Paolo Giannozzi Scuola Normale Superiore, Piazza dei Cavalieri 7 I Pisa, Italy

Transcription

1 Lecture Notes per l Corso d Struttura della Matera (Dottorato d Fsca, Unverstà d Psa, 2002): DENSITY FUNCTIONAL THEORY FOR ELECTRONIC STRUCTURE CALCULATIONS Paolo Gannozz Scuola Normale Superore, Pazza de Cavaler 7 I Psa, Italy 1

2 Contents 1 Densty Functonal Theory The Hohenberg-Kohn Theorem The Kohn-Sham equatons Kohn-Sham equatons and the varatonal prncple DFT, Hartree-Fock, and Slater s exchange Local Densty Approxmaton for the exchange-correlaton energy Successes and falures of LDA On the physcal meanng of Kohn-Sham egenvalues and egenvectors The dscontnuty of exchange-correlaton potental Band gaps and dscontnuty of exchange-correlaton potental Adabatc contnuaton formula, exchange-correlaton hole, and LDA The exact exchange-correlaton potental from many-body theory Practcal DFT calculatons Atoms Molecules Extended systems: unt cells and supercells Plane wave bass set Pseudopotentals Another way of lookng at pseudopotentals Brlloun-Zone samplng Fndng the electronc ground state Iteraton to self-consstency Dagonalzaton of the Hamltonan Drect mnmzaton Movng atoms - complex materals Optmzaton of lattce parameters Optmzaton of atomc postons Hellmann-Feynman forces Pulay forces DFT and Molecular Dynamcs Classcal Molecular Dynamcs Dscretzaton of the equaton of moton Thermodynamcal averages Verlet algorthm as untary dscretzaton of the Louvllan Canoncal ensemble n MD Constant-pressure MD Car-Parrnello Molecular Dynamcs Why Car-Parrnello works Choce of the parameters Appendx Functonals and functonal dervatves Iteratve dagonalzaton Fast-Fourer Transform Conjugate Gradent Essental Bblography

3 1 Densty Functonal Theory Densty Functonal Theory (DFT) s a ground-state theory n whch the emphass s on the charge densty as the relevant physcal quantty. DFT has proved to be hghly successful n descrbng structural and electronc propertes n a vast class of materals, rangng from smple crystallne solds to more complex solds (ncludng glasses and lquds) to molecules. Furthermore DFT s computatonally very smple. For these reasons DFT has become a common tool n frst-prncples calculatons amed at descrbng or even predctng propertes of molecular and condensed matter systems. 1.1 The Hohenberg-Kohn Theorem Let us consder a system of N nteractng (spnless) electrons under an external potental V (r) (usually the Coulomb potental of the nucle). If the system has a nondegenerate ground state, t s obvous that there s only one charge densty n(r) of the ground state that corresponds to a gven V (r). In 1964 Hohenberg and Kohn demonstrated the opposte, far less obvous result: there s only one external potental V (r) whch yelds a gven ground-state charge densty n(r). The demonstraton s very smple and uses a reducto ad absurdum argument. Let us consder two dfferent many-electron Hamltonans H = T +U +V and H = T +U +V, whose respectve ground state wavefunctons are Ψ and Ψ. T s the knetc energy, U the electronelectron nteracton, V and V do not dffer smply by a constant: V V const. The charge densty n(r) s defned as n(r) = N Ψ(r, r 2, r 3,..., r N ) 2 dr 2...dr N (1) and we assume that n[v ] = n[v ]. We have the followng nequalty: E = Ψ H Ψ < Ψ H Ψ = Ψ H + V V Ψ, (2) that s, E < E + (V (r) V (r))n(r)dr. (3) The nequalty s strct because Ψ and Ψ are dfferent, beng egenstates of dfferent Hamltonans. By reversng the prmed and unprmed quanttes, one obtans an absurd result. A subtle pont about the exstence of the potental correspondng to a gven ground state charge densty (the v-representablty problem), and varous extensons of the Hohenberg and Kohn theorem, are dscussed n the specalzed lterature. A straghtforward consequence of the frst Hohenberg and Kohn theorem s that the ground state energy E s also unquely determned by the ground-state charge densty. In mathematcal terms E s a functonal E[n(r)] of n(r). We can wrte E[n(r)] = Ψ T + U + V Ψ = Ψ T + U Ψ + Ψ V Ψ = F [n(r)] + n(r)v (r)dr (4) where F [n(r)] s a unversal functonal of the charge densty n(r) (and not of V (r)). For ths functonal a varatonal prncple holds: the ground-state energy s mnmzed by the ground-state charge densty. In ths way, DFT exactly reduces the N-body problem to the determnaton of a 3-dmensonal functon n(r) whch mnmzes a functonal E[n(r)]. Unfortunately ths s of lttle use as F [n(r)] s not known. 1.2 The Kohn-Sham equatons One year later, Kohn and Sham (KS) reformulated the problem n a more famlar form and opened the way to practcal applcatons of DFT. The system of nteractng electrons s mapped on to an auxlary system of non-nteractng electrons havng the same ground state charge densty n(r). For a system of non-nteractng electrons the ground-state charge densty s representable as a sum over one-electron orbtals (the KS orbtals) ψ (r): n(r) = 2 ψ (r) 2, (5) 3

4 where runs from 1 to N/2 f we assume double occupancy of all states, and the KS orbtals are the solutons of the Schrödnger equaton ) ( h2 2m 2 + V KS (r) ψ (r) = ɛ ψ (r) (6) (m s the electron mass) obeyng orthonormalty constrants: ψ (r)ψ j (r)dr = δ j. (7) The exstence of a unque potental V KS (r) havng n(r) as ts ground state charge densty s a consequence of the Hohenberg and Kohn theorem, whch holds rrespectve of the form of the electron-electron nteracton U. 1.3 Kohn-Sham equatons and the varatonal prncple The problem s now to determne V KS (r) for a gven n(r). Ths problem s solved by consderng the varatonal property of the energy. For an arbtrary varaton of the ψ (r), under the orthonormalty constrants of Eq. (7), the varaton of E must vansh. Ths translates nto the condton that the functonal dervatve (see appendx) wth respect to the ψ of the constraned functonal E = E ( ) λ j ψ (r)ψ j (r)dr δ j, (8) j where λ j are Lagrange multplers, must vansh: δe δψ (r) = δe = 0. (9) δψ (r) It s convenent to rewrte the energy functonal as follows: E = T s [n(r)] + E H [n(r)] + E xc [n(r)] + n(r)v (r)dr. (10) The frst term s the knetc energy of non-nteractng electrons: T s [n(r)] = h2 2m 2 ψ (r) 2 ψ (r)dr. (11) The second term (called the Hartree energy) contans the electrostatc nteractons between clouds of charge: E H [n(r)] = e2 n(r)n(r ) 2 r r drdr. (12) The thrd term, called the exchange-correlaton energy, contans all the remanng terms: our gnorance s hdden there. The logc behnd such procedure s to subtract out easly computable terms whch account for a large fracton of the total energy. Usng δn(r) δψ (r ) = ψ (r)δ(r r ) (13) and the formulae gven n the appendx, one fnds δt s h2 = (r) 2m 2 δψ δe H = e2 (r) δψ 2 ψ (r), (14) n(r ) r r dr ψ (r) (15) 4

5 and fnally ) ( h2 2m 2 + V H (r) + V xc [n(r)] + V (r) ψ (r) = j λ j ψ j (r) (16) where we have ntroduced a Hartree potental V H (r) = e 2 n(r ) r r dr (17) and an exchange-correlaton potental V xc [n(r)] = δe xc δn(r). (18) The Lagrange multpler λ j are obtaned by multplyng both sdes of Eq.16 by ψk (r) and ntegratng: ) λ k = ψk(r) ( h2 2m 2 + V H (r) + V xc [n(r)] + V (r) ψ (r)dr. (19) For an nsulator, whose states are ether fully occuped or completely empty, t s always possble to make a subspace rotaton n the space of ψ s (leavng the charge densty nvarant). We fnally get the KS equatons: (H KS ɛ ) ψ (r) = 0, (20) where λ j = δ j ɛ j and the operator H KS, called KS Hamltonan, s defned as H KS = h2 2m 2 + V H (r) + V xc (r) + V (r) h2 2m 2 + V KS (r) (21) and s related to the functonal dervatve of the energy: 1.4 DFT, Hartree-Fock, and Slater s exchange δe δψ (r) = H KSψ (r). (22) The KS equatons are somewhat remnscent of the Hartree-Fock (HF) equatons. Both are derved from a varatonal prncple: the mnmzaton of the energy functonal for the latter, of the energy for a sngle Slater determnant wavefuncton for the former. Both are self-consstent equatons for one-electron wavefunctons. In the HF equatons the exchange term appears n the place of the exchange-correlaton potental of KS equatons: ) ( h2 2m 2 + V H (r) + V (r) ψ (r) + e 2 j, ψj (r)ψj (r ) r r ψ (r )dr = ɛ ψ (r) (23) where the sum over j extends only to states wth parallel spns. Tradtonally, one defnes the correlaton energy as the dfference between the HF and the real energy. The name exchangecorrelaton n DFT reflects such tradton, although the exchange-correlaton energy of DFT s not exactly the same as HF exchange plus correlaton energy: n fact the former contans a contrbuton comng from the dfference between the true many-body knetc energy Ψ T Ψ and the knetc energy T s [n(r)] of non-nteractng electrons. The exchange term n the HF equatons s a nonlocal operator one actng on a functon φ as (V φ)(r) = V (r, r )φ(r )dr, and s qute dffcult to compute. In earler calculatons, done wth prmtve computer machnery (or even wthout any computer machnery), an approxmated form was often used. In the homogeneous electron gas, the average exchange energy and exchange potental for an electron are ɛ x = 3 e 2 k F 4 π, v x = 3 e 2 k F 2 π (24) 5

6 where k F s the Ferm wavevector: k F = (3π 2 n) 1/3. In 1951 Slater proposed to replace the nonlocal exchange potental wth the above form vald for the homogeneous electron gas, wth k F evaluated at the local densty. Ths procedure yelds a local (multplcatve) exchange potental V x (r) = 3e2 2π [ 3π 2 n(r) ] 1/3, (25) sometmes multpled by coeffcent α varyng between 2/3 and 1 as an adjustable parameter. Ths approxmaton was rather popular n early sold-state physcs but was never regarded as an especally good one (and t wasn t, actually). 1.5 Local Densty Approxmaton for the exchange-correlaton energy We stll don t have a reasonable estmate for the exchange-correlaton energy E xc [n(r)]. Kohn and Sham ntroduced, as early as 1965, the Local Densty Approxmaton (LDA): they approxmated the functonal wth a functon of the local densty n(r) : E xc [n(r)] = ɛ(n(r))n(r)dr, δe xc δn(r) µ xc(n(r)) = ( ɛ(n) + n dɛ(n) ) dn n=n(r) and for ɛ(n(r)) used the same dependence on the densty as for the homogeneous electron gas (also known as jellum) for whch n(r) s constant. Even n such smple case the exact form of ɛ(n) s unknown (except at the HF level, see above). However, approxmate forms have been known for a long tme, gong back to Wgner (1931). Numercal results from Monte-Carlo calculatons (n prncple exact) by Ceperley and Alder have been parameterzed by Perdew and Zunger wth a smple analytcal form: (26) ɛ xc (n) = /r s /( r s r s ), r s 1 = /r s ln r s r s r s ln r s, r s 1 (27) where r s s the usual parameter appearng n the theory of metals: r s = (3/4πn) 1/3, and atomc unts are used (e 2 = h = m = 1: lengths n Bohr rad, energes n Hartree=27.2 ev). Followng HF tradton, the frst term s called exchange (t has the same form as Slater s local approxmaton to exchange), the remanng terms correlaton. We note however that such dstncton s to some extent arbtrary. Actually t has been shown that LDA contans a far amount of error compensaton between exchange and correlaton. The Perdew-Zunger form for ɛ xc s often used. Several other expressons have appeared n the lterature. All forms yeld very smlar results n condensed-matter calculatons, whch s not surprsng, snce all parameterzatons are very smlar n the range of r s applcable for sold-state phenomena. 1.6 Successes and falures of LDA LDA has turned out to be much more successful than expected. LDA s computatonally much smpler than HF, yet t yelds results of smlar or better qualty, even n atoms and molecules hghly nhomogeneous systems for whch an approxmaton based on the homogeneous electron gas would hardly look approprate. Structural and vbratonal propertes of solds are n general accurately descrbed: the correct crystal structure s usually found to have the lowest energy, bond lengths, bulk modul and phonon frequences are accurate wthn a few percent. LDA also has some well-known drawbacks. The followng s a lst of just a few of the more serous: self-nteracton (the nteracton of an electron wth the feld t generates) should cancel exactly (t does n HF by constructon) but t does not n LDA. In fnte systems the presence of self-nteracton s reflected n an ncorrect long-range behavor of the potental felt by an electron. For an atom, we should have V xc (r) 1/r for r, but LDA yelds nstead a potental that decays exponentally. 6

7 LDA tends to badly overestmate ( 20% and more) cohesve energes n molecules and solds. As a general rule, LDA tends to over-bnd. Ths has some nterestng consequences n systems bound by van der Waals (dspersve) forces. The van der Waals nteracton s absent from LDA by constructon: t s due to charge fluctuatons, not to charge overlap. LDA however overestmates the attractve potental comng from the overlap of the tals of the charge densty, thus yeldng apparently good results for the bndng energy (but wrong dependence on the separaton dstance, of course), for the wrong reason. LDA grossly underestmate ( 50%) band gaps n nsulators (see below for ther exact defnton). The study of reasons for the good performances and falures of LDA, as well as the search for better functonals, s stll a very actve feld. More accurate gradent-corrected functonals have been proposed and have found wdespread acceptance. Some mportant results have been acheved n the last years and wll be brefly descrbed n the next paragraphs. 1.7 On the physcal meanng of Kohn-Sham egenvalues and egenvectors One would lke very much to be able to calculate one-electron energes havng the meanng of removal (or addton) energes, as for a non nteractng system (n the language of many-body theory, quaspartcle energes). If one electron n the state v s removed from the system, E N E N 1 = ɛ v, where E N s the energy of the system wth N electrons. If one electron s added to the system n the state c, E N+1 E N = ɛ c. The dfference between the largest addton energy and the smallest removal energy defnes the energy band gap: E g = ɛ c ɛ v = E N+1 + E N 1 2E N. In solds ths s the onset of the contnuum of optcal transtons, f the gap s drect (f the lowest empty state and the hghest flled state have the same k vector). From atomc and molecular physcs, the hghest occuped and lowest unoccuped states are respectvely called HOMO (Hghest Occuped Molecular Orbtal) and LUMO (Lowest Unoccuped MO), whle addton and removal energy are respectvely called electron affnty, A, and onzaton potental, I. In HF the one-electron energes have the meanng of removal (or addton) energes for extended systems (Koopman s theorem). If the world were descrbed by sngle Slater determnants, the dfference between the LUMO and HOMO one-electron HF energes would yeld the real energy gaps n solds (neglectng polarzaton effects,.e. the change n the one-electron states upon addton or removal of an electron). Snce the world s not well descrbed by sngle Slater determnants, the band gap s usually qute overestmated n HF (wth the true exchange potental, not Slater s local approxmaton). In DFT, the one-electron energes have acqured a rather bad reputaton, mostly due to the falure of KS band gaps (that s: calculated as the dfference between LUMO and HOMO KS energes) to reproduce wth an acceptable accuracy the true band gap n solds: gaps n DFT are strongly underestmated. It s not correct however to rule out KS egenvalues as purely mathematcal quanttes wthout any physcal meanng. In partcular, t can be demonstrated that n exact DFT, I = ɛ HOMO holds. Of course, n fnte systems onzaton potentals and electron affntes can be calculated as energy dfferences between the ground state and a state wth one electron added or removed. In extended systems (solds) ths s of course not possble. In recent years the reasons for the band gap fasco have been clarfed. The problem s n the dependence of the exact energy functonal upon the number of electrons and n the nablty of approxmate functonals to reproduce t The dscontnuty of exchange-correlaton potental The basc varatonal property of the densty functonal can be expressed by the statonary condton ( ( )) δ E µ n(r)dr N = 0 (28) δn(r) 7

8 where µ s a Lagrange multpler and N an nteger number. The formulaton of DFT can be extended to nonnteger number of partcles N + ω (ω > 0) va the followng defnton: E[n(r)] = F frac [n(r)] + V (r)n(r)dr (29) and F frac [n(r)] = mn tr{d(t + U)}, D = (1 ω) Ψ N Ψ N + ω Ψ N+1 Ψ N+1 (30) where the mnmum must be searched on all densty matrces D that yeld the prescrbed densty n(r). It s easly verfed that ntegraton of n(r) over all space yelds N + ω electrons. Wth ths defnton the varatonal prncple, Eq. 28, s defned for any number of electrons and yelds the Euler equatons δe δn(r) = µ (31) and that µ s really the chemcal potental: f we call E N the energy at the ground state for N electrons, one has µ(n) = E N N. (32) There s an obvous problem f we consder µ(n) a contnuous functon of N for all values of N. Consder two neutral solated atoms: n general, they wll have two dfferent values for µ. As a consequence the total energy of the two atoms wll be lowered by a charge transfer from the atom at a hgher chemcal potental to the one at lower chemcal potental. In realty there s no paradox, because the E N curve s not contnuous. If we wrte down explctly E N+ω, we fnd that both energy and mnmzng charge densty at fractonary number of electrons are smply a lnear nterpolaton between the respectve values at the end ponts wth N and N + 1 electrons: E N+ω = (1 ω)e N + ωe N+1, n N+ω (r) = (1 ω)n N (r) + ωn N+1 (r) (33) wth obvous notatons. The nterestng and far-reachng consequence s that there s a dscontnuty of the chemcal potental µ(n) and of the functonal dervatve δe/δn(r) at nteger N. Ths s an mportant and essental characterstc of the exact energy functonal that smply reflects the dscontnuty of the energy spectrum. Comng back to our paradox: for an atom wth nuclear charge Z, onzaton potental I(Z) and electron affnty A(Z) n the ground state, µ(n) = I(Z) Z 1 < N < Z (34) = A(Z) Z < N < Z + 1. (35) For a system of two neutral atoms wth nuclear charges X and Y, n whch ω electrons are transferred from the frst to the second atom: µ(ω) = µ(0) + I(Y ) A(X) 1 < ω < 0 (36) = µ(0) + I(X) A(Y ) 0 < ω < 1. (37) Snce the largest A (3.62 ev, for Cl) s stll smaller than the smallest I (3.89 ev, for Cs), the neutral ground state s stable Band gaps and dscontnuty of exchange-correlaton potental A consequence of the results of the prevous secton s that the true band gap of a sold, E g = I A, can be wrtten as E g = µ(n δ) + µ(n + δ) = δe δn(r) δe N+δ δn(r) (38) N δ wth δ 0. 8

9 Let us substtute to E[n(r)] the explct KS form, Eq.10. The Hartree and external potental terms of the functonal wll yeld no dscontnuty and no contrbuton to E g. Only the knetc and exchange-correlaton terms may have a dscontnuty and contrbute to E g. For a non nteractng system, only the knetc term contrbutes, and the gap s exactly gven by the KS gap: Eg KS = δt s δn(r) δt s N+δ δn(r) = ɛ LUMO ɛ HOMO. (39) N δ We remark that even the knetc energy of non nteractng electrons, consdered as a functonal of the densty, must have a dscontnuous dervatve when crossng an nteger number of electrons. Ths s one reason why t s so dffcult to produce explct functonals of the charge densty for T s that are able to yeld good results: no smple functonal form wll yeld the dscontnuty, but ths s needed n order to get the correct energy spectrum. For the nteractng system: E g = δt s δn(r) δt s N+δ δn(r) + δe xc N δ δn(r) δe xc N+δ δn(r) = Eg KS + Eg xc. (40) N δ Note that the knetc term s evaluated at the same charge densty as for the non nteractng system, so t concdes wth the KS gap. In concluson: the KS gaps are not, by constructon, equal to the true gap, because they are mssng a term (Eg xc ) comng from the dscontnuty of dervatves of the exchange-correlaton functonal. Ths s absent by constructon from any current approxmated functonal (be t LDA or gradent-corrected or more complex). There s some evdence that ths mssng term s responsble for a large part of the band gap problem, at least n common semconductors. 1.8 Adabatc contnuaton formula, exchange-correlaton hole, and LDA The exchange-correlaton energy can be recast nto a form that sheds some lght on the unexpected success of LDA and gves a possble path for the producton of better functonals. One consders a system n whch the Coulomb nteracton between electrons s adabatcally swtched on: U λ = λ e2 2,j 1 = λu (41) r r j where λ s a parameter that goes from λ = 0, for the nonnteractng system, to λ = 1, for the true nteractng system. The charge densty s forced to reman equal to the charge densty of the nteractng system: n λ (r) = n(r), (42) whle the potental V λ wll depend on λ. At λ = 0 the potental s nothng but the KS potental: and the energy functonal at λ = 0 has the smple form: E 0 = T s [n(r)] + n(r)v KS (r)dr. (43) The followng step s to wrte the energy functonal for the true nteractng system as an ntegral of the dervatve wth respect to λ: 1 E 1 = E de λ dλ. (44) dλ The dervatve can be smply expressed usng the Hellmann-Feynman theorem: de λ dλ = Ψ λ H λ Ψ λ (45) (see secton on Hellmann-Feynman forces for the demonstraton). Explctely: de λ dλ = Ψ λ U Ψ λ + Ψ λ V λ λ Ψ λ. (46) 9

10 By performng the ntegraton, one fnally fnds E xc = 1 fxc (r, r ) 2 r r n(r)drdr (47) where f xc (r, r ) s the exchange-correlaton hole: the charge mssng around a pont r due to exchange effects (Paul antsymmetry) and to Coulomb repulson. The exchange-correlaton hole obeys the sum rule f xc (r, r )dr = 1. (48) The exchange-correlaton hole s related to the par correlaton functon g(r, r ), gvng the probablty to fnd an electron n r f there s already one n r. Its exact defnton s: f xc (r, r ) = n(r ) 1 0 (g λ (r, r ) 1) dλ (49) where g λ (r, r ) s the par correlaton functon the system havng the electron-electron nteracton multpled by λ, Eq.(41). In homogeneous systems f xc (r, r ) and g(r, r ) are well known and studed functons. It has been shown that n nhomogeneous systems LDA does not gve a good approxmaton for f xc (r, r ). However LDA yelds a very good approxmaton for ts sphercal part f xc (r, s): f xc (r, s) = f xc (r, r + sˆr) dˆr 4π. (50) It s easly shown the Eq.47 depends only on the sphercal part of the exchange-correlaton hole: E xc = 1 fxc (r, s) n(r)drds. (51) 2 s Ths explans at least partally the good performances of LDA. The above procedure s a good startng pont n the search for better functonal, va better modelng of the exchange-correlaton hole. 1.9 The exact exchange-correlaton potental from many-body theory Many-body perturbaton theory yelds the followng exact soluton for the many-body problem: ) ( h2 2m 2 + V (r) + V H (r) ɛ ψ (r) + Σ(r, r, ɛ )ψ (r )dr = 0 (52) where the self-energy Σ(r, r, ɛ) s a complex, nonlocal, energy-dependent operator, the ψ (r) and ɛ have the physcal meanng of quaspartcle states and energes. The energes ɛ are also complex and ther magnary part s related to the lfetme of the state. Both DFT and many-body perturbaton theory are exact on the ground state (and the latter also on excted states). Ths mples n(r) = ImG DFT (r, r, ɛ)dɛ = ImG(r, r, ɛ)dɛ (53) where G(r, r, ɛ) s the Green s functon of the system, G DFT (r, r, ɛ) s the same n DFT, and the ntegraton extends to the energes of occuped states. The Dyson equaton must also apply between G and G DFT : G = G DFT + G DFT (Σ V xc ) G. (54) By combnng the above equatons, one fnally gets the followng result: Im [G DFT (Σ V xc ) G] r=r = 0. (55) Ths equaton can be used to deduce the exact exchange-correlaton potental. Practcal manybody perturbaton theory calculatons are very dffcult but not mpossble. Some test calculatons on smple systems have shown that the LDA V xc s a good approxmaton to the true V xc. 10

11 2 Practcal DFT calculatons 2.1 Atoms Atomc DFT calculatons are usually performed assumng a sphercally averaged charge densty. For closed-shell atoms, such procedure does not ntroduce any approxmaton, whle for open-shell atoms, t ntroduces an error that turns out to be qute small (t can be accounted for usng perturbaton theory f a hgher accuracy s desred). Under such assumpton, an atom can be descrbed as n elementary Quantum Mechancs by an electronc confguraton 1s 2 2s 2 2p 6...: the KS equaton has sphercal symmetry and s separable nto a radal equaton and an angular part (whose solutons are the sphercal harmoncs). The soluton of the KS equatons for an atom proceeds as follows. For a gven electronc confguraton, and startng from some ntal guess of the KS potental, the radal KS equatons are solved for those radal orbtals that correspond to occuped states; the (sphercally averaged) charge densty s recalculated; a new KS potental s calculated from the charge densty, and the procedure s terated untl self-consstency s reached. The mnmum energy s obtaned for the ground state electronc confguraton, that s well known for all atoms. The soluton of the radal KS equaton (step 1 above) s typcally done by numercal ntegraton on a grd, usng any of the many well-known technques that have been developed for one-dmensonal dfferental equatons. The teraton to self-consstency (step 3) s done usng the methods explaned n Sec. Iteraton to self-consstency. One may wonder why we fx the electronc confguraton nstead of fllng the one-electron state startng from the lowest energes and up. For many atoms there s no dfference between the two approaches. Atoms wth ncomplete d and f states however present a problem. The ncomplete d and f shells may have KS energes that are lower than those of outer s and p states; f however we try to move one more electron from s and p states nto the d or f shell, the KS level s pushed up by strong Coulomb repulson between hghly localzed electrons. Ths s a manfestaton of strong correlaton that s responsble for a wealth of nterestng phenomena (such as magnetsm). Currently avalable functonals are unable to reproduce ths behavor and may produce an ncorrect occupancy of state f ths s assgned n the one-electron way. Fxng the electronc confguraton solves the problem (unfortunately only n atoms) by mposng the correct occupancy of the hghly localzed (correlated) d and f states. 2.2 Molecules In molecules, KS equatons are usually solved by expandng KS orbtals nto some sutable bass set. Methods of solutons based on the dscretzaton of the problem on a 3-d grd have also been proposed, though. Localzed bass sets (atomc-lke wavefunctons centered on atoms) are often used, especally n Quantum Chemstry. The most common bass sets are Lnear Combnatons of Atomc Orbtals (LCAO), Gaussan-type Orbtals (GTO), Slater-type Orbtals (STO). These atomc-lke functons are talored for fast convergence, so that only a few (some tens at most) functons per atom are needed. An mpressve body of technque has been developed durng the years on the use of localzed bass sets. Localzed orbtals are qute delcate to use. One problem s the dffcult to check systematcally for convergence. Another problem s the dffculty of calculatng the Hellmann-Feynman forces actng on atoms, due to the presence of Pulay forces (see later). In the followng we wll concentrate on the opposte approach, that s, choosng extended, atomc-ndependent Plane Waves (PW) as bass set. 2.3 Extended systems: unt cells and supercells The atomc arrangement n perfect crystals s descrbed by a perodcally repeated unt cell. For many nterestng physcal systems, however, perfect perodcty s absent, but the system s ether 11

12 approxmately perodc or perodc n one or two drectons or perodc except for a small part. Examples of such systems nclude surfaces, pont defects n crystals, substtutonal alloys, heterostructures ( superlattces and quantum wells). In all such cases t s convenent to smulate the system wth a perodcally repeated fcttous supercell. The form and the sze of the supercell depend on the physcal system beng studed. The study of pont defects requres that a defect does not nteract wth ts perodc replca n order to accurately smulate a truly solated defect. For dsordered solds, the supercell must be large enough to guarantee a sgnfcant samplng of the confguraton space. For surfaces, one uses a crystal slab alternated wth a slab of empty space, both large enough to ensure that the bulk behavor s recovered nsde the crystal slab and that the surface behavor s unaffected by the presence of the perodc replca of the crystal slab. In the examples mentoned above, the supercell approach s usually more convenent than the cluster approach, that s, smulatng an extended system by takng a fnte pece of materal (the more tradtonal approach n Quantum Chemstry). The reason s the absence of an abrupt termnaton n the supercell approach. Even fnte systems (molecules, clusters) can be studed usng supercells. Enough empty space between the perodc replcas of the fnte system must be left so that the nteractons between them are weak. The use of supercells for the smulaton of molecular or completely aperodc systems (lquds, amorphous systems) has become qute common n recent years, n connecton wth frstprncples smulatons (especally molecular dynamcs smulatons) usng a PW bass set. In fact there are mportant computatonal advantages n the use of PW s that may offset the dsadvantage of nventng a perodcty where there s none. The sze of the unt cell the number of atoms and the volume s very mportant. Together wth the type of atoms t determnes the dffculty of the calculaton: large unt cells mean large calculatons. Unfortunately many nterestng physcal systems are descrbed exactly or approxmately by large unt cells. 2.4 Plane wave bass set In the followng we wll assume that our system s a crystal wth lattce vectors R and recprocal lattce vectors G. It s not relevant whether the cell s a real unt cell of a real perodc crystal or f t s a supercell descrbng an aperodc system. The KS wavefunctons are classfed by a band ndex and a Bloch vector k n the Brlloun Zone (BZ). A PW bass set s defned as r k + G = 1 V e(k+g) r, h 2 2m k + G 2 E cut, (56) where V s the crystal volume, E cut s a cutoff on the knetc energy of PW s (from now on, smply the cutoff ). PW s have many attractve features: they are smple to use (matrx elements of the Hamltonan have a very smple form), orthonormal by constructon, unbased (there s no freedom n choosng PW s: the bass s fxed by the crystal structure and by the cutoff) and t s very smple to check for convergence (by ncreasng the cutoff). Unfortunately the extended character of PW s makes t very dffcult to accurately reproduce localzed functons such as the charge densty around a nucleus or even worse, the orthogonalzaton wggles of nner (core) states. In order to descrbe features whch vary on a length scale δ, one needs Fourer components up to q 2π/δ. In a sold, ths means 4π(2π/δ) 3 /3Ω PW s (where Ω s the dmenson of the BZ). A smple estmate for damond s nstructve. The 1s wavefuncton of the carbon atom has ts maxmum around 0.3 a.u., so δ 0.1 a.u. s a reasonable value. Damond has an fcc lattce (Ω = (2π) 3 /(a 3 0/4)) wth lattce parameter a 0 = 6.74 a.u., thus yeldng 250, 000 PW s. Ths s clearly too much for practcal use. 2.5 Pseudopotentals Core states prevent the use of PW s. However they do not contrbute n a sgnfcant manner to chemcal bondng and to sold-state propertes. Only outer (valence) electrons do, whle core electron are frozen n ther atomc state. Ths suggests that one can safely gnore changes n core states (frozen core approxmaton). However the soundness of ths approach was challenged by a 1976 paper by Janak, showng that large varatons n the energy of core states can be nduced by 12

13 changes n the chemcal envronment. The controversy was solved n 1980 by Von Barth and Gelatt. Ther argument s brefly sketched here. Let us ntroduce the notatons n c and n v for the true selfconsstent core and valence charge; n 0 c and n v for the frozen-core charge and the correspondng valence charge. A frozen-core functonal E[n c, n v ] s ntroduced. The frozen-core error s δ = E[n 0 c, n v] E[n c, n v ]. (57) By expandng around n c and n v one fnds δe δ (n 0 c n c )dr + δn c δe δn v (n v n c )dr + 2nd order terms. (58) The mportant pont s that the followng statonary condtons hold: δe δn c = µ c, δe δn v = µ v (59) where µ c and µ v are constants, so that the frst-order terms n the error vansh. The dea of replacng the full atom wth a much smpler pseudoatom wth valence electrons only arses naturally (apparently n a 1934 paper by Ferm for the frst tme). Pseudopotentals (PP s) have been wdely used n sold state physcs startng from the 1960 s. In earler approaches PP s were devsed to reproduce some known expermental sold-state or atomc propertes such as energy gaps or onzaton potentals. Other types of PP s were obtaned from band structure calculatons wth the OPW (orthogonalzed PW) bass set, by separatng the smooth (PW) part from the orthogonalzaton part n the wavefunctons. Modern PP s are called norm-conservng. These are atomc potentals whch are devsed so as to mmc the scatterng propertes of the true atom. For a gven reference atomc confguraton, a norm-conservng PP must fulfll the followng condton: 1) all-electron and pseudo-wavefunctons must have the same energy, and 2) they must be the same beyond a gven core radus r c, whch s usually located around the outermost maxmum of the atomc wavefuncton; 3) the pseudo-charge and the true charge contaned n the regon r < r c must be the same. Ths last condton explans the name norm-conservng. There s an hstorcal reason for ths: some earler PP s volated condton 3 (ths was known as the orthogonalty hole problem). Note that the defnton all-electron, here and n the followng, refers to a KS calculaton that ncludes core electrons, not to a many-electron wavefunctons. Norm-conservng PP are relatvely smooth functons, whose long-range tal goes lke Z v e 2 /r where Z v s the number of valence electrons. They are nonlocal because t s usually mpossble to mmc the effect of orthogonalzaton to core states on dfferent angular momenta l wth a sngle functon. There s a PP for every l: V ps = V loc (r) + l V l (r) P l = V loc (r) + lm Y lm (r)v l (r)δ(r r )Y lm(r ), (60) where V loc (r) Z v e 2 /r for large r and P l = l l s the projecton operator on states of angular momentum l. They are however seldom used n ths form. For computatonal reasons, they are recast nto a separable form (see appendx). The nonlocalty of PP s ntroduces some addtonal but lmted complcatons n the calculaton. In partcular, one has to do the followng generalzaton: V (r)n(r)dr ψ V ψ = ψ (r)v (r, r )ψ (r )drdr. (61) Experence has shown that PP s are practcally equvalent to the frozen core approxmaton: PP and all-electron calculatons on the same systems yeld almost ndstngushable results (except for those cases n whch core states are not suffcently frozen). It should be remarked that the use of PP s s not lmted to PW bass sets: PP s can be used n conjuncton wth localzed bass sets as well. 13

14 2.6 Another way of lookng at pseudopotentals Norm-conservng PP s are stll hard that s, they contan a sgnfcant amount of Fourer components wth large q for a number of atoms, such as N, O, F, and the frst row of transton metals. For these atoms lttle s ganed n the pseudzaton, because there are no orthonormalty wggles that can be removed n the 2p and 3d states, respectvely. More complex Ultrasoft PP s have been devsed that are much softer than ordnary norm-conservng PP s, at the prce of a consderable addtonal complexty. The heavy formalsm of ultrasoft PP s tends to hde the underlyng logc (and physcs). An alternatve approach, called Projector Augmented Waves (PAW), s much more transparent. Moreover PAW ncludes as specal cases a number of other methods and provdes a smple and consstent way to reconstruct all-electron wavefunctons from pseudo-wavefunctons. These are needed for relable calculaton of a number of observables, such as NMR chemcal shfts and hyperfne couplng coeffcents. The dea of PAW s to fnd a mappng between the complete wavefuncton and the pseudowavefuncton va a sutable lnear operator. The pseudo-wavefuncton must be a smooth object that can be expanded nto PW s. Let us consder for smplcty the case of a sngle atom n the system. In a regon R centered around the atom, the mappng s defned as φ l = (1 + T ) φ l (62) where the functons φ l are solutons, regular at the orgn but not necessarly bound, of the allelectron atomc KS equaton; the functons φ l are correspondng pseudo-functons, that are much smoother n the regon R and jon smoothly to the φ l at the border of regon R. Outsde the regon R, we set T = 0. In the regon R, we assume that we may wrte a pseudo-wavefuncton ψ for our molecular or sold-state system as a sum over the atomc pseudo-waves φ l : ψ = l c l φ l (63) By applyng the operator (1 + T ) to both sdes of the above expanson we fnd ψ = l c l φ l (64) where ψ s the all-electron wavefuncton. The above result can be recast nto the form ψ = ψ + c l ( φ ) l φ l. (65) It remans to defne the c l coeffcents. Let us ntroduce the projectors β l wth the followng propertes: β l φ m = δ lm, φ l β l = I. (66) It s easy to verfy that c l = β l ψ and that we can wrte ψ = ψ + ( β l φ φ ) l φ l (67) l [ = I + ] ( φ ) l φ l β l ψ. (68) l l The quantty between square brackets s our 1 + T operator. Ths replaces the pseudo-states φ from the pseudo-wavefunctons around the atoms and replaces them wth the all-electron states φ. The 1 + T operator s a purely atomc quantty that s obtaned from a judcous choce of the φ l all-electron atomc states, the correspondng pseudo-states φ l, and the projectors β l. The equatons to solve n the PAW method are then obtaned by nsertng the above form for ψ n the energy functonal and by fndng ts mnmum wth respect to the varaton of the smooth part 14

15 only, ψ. Rather cumbersome expressons results. An mportant feature of the resultng equatons s that the charge densty s no longer gven smply by the square of the orbtals, but t contans n general an addtonal (augmentaton) term: n(r) = ψ (r) 2 + ψ β l q lm (r) β m ψ (69) where lm q lm (r) = φ l (r) φ m (r) φ l (r)φ m (r) (70) (usng the completeness relaton, Eq.(66)). Conversely the pseudo-wavefunctons are no longer orthonormal, but obey nstead a generalzed orthonormalty relaton: ψ S ψ j = δ j, S = I + β l Q lm β m Q lm = q lm (r)dr. (71) lm R Ultrasoft PP s can be derved from PAW assumng a pseudzed form for q lm (r). Norm-conservng PP s n the separable form can be derved from PAW f the atomc states φ l and φ l obey the norm-conservaton rule (thus S = 1). The LAPW method can also be recast under ths form. The careful reader wll also remark some smlarty between the PAW approach and the venerable PP s based on the OPW method (those wth the nfamous orthogonalty hole : PAW plugs the hole by defnng the charge densty n the correct way). 2.7 Brlloun-Zone samplng In order to calculate the charge densty n(r) n a perodc system one has to sum over an nfnte number of k-ponts: n(r) = ψ k, (r) 2 (72) k where the ndex runs over occuped bands. Assumng perodc (Born-Von Kàrmàn) boundary condtons ψ(r + L 1 R 1 ) = ψ(r + L 2 R 2 ) = ψ(r + L 3 R 3 ) = ψ(r), (73) a crystal has L = L 1 L 2 L 3 allowed k-ponts (L s also the number of unt cells). In the thermodynamc lmt of an nfnte crystal, L, the dscrete sum over k becomes an ntegral over the BZ. Experence shows that ths ntegral can be approxmated by a dscrete sum over an affordable number of k-ponts, at least n nsulators and semconductors. When present, symmetry can be used to further reduce the number of calculatons to be performed. Only one k-pont s left to represent each star the set of k-ponts that are equvalent by symmetry wth a weght w that s proportonal to the number of k-ponts n the star. The nfnte sum over the BZ s replaced by a dscrete sum over a set of ponts {k } and weghts w : 1 L f k (r) k w f k (r). (74) The resultng sum s then symmetrzed to get the charge densty. Sutable sets for BZ samplng n nsulators and semconductors are called specal ponts. Ths name s somewhat msleadng: n most cases those sets just form unform grds n the BZ. In metals thngs are more dffcult because one needs an accurate samplng of the Ferm surface. A sutable extenson of DFT to fractonary occupaton numbers s needed. The Gaussan broadenng and the tetrahedron technques, or varatons of the above, are generally used. In supercells, the k-pont grd s often lmted to the Γ pont (k = 0). A better samplng may be needed only f t s mportant to accurately descrbe the band structure of a subjacent crystal structure. Ths s the case of pont defects n solds and of surfaces. If, on the contrary, supercells are used to smulate completely aperodc or fnte systems, the Γ pont s the good choce: a better k-pont grd would better account for the perodcty of the system, but ths s fcttous anyway. 15

16 3 Fndng the electronc ground state There are two possble ways to fnd the electronc ground state, for fxed atomc postons. The frst s to solve self-consstently the KS equatons, by dagonalzng the Hamltonan matrx and teratng on the charge densty (or the potental) untl self-consstency s acheved. The second s to drectly mnmze the energy functonal as a functon of the coeffcents of KS orbtals n the PW (or other) bass set, under the constrant of orthonormalty for KS orbtals. The basc ngredents are n both cases the same. 3.1 Iteraton to self-consstency In the followng I wll consder the charge densty as the quantty to be determned self-consstently, but smlar consderatons apply to the self-consstent potental V KS as well. We supply an nput charge densty n n (r) to the KS equatons and we get an output charge densty n out (r). Ths defnes a functonal A: At self-consstency, n out (r) = A[n n (r)]. (75) n(r) = A[n(r)]. (76) The frst algorthm that comes to the mnd s to smply use n out (r) as the new nput charge densty: n (+1) n = n () out, (77) where the superscrpts ndcate the teraton number. Unfortunately there s no guarantee that ths wll work, and experence shows that t usually does not. The reason s that the algorthm wll work only f the error on output s smaller than the error on nput. If you have an error δn n (r) on nput, the error on output, close to self-consstency, wll be δn out (r) δa δn(r) δn n(r)dr Jδn n (78) whch may or may not be smaller than the nput error: t depends on the sze of the largest egenvalue, e J, of the operator J, whch s related to the delectrc response of the system. Usually, e J > 1 and the teraton does not converge. A smple algorthm that generally works, although sometmes slowly, s the smple mxng. A new nput charge densty s generated by mxng the nput and output charges: n (+1) n = (1 α)n () n + αn() out (79) The value of α must be chosen emprcally n order to get fast convergence. The error wth respect to self-consstency becomes δn out = [(1 α) + αj] δn n (80) and t s easly seen that the teraton converges f α < 1/e J. In general, the convergence s easer for small cells and symmetrc systems, more dffcult for larger cells, low symmetry, cells elongated along one drectons, surfaces. Relatvely bg values (α = ) can be chosen n easy systems, smaller values are approprate for cases of dffcult convergence. Better results are obtaned wth more sophstcated algorthms (to name a few: Anderson, Broyden, Drect Iteraton n Inverse Space, DIIS) that use nformatons collected from several precedng teratons. Let us sketch the logc of such algorthms. We have a sequence of n () n producng n () out from precedng teratons. We look for the lnear combnaton of nput n new n : n new n = c l n (l) n, c l = 1 (81) l l that mnmses an approprate norm n new n n new n nnew out. Close to self-consstency, n new out l c l (n (l) n n(l) out) (82) and the coeffcents c l are determned by mposng that such norm s mnmum. Then we mx n new n wth n new out = l c ln (l) out (usng smple mxng or whatever algorthm s approprate). 16

17 3.2 Dagonalzaton of the Hamltonan When the wavefunctons are expanded on a fnte bass set the KS equatons take the form of a secular equaton: G H(k + G, k + G )ψ k, (G ) = ɛ k, ψ k, (G), (83) where the matrx elements of the Hamltonan have the form H(k + G, k + G ) = h2 2m (k + G)2 δ G,G + V scf (G G ) + V loc (G G ) + V NL (k + G, k + G ). (84) The term V scf (G G ) s the Fourer transform of the the screenng potental: V scf (G G ) = 1 V scf (r)e (G G )r dr. (85) V (V s the volume of the crystal: the ntegraton extends over the entre crystal) and the same apples to V loc that comes from the local term n the PP s. The nonlocal contrbuton V NL comes from the nonlocal part of the PP s: V NL (k + G, k + G ) = 1 V NL (r, r )e (k+g)r e (k+g )r drdr. (86) V The problem s reduced n ths way to the well-known problem of fndng the lowest egenvalues and egenvectors (only the valence states for nsulators, a few more for metals) of an N pw N pw Hermtan matrx (where N pw s the number of PW s). Ths task can be performed wth well-known bsecton-trdagonalzaton algorthms, for whch very good publc-doman computer packages (for nstance, LAPACK) exst. Unfortunately ths straghtforward procedure has serous lmtatons. In fact: ) the computer tme requred to dagonalze a N pw N pw matrx grows as N 3 pw; ) the matrx must be stored n memory, requrng O(N 2 pw) memory. As a consequence a calculaton requrng more than a few hundred PW s becomes exceedngly tme- and memory-consumng. As the number of PW s ncreases wth the sze of the unt cell t s very hard to study systems contanng more than a few (say 5-10) atoms. Both lmtatons can be pushed much further usng teratve technques (see Appendx). 3.3 Drect mnmzaton It s not necessary to go through KS equatons and self-consstency to fnd the electronc ground state. The energy functonal can be wrtten as a functon of the coeffcents n the bass set of the KS orbtals and drectly mnmzed, under the usual orthonormalty constrants. One has to fnd the mnmum of E (ψ k, (G)) = E(ψ k, (G)) ( ) λ j ψk,(g)ψ k,j (G) δ j, (87) j G wth respect to the varables ψ k, (G) and the Lagrange multplers λ j. The problem s made much smpler by the knowledge of the gradents of the functon to be mnmzed. In fact, rememberng Eq.22, one easly fnds E ψ k, (G) = H(G, G )ψ k,(g ) j λ j ψ k,j(g). (88) Note that, as n teratve dagonalzaton, the basc ngredents are Hψ products. Note also that the Hamltonan depends on the varables ψ k, (G) through V scf and the charge densty. The problem of mnmzng a functon of many varables whose gradents are known, wth the addtonal complcaton due to the presence of constrants, can be solved usng approprate extensons to textbook algorthms, or specalzed algorthms, such as steepest descent (bad) or conjugate gradent (better) or DIIS (even better). 17

18 4 Movng atoms - complex materals Untl now we have assumed that the atomc postons were known and fxed. Ths s the case for smple crystals (slcon for nstance), but n more complex crystals (for nstance, SO 2 ) the equlbrum postons are not fxed by symmetry. In even more complex materals we smply don t know the equlbrum atomc postons and would lke to calculate them. In the followng we assume that ons are classcal objects. At zero temperature the equlbrum atomc postons R, = 1,..., N (N = number of atoms n the unt cell) are determned by the mnmum of the total energy E tot of the system, that s, the sum of the electronc (DFT) energy E and of the on-on nteracton (electrostatc) energy E II. If we consder the electrons n ther ground state for any gven confguraton of R (collectvely ndcated by {R}), the total energy wll be a functon of the atomc postons: E tot ({R}) = E({R}) + E II ({R}). (89) The procedure to fnd the atomc confguraton yeldng the mnmum energy s usually called structural optmzaton or relaxaton. For an nfnte system we must dstngush between atomc dsplacements that change the form and volume of the unt cell (related to elastc modes) and atomc dsplacements nternal to the unt cell (related to phonon modes). Such dstncton does not exst for a fnte system. The optmzaton of the lattce and that of atomc postons have to be done separately, or n any case, usng dfferent procedures (Unless we use varable-cell molecular dynamcs, a very powerful but very complex technque). 4.1 Optmzaton of lattce parameters The determnaton of the equlbrum lattce parameters and of the relatve stablty of dfferent structures for smple semconductors was one of the frst remarkable applcatons of the LDA PW- PP approach (around 1980). The total energy s calculated as a functon of the volume V of the unt cell for varous dfferent canddate structures. The lowest-energy structure wll be the equlbrum structure at zero temperature and at zero pressure. The E(V ) curve can n prncple be drectly calculated. However t s much more convenent to ft an equaton of state to a few calculated ponts. Emprcal equatons of state dependng on a few parameters and coverng a wde range of volumes around the equlbrum are well known and wdely used n geology and geophyscs. The most famous s possbly the Murnaghan equaton of state: [ P (V ) = B (V0 ) B ] B 1 (90) V where the ft parameters are the equlbrum volume V 0, the bulk modulus B: B = V P V = V 2 E V 2 (91) and ts dervatve wth respect to the pressure, B = db/dp, an admensonal quantty rangng from 3 to 10 for almost all solds. The Murnaghan E(V ) s obtaned by ntegratng the former expresson. All these quanttes are drectly comparable to expermental results (at zero temperature). The reason for ths ft procedure s that the straghtforward calculaton of E(V ) suffers from mportant errors. In partcular, when usng PW s wth a gven energy cutoff, the number of PW s depends on V. As most calculatons are done far from convergence, ths wll cause large oscllatons n the calculated E(V ) (ths s remnscent of the Pulay force problem). Experence show that the ft to an equaton of state effectvely smoothes the oscllatons and yelds very good results even f the cutoff of PW s s low. The statement most calculatons are done far from convergence s not as alarmng as t may seem: n fact the slow convergence s due to the regon of charge close to the atomc cores. Ths s an essentally atomc-lke charge that changes lttle from one structure to another. If we are nterested n comparng dfferent structures of the same materals, the relatve energy dfferences wll converge wth the cutoff well before the absolute energy values. Of course, one has to check carefully the relatve convergence wth respect to the BZ samplng as well. 18