5. Response properties in ab initio schemes

Transcription

1 5. Response propertes n ab nto schemes A number of mportant physcal observables s expressed va dervatves of total energy (or free energy) E. Examples are: E R 2 E R a R b forces on the nucle; crtcal ponts on the potental energy surface (mnma and saddle ponts) force constants; vbratonal frequences; nfrared and Raman spectra 3 E R a R b R c E E 2 E R a E anharmonc constrbutons to vbratonal frequences (E electrc feld): dpole moment nfrared ntenstes 2 E E E j polarzablty; lght scatterng 3 E E E j E k hyperpolarzablty; second harmonc generaton 3 E R a E E k Raman ntensty etc. If total energy s avalable n a calculaton wth {R} or {E} as parameters, the above propertes can be obtaned by three methods: 1. A ft of sample ponts to an analytcal form and subsequent dfferentaton. Ths approach s very often used, but the results are subject to numercal nstablty (much more data ponts are needed than there are parameters n the fttng functon, for a numercally stable ft). 2. Fnte dfferences. Ths method s good f calculaton algorthm s accurate, so that numercal nose can be neglected. The followng formulae are for the frst dervatve of 79

2 a numercall functon f(x 1,...,x N ), defned on a mesh wth a step h;: ( ) f = 1 [ f(...,x 0 x 2h +h,...) [f(...,x 0 h,...)] + 1 d 3 f 6 3 h2...and for the second dervatves, wth obvous notaton of f(+) f(...,x 0 +h,...) etc. : 2 f x 2 = 1 h 2 [f(+) + f( ) 2f(0)] f h 2 x 4 2 f x x j = 1 2h h j [f(++) + f( ) + 2f(00) f(+0) f( 0) f(0+) f(0 )] + O(f IV h 2 ). 3. Analytcal dfferentaton, bult n nto the code. Ths typcally means a consderable programmng load and an addtonal amount of calculaton. However, the dfferentaton s exact, and there exst a possblty to calculate many response propertes (all components of force constants etc.) n a sngle run. Also ths may be advantageous n reducng a number of dfferent calculatons to be done. For example, calculaton of a second-order dynamcal matrx of a system wth N atoms demands at least 9N(N 1)/2 + 1 calculatons for dfferent dsplacement patterns f only the total energes are avalable, but [9N(N 1) + 2]/[2(3N 2)],.e. much less, f the forces are avalable as well. 5.1 General consderatons In the followng, we consder the problem of dfferentatng an expresson whch s varatonal (lke, e.g., energy) and depends mplctly and explctly on some parameters. Examples of explct parameters are, e.g., postons of nucle (and possbly of bass functons pnned to them); the mplct dependence of the total energy s on varatonal parameters those descrbng the decomposton of orbtals over bass functons. Moreover, we ntroduce a set of constarnts that could be target pressure, or fxed spn moment. So we deal wth constraned mnmzaton, usng Lagrange multplers. The functon to be mnmzed has a form W = W (C, λ,r) m f m (C,R)λ m, (5.1) where W s e.g. the total energy expresson, C varatonal parameters, λ Lagrange multplers, R explct parameters (e.g., postons of nucle), f constrants, formulated as f m (C,R) = 0 (5.2) (lke orthonormalty of orbtals, fxaton of total magnetc moment etc.) In less general case, one can omt the constrants. Varatonal parameters wll be determned from the equatons 80

3 In the followng, we use the notaton: The optmzed energy wll be W C = 0, f m (C,R) = 0, W C W ; m E = W ( C(R), λ(r),r ), (5.3) W R a W a (5.4) and ts gradent E a = W a + W C a f m λ a m = W a, (5.5) m where W a can be obtaned by drect dfferentatng n R a, and W =0, f m =0 at equlbrum. Smlarly, the second dervatve yelds: E ab = W ab + W a Cb m f a m λb m, (5.6) and other terms, W C ab and f m λ ab m, dsappear for the same reason. Snce W s varatonal, t must be statonary wth respect to varatons of C: W + j W j C j m f m λ m = 0. (5.7) Ths s somethng lke Hartree-Fock, or Kohn Sham equaton equaton, wrtten down for fxed postons of nucle. Dfferentatng n R a yelds: dw dr a = W a + j and snce f m = 0 n a statonary state, W j C a j m f m λ a m = 0, (5.8) df m dr a = f a + j f mj C a j = 0. (5.9) So we arrve at a system of response equatons W j Cj a f m λ a m = W a, j m f mj Cj a = fm a, (5.10) j whch has to be solved n addton to zero-order equaton (5.7). Once ths s done, W a and f a m can be nserted n Eq. (5.6) for E ab. A smlar analyss, but ncludng more coupled terms, can be done for thrd dervatves. Ths approach s descrbed by Pulay Peter Pulay, Analytcal Dervatve Technques and the Calculaton of Vbratonal Spectra, n: Modern Electronc Structure Theory, Part II, ed. by Davd R. Yarkony, World Scentfc, Sngapore (1995), pp

4 5.2 Forces Let us dscuss n more detal the evaluaton of force, that must be, accordng to Eq. (5.5), E a = W a. In practcal calculatons, evaluaton of ths dervatve may be consderably complcated by the necessty to take nto account dervatves of the bass functons. Ths s necessary f e.g. bass functons are centered on atoms and are dsplaced wth them. Ths s not necessary f a poston-ndependent bass set (e.g., planewaves) s used. The reason s the followng. For the exact wavefuncton, E a = ψ H a ψ by the power of the Hellmann Feynman theorem, as was shown n Eq. (3.19). We reproduce here ths argumentaton for completeness: E a d dr a ψ H ψ = ψa H ψ + ψ H a ψ + ψ H ψ a = }{{}}{{} E ψ E ψ = ψ H a ψ + E [ ψ a ψ + ψ ψ a ] = = ψ H a ψ + E [ ] a ψ ψ = ψ H a ψ. (5.11) }{{} =const However, f the wavefuncton contans parameters p t dependent on perturbaton (.e., dsplacement of ons) ether mplctly or explctly, then (5.11) s not generally vald, because ψ a H ψ + ψ H ψ a = 2 Re ψ a H ψ = ψ H ψ p a t 0. (5.12) t p t The valdty of the Hellmann Feynman theorem can be restored f ether p a t = 0,.e. the bass s ndependent on postons of nucle, as s for nstance the case for plane waves, or ψ H ψ p t = 0,.e. all parameters are fully optmzed, that s normally the case f the bass s complete. Or, at least, for each bass functon ts dervatve wth respect to perturbaton must also be present n the bass. In practcal terms, Hellmann Feynman forces are seldom useful, because even bass sets extended to nclude ther gradents are not complete enough. The total energy n the densty functonal theory s gven by Eq. (3.17): Etot el. = ( occuped) ε e2 2 ρ(r)ρ(r ) r r dr dr V XC (r) ρ(r) dr + E XC [ρ]. When consderng a dynamcal problem, one must add here a contrbuton from atomc cores that s not a constant anymore: E tot [ρ] = Etot el. + e2 Z α Z β 2 αβ R α R β. (5.13) 82

5 Gven a fxed confguraton of atoms {R α }, the dsplacement of one atom R α R α + α changes the total energy by δe = δ ε + δ ɛ (core) β dr ρ(r) δv KS (r) F HF α α. (5.14) ( occuped) β(core) The Hellmann Feynman force s purely electrostatc, due to the dsplacement of densty: F HF d α = Z α ez β d α β α τ R α + α R β +τ + e ρ(r) r R α α dr, (5.15) τ are lattce translaton vectors. Other contrbutons addng up to the total force are: 36 F α d E tot = F HF α + F IBS α + F core α. (5.16) d α F IBS s ncomplete bass set, or Pulay force, due to the use of fnte number of postondependent bass functons: F IBS α = 2 [ δϕ Re δ ˆT ] ϕ + V eff ε 1 ϕ δ α δ ˆT ϕ. (5.17) α (occuped) (occuped) F core s a core correcton, due to the fact that for core electrons (at least n the FLAPW method) only sphercal part of the potental s taken nto account. = ρ α (r) V KS (r). (5.18) F core α To gve an dea of relatve mportance of dfferent contrbutons to the force, the values (n mry/a.u., for α =0.089 a.u.) as calculated by the FLAPW method for S are shown on the rght. (after C. Ambrosch-Draxl, lecture at the Workshop The Physcs of the Electronc Behavour n the Core Regon: All-Electron LAPW Electronc Structure Calculatons, , Treste. ) F numercal dff. = F by Eq. (5.16) = F HF = F IBS = 3.06 F core = Dynamcal matrx We dscuss now the calculaton of second dervatves of energy n on dsplacements, accordng to Eq. (5.6). For ths, we need to know C b,.e., Kohn Sham orbtals to 1st order n dsplacements. The Hellmann Feynman theorem holds up to 3d order n perturbatons: 2 E [ ] ρ R V R = + ρ 2 V R dr. (5.19) R a R b R a R b R a R b 36 B. Kohler, S. Wlke, M. Scheffler, R. Kouba and C. Ambrosch-Draxl, Force calculaton and atomcstructure optmzaton for the full-potental augmented plane-wave code WIEN, Comp. Phys. Commun. 94, 31 (1996). 83

6 The set of Kohn Sham equatons (3.16) ] [ h2 2m 2 + V SCF (r) ϕ (r) = ε ϕ (r) ; V SCF (r) = e 2 α ρ(r) = ϕ (r) 2. (occuped) Z α r R α + e2 ρ(r ) r r dr + δe XC δρ(r) ; (5.20) can be lnearzed, ntroducng small parameter λ: ϕ (r) = ϕ (0) (r) + λ ϕ (1) (r) ; V SCF (r) = V (0) (1) SCF (r) + λ V SCF (r) ; (5.21) ρ(r) = ρ (0) (r) + λ ρ (1) (r). The zero-order system for ϕ (0), V (0) SCF and ρ (0) s (5.20), and the frst-order system (where terms lnear n λ are retaned) can be easly wrtten once the nature of perturbaton (that must be proportonal to λ) s specfed. As we are nterested at present n dsplacements of ons λw α as an example of such perturbatons, the frst-order equatons aqure the form: ] [ h2 2m 2 + V SCF(r) (0) ε ϕ (1) (r) = V SCF(r) (1) ϕ (0) (r) ; V (1) SCF (r) = e2 α ρ (1) (r) = 2 Re (occuped) Z α w α (r R α ) ρ + e 2 (1) (r ) r R α 3 r r dr + ρ (1) (r) ϕ (0) (1) (r) ϕ (r). [ ] dvxc (r) dρ ρ (0) ; (5.22) The frst of ths equatons s Sternhemer equaton,.e. the Schrödnger equaton to the frst order. The perturbaton λw α could be constraned to just one atom, or t could be a phonon wth gven wavevector q and polarzaton A, w α = Ae qrα + A e qrα. The system of equatons (5.22) has to be solved self-consstently together wth (5.20). ρ (1) and V (1) SCF can be drectly used n Eq. (5.19), and the second term on the rght sde of Eq. (5.19) can be obtaned from analytcal dfferentaton of the second equaton n (5.22), smlarly to how the forces were obtaned from zero-order Kohn Sham results. 84