Ref. 9, inwhich the linear responses are calculated self-consistently. Itcombines the advantages of the two previous methods, with-out their drawbacks. Baroni and coworkers, as well as otherresearch groups17–19 used this formalism with plane wavesand pseudopotentials. Linear muffin tin orbital20–22 !LMTO"and linear augmented plane-wave23 !LAPW" versions of thislinear-response approach have also been proposed andimplemented. Applications have been numerous, and in-cluded computations of dynamical matrices, Born effectivecharge tensors and dielectric permittivity tensors for bulkmaterials, surfaces or large molecules, as well as computa-tion of elastic constants, piezoelectric tensors, photoelastictensors, internal strain, deformation potential, electron-phonon coupling, thermodynamical properties, atomic tem-perature factors, and phase transitions.18,24–35It was also realized that the BGT approach was derived

from an interesting merging of DFT and perturbation theory,and that it could be extended very efficiently to nonlinearresponses, thanks to the 2n!1 theorem.36–43 From the samepoint of view, often referred to as density-functional pertur-bation theory !DFPT", one can infer the existence of a varia-tional principle for the second-order derivatives of the totalenergy.12,20 This variational principle, its higher-order gener-alizations, and the 2n!1 theorem for DPFT, are thoroughly

PHYSICAL REVIEW B 15 APRIL 1997-IIVOLUME 55, NUMBER 16

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discussed in Ref. 42. Thanks to the second-order variationalprinciple, efficient minimization techniques, like theconjugate-gradient algorithm,12,44,45 can be used for theevaluation of the first-order changes of the wave functions,charge densities, and self-consistent potentials. Moreover,second-order derivatives of the total energy converge morerapidly to the correct answers than in the BGT approach.Like the BGT technique, the variational approach has beenwidely used for studies of dynamical and dielectric proper-ties of various materials #SiO2-quartz,12,46–48SiO2-stishovite,48–52 Si,53 TiO2,54,55 BaTiO3,56–60PbTiO3,61 PbZrO3,62 Al 2Ru,63 and PbTe !Ref. 64"$. The useof a variational principle was also instrumental in the LMTOimplementation of the linear-response approach.20,22Since the presentation of the technique that was used for

the computation of dielectric and dynamical properties ofsolids was rather limited in these previous papers, an exten-sive description of the method will be given in the presentpaper and in P2,4 from the basic determination of the re-sponses, to the computation of phonon band structures. Incommon with BGT,16 the implementation of this scheme isdone using plane waves and pseudopotentials, but, at vari-ance, fast Fourier transforms are used !in the spirit of theCar-Parrinello technique65" as well as elaborate separablepseudopotentials,66 in order to enhance the efficiency of thecalculation. Since nonlinear exchange-correlation corecorrections67,31 have been used in the above-mentionedapplications12,46–50,53–60 the consequences of this technicalfeature will also be described. Specific aspects related tometals that cannot sustain static electric fields, and for whicha Fermi surface is present, will be briefly mentioned whenappropriate. By definition, all nonadiabatic as well as non-harmonic effects are ignored.This paper is organized as follows. In Sec. II, some basic

results of DFPT are recalled. In Sec. III, I describe the pre-conditioned conjugate-gradient gradient algorithm that canbe used to find the first-order changes in wave functions,charge densities, and the self-consistent potential with re-spect to a generic perturbation. Section IV contains a generalanalysis of response to perturbations that are characterizedby a wave vector incommensurate with the periodicity of theunderlying lattice. The factorization of the incommensurabil-ity is shown. In Sec. V, one finds the developments neededto obtain the first-order change in wave functions and densi-ties with respect to collective atomic displacements, in theplane-wave implementation with separable pseudopotentialsand a nonlinear exchange-correlation core correction. Sec-tion VI describes the treatment of homogeneous electricfields, using the long-wave method, then its implementation.In Sec. VII, I mention some technical considerations, andpresent perspectives in Sec. VIII. Atomic !Hartree" units areused throughout this paper.

II. BASICS OF DENSITY-FUNCTIONAL PERTURBATIONTHEORY

A. DFT equations

In the DFT, the ground-state energy of the electronic sys-tem is derived from the following minimum principle:2

Eel%&'(")'

occ

*&'!T!vext!&'+!EHxc#n$ , !1"

where the &'’s are the Kohn-Sham orbitals !to be varieduntil the minimum is found", T is the kinetic energy operator,vext is the potential external to the electronic system thatincludes the one created by nuclei !or ions", EHxc is the Har-tree and exchange-correlation energy functional of the elec-tronic density n(r), and the summation runs over the occu-pied states ' . The occupied Kohn-Sham orbitals are subjectto the orthonormalization constraints,

" &'*!r"&,!r"dr"*&'!&,+"-', , !2"

where ' and , label occupied states. The density is gener-ated from

n!r"")'

occ

&'*!r"&'!r". !3"

The minimization of Eel%&( under the orthonormality con-straints Eq. !2" can be achieved using the Lagrange multi-plier method. The problem turns into the minimization of

Eel!%&(")

'

occ

*&'!T!vext!&'+!EHxc#n$

#)',

occ

.,'!*&'!&,+#-',", !4"

where .', are the Lagrange multipliers corresponding to theset of constraints Eq. !2". The canonical Euler-Lagrangeequations are

H!&'+"),

occ

.,'!&,+, !5"

where the Hamiltonian operator is

H"T!vext!-EHxc

-n "T!vext!vHxc . !6"

Since H is Hermitian, it is always possible to make a unitarytransformation of Eq. !5" in such a way that

H!&'+".'!&'+. !7"

B. Perturbation expansion

Having defined the DFT equations for the possible exter-nal potentials vext , we now choose a reference !unperturbed"external potential vext

(0) and expand the perturbed potentialvext in terms of a small parameter / , as follows:

vext!/""vext!0 "!/vext

!1 "!/2vext!2 "!••• . !8"

We are interested in the change of physical quantities, due tothe perturbation of the external potential.68 So, we expand allof the exact perturbed quantities X(/) in the same form asvext(/),

10 338 55XAVIER GONZE

X!/""X !0 "!/X !1 "!/2X !2 "!••• , !9"

where X can be Eel , &'(r), n(r), .', , or H . For example,the lowest-order expansion of Eq. !7" is simply

H !0 "!&'!0 "+".'

!0 "!&'!0 "+. !10"

Because Eel satisfies a variational principle under con-straints, it is possible to derive a constrained variational prin-ciple for the 2nth order derivative of Eel with respect to thenth order derivative of &' :69,70,42 when the expansion of thewave function of up to an order of n#1 is known, the varia-tional !minimum" principle for the 2nth order derivative ofEel is as follows:

Eel!2n ""min

&'!n "

# Eel!$ )i"0

n

/ i&'! i "% & !2n "

, !11"

under constraints !in the parallel transport gauge42"

)i"0

n

*&'!n#i "!&,

! i "+"0 !12"

for all occupied states ' and , . The explicit expressions forEel(2n) can be worked out by introducing Eq. !4" into Eq. !11".For the second order, we obtain that Eel

(2) is the minimumof the following expression:12,20,42

Eel!2 "%&!0 ";&!1 "(")

'

occ

#*&'!1 "!H !0 "#.'

!0 "!&'!1 "+!!*&'

!1 "!vext!1 "!&'

!0 "+!*&'!0 "!vext

!1 "!&'!1 "+ "!*&'

!0 "!vext!2 "!&'

!0 "+$

!12" " -2EHxc

-n!r"-n!r!" 'n!0 "

n !1 "!r"n !1 "!r!"dr dr!!" dd/

-EHxc-n!r" 'n!0 "

n !1 "!r"dr!12d2EHxcd/2 '

n!0 "

, !13"

where the first-order changes in wave functions &'(1) !these

quantities will be referred to as the first-order wave func-tions, for brevity" are varied under the constraints

*&'!0 "!&,

!1 "+"0 !14"

for all occupied states ' and , , while the first-order densityis given by

n !1 "!r"")'

occ

&'*!1 "!r"&'!0 "!r"!&'*!0 "!r"&'

!1 "!r". !15"

By virtue of Eq. !14", the first-order wave functions are or-thogonal to the unperturbed wave functions of the occupiedstates. This is a specific advantage of the parallel-transportgauge.71Since Eel

(2)%& (0);& (1)( is variational with respect to& (1),72 we deduce the Euler-Lagrange equations !also calledself-consistent Sternheimer equations in this particularcase",73,42

Pc!H !0 "#.'!0 ""Pc!&'

!1 "+"#PcH !1 "!&'!0 "+ , !16"

where Pc is the projector upon the unoccupied states !con-duction bands", H (0), .'

(0) , and & (0) are obtained from Eq.!10", and the first-order Hamiltonian H (1) is given by

H !1 ""vext!1 "!vHxc

!1 " "vext!1 "!" -2EHxc

-n!r"-n!r!" 'n!0 "

n !1 "!r!"dr!

!dd/

-EHxc-n!r" 'n!0 "

. !17"

One defines also

vHxc0!1 " "

dd/

-EHxc-n!r" 'n!0 "

, !18"

not to be confused with vHxc(1) , that contains one more term,

see Eq. !17". vext(1) and vHxc0

(1) do not depend on the first-orderwave functions. Equation !16" can be solved by algorithmsbased on Green’s functions or, within some basis set, bystandard algorithms for dealing with inhomogeneous systemsof equations.16In Eq. !13", the contributions

)'

occ

*&'!0 "!vext

!2 "!&'!0 "+!

12d2EHxcd/2 '

n!0 "

, !19"

that will be denoted by Enon-var(2) , also do not depend on the

first-order wave functions, and will not change in the courseof the minimization procedure or in the self-consistent pro-cedure.For the first-order wave functions that satisfy Eqs. !14"–

!17", or equivalently minimize Eq. !13" under constraints Eq.!14", Eel

(2) can also be computed from

Eel!2 ""

12)'occ

!*&'!1 "!vext

!1 "!vHxc0!1 " !&'

!0 "+

!*&'!0 "!vext

!1 "!vHxc0!1 " !&'

!1 "+ "!Enon-var!2 " , !20"

instead from Eq. !13". Taking into account the time-reversalsymmetry, other expressions for Eel

(2) can be found:

Eel!2 "")

'

occ

*&'!0 "!vext

!1 "!vHxc0!1 " !&'

!1 "+!Enon-var!2 " , !21"

or its Hermitian conjugate. However, if the wave functionsare not exactly the ones that minimize Eq. !13" and satisfy

55 10 339FIRST-PRINCIPLES RESPONSES OF SOLIDS TO . . .

Eqs. !14"–!17", the error in Eqs. !20" and !21" is larger thanthe error in Eq. !13", since Eq. !13" is variational, while Eqs.!20" and !21" are not.

III. A CONJUGATE-GRADIENT ALGORITHM FOR THECOMPUTATION OF THE FIRST-ORDER RESPONSES

Equations !14"–!17" are used in the BGT approach.9,10,16They can be solved self-consistently: one fixes first a basisset, then, supposing H (1) to be known, Eq. !16" is treated asa linear system of equations for & (1), whose solution can beobtained by different standard numerical techniques; once& (1) is found, it can be used to build a new H (1) through Eqs.!15" and !17".In a different spirit, direct minimization of Eq. !13" is

performed in the variational approach described here. Thesetwo possible approaches are directly connected to the twoapproaches that have been used to compute the ground-stateproperties of materials. Indeed, until 1985, the DFT ground-state wave functions were usually computed using a two-level procedure: at the lower level, Eq. !7" was solved usingstandard diagonalization procedures, while at the upper level,a loop used the output of this diagonalization to generate anew density through Eq. !3", and a new Hamiltonian throughEq. !6", thus a new Eq. !7". Self-consistency was enforced, ifneeded, by the use of some convergence accelerator.74 Thisis, in spirit, similar to the BGT technique.By contrast, Car and Parrinello65 suggested the insertion

of the quantity defined in Eq. !4" in a fictitious Lagrangian!that also included the classical kinetic energy of nuclei",giving a unified approach to molecular dynamics and DFT.As a further step, the direct minimization of the functionalEq. !4" under the orthonormalization constraints Eq. !2" wasproposed. For this purpose, Teter, Payne and Allan45 !TPA"

designed a band-by-band !or state-by-state" conjugate-gradient algorithm. Other global minimization algorithmswere proposed in Refs. 75–78. The Car-Parrinello and theband-by-band conjugate-gradient algorithms have been pre-sented in considerable detail in a review article by Payneet al.44 All these techniques are particularly effective withplane-wave basis sets, since the corresponding Hamiltonianis sparse when fast Fourier transforms and separable pseudo-potentials are used.79Since Eq. !13" is also a minimum principle, it is possible

to use the same global minimization techniques for thesecond-order derivative of the energy Eel

(2) as for the ground-state energy. Moreover, the expression for Eel

(2) , Eq. !13", isan exact quadratic form in the space of the first-order wavefunctions, unlike the ground-state energy functional Eq. !4".This feature leads to an easier implementation of the state-by-state conjugate-gradient algorithm for response functionsthan for ground-state energy, that we will now describe.

A. State-by-state decomposition of the energy functional

In a state-by-state conjugate-gradient algorithm, eachwave function is considered successively, and the energyfunctional is minimized with respect to variations of thiswave function, in the potential created by the density of theothers, the latter being temporarily frozen. Let us supposethat the state , , with first-order wave function &,

(1) , is var-ied. One writes the minimum principle Eq. !13" as

Eel!2 ""E,

!2 "!Eno ,!2 " !Enon-var

!2 " , !22"

where the term E,(2) is the only one that depends on &,

(1) .This first term is

E,!2 ""*&,

!1 "!H !0 "#.,!0 "!&,

!1 "+!!*&,!1 "!vext

!1 "!vHxc0!1 " !&,

!0 "+!*&,!0 "!vext

!1 "!vHxc0!1 " !&,

!1 "+ "

!" " -2EHxc-n!r"-n!r!" 'n!0 "

# 12 n,!1 "!r"n,

!1 "!r!"!n,!1 "!r" )

'0,

occ

n'!1 "!r!"& dr dr! , !23"

where n,(1) , the density change due to the first-order wave function &,

(1) , is

n,!1 "!r""&,*!1 "!r"&,

!0 "!r"!&,*!0 "!r"&,!1 "!r". !24"

The second term of Eq. !22",

Eno ,!2 " " )

'0,

occ # *&'!1 "!H !0 "#.'

!0 "!&'!1 "+!!*&'

!1 "!vext!1 "!vHxc0

!1 " !&'!0 "+!*&'

!0 "!vext!1 "!vHxc0

!1 " !&'!1 "+ "&

!12" " -2EHxc

-n!r"-n!r!" 'n!0 "# )

'0,

occ

n'!1 "!r"& # )

'0,

occ

n'!1 "!r!"& dr dr!, !25"


does not depend on &,(1) , but on all the first-order wave

functions for the states other than , . The third term of Eq.!22", Enon-var

(2) , is given by Eq. !19" and does not depend onfirst-order wave functions. Thus, only the first term must beconsidered in the minimization procedure with respect to&,(1) . Once the first-order wave function &,

(1) is sufficientlyconverged !via the conjugate-gradient algorithm described inthe next paragraph", another first-order wave function is var-ied, and so on. As soon as the last wave function has beenvaried, the algorithm goes on with varying the first wavefunction again, and then the others as well. Indeed, the wavefunctions were minimized in the potential created by the oth-ers while these were not yet converged. Thus, there is still aself-consistency loop, although the second-order electronicenergy is always decreased. Because of the latter property,the algorithm is unconditionally stable.

B. Optimization of one state

A basic description of the conjugate-gradient algorithmscan be found in Ref. 80. The procedure of minimization ofEq. !23" follows closely Eqs. !5.10"–!5.38" of Ref. 44, validfor the minimization of the ground-state functional !the TPAalgorithm". Except for a few details, the flow diagram of thisupdate of a single state is very similar to the flow diagram ofthe TPA algorithm, shown in Fig. 17 of Ref. 44.In the initial step (m"1), a trial &,

m(1) is used, which iseither the null vector #it obviously satisfies the constraintsEq. !14"$, or the vector determined from the previous self-consistency loop.Then, an improved &,

m!1(1) is determined by the follow-ing operations !iterated over m , which runs from 1 tommax).

!1" The steepest-descent vector 1,m #minus half the gradi-

ent of Eq. !23" with respect to changes of &,m(1) , projected

on the conduction bands$ is

1,m"#Pc#!H !0 "#.,

!0 ""!&,m!1 "+!Hm!1 "!&,

!0 "+], !26"

with

Hm!1 ""vext!1 "!vHxc0

!1 " !" -2EHxc-n!r"-n!r!" 'n!0 "

(n,m!1 "!r!"

! )'0,

n'!1 "!r!")dr!, !27"

where n,m(1)(r) is inferred from Eq. !24", updated for each

m .!2" The preconditioned steepest-descent vector 2,

m is asfollows: 2,

m"PcK1,m , where the preconditioning operator

K is similar to the one defined in Eq. !5.16" of Ref. 44.!3" The conjugate-gradient !or search" direction 3,

m is3,m"2,

m!4,m 3,

m#1 , with 4,m"*2,

m!3,m+/*2,

m#1!3,m#1+ and

4,1"0.

!4" An improved trial first-order &,m!1(1) will be obtained

by mixing &,m(1) with 3,

m : &,m!1(1)"&,

m(1)!5 3,m , where

the parameter 5 is to be determined by the minimizationrequirement, as follows. The energy Eq. !23", as a functionof 5 , becomes

E,!2 "!5""E,

!2 "!0 "!dE,

m!2 "

d5 '5"0

5!12d2E,

m!2 "

d52 '5"0

52,

!28"

where

dE,m!2 "

d5 '5"0

"#2 Re *3,m!1,

m+, !29"

d2E,m!2 "

d52 '5"0

"2 *3,m!H !0 "#.,

!0 "!3,m+

!" " -2EHxc-n!r"-n!r!" 'n!0 "

6n,m!r"

$6n,m!r!"dr dr! , !30"

with 6n,m(r)"3,

m*(r)&,(0)(r)!&,*(0)(r)3,

m(r). The expres-sion Eq. !28" is minimal when

5"#dE,

m!2 "

d5 '5"0

# d2E,m!2 "

d52 '5"0

& #1

, !31"

a value that allows one to generate &,m!1(1) .

In step 3, unlike for the ground-state minimization algo-rithm, no further orthonormalization or orthogonalization isneeded #compare with Eqs. !5.21" and !5.22" of Ref. 44$,since the constraint Eq. !14" is linear in &,

(1) and fulfilled by3,m . In step 4, the search for the minimum of the second-

order energy is simplified with respect to the ground-stateminimization algorithm, because Eq. !23" is a quadratic formin &,

(1) .Usually, mmax is on the order of 4–6, while the error in

the second derivative of the energy decreases by a factor of 3or more after each set of all-state optimizations. Figure 2 ofRef. 42 shows a typical convergence of this algorithm.In this state-by-state conjugate-gradient algorithm, only

one wave function is varied at a time. Giannozzi andBaroni33 have recently proposed to treat the minimization ofEq. !13" by simultaneous variation of all the first-order wavefunctions. No supplementary self-consistency loop is thenneeded. The latter approach requires more disk space, and acomparative study would be useful in order to assert therelative merits of both methods in terms of computationalefficiency.

IV. RESPONSE TO INCOMMENSURATEPERTURBATIONS OF PERIODIC SYSTEMS

The conventions for the treatment of the unperturbed pe-riodic system !e.g., normalization of Bloch’s wave functions"are presented in Appendix A.

A. Incommensurate perturbations

One considers perturbations of the system that are incom-mensurate with the unperturbed periodic lattice, and charac-terized by a wave vector q. In order to be able to treat non-local pseudopotentials, the general form of a perturbation !a


linear operator and not simply a local function in space" istaken into account. The ground-state potential operator isperiodic, with

vext!0 "!r!Ra ,r!!Ra""vext

!0 "!r,r!", !32"

where Ra is a vector of the real space lattice, while the per-turbing potential operator is such that

vext,q!1 " !r!Ra ,r!!Ra""eiq•Ra vext,q

!1 " !r,r!". !33"

Actually, when q is not equal to half a vector of the recip-rocal lattice, such a perturbing potential is non-Hermitian,and should be always used in conjunction with its Hermitianconjugate counterpart. However, at the level of the linearresponse, there is no consequence of working only with thenon-Hermitian vext,q

(1) , since the response to the sum ofvext,q(1) and its Hermitian conjugate is simply the sum of theresponse to each perturbation separately. Nevertheless we arealso interested in the variational property of the second-orderchange in energy, for which we cannot afford a non-Hermitian external potential. This difficulty is solved as fol-lows. One considers both vext,q and its Hermitian conjugate,written vext,#q !since its wave vector is #q), as well as acomplex expansion parameter / , such that

vext!/""vext!0 "!!/vext,q

!1 " !/*vext,#q!1 " "! !/2vext,q,q

!2 "

!//*vext,q,#q!2 " !/*/ vext,#q,q

!2 " !/*2vext,#q,#q!2 " "

!••• . !34"

This definition is a generalization of Eq. !8". Since bothvext(/) and vext

(0) are Hermitian, the Hermitian conjugates ofvext,q(1) and vext,q,q

(2) are vext,#q(1) and vext,#q,#q

(2) , respectively.One also has the freedom to impose that vext,q,#q

(2) is Hermit-ian and equal to vext,#q,q

(2) !sometimes they will be notedvext,0(2) in what follows".Applying a translation to the first-order wave functions

and densities, one observes the following behaviors:

&m ,k,q!1 " !r!Ra""ei!k!q"•Ra &m ,k,q

!1 " !r" !35"

and

nq!1 "!r!Ra""eiq•Ra nq!1 "!r". !36"

An expansion similar to Eq. !34" applies also to the en-ergy, with

E!/""E !0 "!!/Eq!1 "!/*E#q

!1 " "!!/2Eq,q!2 "!2//*Eq,#q!2 "

!/*2E#q,#q!2 " "!••• . !37"

Due to the requirement of invariance under translation ofthe whole system, one derives, when q is not a vector of thereciprocal lattice,81

Eq!1 ""E#q

!1 ""0, !38"

while for 2q which is not a vector of the reciprocal lattice,81

Eq,q!2 ""E#q,#q

!2 " "0. !39"

Thanks to these equations, it can be shown that Eq,#q(2) is a

real quantity, variational with respect to change in the first-order wave functions. This property will allow one to applythe above-mentioned minimization algorithm to the case ofincommensurate perturbations.

B. Factorization of the phase

The factorization of the phase, in order to map the incom-mensurate problem to an equivalent one presenting the peri-odicity of the unperturbed problem is the crucial point in thetreatment of perturbations characterized by a wave vectorq, like vext,q

(1) . For this purpose, inspired by Eqs. !35" and!36", one defines the periodic functions !see Appendix A forthe notations"

um ,k,q!1 " !r""!N70"

1/2e#i!k!q"•r &m ,k,q!1 " !r" !40"

and

nq!1 "!r""e#iq•r nq!1 "!r", !41"

in which case Eq. !13" becomes

Eel,#q,q!2 " %u !0 ";u !1 "("

70

!28"3"BZ)mocc

s !*umk,q!1 " !Hk!q,k!q

!0 " #.mk!0 "!umk,q

!1 " +!*umk,q!1 " !vext,k!q,k

!1 " !vHxc0,k!q,k!1 " !umk

!0 "+

!*umk!0 "!vext,k,k!q

!1 " !vHxc0,k!q,k!1 " !umk,q

!1 " +!*umk!0 "!vext,k,k

!2 " !umk!0 "+)dk

!12"70

" -2EHxc-n!r"-n!r!" 'n!0 "

nq!1 "*!r" nq

!1 "!r!"e#iq•!r#r!"dr dr!!12d2EHxcd/ d/* 'n!0 "

, !42"

satisfying a minimum principle with respect to variations ofthe first-order wave functions un ,k,q

(1) under constraints

*um ,k!q!0 " !un ,k,q

!1 " +"0, !43"

where the index n runs over the occupied states, while thefirst-order change in density is given by82

nq!1 "!r""

2!28"3"BZ)m

occ

sumk!0 "*!r"umk,q

!1 " !r"dk. !44"

The Euler-Lagrange equations associated with the mini-mization of Eq. !42" under constraints Eq. !43" are


Pc ,k!q!Hk!q,k!q!0 " #.m ,k

!0 " "Pc ,k!q !um ,k,q!1 " +

"#Pc ,k!qHk!q,k!1 " !um ,k

!0 " +, !45"

with

Hk!q,k!1 " "vext,k!q,k

!1 " !vHxc0,k!q,k!1 " !" -2EHxc

-n!r"-n!r!" 'n!0 "

$ nq!1 "!r!"e#iq•!r!r!" dr!. !46"

Finally, there are simpler, but nonvariational, expressionsfor Eel,#q,q

(2) , in the spirit of Eq. !21":82

Eel,#q,q!2 " %u !0 ";u !1 "(

"70

!28"3"BZ)mocc

s !*umk,q!1 " !vext,k!q,k

!1 " !vHxc0,k!q,k!1 " !umk

!0 "+

!*umk!0 "!vext,k,k

!2 " !umk!0 "+ "dk!

12d2EHxcd/ d/* 'n!0 "

, !47"

its Hermitian conjugate, or their mean. All the quantities thatappear in Eqs. !42"–!47" have the periodicity of the unper-turbed lattice. Note that the ground-state wave functions atk and at k!q are needed to determine umk,q

(1) : umk(0) appear in

Eqs. !42", !44", and !45", while the orthonormalization con-straint Eq. !43" makes use of umk!q

(0) .

V. RESPONSE TO COLLECTIVE ATOMICDISPLACEMENTS

We now focus on a first class of perturbations, directlyconnected to phonons. In this section and the next ones, theresponses are treated in view of the implementation of theformalism with a plane-wave basis set, separable pseudopo-tentials, and taking into account the nonlinear exchange-correlation core correction. The notations !definitions of lo-cal and separable potentials, exchange-correlationfunctional" are described in Appendix A. The different quan-tities will be given either in the real space or in the reciprocalspace, whichever is the most appropriate.83

A. First- and second-order changes in potential operators

One considers unit displacements of atoms in sublattice9 , along the ' axis, multiplied by the infinitesimal / !even-tually, a complex quantity" and by a phase varying with thecell to which the atoms belong: the ' component of theirvector position is changed from :9 ,'!Ra ,' to:9 ,'!Ra ,'!/eiq•Ra. Atoms in the other sublattices are notdisplaced. Note that all of these collective displacements canbe generated from q wave vectors restricted inside the Bril-louin zone, the only ones that will be considered. Also, forreasons given in Sec. VIIA and in the next paper !P2", weconsider nonzero q wave vectors.The first-order change in the potential operator Eq. !A12"

!see Appendix A" is

vext,q!1 " !r,r!"")

aeiq•Ra ;

;:9 ,'v9!r#"9#Ra ,r!#"9#Ra",

!48"

while the second-order vext,0(2) (r,r!), needed in Eqs. !42" and

!47", is as follows:

vext!2 "!r,r!"")

a

12

;2

;:9 ,'2 v9!r#"9#Ra ,r!#"9#Ra".

!49"

Their evaluation, based on Eqs. !A15" and !A19", leads tothe following expressions

!1" The first-order change of the local potential is

v loc,q!1 " !G""

170

;

;:9'!e#i!G#q"•"9" v9

loc!G!q"

"1

70!#i "!G!q"'e#i!G!q"•"9 v9

loc!G!q", !50"

where v9loc(G#q) can be found from Eq. !A16", and where

the definition of the phase-factorized, periodic potential

v loc,q!1 " !r""e#iq•r v loc,q

!1 " !r", !51"

has been used #compare with Eq. !41"$. Note that, unlike inEqs. !A15" and !A27", v loc,q

(1) includes the G"0 contribution:there is no associated divergence since we have supposed qto be nonzero.

!2" The second-order change of the local potential is84,85

v loc!!2 "!G""#1

70

12 G'

2 !e#iG•"9" v9loc!G" when G00

"0 when G"0. !52"

!3" The first-order change of the nonlocal potential is

vsep,k!q,k!1 " !G,G!""

170

)<

e<9

;

;:9'

$( # )G e#i!k#q#G"•"91<9!k!q!G"&$# )

G!ei!k!G!"•"91<9* !k!G!"& ) . !53"

!4" The second-order change of the nonlocal potential is

vsep,k,k!2 " !G,G!""

170

)<

e<9

12

;2

;:9'2

$( # )G e#i!k#G"•"91<9!k!G"&$# )

G!ei!k!G!"•"91<9* !k!G!"& ) . !54"

The efficient use of vsep,k!q,k(1) as well as vsep,k,k

(2) is accom-plished thanks to a further manipulation, an example ofwhich is given here for the first-order separable part: Eq. !53"becomes


vsep,k!q,k!1 " !G,G!""

170

)<

e<9( # )G !#i "!k!q!G"'e#i!k!q!G"•"91<9!k!q!G"& # )G!

ei!k!G!"•"91<9* !k!G!"&!# )G e#i!k!G"•"91<9!k!q!G"& # )

G!i!k!q!G!"'ei!k#G!"•"91<9* !k!G!"& ) . !55"

In this way, the sums on the reciprocal vectors G or G! arewell separated. The same manipulation can be performed onEq. !54".

B. First- and second-order changes in the exchangeand correlation energy functional

The phase-factorized first-order change of pseudocorecharge density is given by

nc,q!1 "!r""e#iq•r )

aeiq•Ra ;

;:9 ,'nc ,9!r#"9#Ra", !56"

while the second-order change is

nc!2 "!r"")

a

12

;2

;:9 ,'2 nc ,9!r#"9#Ra". !57"

These expressions allow us to build vxc0,q(1) and

12(d2Exc /d/ d/*)!n(0):31

vxc0,q!1 " !r""

dvxcdn '

n!0 "!r"nc,q

!1 "!r" !58"

and

12

d2Excd/ d/* 'n!0 "

"12"70

dvxcdn '

n!0 "!r"!nc,q

!1 "!r"!2dr

!"70

vxc„n !0 "!r"…nc!2 "!r"dr. !59"

C. Variational principle

Having obtained the first and second derivatives of thepotentials and exchange-correlation energy functional, weare able to write the second-order electronic energy:

Eel,#q,q!2 " %u !0 ";u !1 "("

70

!28"3"BZ)mocc

s !*umk,q!1 " !Hk!q,k!q

!0 " #.mk!0 "!umk,q

!1 " +!*umk,q!1 " !vsep,k!q,k

!1 " !umk!0 "+!*umk

!0 "!vsep,k,k!q!1 " !umk,q

!1 " +

!*umk!0 "!vsep,k,k

!2 " !umk!0 "+ "dk!

12"70

!# nq!1 "!r"$*# v loc,q

!1 " !r"! vxc0,q!1 " !r"$!# nq

!1 "!r"$# v loc,q!1 " !r"! vxc0,q

!1 " !r"$*"dr

!12"70

dvxcdn '

n!0 "!r"!nq

!1 "!r"!2dr!2870)G

!nq!1 "!G"!2

!q!G!2 !"70

!n !0 "!r"v loc!!2 "!r""dr!12

d2Excd/ d/* 'n!0 "

.

!60"

In this expression, as for v loc,q(1) in Eq. !50", the G"0 contri-

bution is included in the Hartree term: there is no associateddivergence since we have supposed q to be nonzero.Equation !60" is to be minimized under the constraints Eq.

!43", with the first-order change in density given by Eq. !44".The associated Euler-Lagrange equation is still Eq. !45",with a more explicit first-order Hamiltonian operator:

Hk!q,k!1 " "vsep,k!q,k

!1 " ! v loc,q!1 " ! vH,q

!1 " ! vxc,q!1 " , !61"

where

vH,q!1 " !G""48

nq!1 "!G"

!G!q!2 !62"

and

vxc,q!1 " !r""# dvxc

dn 'n!0 "!r"

nq!1 "!r" & ! vxc0,q

!1 " !r". !63"

Finally, there exists simpler, but nonvariational expres-sions for Eel,#q,q

(2) , derived from Eq. !47": for example,


Eel,#q,q!2 " "

70

!28"3"BZ)mocc

s !*umk,q!1 " !vsep,k!q,k

!1 " !umk!0 "+

!*umk!0 "!vsep,k,k

!2 " !umk!0 "+ "dk

!12"70

!# nq!1 "!r"$*# v loc,q

!1 " !r"! vxc0,q!1 " !r"$ "dr

!"70

!n !0 "!r"v loc!!2 "!r""dr!12

d2Excd/ d/* 'n!0 "

.

!64"

At this stage, we have written all the theoretical ingredi-ents needed for the computation of the response to a collec-tive displacement of atoms on one sublattice, in which thevector position :9 ,'!Ra ,' is changed into :9 ,'!Ra ,'!/eiq•Ra. In further sections, in order not to confusethem with the response to other perturbations, the first-orderquantities related to this perturbation will be written Xq

:9'

instead of Xq(1) , while the corresponding second-order quan-

tities will be denoted by X#q,q:9'* :9' instead of X#q,q

(2) .

VI. RESPONSE TO AN HOMOGENEOUS, STATICELECTRIC FIELD

Two important problems arise when one attempts to dealwith the response to an homogeneous, static electric field.The first problem comes from the fact that the potential en-ergy of the electron, placed in such a field, is linear in space,and breaks the periodicity of the crystalline lattice:86

vscr!r"")'Emac,' r' . !65"

Second, this macroscopic electric field corresponds to ascreened potential: the change of macroscopic electric field isthe sum of an external change of field and an internal changeof field, the latter being induced by the response of the elec-trons !the polarization of the material". In order to indicatethis fact, the subscript ‘‘scr’’ has been used in Eq. !65". Inthe theory of classical electromagnetism,87 the connectionbetween the macroscopic displacement, electric, and polar-ization fields is

Dmac!r""Emac!r"!48Pmac!r", !66"

where Pmac(r) is related to the macroscopic charge densityby

nmac!r""#=Pmac!r". !67"

It is important to emphasize that these fields are macroscopicfields: the microscopic fluctuations !local fields" have beenaveraged out in this description.87The long-wave method is commonly used to deal with the

first problem: a potential linear in space is obtained as thelimit for q tending to 0 of

v!r"" limq!0

/2sinq•r

!q! " limq!0

/# eiq•ri!q! #

e#iq•ri!q! & , !68"

where q is in the direction of the homogeneous field. Thismethod has its drawback: the homogeneous field and thewave vector are always parallel. In other words, the electricfield is longitudinal. The treatment of transverse fields shouldbe done by considering not only a scalar potential, but avector potential. However, for our purpose, the scalar theorywill be sufficient.26The detailed theoretical treatment of the response to an

electric field, using the long-wave method, and treating thescreening adequately !in order to solve the above-mentionedsecond problem" is given in Appendix B. It is found that anauxiliary quantity is needed: the derivative of the ground-state wave functions with respect to their wave vector. Oncethis quantity has been obtained, the computation of the re-sponse to an homogeneous electric field per se can be per-formed.

A. Derivative of the wave functions with respectto their wave vector

We will use the shorthand notation

umkk' "

dumk!0 "

dk'. !69"

In the parallel-transport gauge,42 the umkk' at each k can be

determined by the minimization of the following expression:

Em ,kk'k'"*umk

k' !Hk,k!0 "#.mk

!0 "!umkk' +!*umk

k' !Tk,kk' #vsep,k,k

k' !umk!0 "+

!*umk!0 "!Tk,k

k' #vsep,k,kk' !umk

k' +, !70"

with the constraints

*umk!0 "!unk

k'+"0. !71"

Tk,kk' and vsep,k,k

k' are the first derivative of kinetic energyoperator and external potential with respect to the wave vec-tor k' :

Tk,kk' !G,G!""!G'!k'"-GG! !72"

and

vsep,k,kk' !G,G!""

170

)<9

e<9

;

;k'

$( # )G e#i!k#G"•"91<9!k!G"&$# )

G!ei!k!G!"•"91<9* !k!G!"& ) . !73"

The Euler-Lagrange equation associated with the minimi-zation procedure Eq. !70" is

Pc ,k!Hk,k!0 "#.m ,k

!0 " "Pc ,k !um ,kk' +"#Pc ,k!Tk,k

k' !vsep,k,kk' "!um ,k

!0 " +.!74"


B. Derivative of the wave functions with respectto an electric field

Having obtained the derivative of the wave functions withrespect to their wave vector, one is able to compute the re-sponse of the system with respect to the change of the long-wave screened potential

6vscr!r"" limq!0

# E'

eiq•ri!q! !E'*

e#iq•r!#i "!q! & . !75"

In the parallel-transport gauge,42 the derivative of thewave functions with respect to E' is obtained through theminimization of the following expression !for which, as atthe end of Sec. V, we rationalize our notation in view of themultiplicity of perturbations that are examined":

EelE'*E'%u !0 ";uE'("

70

!28"3"BZ)mocc

s!*umkE' !Hk,k

!0 "#.mk!0 "!umk

E' +

!*umkE' !iumk

k' +!*iumkk' !umk

E' +"dk

!12"70

dvxcdn '

n!0 "!r"!nE'!r"!2

!2870)G00

!nE'!G"!2

!G!2 , !76"

with

nE'!r""2

!28"3"BZ)mocc

sumk!0 "*!r"umk

E' !r"dk, !77"

under the constraints

*um ,k!0 " !unk

E'+"0, !78"

where the indices m and n run over occupied states, and*iumk

k' !"(#i)*umkk' !.88

The associated Euler-Lagrange equations are

Pc ,k!Hk,k!0 "#.m ,k

!0 " "Pc ,k !um ,kk' +

"#Pc ,k# i ;

;k'!vH!

E'!vxcE'& !um ,k

!0 " + !79"

with84,85

vH!E'!G""48

nE'!G"

!G!2 when G00

"0 when G"0 !80"

and

vxcE'!r""

dvxcdn '

n!0 "!r"nE'!r". !81"

The nonvariational expressions for EelE'*E' can be obtained

as well: for example,

EelE'*E'"

70

!28"3"BZ)mocc

s *iumkk' !umk

E' +dk. !82"

In Eqs. !76" and !79", the operator i(;/;k') acts on thewave functions um ,k

(0) to replace the operator r' , whichshould have been considered if there was no problem ofcompatibility between the linear potential Eq. !65" and theperiodicity of the crystal. This result had been also obtainedby other mathematical transformations.89

VII. TECHNICALITIES

A. Linear combination of perturbations

In Secs. V and VI, we have examined the responses withrespect to two important classes of perturbations: wave-vector-characterized collective atomic displacements, andhomogeneous electric fields. We were also lead to considerderivatives of the wave functions with respect to their wavevector. These perturbations will be considered as basic per-turbations. Since we are at the level of the linear response,the response of the system to a linear combination of theseperturbations will be the linear combination of the responsesof the system to each perturbation: for example, if a first-order change of potential is described by

vq!1 ""C1vq

:9 ,'!C2vq:9!,, , !83"

then the density response will be

nq!1 ""C1nq

:9 ,'!C2nq:9!,, . !84"

This rule also applies to a linear combination of perturba-tions incommensurate with each other, or of different types.As such, it gives a powerful approach to the treatment of

the q!0 limit of the response to collective atomic displace-ments. Indeed, the singularities observed in Eqs. !60" and!62" in the q!0 limit can be treated separately, as an homo-geneous electric field associated with the collective atomicdisplacements. Thus, for q!0, we will first compute theresponse to a collective atomic displacement without associ-ated electric field #by considering Eq. !60" without theG"0 contribution in the local potential and the Hartreeterm$, and then combines it with the response to an electricfield, as elaborated on in Ref. 16 and P2.4As a complementary advantage of working with linear

combination of perturbations, in the case where one does notwant to compute the response to these basic perturbations,but would like to compute directly the response to one, spe-cific perturbation not contained in this set of basic perturba-tions, one can tailor a variational expression by inserting thecorrect first-order change of potential into the expression Eq.!42", the latter being eventually worked out along the linesdeveloped in Sec. VI.Interestingly, the computations of the responses to a dif-

ferent set of perturbations are completely independent ofeach other, and offer a trivial way to parallelize the code.The amount of computation to be done to get the response toone perturbation is rather large compared to the time neededto initialize or gather the results of the different responsecomputations, and there is no communication during thecomputation of responses. Thus the parallelization will be


rather efficient, as soon as the number of perturbations to betreated is sufficiently large compared to the number of inde-pendent processors.90 This part of the overall process ofcomputing dynamical and dielectric properties is by far themost computing intensive. The steps explained in the P2!Ref. 4" are at least one order of magnitude less time con-suming, and can be easily parallelized as well.

B. Method of solution

Although the atomic displacement and electric field typesof perturbations are different, we arrive at strikingly similarvariational principles #Eqs. !60" and !76"$ under the sametype of constraints #Eqs. !43" and !78"$, and the same rule forthe formation of the density change from wave functionschanges #Eqs. !44" and !77"$. The state-by-state conjugate-gradient algorithm previously described !Sec. III" can be ap-plied straightforwardly to all of these minimization prob-lems. Most of the routines of the code will be common to allthe perturbations, because of the common form of the varia-tional principle.The computation of the steepest-descent vector, Eq. !26",

will be the most time-consuming step in the state-by-stateiterations. As already mentioned, the use of the fast Fouriertransform allows us to make it tractable. In particular, whenthe zero-order Hamiltonian H (0) is to be applied to thechange in first-order wave function, !a" the latter, available inthe reciprocal space, is Fourier transformed to the real space;!b" the local potential part of H (0) is applied to it; !c" theremaining of the right-hand side of Eq. !26" is added; !d" thesum is backtransformed to the reciprocal space; !e" finally,one adds the result of applying the kinetic operator and sepa-rable potentials to the first-order wave function, since theseoperations are less time consuming !at the level of a fewatoms per unit cell". Fast Fourier transforms are also neededto evaluate and update the Hartree potential Eqs. !27" and!30".The computation of the derivative of the wave functions

with respect to their wave vector can also be done using thesame methodology. The quantity to be minimized, Eq. !70",is even simpler than those contained in Eqs. !60" and !76".The routines will be also similar.

C. Sampling of the Brillouin zone and symmetries

In view of the practical implementation of Eqs. !60" and!76", other comments must still be made. First, the integralover the Brillouin zone must be replaced by a summation,through discretization of the k space. For insulators, one canuse the special k point technique of Ref. 91. This replace-ment is well-known in ground-state calculations. Its use inperturbed situations does not lead to technical problems. Formetals, the existence of the Fermi surface, and occupied orunoccupied states below or above it, raises interesting ques-tions. The techniques described by de Gironcoli32 andSavrasov22 can be adapted to the present variational ap-proach.Although it is a usual practice to fix the grid of k points,

and then perform the computation of phonons only for theq-wave vectors that are differences between two k points inthis grid, this attitude is not mandatory, as shown by thepresent theory. Though, some additional small error might

appear when this rule is not followed,92 although likely notlarger than the error associated with the replacement of theintegral over the Brillouin zone by a summation on a discreteset of points.In the ground-state calculations, one is able to reduce the

number of k points by folding the Brillouin zone to its irre-ducible part, using the spatial symmetries and the time-reversal symmetry, with a considerable reduction in comput-ing time. In general, it will not be possible to achieve thesame gain in response calculations with respect to a pertur-bation of the atomic displacement or electric field type, be-cause some symmetries will usually be broken by the pertur-bation. The collective atomic displacements arecharacterized by their q wave vector, the sublattice 9 that isdisplaced, and the direction of the displacement ' . Onlywhen all of these elements are left invariant by some sym-metry operation, will it be possible to reduce the number ofk wave vectors for the summation on the Brillouin zone, forthis perturbation. Nevertheless, all the other symmetry opera-tions can be used to deduce the response with respect toanother, symmetry-related one.By the time-reversal symmetry, the wave vector q is

mapped to #q. So, the response to a #q perturbation can bededuced from the response to a q perturbation. Also, thenumber of k points can be decreased by a factor of two,when 2q is equal to a reciprocal-space G vector.The electric field perturbation is characterized by its di-

rection ' . It is left invariant by the time-reversal symmetry,so that the number of k wave vectors for the summation onthe Brillouin zone can usually be halved. If, moreover, point-group symmetries leave the ' direction invariant, the numberof k wave vectors can be further decreased. If not, the othersymmetries can be used to relate the response with respect toan electric field along some direction with the response withrespect to an electric field along another direction.The computation of the derivative of wave functions with

respect to their wave vector k can also benefit from symme-try operations. However, this computation is rather fast,since there is no self-consistency step in the conjugate-gradient minimization.

VIII. PERSPECTIVES

The aim of the present paper was to present a formalism!theory and algorithm" based on a variational principlewithin the DFPT, which allows one to obtain responses toatomic displacements and homogeneous electric fields. P2!Ref. 4" describes the further manipulations needed to deter-mine the dynamical matrices, interatomic force constant,Born effective charges, and dielectric permittivity tensors.Other papers mentioned in the Introduction give examples ofthe application of this technique.The technical advantages that are characteristics of the

perturbative approaches, like the one developed by Baroni,Giannozzi, and Testa,10,16 have been shown to be also validin the present variational approach: a periodic problem towhich an incommensurable perturbation is imposed can bemapped on an unperturbed unit-cell problem, amounting to aconsiderable reduction of computing time; an homogeneouselectric field, whose potential breaks the periodicity of thelattice, can be treated by the long-wave method; because the


formalism is very similar for the different perturbations, theimplementation of codes for all these properties requires rea-sonable human work. The state-by-state conjugate-gradientalgorithm has the nice property of being unconditionallyconvergent, since the trial second-order derivative of the en-ergy is always decreased, and its convergence is easy tomonitor, the right value being approached from above.The present technique can be extended to cover more per-

turbations, especially those derived from modifications of theunit-cell size and shape,93 or from alchemical transformationof atoms.94,28 A variational principle for second-order deriva-tives of the total energy will be equally valid for these otherperturbations, and the same conjugate-gradient algorithm canbe used efficiently for these.

ACKNOWLEDGMENTS

I would like to thank C. Lee, Ph. Ghosez, and G.-M.Rignanese for a careful reading of the manuscript as well asinteresting discussions, and D. C. Allan, M. P. Teter, K.Rabe, and U. Waghmare for their interest in this work. Iacknowledge numerous interesting discussions with the par-ticipants of the ’96 CECAM workshop on Ab InitioPhonons, especially P. Giannozzi, H. Krakauer, and S.Baroni. I thank S. Savrasov for sending me the preprint Ref.22 prior to publication. I also acknowledge financial supportfrom FNRS-Belgium.

APPENDIX A: CONVENTIONS FOR THE UNPERTURBEDPERIODIC SYSTEM

The present appendix describes the conventions on whichthe perturbed expressions developed in this paper are based:Fourier transform, the relation between real and reciprocalspace, the local and separable parts of pseudopotentials,exchange-correlation functional in the LDA with nonlinearcore correction.

1. Periodic system: Real and reciprocal space

By Bloch’s theorem, each wave function can be decom-posed in a product of a phase factor by a periodic function.Explicitly, we write the ground-state unperturbed wave func-tions as

&mk!0 "!r""!N70"

#1/2eik•rumk!0 "!r", !A1"

where N is the number of unit cells repeated in the Born–von Karman periodic box, and 70 the volume of the unper-turbed unit cell. m and k label the number of the band andthe wave vector of the wave function, respectively. The pe-riodic function can be expanded in terms of plane waves asfollows:95

umk!0 "!r"")

GeiG•rumk!0 "!G", !A2"

where the coefficients umk(0)(G) are the Fourier transform of

umk(0)(r), defined for each vector G of the reciprocal lattice,

umk!0 "!G""

170

"70e#iG•rumk!0 "!r"dr. !A3"

Equation !A1" is such that the orthonormalization condi-tion of the umk

(0)(r) functions is

*umk!0 "!unk

!0 "+"-mn , !A4"

where the scalar product of periodic functions, either repre-sented in real space or in reciprocal space, is defined as

* f !g+"1

70"

70f*!r"g!r"dr")

Gf*!G"g!G". !A5"

This definition of scalar product for periodic functions isdifferent from Eq. !2" which was valid for nonextended wavefunctions. Equation !A4" must be fulfilled only between pe-riodic functions characterized by the same wave vector k.The density of the electronic system is obtained by per-

forming an integral over the whole Brillouin zone, and sum-ming on all the occupied bands:

n !0 "!r""1

!28"3"BZ)mocc

sumk!0 "*!r"umk

!0 "!r"dk. !A6"

Since we will consider only nonmagnetic materials, the spindegeneracy factor s is 2. For insulators, the number of occu-pied bands is independent of the wave vector k, which sim-plifies the practical implementation of these calculations.For metals, the number of occupied bands will depend on

the wave vector k. Instead of making a sharp transition fromthe occupied states and the unoccupied ones, a convenientpractice involves introducing a smeared occupationfunction.97 An alternative approach invokes the linear tetra-hedron method.98 The present formalism could be modifiedin order to incorporate the effect of these modifications, fol-lowing de Gironcoli32 or Savrasov.22The ground-state wave functions can be obtained from the

minimization of the electronic energy per unit cell; Eq. !1"becomes

Eel%u !0 "("70

!28"3"BZ)mocc

s *umk!0 "!Tk,k

!0 "!vext,k,k!0 " !umk

!0 "+dk

!EHxc#n !0 "$ . !A7"

In order to keep the amount of different symbols sufficientlylow, in Eq. !A7" and in Secs. IV–VIII, we redefine Eel andEHxc to be energies per unit cell, unlike in Eq. !1" wherethese quantities were defined for the whole system. We havealso redefined the kinetic and potential operators that act nowon the periodic part of the Bloch functions, according to thefollowing rule, valid for a generic operator O #this definitionis coherent with Eq. !A1"$:

Ok,k!"e#ik•r O eik!•r!. !A8"

For example, we obtain the following expression for the ki-netic operator in reciprocal space, for k"k!:

Tk,k!0 "!G,G!""

12 !G!k"2-GG! . !A9"

The Euler-Lagrange equations associated with the minimiza-tion of Eq. !A7" under constraint Eq. !A4", followed by aunitary transformation, as to get Eq. !7", give


Hk,k!0 "!um ,k

!0 " +".m ,k!0 " !um ,k

!0 " +, !A10"

where

Hk,k!0 ""Tk,k

!0 "!vext,k,k!0 " !

-EHxc-n '

n!0 "

. !A11"

2. Description of the potential operator

In applications based on plane waves, the bare nuclearpotential operator is replaced by a pseudopotential made oflocal and nonlocal contributions from all atoms inside eachrepeated cell with lattice vector Ra :

vext!r,r!"")a9

v9!r#"9#Ra ,r!#"9#Ra", !A12"

where "9 is the vector position of the atoms inside the cell,and each atom contribution is

v9!r,r!""v9loc!r"-!r#r!"!v9

sep!r,r!". !A13"

I consider here only nonlocal parts of the separabletype,66,96,44

v9sep!r,r!"")

<e<91<9!r"1<9* !r!", !A14"

where only a few separable terms, labeled by < , are present.The functions 1<9 are short ranged, and should not overlapfor adjacent atoms. Because of their different mathematicalexpressions, the local and nonlocal parts are treated in dif-ferent ways. A local potential is naturally applied on thewave functions in the real space, since it is a diagonal opera-tor in that representation. A separable potential could betreated efficiently either in reciprocal space or in real space.For small systems !on the order of ten atoms, or less", it ismore efficient to apply the separable potential in the recipro-cal space. The transformations of the wave functions be-tween the real and reciprocal space are carried out by meansof fast Fourier transforms.65Let us first treat the local part. For each atom, this local

part is long ranged, with asymptotic behavior #Z9 /r , whereZ9 is the charge of the !pseudo" ion. It is well known that, ina periodic geometry, this long-ranged part creates a diver-gence in the ionic potential that must be treated together witha similar divergence in the Hartree potential !the divergencescancel each other, but give also a residue, usually incorpo-rated in the ion-ion energy".2 In the reciprocal space, thesedivergences are associated with terms at G"0, constant inreal space. Thus in any case these compensating divergencesare of no importance for the generation of the wave functionsand the density, since only the mean of the potential is af-fected. Although the local potential operator as well as itsderivatives are applied to the wave function in the real space,we will give their !simpler" expression in the reciprocalspace. Their expression in the real space can be obtained bya Fourier transform, similar to Eq. !A2". We define84,85

v loc,k,k! !G""1

70)9e#iG•"9 v9

loc!G" when G00

"0 when G"0. !A15"

In this expression,

v9loc!K""" e#iK•rv9

loc!r"dr, !A16"

where the latter integral is performed throughout all thespace. The limiting behavior of v9

loc(K) for K tending to zerodiverges

v9loc!K!0 ""#

48Z9

K2 !C9!O!K2", !A17"

with

C9"" # v9loc!r"!

Z9

r & dr. !A18"

For the separable part, one obtains

vsep,k,k!0 " !G,G!""

170

)<9

e<9# )G e#i!k#G"•"91<9!k!G"&$# )

G!ei!k!G!"•"91<9* !k!G!"& , !A19"

where

1<9!K""" e#iK•r1<9!r"dr. !A20"

The special form of the matrix of separable potential, Eq.!A19", allows for its efficient application to any wave func-tion.

3. Exchange and correlation energy and potential in the LDA

In the LDA, the exact exchange-correlation energy, afunctional of the density everywhere, is replaced by the in-tegral of the density n(r) times the mean exchange-correlation energy per particle .xc(r) of the homogeneouselectron gas at the point r. However, when combined withpseudopotentials, this simple definition is to be modified, inorder to take into account that only valence states are used tobuild the density: the contribution of the core electronsshould be included, because of the nonlinear character of theexchange-correlation energy functional.67 The functionalthen is

Exc#n!r"$""70

#n!r"!nc!r"$.xc#n!r"!nc!r"$dr,

!A21"

where the pseudocore density nc is made of nonoverlappingcontributions from each atom,

nc!r"")a9

nc ,9!r#"9#Ra". !A22"

The pseudocore density from each atom nc ,9 is built at thesame time as the pseudopotential.67,99 It has spherical sym-metry, and is specified by a one-dimensional function. Thecorresponding exchange-correlation potential is


vxc„n!r"…"d#!n!nc".xc!n!nc"$dn '

n"n!r";nc"nc!r",

!A23"

and its derivative with respect to the density

dvxc„n!r"…dn "

d2#!n!nc".xc!n!nc"$dn2 '

n"n!r";nc"nc!r".

!A24"

4. Unperturbed energy and Hamiltonian in the LDA

With these definitions, the electronic energy is

Eel%u !0 "("70

!28"3"BZ)mocc

s *umk!0 "!Tk,k

!0 "!vsep,k,k!0 " !umk

!0 "+dk!"70n !0 "!r"v loc!!0 "!r"dr

!"70

#n !0 "!r"!nc!0 "!r"$„.xc#n !0 "!r"!nc

!0 "!r"$…dr!2870 )G00

!n !0 "!G"!2

!G!2 . !A25"

The Hartree energy #last term of Eq. !A25"$ can also becomputed as

EH""70n !0 "!r"vH!

!0 "!r"dr, !A26"

with the Hartree potential being defined as84,85

vH!!0 "!G""2870 )

G00

n !0 "!G"

!G!2 when G00

"0 when G"0. !A27"

The Hamiltonian is given by

Hk,k!0 ""Tk,k

!0 "!vsep,k,k!0 " !!v loc!!0 "!vH!

!0 "!vxc!0 "". !A28"

The local, Hartree, and exchange-correlation !XC" potentialsare operators local in the real space which are independent ofk.

APPENDIX B: RESPONSE TO AN HOMOGENEOUS,STATIC ELECTRIC FIELD BY THE

LONG-WAVE METHOD: DETAILED TREATMENT

1. Small-wave-vector limit of the responseto an incommensurate perturbation

Inspired by Eq. !68", we write the following first-orderpotential operator change:

vext,q!1 " !r,r!""eiq•r-!r#r!", !B1"

which corresponds to the simple change of phase-factorizedlocal potential

v loc,q!1 " "1. !B2"

The supplementary constant 1/i!q!, present in Eq. !68", willbe taken into account afterwards. No second-order potentialchange, first-order separable potential change, or nonlinearexchange-correlation core correction is present. This leads toimportant simplifications of the variational principle #com-pare with Eq. !60", which was obtained for collective atomicdisplacements$:

Eel,#q,q!2 " %u !0 ";u !1 "(

"70

!28"3"BZ)mocc

s *umk,q!1 " !Hk!q,k!q

!0 " #.mk!0 "!umk,q

!1 " +dk

!70

2 # nq*!1 "!G"0"! nq!1 "!G"0"$

!12"70

dvxcdn '

n!0 "!r"!nq

!1 "!r"!2 dr

!2870)G

!nq!1 "!G"!2

!q!G!2 , !B3"

where we have taken advantage of

"70nq

!1 "!r"dr"70nq!1 "!G"0". !B4"

The corresponding nonvariational expressions are simple:

Eel,#q,q!2 " "

70

2 nq*!1 "!G"0""70

2 nq!1 "!G"0". !B5"

The associated first-order Hamiltonian operator is a local po-tential operator, here written in the reciprocal space,

Hq!1 "!G""-G,0!48

nq!1 "!G"

!G!q!2 ! vxc,q!1 " !G". !B6"

The exchange-correlation contribution vxc,q(1) (G) is obtained

from the knowledge of nq(1) through Eq. !63". This local po-

tential operator includes a long-wave part !for G"0), butalso local fields !for G00), the latters being of electrostaticorigin !the Hartree part" as well as exchange-correlation ori-gin.From Eq. !B6" one can infer that the long-wave !macro-

scopic" potential, for G"0 in the limit of q!0, is made ofthe bare applied potential and the electronic screening, due tothe Hartree term:


Hq!1 "!G"0""1!48

nq!1 "!G"0"

!q!2 . !B7"

In the derivation of this equation, we have neglected theexchange-correlation contribution in comparison to the Har-tree term in the limit of q!0, because of the 1/q2 divergenceof the latter while in the LDA, the exchange-correlation en-ergy functional is well behaved in this limit.100

2. Analytical treatment of the Hartree divergence

The divergence of the Hartree term in Eq. !B3" can betreated analytically, thanks to the Sherman-Morrison formulafor connecting two inversion !or minimization" problems.101The following auxiliary variational expression is first mini-mized #this expression would be equal to Eq. !B3" if it werenot for the removal of the divergent G"0 Hartree contribu-tion$:

Eel,#q,q!2 " %u !0 "; u !1 "(

"70

!28"3"BZ)mocc

s *u mk,q!1 " !Hk!q,k!q

!0 " #.mk!0 "!u mk,q

!1 " +dk

!70

2 # n q*!1 "!G"0"! n q!1 "!G"0"$

!12"70

dvxcdn '

n!0 "!r"!n q

!1 "!r"!2!2870)G00

!n q!1 "!G"!2

!q!G!2 ,

!B8"

with

n q!1 "!r""

2!28"3"BZ)m

occ

sumk!0 "*!r"u mk,q

!1 " !r"dk !B9"

under constraints

*um ,k!q!0 " !u n ,k,q

!1 " +"0, !B10"

where the indices m and n runs over occupied states. Theassociated nonvariational expressions give

Eel,#q,q!2 " "

70

2 n q*!1 "!G"0""70

2 n q!1 "!G"0". !B11"

The quantity nq(1)(G"0) has the following expression in

terms of the scalar products of zeroth- and first-order wavefunctions:

n q!1 "!G"0""

2!28"3"BZ)m

occ

s *umk!0 "!u mk,q

!1 " +dk.

!B12"

The associated first-order Hamiltonian operator, in the limitq!0 is

Hq!1 "!G""48

n q!1 "!G"

!G!q!2 ! v xc,q!1 " !G" when G00

"1 when G"0. !B13"

That is, the screening by the Hartree term has been removedwhen G"0.By the Sherman-Morrison formula, at the minimum of

both Eq. !B8" and Eq. !B3" one gets the following relation-ships:

n q!1 "!G"0""

nq!1 "!G"0"

1!48

q2 nq!1 "!G"0"

, !B14"

nq!1 "!G"0""

n q!1 "!G"0"

1#48

q2 n q!1 "!G"0"

, !B15"

Eel,#q,q!2 " "Eel,#q,q

!2 " # 1!48

q2 nq!1 "!G"0" & , !B16"

!umk,q!1 " +"!u mk,q

!1 " +# 1!48

q2 nq!1 "!G"0" & . !B17"

These equations are especially important in that, whencompared with Eq. !B7", they show that the rate of change ofEel,#q,q(2) with respect to an electric field #the long-wave partof Hq

(1)(G"0)], is the same as the rate of change ofEel,#q,q(2) with respect to a bare applied field Hq

(1)(G"0).

3. The limit q˜0

We now chose a particular direction q along which thelimit q!0 is taken. Let it be the ' direction. We haveq"qe' , where e' is a unit vector along direction ' , and q isthe norm of the q vector, tending to zero. We expand thezeroth-order and first-order wave functions in powers of thesmall parameter q:

umk!q!0 " "umk

!0 "!qdumk

!0 "

dk'!O!q2" !B18"

and

u mk,q!1 " " u mk,0

!1 " !qdu mk,0

!1 "

dq'!O!q2". !B19"

At the lowest order in q , the Euler-Lagrange equationderived from Eq. !B8", with the Hamiltonian Eq. !B13", issatisfied by taking

u mk,0!1 " "0, !B20"

which constitutes the unique solution of them, because thewhole quadratic form in Eq. !B8" is definite positive !see the


discussion in Sec. IV B of Ref. 42". This means that theexpansion of n q

(1) , v xc,q(1) , or Hq

(1) in powers of q only be-gins with the linear term.From Eq. !B20", at the first order in q , the constraints Eq.

!B10" give

* um ,k!0 " 'du nk,0!1 "

dq'+ "0, !B21"

where the indices m and n run over occupied states. Becauseof Eqs. !B9", !B20", and !B21", the expansion ofn q(1)(G"0) will even begin at the second order only. The

same is true for the expansion of Eel,#q,q(2) .

Finally, taking into account the second-order expansion ofEq. !B10",

* um ,k!0 " 'd2u nk,0!1 "

dq'2 + !* du m ,k

!0 "

dk''du nk,0

!1 "

dq'+ "0, !B22"

as well as its complex conjugate, we obtain a variationalexpression with respect to the quantities du nk,0

(1) /dq' :

12d2Eel,#q,q

!2 "

dq'2 '

q!0$ u !0 ";

du !1 "

dq'%

"70

!28"3"BZ)mocc

s# * du mk,q!1 "

dq''Hk!q,k!q

!0 " #.mk!0 "'du mk,q

!1 "

dq'+

#* du mk,q!1 "

dq''dumk!0 "

dk'+ #* dumk!0 "

dk''du mk,q

!1 "

dq'+ dk

!12"70

dvxcdn '

n!0 "!r"' dn q

!1 "!r"dq'

'2

!2870)G00

' dn q!1 "!G"

dq''2

!G!2 , !B23"

with

dn q!1 "!r"dq'

"2

!28"3"BZ)mocc

sumk!0 "*!r"

du mk,0!1 " !r"dq'

dk

!B24"under constraints Eq. !B21".The connection with the equations presented in Secs. VIA

and VIB is now possible, thanks to the identification

umkE' "

q'

i!q!du mk,0

!1 " !r"dq'

. !B25"

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47X. Gonze, J.-C. Charlier, D. C. Allan, and M. P. Teter, Phys. Rev.B 50, 13 035 !1994".

48C. Lee and X. Gonze, Phys. Rev. B 51, 8610 !1995".49C. Lee and X. Gonze, Phys. Rev. Lett. 72, 1686 !1994".50C. Lee and X. Gonze, J. Phys. Condens. Matter 7, 3693 !1995".51C. Lee and X. Gonze !unpublished".52C. Lee and X. Gonze !unpublished".53G.-M. Rignanese, J.-P. Michenaud, and X. Gonze, Phys. Rev. B53, 4488 !1996".

54C. Lee and X. Gonze, Phys. Rev. B 49, 14 730 !1994".55C. Lee, Ph. Ghosez, and X. Gonze, Phys. Rev. B 50, 13 379

!1994".56Ph. Ghosez, X. Gonze, and J.-P. Michenaud, Ferroelectrics 153,91 !1994".

57Ph. Ghosez, X. Gonze, Ph. Lambin, and J.-P. Michenaud, Phys.Rev. B 51, 6765 !1995".

58Ph. Ghosez, X. Gonze, and J.-P. Michenaud, Ferroelectrics 164,113 !1995".

59Ph. Ghosez, X. Gonze, and J.-P. Michenaud, Ferroelectrics 186,73 !1996".

60Ph. Ghosez, X. Gonze, and J.-P. Michenaud, Europhys. Lett. 33,713 !1996".

61K. Rabe and U. Waghmare, Ferroelectrics 164, 15 !1995".62K. Rabe and U. Waghmare, J. Phys. Chem. Solids 57, 1397

!1996".63S. Ogut and K.M. Rabe, Phys. Rev. B 54, R8297 !1997".64E. J. Cockayne and K. Rabe !unpublished".65R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 !1985".66L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425

!1982".67S. G. Louie, S. Froyen, and M. L. Cohen, Phys. Rev. B 26, 1738

!1982".68 In general, the perturbation could also directly affect the kineticoperator, as well as the exchange-correlation functional or even


the Hartree functional. The case of an exchange-correlationfunctional that is affected by atomic displacements is treated inAppendix A !nonlinear exchange-correlation core correction",but direct effects on the kinetic operator and on the Hartreefunctional are not considered here. They do not appear for thetwo types of perturbations considered in the present paper. Theyare treated at a general level in Ref. 42.

69O. Sinanoglu, J. Chem. Phys. 34, 1237 !1961".70X. Gonze, Phys. Rev. A 52, 1086 !1995".71Unfortunately, the perturbation expansion of band-specific quan-tities like the eigenvalues, or any band-by-band decomposition,cannot be performed in the parallel transport gauge, but in thediagonal gauge, which we will not describe here !Ref. 42".

72But not with respect to & (0).73R. M. Sternheimer, Phys. Rev. B 96, 951 !1954".74P. H. Dederichs and R. Zeller, Phys. Rev. B 28, 5462 !1983".75 I. Stich, R. Car, M. Parrinello, and S. Baroni, Phys. Rev. B 39,4997 !1989".

76M. J. Gillan, J. Phys. Condens. Matter 1, 689 !1989".77C. H. Park, I.-H. Lee, and K. J. Chang, Phys. Rev. B 47, 15 996

!1993".78R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 49, 5828

!1994".79Actually, iterative diagonalization techniques #see, for example,D. M. Wood and A. Zunger, J. Phys. A 18, 1343 !1985"; or J.Hutter, H. P. Luthi, and M. Parrinello, Comp. Mater. Science 2,244 !1994"$ can also take advantage of the sparseness of theHamiltonian in the case of the two-level procedure, but never-theless some advantages of the conjugate-gradient scheme overthese improved two-level procedures should remain: in theconjugate-gradient algorithm the energy always decreases,which gives a very robust algorithm, and the merge of the twolevels to one can lead to a decrease of the number of operationsneeded to reach the same accuracy.

80W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetter-ling, Numerical Recipes: The Art of Scientific Computing !Cam-bridge University Press, Cambridge, England, 1989".

81For such a q wave vector, the perturbing potential can beHermitian, since the phase factor in Eq. !33" is a real number.Thus, it is not required to resort to the theory of Sec. IVB totreat them. Moreover, one can show that the results will be thesame by either treating these commensurate perturbations assuch, or using the limit of incommensurate perturbations tendingto the commensurate wave vector, with the exception of theeffects connected to the presence of an electric field !see Sec.VI".

82 In order to obtain Eqs. !44" and !47", the time-reversal symmetryhas been used in order to get rid of the first-order wave functionsfor #q using the following equalities:

umk,q!1 " !r""2 um!#k",!#q"

!1 "* !r" !B26"

and

umk!0 "!r""2 um!#k"

!0 "* !r" !B27"

!where 2 is a complex number of module unity", or using ageneralization of these equalities when degeneracies are present.Other equations were also simplified thanks to this symmetry:all those giving the first-order densities in terms of the first-orderwave functions, and the nonvariational expressions for thesecond-order derivatives of the energy.

83The transformation of wave functions, the density, and potentialbetween the real space and the reciprocal space will be oftenperformed in an efficient way thanks to the fast Fourier trans-form algorithm.

84The local potential v loc,k,k! does not depend on k and will beusually written v loc! . This shortening of the notation will be usedfor all occurrences of a k independent quantity.

85The prime symbol indicates that the G"0 term is excluded.86 In this expression, it is taken into account that the electroniccharge is #1 in the atomic units.

87L. D. Landau and E. M. Lifshits, Electrodynamics of ContinuousMedia !Pergamon, London, 1960".

88These equations differ from those given in Ref. 55, because ofdifferent choices of conventions.

89See, for example, E. I. Blount, in Solid State Physics: Advances inResearch and Applications, edited by F. Seitz and D. Turnbull!Academic, New York, 1962", Vol. 13, p. 306; S. Baroni and R.Resta, Phys. Rev. B 33, 7017 !1986".

90G.-M. Rignanese, J.-M. Beuken, J.-P. Michenaud, and X. Gonze!unpublished".

91H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 !1976".92K. Rabe and U. Waghmare !private communication".93S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 59, 2662

!1987".94 J. Baur, K. Maschke, and A. Baldereschi, Phys. Rev. B 27, 3720

!1983".95These notations do not distinguish between the functions whosearguments are in real space or in reciprocal space, and that are aFourier transform of each other. However, no confusion shouldhappen, since the arguments belong clearly to one space or theother.

96X. Gonze, R. Stumpf, and M. Scheffler, Phys. Rev. B 44, 8503!1991".

97C.-L. Fu and K.-M. Ho, Phys. Rev. B 28, 5480 !1983"; R. J.Needs, R. M. Martin, and O. H. Nielsen, ibid. 33, 3778 !1986";M. Methfessel and A. T. Paxton, ibid. 40, 3616 !1989"; M. Ped-erson and K. Jackson, ibid. 40, 7312 !1991"; R. M. Wentzco-vitch, J. L. Martins, and P. B. Allen, ibid. 45, 11 372 !1992"; A.De Vita, Ph.D. thesis, Keele University, 1992.

98O. Jepsen and O. K. Andersen, Phys. Rev. B 29, 5965 !1984".99M. P. Teter, Phys. Rev. B 48, 5031 !1993".100However, in the ‘‘exact’’ DFT, an exchange-correlation electricfield exists in the limit q!0, which is totally absent in the LDA!Ref. 3".

101Page 67 in Ref. 80.


Dynamical matrices, Born effective charges, dielectric permittivity tensors,and interatomic force constants from density-functional perturbation theory

Xavier GonzeUnite de Physico-Chimie et de Physique des Materiaux, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

Changyol LeeDepartment of Physics, Ewha Womans University, Seoul, 120-750, South Korea

!Received 7 November 1996"

Starting from the knowledge of first-order changes of wave functions and density with respect to smallatomic displacements or infinitesimal homogeneous electric fields within the density-functional theory, wewrite the expressions for the diagonal or mixed second-order derivatives of the total energy with respect tothese perturbations: dynamical matrices for different wave vectors, Born effective-charge tensors and elec-tronic dielectric permittivity tensors. Interatomic force constants and the phonon-band structure are then ob-tained by computing the Fourier transform of dynamical matrices on a regular mesh of wave vectors, with aneventual, separate treatment of the long-range dipole-dipole interaction. The same ingredients also allow one tocompute the low-frequency response of the crystal to homogeneous electric fields. #S0163-1829!97"05116-3$

I. INTRODUCTION

Nowadays, the density-functional theory1,2 !DFT" is con-sidered as the method of choice for simulating solids fromthe first principles. The present paper focuses on the compu-tation, from perturbation theory within DFT !actually withinthe local-density approximation to DFT", of second deriva-tives of the total energy of periodic solids with respect to !a"collective displacements of atoms with different wave vec-tors, either commensurate or incommensurate with the un-derlying lattice; and !b" homogeneous static electric fields. Inthe preceding paper !P1",3 it was shown how to compute thecorresponding first-order changes in density, wave functions,and self-consistent potentials, thanks to a conjugate-gradientalgorithm, with plane waves and pseudopotentials. The fur-ther second-order derivatives of the energy are directlylinked to the dynamical matrices at any wave vector, thelow-frequency !ion-clamped" dielectric permittivity tensor%&, and the Born effective-charge tensors Z* !mixed second-order derivative with respect to atomic displacement andelectric field".As for the calculation of the first-order responses, the

methods used for the second-order derivatives of the energyare of two types: the direct approaches, and the perturbativeapproaches. In the frozen-phonon method !a direct ap-proach", a small, but finite, perturbation is frozen in the sys-tem, allowing us to compute, e.g., interatomic forceconstants.4 It is also possible to extract phonon eigenfrequen-cies and eigenmodes from molecular dynamic trajectories,5which is another direct method. Recent progress in polariza-tion theory has open the way to direct approaches of Borneffective charges,6 and dielectric permittivity tensors.7However, in the frozen-phonon or the molecular dynam-

ics methods, one has to deal with supercells, whose size de-pends on the commensurability of the perturbation with theunperturbed periodic cell. When the original cell is small !afew atoms", the supercells to take into account will be typi-cally four or eight times larger, with a considerable increase

in computer time. A recently proposed order N approach tothe computation of phonon bands and interatomic force con-stant could partially waive this drawback.8By contrast, for wave-vector-characterized perturbations,

perturbation theory allows one to map the computation of theresponses onto an equivalent problem presenting the period-icity of the unperturbed periodic ground state, which is anobvious advantage over direct methods. Baroni, Giannozziand Testa9 !BGT" have popularized this type of method, asdescribed in P1.10 Many perturbative implementations of thecomputation of the first-order responses have been realized.When the first-order responses have been obtained, the gen-eration of the diagonal !two derivatives with respect to thesame perturbation" or mixed !one derivative with respect toone perturbation, one more derivative with respect to an-other" second-order derivatives of the total energy, can beperformed. This supplementary step is rather easy, comparedto the computation of the first-order responses. Actually,from the latter, even the mixed third-order derivatives of theenergy can be computed easily.11There are different formulas connecting the first-order re-

sponses to second-order derivatives of the energy. Some ofthem are stationary with respect to the errors made in thefirst-order responses. Others, inherently less accurate, have,in the case of mixed second-order derivatives of the totalenergy, the advantage of using the knowledge of the first-order responses with respect to only one of the two pertur-bations. This property, in a different context, was called the‘‘interchange theorem.’’12 The stationarity of some formulas,as well as the interchange theorem, is a consequence of theexistence of the variational principle for the total energy ofthe system.2 In this paper, the different formulas,nonstationary,13 as well as stationary, will be developed, forthe above-mentioned perturbations, in the framework of aplane-wave-pseudopotential method. Like in P1, efficientseparable pseudopotentials,14 as well as the nonlinearexchange-correlation core correction,15,16 are considered. Re-sults obtained with these formulas were exhibited in Refs.

PHYSICAL REVIEW B 15 APRIL 1997-IIVOLUME 55, NUMBER 16

550163-1829/97/55!16"/10355!14"/$10.00 10 355 © 1997 The American Physical Society

17–22, with restricted presentation of the underlying theory.For insulators, once the analytic part of the dynamical

matrix at q!0 as well as %& and Z* are available, it ispossible to compute the LO-TO splitting of phonon frequen-cies at q!0, the low-frequency dielectric permittivity tensor,including the effect of ionic motion, and also the infraredreflectivity. These formulas will be derived in the presentcontext, explicitly taking into account the anisotropy of %&

and Z*.When the dynamical matrices are known for a sufficiently

fine grid of wave vectors in the irreducible Brillouin zone,one can generate easily the interatomic force constants!IFC’s" using a Fourier transformation, as well as dynamicalmatrices and phonon frequencies interpolated for any wavevectors. The efficiency of these transformations can benefitfrom a separate, analytic, treatment of the long-rangeddipole-dipole interactions, made possible by the knowledgeof %& and Z*. This treatment, in the case of isotropic %& andZ* is rather easy to formulate, while the generalization toanisotropic quantities was only recently proposed.17 It hasbeen used to compute the phonon band structure of SiO2stishovite,18–20,23,24 SiO2 quartz,17,19 and to analyze the insta-bilities in cubic and rhombohedral BaTiO3.21 A comprehen-sive description of this technique is presented here.Once the complete phonon-band structure is available,

one can compute the phonon density of states, some thermo-dynamical properties, and the atomic temperature factors.The corresponding formulas have been recalled in Ref. 19.This paper is organized as follows. In Sec. II, we present

the different generic formulas !stationary and nonstationary",that allow us to compute mixed second derivatives of thetotal energy from the knowledge of the first-order responses.In Sec. III, the second-order derivatives of the total energywith respect to atomic displacements are presented, withinthe plane-wave-pseudopotential implementation: they allowus to compute the dynamical matrices and phonon frequen-cies. The derivatives with respect to homogeneous electricfields, which allow us to compute dielectric permittivity ten-sors are developed in Sec. IV. Section V focuses on mixedderivatives with respect to atomic displacements and homo-geneous electric fields: the Born effective charges. Then, wediscuss the implementation of these equations !Sec. VI" anddetail the sum rules to be checked for accuracy !Sec. VII".The two last sections of the paper build upon the resultsobtained in the previous sections: the computation of thelow-frequency dielectric permittivity tensor and the associ-ated LO-TO splitting !Sec. VIII", and the computation ofinteratomic force constants and phonon-band structures !Sec.IX". Some perspectives are presented in Sec. X. Appendix Adescribes briefly the computation of the dielectric permittiv-ity tensors using different approximations: the ‘‘local densityplus scissors’’ approach, the random phase approximation,and the neglect of local fields.Throughout this paper, we use the atomic !Hartree" units.

The notations and conventions are described in P1 !Ref. 3"and Ref. 25.

II. MIXED DERIVATIVES OF THE TOTAL ENERGY

We consider two or more simultaneous Hermitian pertur-bations, combined in a Taylor-like expansion of the follow-ing type:

vext!!"!vext!0 ""'

j1( j1vext

j1 "'j1 j2

( j1( j2vextj1 j2"••• !1"

!the indices j1 and j2 are not exponents, but label the differ-ent perturbations". The mixed derivative of the energy of theelectronic system

Eelj1 j2!

12

)2Eel)( j1)( j2

!2"

is obtained in the local-density approximation to DFTfrom25,26

Eelj1 j2!

12 ! Eel

j1 j2"Eelj2 j1", !3"

with

Eelj1 j2*+!0 ";+ j1,+ j2,

!'-

#.+-j1!H !0 "#%-

!0 "!+-j2/"!.+-

j1!vextj2 "vHxc0

j2 !+-!0 "/

".+-!0 "!vext

j1 "vHxc0j1 !+-

j2/"".+-!0 "!vext

j1 j2!+-!0 "/$

"12" " 02EHxc

0n!r"0n!r!" #n!0 "

n j1!r"n j2!r!"dr dr!

"12

d2EHxcd( j1d( j2

#n!0 "

. !4"

The derivatives of the wave functions and density with re-spect to one perturbation can be obtained from the techniqueexplained in P1, or from the BGT technique,27–29 applied tothe case of that particular perturbation.Supposing that the first-order wave functions and densi-

ties are not exact, then Eqs. !3" and !4" give an estimation ofEelj1 j2 that has an error proportional to the product of errors

made in the first-order quantities for the first and secondperturbations. It is a stationary expression. If these errors aresmall, their product will be much smaller. However, the signof the error is undetermined, unlike for the variationalexpressions30,31 presented in detail in P1 #see Eq. !13" of Ref.3$.The following expressions do not have these interesting

properties !their error is on the order of the errors made onthe first-order wave functions or densities, and not theirproduct", but allows us to evaluate Eel

j1 j2 from the knowledgeof the derivative of wave functions with respect to only oneof the perturbations:

10 356 55XAVIER GONZE AND CHANGYOL LEE

Eelj1 j2!

12'-occ

!.+-j2!vext

j1 "vHxc0j1 !+-

!0 "/

".+-!0 "!vext

j1 "vHxc0j1 !+-

j2/""Enon-varj1 j2

!'-

occ

.+-j2!vext

j1 "vHxc0j1 !+-

!0 "/"Enon-varj1 j2

!'-

occ

.+-!0 "!vext

j1 "vHxc0j1 !+-

j2/"Enon-varj1 j2 , !5"

where

Enon-varj1 j2 !'

-

occ

.+-!0 "!vext

j1 j2!+-!0 "/"

12

d2EHxcd( j1d( j2

#n!0 "

. !6"

In the expressions Eq. !5", !+-j1/ is not needed, while the

computation of vextj1 and vHxc0

j1 takes little time. Similar ex-pressions that do not involve !+-

j2/ but !+-j1/ are also avail-

able.The time-reversal symmetry allows us to simplify further

these expressions. For example,

Eelj1 j2!'

-

occ

.+-j2!vext

j1 "vHxc0j1 !+-

!0 "/"Enon-varj1 j2 . !7"

These results Eqs. !5"–!7" are generalizations of the so-called ‘‘interchange theorem,’’12 and will be exploited in thenext three sections.13 We will, moreover, suppose that wehave been able to compute the first-order responses !i.e.,changes in wave functions and densities" to the basic pertur-bations described previously.27,28,3,29

III. DYNAMICAL MATRIX AND PHONON FREQUENCIES

The total energy of a periodic crystal with small latticedistortions from the equilibrium positions can be expressedas

E tot!*1","!E tot!0 ""'

a2-'b2!3

12 $ )2E tot

)42-a )42!3

b %142-a 142!3

b

"••• , !8"

where 142-a is the displacement along direction - of the

atom 2 in the cell labeled a !with vector R a), from its equi-librium position "2 .The matrix of the IFC’s is defined as

C2- ,2!3!a ,b "!$ )2E tot)42-

a )42!3b % , !9"

its Fourier transform is

C2- ,2!3!q"!1N'

abC2- ,2!3!a ,b "e#iq•!Ra#Rb"

!'bC2- ,2!3!0,b "eiq•Rb , !10"

where N is the number of cells of the crystal in the Born–von Karman approach.32 It is connected to the dynamicalmatrix D2- ,2!3(q) by

D2- ,2!3!q"!C2- ,2!3!q"/!M2M2!"1/2 . !11"

The squares of the phonon frequencies 5mq2 at q are obtained

as eigenvalues of the dynamical matrix D2- ,2!3(q), or assolutions of the following generalized eigenvalue problem:

'2!3

C2- ,2!3!q"Umq!2!3"!M25mq2 Umq!2-" . !12"

From Eqs. !8"–!10", the matrix C2- ,2!3(q) can be linkedto the second-order derivative of the total energy with re-spect to collective atomic displacements of the type de-scribed in P1:

C2- ,2!3!q"!2E tot,#q,q42-* 42!3 . !13"

E tot is made of a contribution from the electron system and acontribution from the electrostatic energy between ions.Similarly, the C matrix is split in two parts:

C2- ,2!3!q"!Cel,2- ,2!3!q""CEw,2- ,2!3!q". !14"

The tools developed in P1 would allow us to build thediagonal part of the Cel(q) matrix, in a plane wave basis,with efficient separable pseudopotentials and a nonlinearexchange-correlation core correction. In the following sub-sections, these results are generalized to the nondiagonal partof this matrix, and the ion-ion term is also computed.

A. The electronic contribution

The use of the mixed derivative formulas, shown in Sec.II, gives the following stationary expression !see P1 for thenotations":

55 10 357DYNAMICAL MATRICES, BORN EFFECTIVE CHARGES, . . .

Eel,#q,q42-* 42!3*u !0 ";uq

42- ,uq42!3,!

60

!27"3"BZ'mocc

s !.umk,q42- !Hk"q,k"q

!0 " #%mk!0 "!umk,q

42!3 /".umk,q42- !vsep,k"q,k

42!3 !umk!0 "/

".umk!0 "!vsep,k,k"q

42- !umk,q42!3 /".umk

!0 "!vsep,k,k42-42!3!umk

!0 "/ "dk

"12"60

*# nq42-!r"$*# v loc,q

42!3!r"" vxc0,q42!3 !r"$" nq

42!3!r"# v loc,q42- !r"" vxc0,q

42- !r"$*,dr

"12"60

dvxcdn #

n!0 "!r"# nq

42-!r"$*nq42!3!r"dr"2760'

G

# nq42-!G"$*nq

42!3!G"

!q"G!2

""60

„n !0 "!r"v loc! 42-* 42!3!r"…dr"12d2Exc

d42- ,#qd42!3 ,q#n!0 "

. !15"

The corresponding nonstationary expressions are

Eel,#q,q42-* 42!3*u !0 ";uq

42-,!60

!27"3"BZ'mocc

s !.umk,q42- !vsep,k"q,k

42!3 !umk!0 "/".umk

!0 "!vsep,k,k42-42!3!umk

!0 "/ "dk

"12"60

*# nq42-!r"$*# v loc,q

42!3!r"" vxc0,q42!3 !r"$,dr

""60

„n !0 "!r"v loc! 42-* 42!3!r"…dr"12d2Exc

d42- ,#qd42!3 ,q#n!0 "

!16"

and

Eel,#q,q42-* 42!3*u !0 ";uq

42!3,!60

!27"3"BZ'mocc

s !.umk!0 "!vsep,k,k"q

42- !umk,q42!3 /".umk

!0 "!vsep,k,k42-42!3!umk

!0 "/ "dk

"12"60

*nq42!3!r"# v loc,q

42- !r"" vxc0,q42- !r"$*,dr

""60

„n !0 "!r"v loc! 42-* 42!3!r"…dr"12d2Exc

d42- ,#qd42!3 ,q#n!0 "

. !17"

Using Eqs. !16" and !17", a whole column or a whole row of the dynamical matrix C2- ,2!3(q) can be obtained from theknowledge of the first-order wave functions with respect to only one perturbation, either uq

42- or uq42!3 , respectively.

The mixed second derivatives of the local and nonlocal potentials !given here in reciprocal space", and the second deriva-tives of the exchange-correlation functional are obtained from

v loc! 42-* 42!3!G"!#022!260

G-G3 e#iG•"2 v2loc!G" when G80

!0 when G!0, !18"

vsep,k,k42-* 42!3!G,G!"!

022!260

'9

e92

)2

)42-)42!3& $ 'G e#i!k"G"•"2:92!k"G"% $ '

G!ei!k#G!"•"2:92* !k"G!"% ' , !19"

12

d2Excd42- ,#qd42!3 ,q

#n!0 "

!022!$ 12"60

dvxcdn #

n!0 "!r"# nc ,q

42-!r"$*# nc ,q42!3!r"$dr""

60vxc„n !0 "!r"…nc42-* 42!3!r"dr% , !20"


with

nc ,042-* 423!r"!'

a

12

)2

)42 ,-)42 ,3nc ,2!r#"2#Ra". !21"

B. The ion-ion contribution

Following the Ewald summation method,2 the ion-ioncontribution to the unperturbed total energy per unit cell !towhich the residue of the cancellation of the divergences men-tioned in Sec. IV B of P1 is added" is obtained as

EEw!12'22!

Z2Z2!& 'G80

47

60G2 eiG•! "2# "2!"exp$ #G2

4;2%#'

a;eiq•RaH!;da ,22!"#

2!7

;022!#7

60;2'

"12'22!

Z2C2!60

, !22"

with H(y)!erfc(y)/y , da ,22!!!da ,22!!, and da ,22!!Ra" "2!# "2 . The parameter ; can assume any value,and is adjusted to obtain the fastest convergence of bothreciprocal- and real-space sums.The contribution of the second derivative of the ion-ion

energy to the matrix CEw(q) can be computed following Ref.33,

CEw,2- ,2!3!q"!CEw,2- ,2!3!q"#022! '2"

CEw,2- ,2!3!q!0".

!23"

CEw,2- ,2!3(q) can be split into three parts: a rapidly con-vergent sum in reciprocal space; a rapidly convergent sum inreal space; and a rather simple residual contribution,34

CEw,2- ,2!3!q"

!Z2Z2!& 'G with K!G"q

47

60

K-!K3!K2 ei K•! "2# "2!"

$exp$ #K2

4;2% #'a

;3 ei q•Ra H-!3!iso !;da ,22!"

#4

3!7;3 022!' , !24"

with

H-3iso !x"!

x-x3

x2 & 3x3erfc!x ""2

!7e#x2 $ 3x2 "2 % '

#0-3 $ erfc!x "

x3 "2

!7

e#x2

x2 % . !25"

The superscript ‘‘iso’’ is used in order to distinguish thisquantity from its anisotropic generalization, needed in Sec.IX.

C. The q$0 case

As mentioned in Sec. VII A of P1, the limit q!0 must beperformed carefully. By the separate treatment of the electricfield associated with phonons in this limit, one sees that a‘‘bare’’ q!0 dynamical matrix must be computed, to whicha ‘‘nonanalytical’’ part will be added, in order to reproducecorrectly the q!0 behavior along different directions !seeSec. VIIIB". The bare dynamical matrix is obtained from thefollowing electronic contribution:

Eel,0,042-* 42!3*u !0 ";uq!0

42- ,uq!042!3,!

60

!27"3"BZ'mocc

s !.umk,q!042- !Hk,k

!0 "#%mk!0 "!umk,q!0

42!3 /".umk,q!042- !vsep,k,k

42!3 !umk!0 "/

".umk!0 "!vsep,k,k

42- !umk,q!042!3 /".umk

!0 "!vsep,k,k42-42!3!umk

!0 "/ "dk

"12"60

*# nq!042- !r"$*# v loc,q!0!42!3 !r"" vxc0,q!0

42!3 !r"$" nq!042!3!r"# v loc,q!0! 42- !r"" vxc0,q!0

42- !r"$*,dr

"12"60

dvxcdn #

n!0 "!r"# nq!0

42- !r"$*nq!042!3!r"dr"2760'

G80

# nq!042- !G"$*nq!0

42!3!G"

!G!2

""60

#n !0 "!r"v loc!42-42!3!r"$dr"12

d2Excd42- ,#q!0d42!3 ,q!0

#n!0 "

, !26"


combined with the modified ion-ion contribution

CEw,2- ,2!3!q!0"

!Z2Z2!& 'G80

47

60

G-!G3!G2 ei G•! "2# "2!"

$exp$ #G2

4;2% #'a

;3H-!3!iso !;da ,22!"

#4

3!7;3 022!' . !27"

Note the absence of the G!0 contributions in the Hartreecontribution to Eq. !26" and in the first term of Eq. !27".Equation !26" is a stationary expression. Simpler, nonstation-ary expressions exist as well, and are similar to Eqs. !16" and!17".

IV. ELECTRONIC DIELECTRIC PERMITTIVITYTENSOR

For insulators, the dielectric permittivity tensor is the co-efficient of proportionality between the macroscopic dis-placement field and the macroscopic electric field, in the lin-ear regime:

Dmac,-!'3

%-3Emac,3 . !28"

It can be obtained as

%-3!)Dmac,-)Emac,3

!0-3"47)Pmac,-)Emac,3

. !29"

In general, the displacement Dmac , or the polarizationPmac , will include contributions from ionic displacements. Inthe present section, we examine only the contribution to thedielectric permittivity tensor from the electronic polarization,and for low frequencies of the applied field. This contribu-tion is usually noted %-3

& . In Sec. VIIIA, we will take care ofthe supplementary contributions from the ionic displace-ments.We connect the dielectric permittivity tensor to the polar-

izability matrix, following Refs. 35 and 36. The polarizabil-ity of a solid describes the density response to an appliedpotential. In real space, one has

n !1 "!r"!" <!r,r!"vext!1 "!r!"dr!. !30"

In the reciprocal space, for a periodic solid,

nq!1 "!G"!'

G!<G,G!!q"vext,q

!1 " !G!". !31"

Since the density is the first-order derivative of the total en-ergy with respect to a change of potential,

n!r"!)E tot

)vext!r", !32"

the polarizability is related to the second-order derivative ofthe total energy:

<!r,r!"!)2E tot

)vext!r")vext!r!". !33"

The connection with the dielectric permittivity tensor, fol-lowing Ref. 35, proceeds through the definition of the in-verse dielectric matrix, with

%G,G!#1 !q"!0G,G!"

47

!q"G!2 <G,G!!q", !34"

and, for q approaching to zero, one finds35

'-3

q-%-3& q3 !

1%0,0

#1!q", !35"

where q is the unit vector in the direction of q.These theoretical definitions give the following approach

to the electronic contribution to the dielectric permittivitytensor:37

%-3& !0-3#

47

602Eel

E-*E3 , !36"

where EelE-*E3 is the mixed derivative generalization of the Eq.

!76" of the preceding paper:3

EelE-*E3*u !0 ";uE-,uE3,

!60

!27"3"BZ'mocc

s!.umkE- !Hk,k

!0 "#%mk!0 "!umk

E3 /

".umkE- !iumk

k3 /".iumkk- !umk

E3 /"dk

"12"60

dvxcdn #

n!0 "!r"#nE-!r"$*nE3!r"dr

"2760'G80

#nE-!G"$*nE3!G"

!G!2 . !37"

A much simpler nonstationary formula also gives EelE-*E3 :

EelE-*E3*u !0 ";uE-,!

60

!27"3"BZ'mocc

s.umkE- !iumk

k3 /dk. !38"

By this last expression, the knowledge of uE-, the first-orderderivative of the wave functions with respect to an electricfield along direction - , allows us to compute the elements ofthe dielectric permittivity tensor %-3 , for any value of 3 ,provided that the derivative of the unperturbed wave func-tion with respect to their wave vector along 3 is also known.

V. BORN EFFECTIVE CHARGES

For insulators, the Born effective charge tensor Z2 ,3-*!Ref. 38" is defined as the proportionality coefficient relating,


at linear order, the polarization per unit cell, created alongthe direction 3 , and the displacement along the direction -of the atoms belonging to the sublattice 2 , under the condi-tion of a zero electric field. The same coefficient also de-scribes the linear relation between the force on an atom andthe macroscopic electric field, because both can be connectedto the mixed second-order derivative of the energy with re-spect to atomic displacements and a macroscopic electricfield:

Z2 ,3-* !60)Pmac,3

)42-!q!0" !)F2 ,-

)E3. !39"

In the present formalism, this quantity can be obtainedfrom

Z2 ,3-* !Z203-"1Z2 ,3- , !40"

where Z2 is the charge of the !pseudo-"ion 2 , and the elec-tronic screening 1Z2 ,3- is

1Z2 ,3-!2& 60

!27"3"BZ'mocc

s!.umk,q!042- !Hk,k

!0 "#%mk!0 "!umk

E3 /

".umk,q!042- !iumk

k3 /".umk!0 "!vsep,k,k

42- !umkE3 /"dk

"12"60

*#v loc,q!0!42- !r""vxc0,q!042- !r"$# nE3!r"$*,dr

"12"60

dvxcdn #

n!0 "!r"#nq!0

42- !r"$*nE3!r"dr

"2760 'G80

#nq!042- !G"$*nE3!G"

!G!2 ' . !41"

In this stationary expression, the basic ingredients are thefirst-order derivative of the wave functions with respect to aq!0 collective displacement, and the first-order derivativesof the wave functions with respect to an electric field and totheir wave vector. By contrast, in the following nonstation-ary expressions, more sensitive to wave function conver-gence errors, the derivative with respect to an electric field isnot needed:

1Z2 ,3-!260

!27"3"BZ'mocc

s.umk,q!042- !iumk

k3 /dk !42"

or only its knowledge is required:

1Z2 ,3-!2& 60

!27"3"BZ'mocc

s.umk!0 "!vsep,k,k

42- !umkE3 /dk

"12"60

*#v loc,q!0!42- !r""vxc0,q!042- !r"$# nE3!r"$*,dr' .

!43"

VI. IMPLEMENTATION NOTES

The stationary formulas Eqs. !15", !26", !37", !41", !A1",!A3", and !A5", to be considered for the computation of the

second-order derivatives of the energy are all similar. Thisfact strongly reduces the time needed to implement this for-malism. The similarity is also observed for the nonstationaryformulas Eqs. !16", !17", !38", !42", and !43".In order to compute the above-mentioned stationary ex-

pressions, one needs to know the first-order derivative of thewave functions with respect to the two perturbations definingthe second-order derivative, and eventually the auxiliary de-rivative of the wave functions with respect to their wavevector. By contrast, for the nonstationary Eqs. !16", !17",!38", !42", and !43", the derivative of the wave vections withrespect to only one of these perturbations is needed !andeventually the derivative with respect to the wave vector".This latter advantage can prove useful if, for example, the setof Born effective charges must be computed, while the dy-namical matrix at q!0 is not needed: even if the number ofatoms on the unit cell is large, all the effective charges canbe easily found from the knowledge of the first-order re-sponses with respect to the electric field only.The parallelization of these formulas is easily achieved by

considering the mixed derivatives one at a time: all the N2mixed derivatives with respect to the N perturbations can becomputed in parallel, when all the first-order wave functionsand densities have been computed. Also, the evaluation ofthe nonstationary expressions could be done at the end of theparallel computation of the first-order wave functions, usingonly one set of the first-order wave functions, since no infor-mation is required from the other processors.In any case, the amount of computational work to evalu-

ate these expressions is rather small compared with the workneeded to obtain the first-order derivatives of the wave func-tions through the conjugate-gradient algorithm described inP1, or from the BGT procedure.27

VII. SUM RULES

A few sum rules are available to monitor whether thecalculation is well converged with respect to numerical pa-rameters, like the number of plane waves, the sampling ofthe Brillouin zone, and the number of points of theexchange-correlation grid.The first is the acoustic-sum rule:32 the dynamical matrix

at the zone center should admit the homogeneous transla-tions of the solid as an eigenvector, with a zero eigenfre-quency, because of the invariance of the total energy withrespect to translation. From Eq. !12", this gives

'2!

C2- ,2!3!q!0"!0. !44"

Since the dynamical matrix is symmetric, the transpose rela-tion is also valid. In the implementation of the formalismexplained here, this relation is slightly broken because of thepresence of the exchange-correlation grid in the real space,on which the exchange-correlation potential and energies areevaluated: if all the atoms are translated by a given vector,while the exchange-correlation grid is unchanged, the ener-gies will slightly change, and induce the breaking of the sumrule. All of the other terms can be implemented in atranslation-invariant way. If needed, this problem can be by-passed by the following simple modification:


C2- ,2!3New !q!0"!C2- ,2!3!q!0"#022!'

2"C2- ,2"3!q!0".

!45"However, by this operation, the eigenfrequencies at q$0 willchange, and will not be the limit of the eigenfrequenciesobtained by making q!0, unless the other dynamical matri-ces, for q80, are also corrected. The generalization of Eq.!45" for q80 will be discussed in Sec. IX.The second sum rule guarantees that the charge neutrality

is also fulfilled at the level of the Born effective charges. Forevery direction - and 3 , one must have39

'2Z2 ,-3* !0, !46"

i.e., the sum of the Born effective charges of all atoms in onecell must vanish, element by element. This sum rule will bebroken because of the finiteness of the number of planewaves or special points, or because of the discretization ofthe real-space integral !needed for the evaluation of theexchange-correlation energies and potentials". This problemcould be corrected as follows. We define the mean effectivecharge excess per atom

Z-3* !1Nat'2

Z2 ,-3* . !47"

This excess can be redistributed equally among the atoms asfollows:27

Z2 ,-3*,New!Z2 ,-3* #Z-3* , !48"

or can be redistributed among them in proportion to theirmean electronic effective charge,

Z2 ,-3*,New!Z2 ,-3* #Nat'-31Z2 ,-3

'2 ,-31Z2 ,-3Z-3* . !49"

Other weighting schemes for the redistribution of this excesscould be designed. Finally, in the case of the response to anelectric field, one can also monitor the fulfillment of thef -sum rule, as described in Ref. 40.

VIII. LOW-FREQUENCY DIELECTRIC PERMITTIVITYTENSOR AND LO-TO SPLITTING

In this section, we discuss two phenomena that arise fromthe same basic mechanism: the coupling between the macro-scopic electric field and the polarization associated with theq!0 atomic displacements. In the computation of the low-frequency !infrared" dielectric permittivity tensor, one has toinclude the response of the ions, whose motion will be trig-gered by the force due to the electric field, and whose polar-ization will be created by their displacement. The Born ef-fective charges are involved in both mechanisms.Also, in the computation of the long-wavelength limit of

phonons, a macroscopic polarization and electric field will beassociated with the atomic displacements. At the simplestlevel, the eigenfrequencies of phonons will depend on thedirection along which the limit is taken as well as on thepolarization of the phonon. This gives birth to the LO-TOsplitting, and to the Lyddane-Sachs-Teller relation.32 Thisphenomenon is also directly described by the Born effectivecharges.

A. Low-frequency dielectric permittivity tensor

The macroscopic low-frequency !static" dielectric permit-tivity tensor %-3(5) is calculated by adding to %-3

& the ioniccontribution, following Maradudin et al.33 In our notations,we obtain

%-3!5"!%-3& "

47

60'22!

'-!3!

Z2 ,--!* #C!q!0"

#M52$2-!,2!3!#1 Z2!,33!

* . !50"

Using the knowledge of the eigendisplacements Umq(23) ofC(q!0) from Eq. !12", normalized as

'23

M2#Umq!23"$*Unq!23"!0mn , !51"

one derives

%-3!5"!%-3& "

47

60'm

„'2-!Z2 ,--!* Umq!0* !2-!"…„'2!3!Z2!,33!

* Umq!0!23!"…5m2 #52 . !52"

We define the components of the mode-effective charge vec-tor Zm* as

Zm ,-* !'23Z2 ,-3* Umq!0!23"

„'23#Umq!0!23"$*Umq!0!23"…1/2 . !53"

Note that with this definition, the mode-effective charge of apure translation of the solid vanishes, unlike with the defini-tion given in Ref. 41. The mode-oscillator strength tensorSm ,-3 is defined as

Sm ,-3!$ '2-!

Z2 ,--!* Umq!0* !2-!"%

$$ '2!3!

Z2!,33!* Umq!0!2!3!"% , !54"

so that

%-3!5"!%-3& "

47

60'm

Sm ,-3

5m2 #52 . !55"


One can also evaluate the value of the dielectric permittivityconstant along the direction q, by

% q!5"!'-3

q-%-3!5"q3

!'-3

q-%-3& q3"

47

60'm

'-3q-Sm ,-3q3

5m2 #52 !56"

!'-3

q-%-3& q3

"47

60'm

!Zm*•q!2

5m2 #52$ '

23#Umq!0!23"$*Unq!0!23" % .

!57"

The reflectivity of optical waves normal to the surface, withtheir electric field along an optical axis of the crystal q, isgiven by

R!5"!# % q1/2

!5"#1

% q1/2

!5""1 #2

. !58"

More general expressions for the reflectivity may be found inclassical textbooks.42Equation !56" shows that, if the vector Zm* is perpendicu-

lar to q, the mode m does not contribute to the dielectricpermittivity constant along q. For each mode m , there willthus be one direction along which the mode contributes tothe dielectric permittivity constant, in which case it is re-ferred to as longitudinal, while for the perpendicular direc-tions, the mode will be referred to as transverse. We recoverthe usual distinction between LO and TO modes, confirmedby the following analysis of the q!0 limit of the dynamicalmatrix.

B. LO-TO splitting

The macroscopic electric field that accompanies the col-lective atomic displacements at q!0 can be treated sepa-rately, as mentioned in Sec. III C and in Sec. VII A of P1.After a careful treatment, one is able to recover the importantresult,39,27

C2- ,2!3!q!0"!C2- ,2!3!q!0""C2- ,2!3NA !q!0",

!59"

where the nonanalytical, direction-dependent termC2- ,2!3NA (q!0) is given by

C2- ,2!3NA !q!0"!

47

60

!'=q=Z2 ,=-* "!'=!q=!Z2!,=!3* "

'-3q-%-3& q3

.

!60"

In general, the eigenvectors of the C(q!0) matrix willnot be identical to those of the C(q!0). However, themodes that are transverse to the direction of q are common to

both. Indeed, the NA term in Eq. !59" acts in a space that isperpendicular to the space spanned by the TO modes !forwhich Z*m•q!0):

'2!3

$ '=!

q=!Z2!,=!3* %Umq!0!2!3"!0. !61"

Sometimes, symmetry constraints will be sufficient toguarantee that some LO eigendisplacements of C(q!0) willbe identical to those of C(q!0), even if the eigenfrequen-cies are not the same. In this case, the following relationship,linking LO and TO modes, holds:

5m2 !q!0"!5m

2 !q!0""$ 47

60% '-3q-Sm ,-3q3

'-3q-%-3& q3

. !62"

By summing on all modes, and using the orthonormaliza-tion of eigenvectors Eq. !51", one gets

'm

5m2 !q!0"#5m

2 !q!0"

!$ 47

60%'

2

1M2

'-3=q-Z2 ,-3* Z2 ,=3* q=

'-3q-%-3& q3

. !63"

Finally, let us mention an interesting generalization of theLyddane-Sachs-Teller relationship, linking dielectric proper-ties and phonon frequencies, in the harmonicapproximation:43

'-3q-%-3!5"q3

'-3q-%-3& q3

!>m

5m2 !q!0"#52

5m2 !q!0"#52 . !64"

IX. INTERATOMIC FORCE CONSTANTS,PHONON-BAND STRUCTURES

If the dynamical matrices were known everywhere in theBrillouin zone, the IFC’s could be built by inverting Eq.!10", which defines the dynamical matrix from the IFC’s:

C2- ,2!3!0,b "!!27"3

60"BZC2- ,2!3!q"eiq•Rbdq. !65"

Unfortunately, the dynamical matrices are not known ev-erywhere in the Brillouin zone: for computational reasonsthey are only obtained for a small set of wave vectors. In thiscase, a numerical integration technique must be used to per-form the integration appearing in Eq. !65". For that purpose,the use of a discrete Fourier transform is tempting: the dy-namical matrices on a regular grid of (l$m$n) points inthe Brillouin zone44 will generate approximate IFC’s in alarge box, made of (l$m$n) periodic cells. Outside of thisbox, the IFC’s, are supposed to vanish:45


C2- ,2!3!0,b "!1Nq '

q!grid! l$m$n "C2- ,2!3!q"eiq–Rb if Rb""2#"2!!box ! l$m$n "

!0 if Rb""2#"2!"box ! l$m$n ". !66"

The vanishing of the IFC’s beyond some distance is intrinsicto this discrete Fourier transform technique. If the integrandin Eq. !65" were infinitely differentiable, then the IFC’swould decrease exponentially fast, and this intrinsic limita-tion would not be a practical concern. However, for insula-tors with nonvanishing effective charges, Eqs. !59" and !60"shows that, close to q!0, the behavior of the dynamicalmatrices is strongly nonanalytical: it depends on the direc-tion along which q!0 is attained. In the real space, it can beseen that this nonanalytical behavior corresponds to long-ranged IFC’s, with an average 1/d3 decay (d being the dis-tance between atoms", corresponding to dipole-dipole inter-actions.Indeed, a dipole is created when an atom is displaced

from its original position, and the proportionality coefficientbetween the dipole and the displacement is the Born effec-tive charge. Even if the Born effective charge vanishes !thismay be imposed by symmetry constraints, in elemental crys-tals", the atomic displacement will create a quadrupole or anoctupole !the latter cannot be forbidden for symmetry rea-sons", with corresponding quadrupole-quadrupole 1/d5 de-cay, or octupole-octupole 1/d7 decay. However, the non-analyticity corresponding to the dipole-dipole interaction isthe strongest, and in the context of the present paper, eventhe dipole-quadrupole interaction, with 1/d4 decay, will beneglected. Thus, if the Born effective charges of all atoms ina crystal vanish, we consider that Eq. !66" will give an ad-equate description of the IFC’s.For metals, the electrostatic interactions are screened for

sufficiently large distances. On the other hand, Friedel oscil-lations, due to the abrupt change of the occupation number atthe Fermi level, cause a long-ranged decay of the IFC’s. In asimple isotropic model, the decay of the IFC’s is given bycos2kFd/kF

3d3, where kF is the Fermi wave vector.46 In morerealistic situations, the decay will still be inversely propor-tional to the cube of the distance, but the oscillatory behaviorwill be more complex, and determined by the shape of theFermi surface. In many practical applications, this long-range decay of metallic interatomic force constants in thereal space, and the associated singularity in the reciprocalspace are of little importance.For insulators with nonvanishing Born effective charges,

the nonanalytical behavior of the dynamical matrices close toq!0 is perfectly defined from the knowledge of the Borneffective charges and the electronic dielectric permittivitytensor, as shown in Eq. !60". This term cannot be neglectedin practical applications. In a homogeneous material with anisotropic dielectric permittivity tensor %0-3 !the superscript& of the %& tensor will be omitted in the remainder of thispaper, for brevity", the dipole-dipole interaction created bythe displacement of atoms with !isotropic" charges Z2 andZ2! will be described by the following force constants:

27

C2- ,2!3!0,a "!Z2Z2!

% $ 0-3

d3 #3d-d3

d5 % , !67"

where

d!Ra""2!#"2 . !68"

The Fourier transform of these force constants exhibits thefollowing nonanalytical behavior:

C2- ,2!3NA !q!0"!

47

60

Z2Z2!

%

q-q3

q2 . !69"

Comparing the nonanalytical behaviors of Eqs. !60" and!69", it appears that, in the former, the % tensor is present asa metric in the reciprocal space. In order to reproduce thenonanalytical behavior of the dynamical matrix in the case ofa material with anisotropic dielectric permittivity tensor andanisotropic effective-charge tensor, the following generaliza-tion of the dipole-dipole force constants Eq. !67" can beused, where the (%)#1 tensor is used as a metric in the realspace:

C2- ,2!3DD !0,a "! '

-!3!Z2 ,--!* Z2!,33!

* $ !%#1"-!3!D3 #3

1-!13!D5 %

$!det%"#1/2, !70"

where 1-!'3(%#1)-3 d3 is the conjugate of the vector drelating nuclei, while the norm of the latter in this metrics isD!! %•d. The supplementary factor (det%)#1/2 is neededto get Eq. !60", and is connected to the Jacobian of the trans-formation between real and reciprocal space. The a!0 and2!2! case is obtained by imposing the acoustic-sum rule onthe first or the second indices. The contribution CEw

DD of thesedipole-dipole IFC’s to the dynamical matrix can becalculated17 using the Ewald summation technique asfollows.47

!1 " CEw,2- ,2!3DD !q"!CEw,2- ,2!3

DD !q"

#022! '2"

CEw,2- ,2!3DD !q!0".

!71"

!2" The effective charge tensors can be factored out fromCEwDD ,

CEw,2- ,2!3DD !q"! '

-!3!Z2 ,-!-* Z2!,3!3

* CEw,2- ,2!3DD !q".

!72"


!3" CEwDD , the remaining quantity, is split into three parts: a

rapidly convergent sum in the reciprocal space; a rapidlyconvergent sum in the real space; and the limiting contribu-tion !as usual in Ewald summation techniques",

CEw,2- ,2!3DD !q"! '

G with K!G"q

47

60

K-!K3!'==!K=%==!K=!

$ei K•! "2# "2!" exp$ #'==!

K=%==!K=!4;2 %

#'a

;3 ei q•Ra H-!3!!;%,;D "

$!det%"#1/2

#4

3!7;3 022! !%#1"-!3! !det%"#1/2,

!73"with

H-3!x,y "!x-x3

y2 & 3y3 erfc!y ""2

!7e#y2 $ 3y2 "2 % '

#!%#1"-3 & erfc!y "

y3 "2

!7

e#y2

y2 ' . !74"

This expression is invariant under the change of the param-eter ; , which can be adjusted to obtain the fastest conver-gence of both the reciprocal- and real-space sums.If ; is made equal to 0, the reciprocal-space sum in Eq.

!73" vanishes, as well as the limiting contribution. Thecomplementary error functions in Eq. !74" will have thevalue 1, while the contributions from the Gaussians in thesame equation will vanish. Altogether, one finds that the dy-namical matrices described by Eqs. !71"–!74" are indeed theFourier transform of the IFC’s, Eq. !70".Alternatively, putting ; to infinity allows us to make the

real-space sum vanish. The limiting behavior is suppressed,due to Eq. !71", and finally one finds

CEw,2- ,2!3DD !q"! '

G with K!G"q

47

60

!'-!K-!Z2 ,-!-* "!'3!K3!Z2!,3!3

* "

'==!K=%==!K=!ei K•! "2# "2!"

#022! '2"

'G80

47

60

!'-!G-!Z2 ,-!-* "!'3!G3!Z2",3!3

* "

'==!G=%==!G=!ei G•! "2# "2"". !75"

The nonanalytical behavior of this expression, for q!0, isfound to be Eq. !60", as expected.48With the help of the dipole-dipole expressions for the dy-

namical matrix and the IFC’s, we are now able to bypass theproblems mentioned at the beginning of the present section.Indeed, the long-range behavior of the IFC’s for real mate-rials should not be different from the long-range behavior ofthe dipole-dipole IFC’s characterized by the same Z* and% . So, we remove, from the dynamical matrices of real ma-terials, determined on a homogeneous set of wave vectors inthe Brillouin zone with the grid (l$m$n), the dynamicalmatrices of the dipole-dipole system for the same wave vec-tors, shown in Eq. !75":

C2- ,2!3SR !q"!C2- ,2!3!q"#CEw,2- ,2!3

DD !q". !76"

It is expected that their inverse Fourier transform, approxi-mated by

C2- ,2!3SR !0,b "!

1Nq '

q!grid ! l$m$n "C2- ,2!3SR !q"e#iq•Rb

if Rb""2#"2!!box ! l$m$n "

!0 if Rb""2#"2!"box ! l$m$n "!77"

decays like 1/d4 or faster. The total IFC’s, following thistechnique, is given by

C2- ,2!3!0,b "!C2- ,2!3SR !0,b ""CEw,2- ,2!3

DD !0,b ", !78"

where the short-ranged part is given by Eq. !77", and thedipole-dipole part is given by Eq. !70".This technique not only allows us to get the IFC’s, but

also allows an easy interpolation of the dynamical matrixacross the full Brillouin zone, with

C2- ,2!3!q"! 'db!box ! l$m$n "

C2- ,2!3SR !0,b "eiq•Rb

"CEw,2- ,2!3DD !q". !79"

Thus, it is possible to build the IFC’s, and the full phononspectrum, from the knowledge of the Born effective chargetensor, the dielectric permittivity tensor, and a few dynami-cal matrices, which sample adequately the whole Brillouinzone. Moreover, the use of the symmetries of the material!spatial operations of symmetries, as well as the time-reversal symmetry", allows us to sample the dynamical ma-trices only in the irreducible part of the Brillouin zone, witha considerable reduction of computing time !see Sec. VI".As mentioned previously, the dynamical matrix at q!0

computed from the formulas of Sec. IIIC does not satisfy


exactly the acoustic-sum rule. In term of the IFC’s, the fol-lowing relationship #Fourier transform of Eq. !44"$, is notsatisfied:

'2!b

C2- ,2!3!a ,b "!0. !80"

This problem can be bypassed by generalizing, to every qwave vector, the recipe of Eq. !45" for q!0:39,27

C2- ,2!3New !q"!C2- ,2!3!q"#022!'

2"C2- ,2"3!q!0".

!81"In this case, a wave-vector-independent, site-diagonal(022!) correction is applied to the dynamical matrices. In thereal space, only the ‘‘on-site’’ IFC’s are affected. For everya and 2 ,

C2- ,23New !a ,a "!# '

!2",b "8!2 ,a "

C2- ,2"3!a ,b ". !82"

Other correction schemes are possible. At this stage, one isable to compute the full phonon-band structure, and use it topredict thermodynamical properties by occupying the pho-non modes following the Bose-Einstein statistics.

X. PERSPECTIVES

In the present paper, different equations that allow one toinvestigate the dynamical matrices, interatomic force con-stants, Born effective charge and dielectric permittivity ten-sor, have been presented in detail. Papers mentioned in theIntroduction give examples of the application of this tech-nique.At the level of the second-order derivatives of the energy,

these equations can be extended to cover more perturbations,especially those derived from the consideration of modifica-tions of the unit cell, like the elastic constants, and the con-nected mixed derivatives, like the internal strain or the pi-ezoelectricity. The extension to nonlinear properties is alsopossible, following the ideas contained in Refs. 25,11. Thenumber of properties covered by such generalizations is verylarge: mode-Gruneisen parameters, nonlinear optical coeffi-cients, phonon-phonon interaction, Raman scattering crosssection, nonlinear piezoelectricity, nonlinear elasticity, etc.The technique of Fourier interpolation of dynamical ma-

trices takes advantage of the known asymptotic behavior ofthe IFC’s. This technical point, combined with the advancesdescribed in P1,3 have allowed us to obtain full phonon-bandstructures for moderately complex materials such as SiO2- quartz and stishovite.18,17A few thermodynamical properties, like constant-volume

specific heats and entropy, have been derived from thisknowledge, based upon the Bose-Einstein occupation of thephonon degrees of freedom. Other properties, like the ther-mal expansion or thermal conductivity, are also in the reachof this method.

ACKNOWLEDGMENTS

The authors acknowledge interesting discussions with D.C. Allan, K. Rabe, Ph. Ghosez, and G.-M. Rignanese, as well

as with participants to the ’96 CECAM workshop on AbInitio Phonons, especially P. Giannozzi, H. Krakauer, and S.Baroni. We also thanks Ph. Ghosez and G.-M. Rignanese fora careful reading of the manuscript. X.G. thanks S. Savrasovfor a copy of Ref. 29 prior to publication. X.G. acknowl-edges financial support from FNRS-Belgium and C.L. ac-knowledges support from Korea Science and EngineeringFoundation Grant No. 961-0207-043-2, Ministry of Educa-tion Grant No. BSRI-96-2428, Center for Theoretical Phys-ics in Seoul National University.

APPENDIX A: THE COMPUTATIONOF THE DIELECTRIC PERMITTIVITY TENSOR

IN DIFFERENT APPROXIMATIONS

In Sec. IV, the computation of the dielectric permittivitytensor in the local-density approximation !LDA" was pre-sented. Other approximate schemes are amenable to similarformulas.

1. Electronic dielectric permittivity tensorin the ‘‘local-density plus scissor’’ approximation

Because the agreement between the LDA dielectric per-mittivity tensor and the experiment was not satisfactory, Le-vine and Allan have introduced the scissor operator correc-tion to the LDA.40 This correction leads to an improvedagreement between theory and experiment for many semi-conductors, although some cases of negative results havebeen reported. The reasons of the partial failure of LDA havebeen discussed in Ref. 49.The modifications of the equations appearing in Sec. VI B

of the preceding paper,3 needed to incorporate a scissor cor-rection, are rather simple. Supposing that the gap betweenthe valence and conduction states must be increased fromEgLDA to Eg

LDA"1 , then Eq. !37" is slightly modified andbecomes

ESCIE-*E3*u !0 ";uSCI,E-,uSCI,E3,

!60

!27"3"BZ'mocc

s!.umkSCI,E-!Hk,k

!0 "#%mk!0 ""1!umk

SCI,E3/

".umkSCI,E-!iumk

k3 /".iumkk- !umk

SCI,E3/"dk

"12"60

dvxcdn #

n!0 "!r"#nSCI

E- !r"$*nSCIE3 !r"dr

"2760'G80

#nSCIE- !G"$*nSCI

E3 !G"

!G!2 . !A1"

The minimization problem appearing in Sec. VI B of thepreceding paper3 is to be replaced by Eq. !A1", where3!- . The associated Euler-Lagrange equation is

Pc ,k !Hk,k!0 "#%m ,k

!0 " "1" Pc ,k !um ,kSCI,E-/

!#Pc ,k $ i )

)k-"vH!

SCI,E-"vxcSCI,E-% !um ,k

!0 " /. !A2"


The other equations are unchanged, except for the replace-ment of !um ,k

E- / by !um ,kSCI,E-/, nE- by nSCI

E- , vH!E- by vH!

SCI,E- ,and vxc

E- by vxcSCI,E- .

Because of the positive definiteness of the term governedby 1 in the quadratic form underlying Eel

E-*E- , it is straight-forward that if 1 is positive, the dielectric permittivity con-stant along any direction is always smaller in the ‘‘local-density plus scissor’’ approximation than in the local-densityapproximation.

2. Electronic dielectric permittivity tensorin the random phase approximation

In the random phase approximation !RPA", described inthe classical papers by Adler and Wiser,50 one neglects theexchange and correlation effects. In the present variationalapproach, the RPA dielectric permittivity tensor is obtainedby the following modifications to the LDA expressions Eq.!37":

ERPAE-*E3*u !0 ";uRPA,E-,uRPA,E3,

!60

!27"3"BZ'mocc

s!.umkRPA,E-!Hk,k

!0 "#%mk!0 "!umk

RPA,E3/

".umkRPA,E-!iumk

k3 /".iumkk- !umk

RPA,E3/"dk

"2760'G80

#nRPAE- !G"$*nRPA

E3 !G"

!G!2 , !A3"

where the first-order uRPA,E- are to be obtained from theminimization of the same expression, for 3!- , and the as-sociated Euler-Lagrange equation is

Pc ,k !Hk,k!0 "#%m ,k

!0 " " Pc ,k !um ,kSCI,E-/

!#Pc ,k $ i )

)k-"vH!

RPA,E-% !um ,k!0 " /. !A4"


E- / by !um ,kRPA,E-/, nE- by nRPA

E- , and vH!E- by

vH!RPA,E- , respectively.

Because the contribution of the LDA exchange-correlation term to the quadratic form underlying Eel

E-*E- isnegative definite, it is straightforward that the dielectric per-mittivity constant along any direction is always smaller inthe RPA than in the LDA.

3. Electronic dielectric permittivity tensor without local fields

It is also possible to neglect the effect of all local fields,not only those connected to the exchange-correlation effects.This approach51 has been also heavily used. In the variationaldensity-functional perturbation theory !DFPT", the computa-tions performed in this approximation rely on the followingmodifications to the LDA expressions Eq. !37":

E00E-*E3*u !0 ";u00,E-,u00,E3,

!60

!27"3"BZ'mocc

s!.umk00,E-!Hk,k

!0 "#%mk!0 "!umk

00,E3/

".umk00,E-!iumk

k3 /".iumkk- !umk

00,E3/"dk, !A5"

where the first-order wave functions u00,E- are to be obtainedfrom the minimization of the same expression, for 3!- , andthe associated Euler-Lagrange equation is

Pc ,k !Hk,k!0 "#%m ,k

!0 " " Pc ,k !um ,k00,E-/!#Pc ,k $ i )

)k-% !um ,k

!0 " /.

!A6"


E- / by !um ,k00,E-/.

Because the contribution of the Hartree term to the qua-dratic form underlying Eel

E-*E- is positive definite and usuallylarger than the negative-definite exchange-correlation contri-bution !this is not always true, see Ref. 52", it is expectedthat the dielectric permittivity constant along any direction islarger in the approximation without local fields than in theLDA.

1P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 !1964"; W.Kohn and L. J. Sham, ibid. 140, A1133 !1965".

2W. Pickett, Comput. Phys. Rep. 9, 115 !1989".3X. Gonze, preceding paper, Phys. Rev. B 55, 10 337 !1997".4K. M. Ho and K. P. Bohnen, Phys. Rev. Lett. 56, 934 !1986"; 38,12 897 !1988"; S. Wei and M. Y. Chou, ibid. 69, 2799 !1992";Y. Chen et al., ibid. 70, 603 !1993"; S. Wei, C. Li, and M. Y.Chou, Phys. Rev. B 50, 14 587 !1994"; W. Frank, C. Elsasser,and M. Fahnle, Phys. Rev. Lett. 74, 1791 !1995"; Y. Miyamoto,M. L. Cohen, and S. G. Louie, Phys. Rev. B 52, 14 971 !1995";G. Kresse, J. Furthmuller, and J. Hafner, Europhys. Lett. 32, 729!1995".

5T. A. Arias, M. C. Payne, and J. D. Joannopoulos, Phys. Rev.Lett. 69, 1077 !1992"; J. Kohanoff, W. Andreoni, and M. Par-

rinello, Phys. Rev. B 46, 4371 !1992"; J. Kohanoff, Comput.Mater. Sci. 2, 221 !1994"; G. Onida, W. Andreoni, J. Kohanoff,and M. Parrinello, Chem. Phys. Lett. 219, 1 !1994"; I. Stich, J.Kohanoff, and K. Terakura, Phys. Rev. B 54, 2642 !1996".

6R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651!1993"; R. Resta, Rev. Mod. Phys. 66, 899 !1994".

7R. W. Nunes and D. Vanderbilt, Phys. Rev. Lett. 73, 712 !1994".8P. Ordejon, D. A. Drabold, R. M. Martin, and S. Itoh, Phys. Rev.Lett. 75, 1324 !1995".

9S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58, 1861!1987".

10See Refs. 16–40,46–62 in Ref. 3.11See Refs. 33–38 in Ref. 3.12A. Dalgarno and A. L. Stewart, Proc. R. Soc. London Ser. A 247,


245 !1958"; J. O. Hirschfelder, W. B. Brown, and S. T. Epstein,Advances in Quantum Chemistry !Academic Press, New York,1964", Vol. 1, p. 289.

13Nonstationary expressions have been mentioned in Refs. 27,28.14L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425

!1982".15S. G. Louie, S. Froyen, and M. L. Cohen, Phys. Rev. B 26, 1738

!1982".16A. Dal Corso, S. Baroni, R. Resta, and S. de Gironcoli, Phys.Rev. B 47, 3588 !1993".

17X. Gonze, J.-C. Charlier, D. C. Allan, and M. P. Teter, Phys. Rev.B 50, 13 035 !1994".

18C. Lee and X. Gonze, Phys. Rev. Lett. 72, 1686 !1994".19C. Lee and X. Gonze, Phys. Rev. B 51, 8610 !1995".20C. Lee and X. Gonze, J. Phys. Condens. Matter 7, 3693 !1995".21Ph. Ghosez, X. Gonze, and J.-P. Michenaud, Europhys. Lett. 33,713 !1996".

22See Refs. 12,46–62 in Ref. 3.23C. Lee and X. Gonze !unpublished".24C. Lee and X. Gonze !unpublished".25X. Gonze and J.-P. Vigneron, Phys. Rev. B 49, 13 120 !1989".26X. Gonze, Phys. Rev. A 52, 1096 !1995".27P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, Phys.Rev. B 43, 7231 !1991".

28R. Yu and H. Krakauer, Phys. Rev. B 49, 4467 !1994".29S. Yu. Savrasov, Phys. Rev. B 54, 16 470 !1996".30X. Gonze, D. C. Allan, and M. P. Teter, Phys. Rev. Lett. 68, 3603

!1992".31S. Yu. Savrasov, Phys. Rev. Lett. 69, 2819 !1992".32M. Born and K. Huang, Dynamical Theory of Crystal Lattices

!Oxford University Press, Oxford, 1954".33A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova,in Solid State Physics: Advances in Research and Applications,edited by H. E. Ehrenreich, F. Seitz, and D. Turnbull !Aca-demic, New York, 1971", Suppl. 3, Chap. 4.

34Only the reciprocal-space part of this expression was mentionedin Eq. !A6" of Ref. 27. However, taking the limit -!& of thelatter formula allows us to recover the correct Eq. !75".

35R. Car, E. Tossatti, S. Baroni, and S. Leelaprute, Phys. Rev. B 24,985 !1981".

36M. S. Hybertsen and S. G. Louie, Phys. Rev. B 35, 5585 !1987".37Because of different conventions, these equations differ fromthose given in C. Lee, Ph. Ghosez, and X. Gonze, Phys. Rev. B50, 13 379 !1994".

38The star superscript of Z2 ,-3* is not the symbol for the complexconjugation operation: Z2 ,-3* is always a real quantity.

39R. M. Pick, M. H. Cohen, and R. M. Martin, Phys. Rev. B 1, 910!1970".

40Z. H. Levine and D. C. Allan, Phys. Rev. Lett. 63, 1719 !1989".41W. Zhong, R.D. King-Smith, and D. Vanderbilt, Phys. Rev. Lett.72, 3618 !1994".

42L. Landau and E. Lifschits, Electrodynamics of Continuous Me-dia !Pergamon Press, New York, 1960".

43This result is an extension of a formula given by W. Cochran andR. A. Cowley, J. Phys. Chem. Solids 23, 447 !1962", to thefrequency-dependent case. It is valid only within the harmonicapproximation.

44H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 !1976".45Equation !66" is to be modified in the case of the hexagonalsymmetry !in order for the grid not to break this symmetry", orwhen d is just on the border of the box.

46See, for example, E. G. Brovman and Yu. M. Kagan, in Dynami-cal Properties of Solids, edited by G. K. Horton and A. A.Maradudin !North-Holland, Amsterdam, 1974", Vol. 1, p. 247.

47The nonanalytical term !for G!0) from the reciprocal-spacesummation in Eq. !73" should not be included through Eq. !72"in the latter part (q!0) of this formula Eq. !71".

48 In Ref. 17, an error of notation is present: the dipole-dipole inter-action and its nonanalytical part are described by the same sym-bol. The equation before Eq. !4" in that paper is to be changedinto Eq. !60" of the present one, the latter being obviously onlya part of Eq. !75".

49X. Gonze, Ph. Ghosez, and R. W. Godby, Phys. Rev. Lett. 74,4035 !1995".

50S. L. Adler, Phys. Rev. 126, 413 !1962"; N. Wiser, ibid. 129, 62!1963".

51H. Ehrenreich and M. H. Cohen, Phys. Rev. 115, 786 !1959".52 I.J. Robertson and B. Farid, Phys. Rev. Lett. 66, 3265 !1991".


Nonlinear optical susceptibilities, Raman efficiencies, and electro-optic tensorsfrom first-principles density functional perturbation theory

M. Veithen,1 X. Gonze,2 and Ph. Ghosez1

1Département de Physique, Université de Liège, B-5, B-4000 Sart-Tilman, Belgium2Unité de Physico-Chimie et de Physique des Matériaux (PCPM), Université Catholique de Louvain,

place Croix du Sud 1, B-1348 Louvain-la-Neuve, BelgiumsReceived 2 September 2004; published 9 March 2005d

The nonlinear response of infinite periodic solids to homogenous electric fields and collective atomic dis-placements is discussed in the framework of density functional perturbation theory. The approach is based onthe 2n+1 theorem applied to an electric-field-dependent energy functional. We report the expressions for thecalculation of the nonlinear optical susceptibilities, Raman scattering efficiencies, and electro-opticsEOd co-efficients. Different formulations of third-order energy derivatives are examined and their convergence withrespect to thek-point sampling is discussed. We apply our method to a few simple cases and compare ourresults to those obtained with distinct techniques. Finally, we discuss the effect of a scissors correction on theEO coefficients and nonlinear optical susceptibilities.

DOI: 10.1103/PhysRevB.71.125107 PACS numberssd: 71.15.Mb, 78.20.Jq

I. INTRODUCTION

Nowadays, density functional theory1,2 sDFTd is consid-ered as a standard method in condensed matter physics, tostudy electronic, structural, and macroscopic properties ofsolids from first principles. Combined with adiabatic pertur-bation theory, it allowsa priori the computation of deriva-tives of the energy and related thermodynamic potentials upto any order. At the second order, this approach has beenapplied to compute linear response functions such as phononfrequencies or Born effective charges with an accuracy of afew percent. The third-order derivatives are related to non-linear properties such as phonon lifetimes, Raman tensors, ornonlinear optical susceptibilities.

The linear-response formalism has been implemented invarious first-principles codes and is routinely applied to anincreasing number of systemsssee, for example, Ref. 3 andreferences thereind. By contrast, the nonlinear response for-malism has been mostly restricted to quantum chemistryproblems. Although the hyperpolarizabilities of a huge num-ber of molecules have been computed, taking into accountboth electronic and vibrationalsionicd contributions,4,5 appli-cations in condensed matter physics have focused on rathersimple cases.6–13

Here, we present a methodology for the computation ofnonlinear response functions and related physical quantitiesof periodic solids from density functional perturbation theorysDFPTd. We focus on perturbations characterized by a zerowave vector and involving either three electric fields, or twoelectric fields and one atomic displacement. FollowingNunes and Gonze,14 our approach makes use of the 2n+1theorem applied to an electric-field-dependent energyfunctional.15 We report the local density approximationsLDA d expressions, as implemented within theABINIT

package.16

Our paper is organized as follows. In Sec. II, we describethe theoretical background related to the 2n+1 theorem andthe electric field perturbation. In Sec. III, we describe the

computation of the nonlinear optical susceptibilities, the non-resonant Raman scattering efficiencies of both transversesTOd and longitudinalsLOd zone-center optical phonons andthe linear electro-opticsEOd tensor. In Sec. IV, we illustratethe validity of the formalism by applying our methodology tosome semiconductors and ferroelectric oxides and we brieflydiscuss the effect of a scissors correction on the EO coeffi-cients and nonlinear optical susceptibilities.

Some of the tensors we consider in this work depend onstatic electric fields: they include contributions of both theelectrons and the ions. Other quantities imply only the re-sponse of the valence electrons: they are defined for frequen-cies of the electric field high enough to get rid of the ioniccontributions but sufficiently low to avoid electronic excita-tions. For clarity, we adopt the following convention. Staticfields will be labeled by Greek indicessa ,b , . . .d while wewill refer to optical fields with Latin symbolssi , j , . . .d. Tosimplify the notation, we will also drop labels such as` forquantities that do not involve the response of the ions. Usingthis convention, we can write«i j and «ab, respectively, forthe optical and static dielectric tensor, respectively, andr ij gfor the linear EOsPockelsd tensor that involves two opticaland one static electric field.

II. FORMALISM

A. Mixed third-order energy derivatives

In this section, we present the general framework of thecomputation of third-order energy derivatives based on the2n+1 theorem.17–19 Using the notation of Refs. 20 and 21,we consider three Hermitian perturbations labeledl1, l2, andl3. The mixed third-order derivatives

El1l2l3 =1

6U ]3E

]l1 ] l2 ] l3U

l1=0,l2=0,l3=0s1d

can be computed from the ground-state and first-order wavefunctions19

PHYSICAL REVIEW B 71, 125107s2005d

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El1l2l3 =1

6sEl1l2l3 + El1l3l2 + El2l1l3 + El2l3l1 + El3l2l1 + El3l1l2d, s2d

with

El1l2l3 = oa

fkcal1usT + vextdl2l3uca

s0dl + kcal1usT + vext+ vHxcdl2uca

l3l + kcas0dusT + vextdl1l2l3uca

s0dl + kcas0dusT + vextdl1l2uca

l3lg

− oa,b

Lbal2 kuca

l1ucbl3l+

1

6E drdr 8dr 9

d3EHxcfns0dgdnsr ddnsr 8ddnsr 9d

nl1sr dnl2sr 8dnl3sr 9d

+1

2E drdr 8

d

dl2U d2EHxcfns0dg

dnsr ddnsr 8dU

l=0nl1sr dnl3sr 8d +

1

2E dr

d2

dl1dl3UdEHxcfns0dg

dnsr dU

l=0nl2sr d +

1

6Ud3EHxcfns0dg

dl1dl2dl3U

l=0.

s3d

T is the kinetic energy andEHxc svHxcd is the sum of theHartree and exchange-correlation energyspotentiald. Thefirst-order potentialvHxc

l2 can be computed as a second-orderfunctional derivative ofEHxc:

19

vHxcl2 =E d2EHxcfns0dg

dnsrddnsr8dnl2sr8ddr8 +

d

dl2UdEHxcfns0dg

dnsrdU

l=0.

s4d

Within the parallel gauge, the first-order Lagrange multipli-ers are given by

Lbal2 = kcb

s0dusT + vext+ vHxcdl2ucas0dl. s5d

As a consequence of the 2n+1 theorem, the evaluation ofEq. s3d requires no higher-order derivatives of the wavefunctions than the first one. These first-order wave functionsare nowadays available in several first-principles codes. Theycan be computed from linear response by minimizing a sta-tionary expression of the second-order energy20 or equiva-lently by solving the corresponding Sternheimer equation.22

It follows that the computation of third-order energy deriva-tives does not require additional quantities other than thecalculation of second-order energy derivatives.

Equations3d is the general expression of the third-orderenergy derivatives. In case at least one of the perturbationsdoes not affect the explicit form of the kinetic energy or theHartree and exchange-correlation energy, it can be simpli-fied: some of the terms may be zero. This is the case for theelectric field perturbations treated in this work as well as forphonon-type perturbations. Further simplifications can bemade in case pseudopotentials without nonlinear exchange-correlation core correction are used.

B. The electric field perturbation

As mentioned in the Introduction, special care is requiredin case one of the perturbationsl j is a macroscopic electricfield E. In fact, as discussed in the literature, for infiniteperiodic solids, usually treated with Born–von Kármánboundary conditions, the scalar potentialE ·r breaks the pe-riodicity of the crystal lattice. Moreover, it is unbound from

below: it is always possible to lower the energy by transfer-ring electrons from the valence states to the conduction statesin a distant regionsZener breakdownd. However, for suffi-ciently small fields, the tunneling current through the bandgap can be neglected and the system is well described by aset of electric-field-dependent Wannier functionsWnsr d. Asshown by Nunes and Vanderbilt,15 these Wannier functionsminimize the energy functional

EfWn;Eg = E0fWng − V0E ·P, s6d

whereE0 is the Kohn-Sham energy under zero field,V0 theunit cell volume, andP the macroscopic polarization thatcan be computed from the Wannier function centers. It isimportant to note that these Wannier functions do not corre-spond to the true ground state of the system but rather to along-lived metastable state.

In practical applications, it is not mandatory to evaluatethe functional equations6d in a Wannier basis. It can equiva-lently be expressed using Bloch functionsunk related toWnby a unitary transform. In this case, the polarization can becomputed as a Berry phase of the occupied bands23

P = −2ie

s2pd3on

occEBZ

dkkunku=kuunkl, s7d

where BZ is the Brillouin zone,e is the absolute value of theelectronic charge, and the factor of 2 accounts for spin de-generacy. The Bloch functions are chosen to satisfy the pe-riodic gauge condition

eiG·Runk+G = unk . s8d

In order to use Eq.s7d in practical calculations, the integra-tion over the BZ, as well as the differentiation with respect tok, has to be performed on a discrete mesh ofNk k points. Incase of the ground-state polarization, the standard approachis to build strings ofk points parallel to a vector of thereciprocal spaceGi. The polarization can then be computedas a string-averaged Berry phase. Unfortunately, the adapta-tion of this method to the computation of the energy deriva-tives is plagued with several difficulties, like the following.The general form of the nonlinear optical susceptibility ten-

VEITHEN, GONZE, AND GHOSEZ PHYSICAL REVIEW B71, 125107s2005d

125107-2

sor of a compound is imposed by its symmetry. For example,in zinc-blende semiconductors, this tensor, expressed inCar-tesian coordinates, reduces toxi jl

s2d=xs2duei jl u, wheree is theLevi-Civita tensor. It follows that thereduced coordinatefor-mulation ofxi jl

s2d satisfies the relation

Uxi jls2d

xiiis2dU =

1

3, s9d

where at least one of the three indicesi, j , andl are differentfrom the two others. When we tried to use strings ofk pointsto computexi jl

s2d, Eq. s9d was not satisfied. However, we wereable to avoid these problems by using the finite differenceformula of Marzari and Vanderbilt24 on a regular grid ofspecialk points sinstead of stringsd,

= fskd = ob

wbbffsk + bd − fskdg, s10d

whereb is a vector connecting ak-point to one of its nearestneighbors andwb is a weight factor. The sum in Eq.s10dincludes as many shells of first neighbors as necessary tosatisfy the condition

ob

wbbabb =gab

s2pd2 , s11d

where ba are the reduced coordinates ofb and gab is themetric tensor associated with the real space crystal lattice.

In the case of the ground-state polarization, we cannotapply the discretization equations10d directly to Eq.s7d. Asshown by Marzari and Vanderbilt, the discretization of Eq.s7d does not transform correctly under the gauge transforma-tion

unksr d → e−ik·Runksr d. s12d

Since Eq.s12d is equivalent to a shift of the origin by onelattice vectorR, P must change accordingly by one polar-ization quantum. In order to obtain a discrete expression thatmatches this requirement, we must combine Eq.s10d withthe King-Smith and Vanderbilt formula23,25

P =2e

NkV0o

ko

bwbb Im ln det fSsk,k + bdg, s13d

whereS is the overlap matrix between Bloch functions atkandk +b:

Sn,msk,k + bd = kunkuumk+bl. s14d

As discussed by Nunes and Gonze,14 when we apply theperturbation expansion of the preceding section to the energyfunctional Eq.s6d, we can adopt two equivalent approaches.The first possibility is the use of Eq.s7d for the polarizationand a discretization after having performed the perturbationexpansion. The second possibility is to apply the 2n+1 theo-rem directly to Eq.s13d in which case no additional discreti-zation is needed. Using the notations of Nunes and Gonze,we will refer to the first case as the discretization after per-turbation expansionsDAPEd formulation and to the second

one as the perturbation expansion after discretizationsPEADd formulation of the third-order energy. In the follow-ing sections, we will discuss both expressions. In addition, inSec. IV B, we will compare their convergence with respect tothek-point sampling on a realistic example. The perturbationexpansion of the first termsE0d of Eq. s6d can easily beperformed, as described in the Sec. II A. In contrast, theexpansion of the second terms−V0E ·Pd is more trickysince it explicitly depends on the polarization. In the twosections that follow, we will focus on the −V0E ·P term ofEq. s6d. It will be referred to asEpol.

C. DAPE expression

According to the formalism of the preceding section, theE ·P term acts as an additional external potential that has tobe added to the ionic one. TheE ·P perturbation is linear inthe electric field and does not depend explicitly on othervariables such as the atomic positions. It just enters the termsof Eq. s3d that involve the first derivative ofvext with respectto E. In other words, the only terms in Eq.s2d that involve

the expansion ofP are of the formEl1Eil3, wherel1 andl3represent an arbitrary perturbation such as an electric field oran atomic displacement.

The DAPE expression of the third-order derivative ofEpolis written as follows14

Epoll1Eil3 =

2ieV0

s2pd3EBZ

dkon

occ

kunkl1uS ]

]kio

m

occ

uumkl3 lkumk

s0d uDuunks0dl,

s15d

whereunkl j are the projections of the first-order wave func-

tions on the conduction bands. The complete expression ofvarious third-order energy derivatives, taking into accountthe expansion of bothE0 and Epol, are reported in Sec. III.Eq. s15d was derived first by Dal Corso and Mauri26 in aslightly different context: they performed the perturbationexpansion of the energy functional equations6d using a Wan-nier basis. The resulting expression of the third-order energywas expressed in terms of Bloch functions by applying aunitary transform to the Wannier orbitals.

Using the finite difference expression of Marzari andVanderbilt equations10d, Eq. s15d becomes

Epoll1Eil3 =

2ie

Nko

ko

bon,m

occ

wbsb ·Gid

3 hkunkl1uumk+b

l3 lkumk+bs0d uunk

s0dl − kunkl1uumk

l3 ldn,mj,

s16d

whereGi is a basis vector of the reciprocal lattice.

NONLINEAR OPTICAL SUSCEPTIBILITIES, RAMAN… PHYSICAL REVIEW B 71, 125107s2005d

125107-3

D. PEAD expression

Applying directly the 2n+1 theorem to Eq.s13d we obtainthe alternative PEAD formulation of the third-order energy:14

Epoll1Eil3 =

− e

NkImo

ko

bwbsb ·Gid

3 F2on,m

occ

kunkl1uumk+b

l3 lQmnsk,k + bd

− on,m,l,l8

occ

Smnl1 sk,k + bdQnlsk,k + bd

3 Sll8l3sk,k + bdQl8msk,k + bdG s17d

whereQ is the inverse of the overlap matrixS and Sl j itsfirst-order perturbation expansion

Snml j sk,k + bd = kunk

l j uumk+bs0d l + kunk

s0duumk+bl j l. s18d

III. NONLINEAR OPTICAL PROPERTIES

In the preceding section we have discussed the generalexpressions of third-order energy derivatives. We now par-ticularize them to the computation of selected nonlinearproperties.

A. Nonlinear optical susceptibilities

In an insulator the polarization can be expressed as a Tay-lor expansion of the macroscopic electric field

Pi = Pis + o

j=1

3

xi js1dE j + o

j ,l=1

3

xi jls2dE jEl + ¯ , s19d

wherePis is the zero-fieldsspontaneousd polarization,xi j

s1d thelinear dielectric susceptibilityssecond-rank tensord, andxi jl

s2d

the second-order nonlinear optical susceptibilitysthird-ranktensord. In the literature on nonlinear optics, one often finds

another definition of the nonlinear optical susceptibility: in-stead ofxi jl

s2d, it is more convenient to rely on thed tensordefined as

dijl =1

2xi jl

s2d. s20d

In general, the polarization depends on valence electronsas well as ions. In the present section, we deal only with theelectronic contribution: we will consider the ionic cores asclamped to their equilibrium positions. This constraint willbe relaxed in the following sections where we allow for ionicdisplacements.

Experimentally, the electronic contribution to the linearand nonlinear susceptibilities corresponds to measurementsfor electric fields at frequencies high enough to get rid of theionic relaxation but low enough to avoid electronic excita-tions. In case of the second-order susceptibilities, this con-straint implies that both the frequency ofE, and its secondharmonic, are lower than the fundamental absorption gap.

The general expression of the electronic nonlinear opticalsusceptibility depends on the frequencies of the optical elec-tric fields ssee, for example, Ref. 27d. In the present contextof the 2n+1 theorem applied within the LDA tosstaticd DFT,we neglect the dispersion ofxi jl

s2d. As a consequence,xi jls2d

satisfies Kleinman’s28 symmetry condition, which means thatit is symmetric under a permutation ofi, j , andl. In order tobe able to investigate its frequency dependence, one wouldneed either to apply the formalism of time-dependent DFT9

or to use expressions that involve sums over excitedstates.29–33 Following the work of Dal Corso andco-workers9,26 we can relate the nonlinear optical suscepti-bilities to a third-order derivative of the energy with respectto an electric field,

xi jls2d = −

3

V0EEiE jEl , s21d

whereEEiE jEl is defined as the sum over the permutations of

the three perturbationsEEiE jEl fEq. s2dg. Using the PEAD for-mulation of Sec. II B we can express these terms as follows:

EEiE jEl =− e

NkImo

ko

bwbsb ·G jdF2o

n,m

occ

kunkEi uumk+b

El lQmnsk,k + bd

− on,m,n8,m8

occ

SmnEi sk,k + bdQnn8sk,k + bdSn8m8

El sk,k + bdQm8msk,k + bdG+

2

Nko

kon,m

occ

fdm,nkunkEi uvhxc

E j uumkEl l − kumk

s0d uvhxcE j uunk

s0dlkunkEi uumk

El lg

+1

6E drdr 8dr 9

d3Excfn0gdnsr ddnsr 8ddnsr 9d

nEisr dnE jsr 8dnElsr 9d. s22d


125107-4

B. Raman susceptibilities of zone-center optical phonons

We now consider the computation of Raman scatteringefficiencies of zone-center optical phonons. In the limit

q→0, the matrix of interatomic force constantsC can beexpressed as the sum of an analytical part and a nonanalyti-cal term21

Cka,k8bsq → 0d = Cka,k8bAN sq = 0d + Cka,k8b

NA sq → 0d.

s23d

The analytical part corresponds to the second-order deriva-tive of the energy with respect to an atomic displacement atq=0 under the condition of vanishing macroscopic electricfield. The second term is due to the long-range electrostaticinteractions in polar crystals. It is at the origin of the so-called LO-TO splitting and can be computed from theknowledge of the Born effective chargesZkab

* and the elec-tronic dielectric tensor21 «i j . The phonon frequenciesvm andeigendisplacementsumskad are solutions of the followinggeneralized eigenvalue problem,

ok8,b

Cka,k8bumsk8bd = Mkvm2 umskad, s24d

whereMk is the mass of atomk. As a convention, we choosethe eigendisplacements to be normalized as

ok,a

Mkumskadunskad = dm,n. s25d

In what follows we consider nonresonant Raman scatter-ing where an incoming photon of frequencyv0 and polariza-tion e0 is scattered to an outgoing photon of frequencysv0

−vmd and polarizationeS by creating a phonon of frequencyvm sStokes processd. The scattering efficiency34,35scgs unitsdcorresponds to

dS

dV= ueS·Rm ·e0u2 =

sv0 − vmd4

c4 ueS· am ·e0u2"

2vmsnm + 1d,

s26d

wherec is the speed of light in vacuum andnm the bosonfactor

nm =1

exps"vm/kBTd − 1. s27d

The Raman susceptibilityam is defined as

ai jm = ÎV0o

k,b

]xi js1d

]tkb

umskbd, s28d

wherexi js1d is the electronic linear dielectric susceptibility ten-

sor. In Eq.s26d, V is the angle of collection in which theoutgoing photon is scattered. As discussed in Ref. 34, wehave to be careful when we compare the theoretical and ex-perimental scattering efficiencies. Due to Snell’s law, theangle of collection is modified at the interface between the

sample and the surrounding medium. Experimentally, thescattering efficiencies are measured with respect to the solidangle of the surrounding medium while Eq.s26d refers to thesolid angle inside the sample. In order to relate theory andexperiment, one has to take into account the different refrac-tive indices of the sample and medium. For example, in caseof an isotropic sample, Eq.s26d has to be multiplied34 bysn8 /nd2 wheren andn8 are, respectively, the refractive indi-ces of the sample and the medium. In contrast to the scatter-ing efficiencies, the Raman susceptibilities defined in Eq.s28d are intrinsic properties of the sample and do not dependon the change in the angle of collection.

For pure transverse optical phonons,]xi js1d /]tkb can be

computed as a mixed third-order derivative of the energywith respect to an electric field, twice, and to an atomic dis-placement under the condition of zero electric field

U ]xi js1d

]tkl

UE=0

= −6

V0EtklEiE j . s29d

In case of longitudinal optical phonons with wave vectorq→0 in a polar crystal, Eq.s28d must take into account theeffect of the macroscopic electric field generated by the lat-tice polar vibration. This field enters the computation of theRaman susceptibilities at two levels. On one hand, it givesrise to the nonanalytical part of the matrix of interatomicforce constants, Eq.s23d, that modifies the frequencies andeigenvectors with respect to pure transverse phonons. On theother hand, the electric field induces an additional change inthe dielectric susceptibility tensor related to the nonlinearoptical coefficientsxi jk

s2d. For longitudinal optical phonons,Eq. s29d has to be modified as follows:36

]xi js1d

]tkl

= U ]xi js1d

]tkl

UE=0

−8p

V0

olZkll

* ql

ol,l8ql«ll8ql8

ol

xi jls2dql . s30d

The mixed third-order derivatives Eq.s29d can be computedfrom various techniques including finite differences of thedielectric tensor37–39 or the second derivative of the elec-tronic density matrix.40,41Here, we follow an approach simi-lar to Deinzer and Strauch10 based on the 2n+1 theorem. Thethird-order energy can be computed as the sum over the 6permutations, Eq.s2d, of tkl, Ei, and E j. According to thediscussion of Sec. II B, we have to distinguish between theterms that involve the discretization of the polarization such

as EtklEiE j or EE jEitkl and those that can be computed from astraightforward application of the 2n+1 theorem such as

EEitklE j. The former ones show an electric field as secondperturbation. They can be computed from an expressionanalogous to Eq.s22d:


125107-5

EtklEiE j =− e

NkImo

ko

bwbsb ·GidF2o

n,m

occ

kunktkluumk+b

E j lQmnsk,k + bd − on,m,l,l8

occ

Smntklsk,k + bdQnlsk,k + bdSll8

E j sk,k + bdQl8msk,k + bdG+

2

Nko

kon,m

occ

fdm,nkunktkluvhxc

Ei uumkE j l − kumk

s0d uvhxcEi uunk

s0dlkunktkluumk

E j lg +1

6E drdr 8dr 9


ntklsr dnEisr 8dnE jsr 9d.

s31d

We obtain a similar expression forEE jEitkl. The remaining terms do not require any differentiation with respect tok. They canbe computed from the first-order change of the ionicspseudo-d potential with respect to an atomic displacementvext

tkl:

EEitklE j =2

Nko

kon,m

occ

fkunkEi uvext

tkl + vhxctkluumk

E j ldn,m − kunks0duvext

tkl + vhxctkluumk

s0dlkumkEi uunk

E j lg

+1

2E drdr 8

d

dtkl

ud2EHxcfn0g

dnsr ddnsr 8dnEisr dnE jsr 8d +

1

6E drdr 8dr 9


ntklsr dnEisr 8dnE jsr 9d. s32d

In pseudopotential calculations, the computation of the first-order ionic potentialvext

tkl requires the derivative of local andnonlocalsusually separabled operators. These operations canbe performed easily without any additional workload as de-scribed in Ref. 20.

In spite of the similarities between Eqs.s31d ands32d andthe expression proposed by Deinzer and Strauch we canquote a few differences. First, our expression of the third-order energy derivatives makes use of the PEAD fomulationfor the expansion of the polarization. Moreover, Eq.s32d ismore general since it allows the use of pseudopotentials withnonlinear core correction through the derivative of thesecond-order exchange-correlation energy with respect totkl

sthird termd.

C. Sum rule

As in the cases of the Born effective charges and of thedynamical matrix,42 the coefficients]xi j

s1d /]tka must vanishwhen they are summed over all atoms in the unit cell:

ok

]xi js1d

]tka

= 0. s33d

Physically, this sum rule guarantees that the macroscopic di-electric susceptibility remains invariant under a rigid transla-tion of the crystal. In practical calculations, it is not alwayssatisfied although the violation is generally less severe than

in case ofC or Z* . Even in calculations that present a lowdegree of convergence, the deviations from this law can bequite weak. They can be corrected using similar techniquesas in the case of the Born effective charges.21 For example,we can define the mean excess of]xi j

s1d /]tka per atom

]xi js1d

]ta

=1

Nato

k

]xi js1d

]tka

s34d

and redistribute it equally between the atoms:

]xi js1d

]tka

→ ]xi js1d

]tka

−]xi j

s1d

]ta

. s35d

D. Electro-optic tensor

The optical properties of a compound usually depend onexternal parameters such as the temperature, electric fields,or mechanical constraintssstress, straind. In the present sec-tion we consider the variations of the refractive index in-duced by a static or low-frequency electric fieldEg. At linearorder, these variations are described by the linear EO coeffi-cientssPockels effectd

Ds«−1di j = og=1

3

r ij gEg, s36d

wheres«−1di j is the inverse of the electronic dielectric tensorand r ij g the EO tensor.

Within the Born-Oppenheimer approximation, the EOtensor can be expressed as the sum of three contributions: abare electronic partr ij g

el , an ionic contributionr ij gion, and a

piezoelectric contributionr ij gpiezo.

The electronic part is due to an interaction ofEg with thevalence electrons when considering the ions artificially asclamped at their equilibrium positions. It can be computedfrom the nonlinear optical coefficients. As can be seen fromEq. s19d, xi jl

s2d defines the second-order change of the inducedpolarization with respect toEg. Taking the derivative of Eq.s19d, we also see thatxi jl

s2d defines the first-order change of thelinear dielectric susceptibility, which is equal tos1/4pdD«i j .Since the EO tensor depends onDs«−1di j rather thanD«i j , wehave to transformD«i j to Ds«−1di j by the inverse of the zero-field electronic dielectric tensor43


125107-6

Ds«−1di j = − om,n=1

3

«im−1D«mn«nj

−1. s37d

Using Eq. s37d we obtain the following expression for theelectronic EO tensor:

r ij gel =U − 8p o

l,l8=1

3

s«−1dilxll8ks2d s«−1dl8 jU

k=g

. s38d

Equations38d takes a simpler form when expressed in theprincipal axes of the crystal under investigation:44

r ij gel =U− 8p

ni2nj

2 xi jks2dU

k=g

, s39d

where theni coefficients are the principal refractive indices.The origin of the ionic contribution to the EO tensor is the

relaxation of the atomic positions due to the applied electricfield Eg and the variations of«i j induced by these displace-ments. It can be computed from the Born effective chargesZk,ab

* and the]xi js1d /]tka coefficients introduced in Sec. III B.

As shown in the Appendixssee also Refs. 36 and 45d, theionic EO tensor can be computed as a sum over the trans-verse optic phonon modes atq=0:

r ij gion = −

4p

ÎV0ni2nj

2om

ai jmpm,g

vm2 , s40d

wheream is the Raman susceptibility of modem fEq. s28dgandpm,g the mode polarity

pm,g = ok,b

Zk,gb* umskbd, s41d

which is directly linked to the mode oscillator strength

Sm,ab = pm,apm,b. s42d

For simplicity, we have expressed Eq.s40d in the principalaxes while a more general expression can be derived fromEq. s37d.

Finally, the piezoelectric contribution is due to a relax-ation of the unit cell shape due to the converse piezoelectriceffect.46,47As it is discussed in the Appendix, it can be com-puted from the elasto-optic coefficientspij mn and the piezo-electric strain coefficientsdgmn:

r ij gpiezo= o

m,n=1

3

pij mndgmn. s43d

In the discussion of the EO effect, we have to specifywhether we are dealing with strain-freesclampedd or stress-free sunclampedd mechanical boundary conditions. Theclamped EO tensorr ij g

h takes into account the electronic andionic contributions but neglects any modification of the unitcell shape due to the converse piezoelectric effect:46,47

r ij gh = r ij g

el + r ij gion. s44d

Experimentally, it can be measured for frequencies ofEg

high enough to eliminate the relaxations of the crystal latticebut low enough to avoid excitations of optical phonon modes

susually above,100 MHzd. To compute the unclamped EOtensorr ij g

s , we have to add the piezoelectric contribution tor ij g

h :

r ij gs = r ij g

h + r ij gpiezo. s45d

Experimentally,r ij gs can be measured for frequencies ofEg

below the sgeometry-dependentd mechanical body reso-nances of the sample47 susually below,1 MHzd.

IV. RESULTS

A. Technical details

Our calculations have been performed within the localdensity approximationsLDA d to the density functionaltheory1,2 sDFTd. We used theABINIT package,16 a plane-wave, pseudopotential DFT code48 in which we have imple-mented the formalism presented above. For reasons that willbecome obvious below, we chose the PEAD formulation, Eq.s17d , to perform the differentiation with respect tok. For theexchange-correlation energyExc we relied on the parametri-zation of Perdew and Wang49 as well as the parametrizationof Goedecker, Teter, and Hutter.50 These expressions havethe advantage of avoiding any discontinuities in the func-tional derivative ofExc.

In case of the semiconductors Si, AlAs, and AlP, we useda 16316316 grid of specialk points, a plane-wave kineticenergy cutoff of 10 hartree, and Troullier-Martins51 norm-conserving pseudopotentials. These calculations have beenperformed at the theoretical lattice constant. To perform thefinite difference calculations of the Raman polarizabilities,changes of the electronic dielectric tensor were computed foratoms displaced by ±1% of the unit cell parameter along theCartesian directions.

In case of rhombohedral BaTiO3, we used a 10310310 grid of specialk points, a plane-wave kinetic energycutoff of 45 hartree, and extended norm-conservingpseudopotentials.52 Since the ferroelectric instability is quitesensitive to the volume of the unit cell and tends to disappeardue to the volume underestimation of the LDA,53 we choseto work at the experimental lattice constants. In contrast tothe lattice parameters, the atomic positions have been re-laxed: the residual forces on the atoms were smaller than 5310−5 hartree/bohr.

It was shown by Gonze, Ghosez, and Godby54 that anaccurate functional for the exchange-correlation energy inextended systems should depend on both the density and thepolarization. The LDA used here neglects this polarizationdependence and may consequently introduce significant rela-tive errors when studying the response of a solid to an elec-tric field. In case of the second-order derivatives, the LDAusually yields an overestimate of the dielectric tensorsaslarge as 20% in BaTiO3d.55 In contrast, no clear trends havebeen reported yet concerning nonlinear optical propertiessuch asxi jl

s2d.9,29

In LDA calculations, it is common practice to apply ascissors correction57 to compensate for the lack of polariza-tion dependence of the exchange-correlation functional. Incase of nonlinear optical properties, such a correction can be


125107-7

applied at different levels. On one hand, we can compute thenonlinear optical susceptibilitiesfEq. s22dg using a scissorsoperator for the first-order wave functions.21 On the otherhand, in the computation of the EO coefficients, we can usea scissors corrected refractive index in Eqs.s39d and s40d.The influence of these corrections will be discussed below.

B. Nonlinear optical susceptibilities and Ramanpolarizabilities of semiconductors

In order to illustrate the computation of third-order energyderivatives, we performed a series of calculations on variouscubic AB semiconductors where the atomA is located at theorigin and B in s1/4,1/4,1/4d. In these compounds, thenonlinear optical susceptibility tensor has only one indepen-dent element,d123,

dijk = d123uei jku, s46d

where«i jk is the Levy-Civita tensor. The Raman susceptibili-ties are also defined by a single number,a12. For phononspolarized along the Cartesian directionl, the Raman suscep-tibility tensors are written as

ai jsld = a12uei j lu. s47d

In cubic semiconductors, it is customary to report the Ramanpolarizability34,60 defined as

a = ÎmV0a12 = V0]x12

s1d

]tA,3, s48d

wherem is the reduced mass of the two atoms in the unit cellandtA,3 a displacement of atomA along thez direction.

The formalism of Sec. II involves an integration over theBZ and a differentiation with respect tok. In practical cal-culations, these operations must be performed on a discretemesh of specialk points. As we explained in Sec. II, thediscretization can either be performed beforesPEADd or af-ter sDAPEd the perturbation expansion of the energy func-tional equations6d. Up until now, the applications of thepresent formalism to real materials9,10made use of the DAPEformula of the third-order energy. The only application of thePEAD formula has been reported by Nunes and Gonze14 ona one-dimensional model system. These authors observedthat the PEAD formula converges better with respect to thek-point sampling than the DAPE formula. In order to com-pare the performance of these two approaches for a realisticcase, we applied both of them to compute the nonlinear op-tical susceptibility d123 of AlAs. We performed a series ofcalculations on an3n3n grid of specialk points. As can beseen in Fig. 1 the PEAD formula converges much faster thanthe DAPE formula. Therefore, the PEAD formulation hasbeen applied to obtain the results presented below. It is theone that is actually available in theABINIT code.

It is interesting to note that a different speed of conver-gence for distinct expressions has also been reported for thelocalization tensor. In Ref. 56 Sgiarovello and co-workerscompared the convergence of two formulations based on dif-ferent discretizations of the samek space integral.They ob-served thatsid both expressions converge to the same value

in the limit of a large number ofk points andsii d that theexpression used in their work converges faster than the ex-pression used in Ref. 24.

In Table I, we report the nonlinear optical susceptibilitiesof the cubic semiconductors AlAs and AlP. Our results are inclose agreement with the values obtained by Dal Corso andco-workers9 who applied the 2n+1 theorem within theDAPE formalism, the results of Levine and Allan29 who useda “sum over excited states” technique, and the values ob-tained by Souza and co-workers,58 who followed a finiteelectric field approach. The values in the lower part of TableI have been obtained using a scissors correction. Our meth-odology provides a correction similar to that reported byLevine and Allan.30

The scissors correction decreases the value of the nonlin-ear optical susceptibilities in agreement with the discussionof Ref. 59. To the authors’ knowledge, no experimental dataare available for AlAs and AlP. For other cubic semiconduc-tors, it is, however, not clear that the use of a scissors cor-rection improves agreement with experiment29 and will evenhave a negative effect when the LDA underestimates the ex-perimental value. In addition, it is not straightforward to iso-late the error of the LDA on the nonlinear response functionsfrom other sources of errors. Other factors have a stronginfluence onxi jl

s2d similar to the scissors correction. For ex-

FIG. 1. Nonlinear optical susceptibility d123spm/Vd of AlAs forvarious grids ofn3n3n specialk points.

TABLE I. Nonlinear optical susceptibilitiesd123spm/Vd of somesemiconductors. The values in the lower part of the table have beenobtained using a scissorssSCId correction.

Method AlAs AlP

2n+1 theoremspresentd 35 21

2n+1 theorema 32 19

Finite fieldsb 32 19

Sum over statesc 34 21

2n+1 theorem + SCIspresentd 21 13

Sum over states + SCIc 21 13

aReference 9.bReference 58.cReference 30.


125107-8

ample, the values of the nonlinear optical susceptibilitiesstrongly depend on the pseudopotential9 or on the error onthe unit cell volume29,59 that is usually underestimated inLDA calculations.

We also computed the Raman polarizabilities of the trans-versesTOd and longitudinal opticalsLOd phonons of varioussemiconductors. In addition, we performed finite differencecalculations of the dielectric tensor with respect to atomicdisplacements. Our results are summarized in Table II wherewe also report the results of Deinzer and Strauch10 sDSd andBaroni and Resta37 sBRd as well as the experimental result ofWagner and Cardona60 for Si. The agreement between ourresults and those obtained in previous works is quite good. Inaddition, the results we obtained from the 2n+1 theoremclosely agree with the finite difference calculations, giving ussome indication of the numerical accuracy of the implemen-tation.

The Raman polarizabilities of the TO and LO modes aredifferent. As it is discussed in Sec. III B, this difference isattributed to the macroscopic electric field associated with alongitudinal polar lattice vibration. On the one hand, thisfield modifies the dynamical matrix atq→0. The eventualrelated modification of the eigenvectors of the LO modesmay imply a first change of the Raman susceptibility. On theother hand, the macroscopic electric field itself may inducean additional change ofa related to the nonlinear opticalcoefficientsxi jl

s2d. In the cubic semiconductors, the eigenvec-tors of the TO and LO modes are identical. The differencebetween the polarizabilities of the TO and LO modes comestherefore exclusively from the second term of Eq.s30d.

C. EO tensor in ferroelectric oxides

In the rhombohedral phase of BaTiO3, the EO tensor hasfour independent elements:r13, r33, r22, andr51. In contrast tothe dielectric tensor, the EO coefficients can either be posi-tive or negative. The sign of these coefficients is often diffi-cult to measure experimentally. Moreover, it depends on thechoice of the Cartesian axes. Experimentally, these axes arechosen according to theStandards on PiezoelectricCrystals.65,71 The z axis is along the direction of the sponta-

neous polarization and they axis lies in a mirror plane. Thez and y axes are both piezoelectric. Their positive ends arechosen in the direction that becomes negative under com-pression. The orientation of these axes can easily be foundfrom pure geometrical arguments. Unfortunately, these argu-ments do not allow us to determine the direction of theyaxis. Therefore, we applied the methodology of Ref. 61 tocompute the piezoelectric tensor from finite differences ofthe Berry phase polarization. Our results are reported in theCartesian axes where the piezoelectric coefficientse22 ande33 are positive.

These coefficients, as well as their decomposition on theindividual phonon modes and their electronic part, are re-ported in Table III. All EO coefficients are positive. As is thecase for the tetragonal phase,46 the modes that have thestrongest overlap with the soft mode of the paraelectric phasedominate the amplitude of the EO coefficients. Moreover, theelectronic contribution is found to be quite small.

As we discussed in the previous sections, linear and non-linear optical susceptibilities are sometimes relatively inac-curate within the LDA. In this context, it is interesting toinvestigate the error due to the use of the LDA optical di-electric constants in the transformation equations37d. Unfor-tunately, we could not find any experimental data on the EOcoefficients in the rhombohedral phase of BaTiO3. In Ref.

TABLE III. Decomposition of the clamped EO tensorspm/Vdin the rhombohedral phase of BaTiO3. Reported are the contribu-tions of individual zone-center phonon modes and the electroniccontribution. The phonon frequencies are reported in cm−1.

A1 modes E modes

v r13h r33

h v r22h r51

h

TO1 168 0.65 2.16 163 0.79 5.15

TO2 253 13.82 27.32 202 5.40 19.16

TO3 509 1.31 2.05 293 0.01 −0.02

TO4 469 0.24 0.65

Elect. 1.15 2.95 0.12 1.24

Tot. 16.93 34.48 6.56 26.18

TABLE II. Raman polarizabilities of the transversesTOd and longitudinalsLOd optical modessÅ2d ofsome semiconductors.

Si AlAs sTOd AlAs sLOd AlP sTOd AlP sLOd

2n+1 theorem

Present 20.02 8.48 12.48 4.30 7.46

DSa 23.56 7.39 5.13

Finite differences

Present 20.17 8.59 4.25

DSa 20.44 5.64 4.44

BRb 26.16

Experiment 23±4c

aReference 10.bReference 37.cReference 60.


125107-9

46, we studied the EO coefficients of ferroelectric LiNbO3and tetragonal BaTiO3 and PbTiO3 and found an overallgood agreement between theory and experiment. In Table IV,we report the EO coefficients of these compounds as well asthe values obtained using a scissors-corrected optical dielec-tric constant. No scissors correction has been applied for thenonlinear optical susceptibilities of these compounds that arerequired to compute the electronic contributions.

The effect of this correction is more important for theperovskite compounds than for LiNbO3, for which the LDAband gap and optical dielectric constants are in reasonableagreement with experiment.62 For BaTiO3, we tested the op-tical dielectric tensor obtained from the scissors correctionthat modifies the LDA band gap to its experimental value:21

we obtainr13h =12.68 pm/V andr33

h =30.84 pm/V, in closeragreement with experimental data. However, such an im-provement is not a general rule. In PbTiO3, a scissors shiftthat corrects the LDA band gap fails to correct the LDAoptical dielectric constantfwe obtain «11=5.81 and «33=5.51, while the experimental values are 6.63 and 6.64sRef.63dg and yieldsr13

h =14.24 pm/V andr33h =8.94 pm/V. Using

the experimental dielectric constants, we obtainr13h

=10.92 pm/V andr33h =6.16 pm/V in better agreement with

the experiment.

V. CONCLUSIONS AND PERSPECTIVES

In this paper, we presented the general framework for thecomputation of third-order energy derivatives within DFT.Our formalism makes use of the 2n+1 theorem and the mod-ern theory of polarization. Focusing on derivatives that arecharacterized by a zero wave vector and that involve eitherthree electric fields or two electric fields and one atomicdisplacement, we described the computation of nonlinear op-tical susceptibilities, of Raman scattering efficiencies of TOand LO phonons, and of the EO tensor.

The computation of the Berry phase polarization involvesa derivative of the wave functions with respect to their wavevector. In practice, this differentiation is computed on a gridof specialk points. The perturbation expansion can either beperformed beforesDAPEd or aftersPEADd the discretization,leading to two mathematically distinct expressions of thethird-order energies. We used both of them to compute thenonlinear optical susceptibility of AlAs, and we have shownthat the PEAD formulation converges faster with respect tothe k-point sampling.

We have computed the nonlinear optical susceptibilitiesand Raman polarizabilities of some cubic semiconductors aswell as the EO tensor in the rhombohedral phase of BaTiO3.

Finally, we have studied the effect of a scissors correctionon the EO coefficients and the nonlinear optical susceptibili-ties. In contrast to the dielectric tensor, we did not find asystematic improvement of the results by using this correc-tion.

We can figure out several applications of the methodologypresented in this work. Combined with the calculation ofphonon frequencies and infrared intensities, the computationof Raman efficiencies can be a useful complementary toolfor the interpretation of experimental spectra. Furthermore,the computation of the EO tensor from first principles canguide the tuning of the EO properties and help in designingnew efficient EO materials. This could be particularly helpfulsince accurate optical measurements require high-qualitysingle crystals not always directly accessible.

ACKNOWLEDGMENTS

The authors are grateful to M. D. Fontana, P. Bourson, B.Kirtman, and B. Champagne for helpful discussions. M. V.and X. G. acknowledge financial support from the FNRSBelgium. This work was supported by the Volkswagen-Stiftung within the project “Nano-sized ferroelectric Hy-brids” sI/77 737d, the Region WallonnesNomade, project115012d, the Communauté Française de Belgique–Actions deRecherche Concertées, the PAI/UIAP Phase 5 “Quantum sizeeffects in nanostructured materials,” FNRS-Belgium throughgrants Nos. 9.4539.00 and 2.4562.03, the European Unionthrough the Research and Training Network “EXCITING”sHPRM-CT-2002-00317d and the European Networks of Ex-cellence FAME sFPG-500159-1d and NANOQUANTAsNMP4-CT-2004-500198d.

APPENDIX: EXPRESSIONS OF THE CLAMPED ANDUNCLAMPED EO TENSORS

1. Macroscopic approach

As discussed in Sec. III D, the optical properties of acompound are modified by an electric fieldEg or a mechani-

TABLE IV. Effect of a scissors correction on the EO coefficientsof LiNbO3 sclamped and unclamped casesd and the tetragonalphases of PbTiO3 and BaTiO3 sclamped cases onlyd. The dielectrictensor required to perform the transformation, Eq.s37d, has beencomputed within the LDAs«LDAd and using the LDA with a scissorscorrections«SCId. No scissors correction has been used to computethe nonlinear optical susceptibilities that determine the electroniccontribution to the EO coefficients. In case of PbTiO3, we also usethe experimental dielectric tensors«exptd to compute the EO coeffi-cients. The values are compared to the experimental results.

r13 r33 r22 r51

LiNbO3 «LDA 9.67 26.93 4.55 14.93

sclampedd «SCI 10.37 28.89 4.88 16.02

Expt.a 8.6 30.8 3.4 28

LiNbO3 «LDA 10.47 27.08 7.53 28.61

sunclampedd «SCI 11.23 29.06 8.08 30.69

Expt.a 10 32.2 6.8 32.6

Expt.b 9.89

PbTiO3 «LDA 8.98 5.88 30.53

sclampedd «SCI 14.24 8.94 47.39

«expt 10.92 6.16 34.45

Expt.c 13.8 5.9

BaTiO3 «LDA 8.91 22.27

sclampedd «SCI 12.68 30.84

Expt.d 10.2 40.6

Expt.e 8 28

aReference 71.bReference 72.cReference 73.dReference 74.eReference 47.


125107-10

cal constraintsa stresssmn or a homogeneous strainhmnd. Atlinear order, the variations of«i j

−1 can be described usingeither the variables64,65 sEg ,hmnd or sEg ,smnd:

Ds«−1di j = og=1

3

r ij gh Eg + o

m,n=1

3

pij mnhmn, sA1ad

Ds«−1di j = og=1

3

r ij gs Eg + o

m,n=1

3

pi j mnsmn, sA1bd

wherer ij gh andr ij g

s are, respectively, the clampedsstrain-freedand unclampedsstress-freed EO coefficients,pij mn are theelasto-optic sstrain-opticd coefficients, andpi j mn are thepiezo-opticalsstress-opticald coefficients. In order to relateEqs. sA1ad and sA1bd, we can express the strain as beinginduced by the stress or by the electric fieldsconverse piezo-electric effectd,

hmn = om8,n8=1

3

Smnm8n8sm8n8 + og=1

3

dgmnEg, sA2d

whereSmnm8n8 are the elastic compliances anddgmn the piezo-electric strain coefficients.

If we assume, for example, that the unit cell is free torelax within the electric fieldsstress-free mechanical bound-ary conditionsd we can either use Eq.sA1bd sin which casethe second term of the right-hand side is zerod or Eq. sA1adto computeDs«−1di j . In the latter case, the strain induced bythe electric field can be obtained from the second term of theright-hand side of Eq.sA2d:

Ds«−1di j = og=1

3

r ij gs Eg = o

g=1

3

r ij gh Eg + o

m,n=1

3

og=1

3

pij mndgmnEg.

sA3d

Using this identity, we obtain the following relation betweenthe unclamped and the clamped EO coefficients:

r ij gs = r ij g

h + om,n=1

3

pij mndgmn. sA4d

2. Microscopic approach

In order to derive the expressions of the clamped andunclamped EO tensor of Sec. III D, we use a Taylor expan-sion of the electric enthalpy66 F. Similar developments havealready been applied to determine the lattice contribution ofthe static dielectric tensor and of the piezoelectric tensor.67,68

They are based on an expansion ofF up to the second orderin the atomic coordinatesRka, the homogeneous strainhmn,and the macroscopic electric fieldEg. In this section, weextend these developments to the third order.

The electric enthalpy of a solid in an electric field is ob-tained by the minimization

FsEd = minR,h

FsR,h,Ed. sA5d

We denoteRsEd andhsEd the atomic positions and the strainthat minimizeF at constantE andR0, h0s=0d their values at

E=0. For small fields, we can expand the functionFsR ,h ,Ed in powers ofE aroundE=0:

FsR,h,Ed = FsR,h,0d − V0oi=1

3

PisR,hdEi

−V0

8poi,j=1

3

«i jsR,hdEiE j

−V0

3 oi,j ,k=1

3

xi jks2dsR,hdEiE jEk + . . . sA6d

where V0 is the volume of the primitive unit cell in realspace andPsR ,hd, «i jsR ,hd, andxi jk

s2dsR ,hd are the macro-scopic polarization, electronic dielectric tensor, and nonlin-ear optical coefficients at zero macroscopic electric field andfor a given configurationsR ,hd. At nonzero field, thesequantities are defined as partial derivatives ofF with respectto E. For example, the electric-field-dependent electronic di-electric tensor can be computed from the expression

«i j„RsEd,hsEd,E… = − U4p

V0

]2F

]Ei ] E jU

RsEd,hsEd,E. sA7d

Let tka=Rka−R0,ka be the displacement of atomk alongdirectiona andtka

l shmnl d the first-order modification of the

atomic positionsstraind induced by a perturbationl:

tkal = U ]tka

]lU

l=0, hmn

l = U ]hmn

]lU

l=0. sA8d

In the discussion that follows, we will study the effect of anelectric field perturbation and a strain perturbation on theelectric enthalpyF in order to obtain the formulas to com-pute the elasto-optic coefficients as well as the clamped andthe unclamped EO tensors.

a. Elasto-optic coefficients„E=0…

The elasto-optic tensor can be computed from thetotalderivative of the dielectric tensor with respect tohmn at zeroelectric field

Ud«i jsR,h,0ddhmn

UR0,h0

= U ]«i jsR,hd]hmn

UR0,h0

+ U4poka

]xi js1dsR,hd]tka

UR0,h0

tkahmn.

sA9d

The derivative in the first term of the right-hand side is com-puted considering the ionic cores as artificially clamped attheir equilibrium positions. The remaining terms representthe ionic contribution to the elasto-optic tensor. They involvederivatives of the linear dielectric susceptibilityxi j

s1d with re-spect to the atomic positions that have to be multiplied by thefirst-order strain-induced atomic displacementstka

hmn fEq.sA8dg. To compute these quantities we use the fact thatF isminimum at the equilibrium for an imposed strainh. Thiscondition implies


125107-11

U ]FsR,hd]tka

URshd,h

= 0. sA10d

Since we are interested in first-order atomic displacementswe can writetkashd=om,n=1

3 tkahmnhmn+Osh2d. Solving the ex-

tremum equationsA10d to the linear order inh, we obtain

ok8,a8

U ]2FsR,hd]tka ] tk8a8

UR0,h0

tk8a8hmn = − U ]2FsR,hd

]hmn ] tkaU

R0,h0

.

sA11d

The second derivatives on the left side of Eq.sA11d definethe matrix of interatomic force constants at zero macroscopicelectric field, which enables the computation of the trans-verse phonon frequenciesvm and eigendisplacementsumskad. By decomposingtka

hmn in the basis of the zone-centerphonon-mode eigendisplacements

tkahmn = o

m

tmhmnumskad sA12d

and using Eqs.s24d ands25d we derive the following expres-sion for the first-order strain-induced atomic displacements:

tmhmn =

− 1

vm2 U ]2FsR,hd

]hmn ] tmU

R0,h0

, sA13d

where

U ]2FsR,hd]hmn ] tm

UR0,h0

= ok,aU ]2FsR,hd

]hmn ] tkaU

R0,h0

umskad.

sA14d

If we introduce Eqs.sA12d andsA13d into Eq. sA9d and usethe definition of the Raman susceptibility, Eq.s28d, and thetransformation, Eq.s37d, we finally obtain the formula tocompute the elasto-optic tensor

pij mn =− 1

ni2nj

2U ]«i jsR,hd]hmn

UR0,h0

+4p

ni2nj

2ÎV0o

m

ai jm

vm2 U ]2FsR,hd

]hmn ] tmU

R0,h0

. sA15d

To simplify, we write Eq.sA15d in the principal axes of thecrystal under investigation. A more general expression can beobtained from Eq.s37d.

EquationsA15d is different from the approach used pre-viously by Detraux and Gonze to study the elasto-optic ten-sor in a-quartz.69 The authors of Ref. 69 used finite differ-ences with respect to strains to compute the total derivativeof «i j . In their approach, the atoms where relaxed to theirequilibrium positions in the strained configurations. In caseof Eq. sA15d, the first term of the right-hand side is com-puted at clamped atomic positions while the effect of thestrain-induced atomic relaxations is taken into account by thesecond term.

b. Clamped EO coefficients„h=0…

The clamped EO tensor can be computed from thetotalderivative of the electric field dependent dielectric tensor,Eq. sA7d, with respect toE

Ud«i jsR,h0,EddEg

UR0,E=0

= U ]«i jsR0,h0,Ed]Eg

UE=0

+ 4poka

U ]xi js1dsR,h0d]tka

UR0

tkaEg .

sA16d

The derivative in the first term is computed considering theionic cores as artificially clamped at their equilibrium posi-tions. This term represents the bare electronic contribution tothe EO tensor that can be computed from the nonlinear op-tical coefficients

UU ]«i jsR0,h0,Ed]Eg

UE=0

= 8pxi jks2dU

k=g

, sA17d

related to a third-order partial derivative ofF:

xi jks2d = xi jk

s2dsR0,h0d = U − 1

2V0

]3FsR0,h0,Ed]Ei ] E j ] Ek

UE=0

. sA18d

The remaining terms in Eq.sA16d represent the ionic contri-bution to the EO tensor. They involve derivatives of thelinear dielectric susceptibilityxi j

s1d with respect to the atomicpositions that have to be multiplied by the first-order electricfield induced atomic displacementstka

Eg fEq. sA8dg. To obtainthese quantities, we proceed the same way as in case of theelasto-optic tensor. Using the equilibrium condition

]F

]tka

= 0 =U ]FsR,h0,0d]tka

URsEd

− V0oi=1

3 U ]PisR,h0d]tka

URsEd

Ei

− UV0

8poi,j=1

3]«i jsR,h0d

]tkaU

RsEd

EiE j + . . . sA19d

and expandingtka to the first order in the electric field, weobtain

ok8,a8

U ]2FsR,h0,0d]tka ] tk8a8

UR0

tk8a8Eg = V0U ]PgsR,h0d

]tkaU

R0

.

sA20d

This expression is similar to Eq.sA11d. The second-orderderivatives ofF on the left side are the interatomic forceconstants and the derivative of the zero field polarizationwith respect totka on the right side is the Born effectivecharge tensorZk,ga

* of atomk. DecomposingtkaEg in the basis

of the zone-center phonon-mode eigendisplacementsfEq.sA12dg and using the orthonormality constraint Eq.s25d wederive the following expression for the first-order electric-field-induced atomic displacements:


125107-12

tmEg =

1

vm2 o

k,aZk,ga

* umskad. sA21d

If we introduce Eqs.sA17d and sA21d into Eq. sA16d wefinally obtain the formula to compute the total derivative ofthe dielectric tensor:

UUd«i jsR,EddEg

UR0,E=0

= 8pxi jks2dU

k=g

+ 4pom

1

vm2 So

k,a

]xi js1dsRd

]tka

umskadD3So

k8,b

Zk8,gb* umsk8bdD . sA22d

Using the definition of the Raman susceptibilityfEq. s28dg,the mode polarityfEq. s41dg and the transformationfEq.s37dg, we obtain the expression of the clamped EO tensor

r ij gh =U− 8p

ni2nj

2 xi jls2dU

l=g

−4p

ni2nj

2ÎV0o

m

ai jmpm,g

vm2 . sA23d

As in the case of the elasto-optic tensorfEq. sA15dg, we havewritten Eq. sA23d in the principal axes of the crystal underinvestigation.

c. Unclamped EO tensor„s=0…

In order to compute the unclamped EO tensor, we have totake into account both the electric-field-induced atomic dis-placmentstka

Eg and the electric-field-induced strainhmnEg when

computing the total derivative of«i j :

Ud«i jsR,h,EddEg

UR0,h0,E=0

= U ]«i jsR0,h0,Ed]Eg

UE=0

+ U4poka

]xi js1dsR,h0d]tka

UR0

tkaEg

+ 4p om,n=1

3 U ]xi js1dsR0,hd]hmn

Uh0

hmnEg . sA24d

The electronic contributionffirst term of Eq.sA24dg is thesame as for the clamped EO tensor. It can be computed fromthe nonlinear optical coefficientsfEq. sA17dg. To computetkaEg andhmn

Eg , we can use an equilibrium condition similar toEq. sA19d where we require that the first-order derivatives ofF with respect totka andhmn vanish. Expandingtka andhmn

to the first order in the electric field, we obtain the system ofcoupled equationsssee also Ref. 70d

ok8,a8

U ]2FsR,h,0d]tka ] tk8a8

UR0,h0

tk8a8Eg + o

m,nU ]2FsR,h,0d

]tka ] hmnU

R0,h0

hmnEg

= UV0

]PgsR,hd]tka

UR0,h0

, sA25ad

om8,n8

U ]2FsR,h,0d]hmn ] hm8n8

UR0,h0

hm8n8Eg

+ ok8,a8

U ]2FsR,h,0d]tk8a8 ] hmn

UR0,h0

tk8a8Eg

= UV0

]PgsR,hd]hmn

UR0,h0

. sA25bd

Because of the coupling betweentkaEg andhmn

Eg , defined by themixed second-order derivatives]2F /]tkahmn, the secondterm of the right-hand side of Eq.sA24d is different from thatof Eq. sA16d. That means that the sum of the first and secondterms of Eq.sA24d is not identical to the clamped EO coef-ficients r ij g

h . Moreover, the third term of Eq.sA24d is differ-ent from the piezoelectric contribution of Sec. A 1.

In order to obtain the decomposition ofr ij gs into elec-

tronic, ionic, and piezoelectric contributions defined previ-ously, we can solve Eq.sA25ad for tka

Eg . In the basis of thezone-center phonon mode eigendisplacements we can write

tnEg =

pn,g

vn2 −

1

vn2o

mn

U ]2FsR,h,0d]tn ] hmn

UR0,h0

hmnEg . sA26d

If we insert this relation into Eq.sA24d and use the transfor-mation equations37d we obtain the following expression ofthe unclamped EO tensor in the principal axes:

r ij gs = U− 8p

ni2nj

2 xi jls2dU

l=g

−4p

ni2nj

2ÎV0o

m

ai jmpm,g

vm2

−4p

ni2nj

2om,nFU ]xi j

s1dsR,h,Ed]hmn

UR0,h0,E=0

−1

ÎV0o

mUai j

m

vm2

]2FsR,h,0d]tm ] hmn

UR0,h0,E=0

GhmnEg

sA27d

The sum of the first and second term of the right-hand side ofEq. sA27d is equal to the clamped EO coefficientr ij g

h . Theproduct of the conversion factor times the bracket in the thirdterm of Eq.sA27d is equal to the elasto-optic coefficientpij mn

fEq. sA15dg. Finally, by definition of the converse piezoelec-tric effect,hmn

Eg is equal to the piezoelectric strain coefficientdgmn. We thus obtain the following expression of theunclamped EO coefficients that is equal to the one derived inSec. A 1 from pure macroscopic arguments:

r ij gs = r ij g

h + om,n=1

3

pij mndgmn. sA28d

It is worth noting that the so-called piezoelectric contributionnot only takes into account the change of the linear opticalsusceptibility with strainfthird term of the right-hand side ofEq. sA24dg but also includes the modification of the ioniccontribution, with respect to the clamped case, that is asso-ciated to the modification of the ionic relaxation induced bythe strain.


125107-13

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