Introduction to Orbital-Free Density-Functional Theory. Ralf Gehrke FHI Berlin, February 8th 2005

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1 Intoduction to Obital-Fee Density-Functional heoy Ralf Gehke FHI Belin, Febuay 8th 005

2 Outline Basics of functional deivatives I Pinciples of Obital-fee Density-Functional heoy basics of Density-Functional heoy motivation fo OF-DF basics of the numeical implementation II Appoximations to the kinetic-enegy functional he homas-femi Appoximation he von-weizsäcke tem Linea-Response heoy Combination of F and vw he Conventional Gadient Expansion Aveage-Density Appoximation

3 Basics of functional deivatives function of multiple agument functional { y, L 0 y N } F[ y( x ] F, analogous { y i } numbe y( x numbe y i df F dyi y(x yi δ F δf δy ( x 0 δy ( x 0 x i dy i x δy( x 0 x 0 x df F y i i dy i δf δf F[ y + δy] F[ y] dx δy x δy ( x (

4 Basics of functional deivatives F [ y( x ] dx f ( y( x some ules δ F δ y [ y ] f ( x y δf δy Poof : δf [ y] F[ y + δy] F[ y] dx f ( y + δy f 44 f ( y + δy+ O( δy y [ y( x ] dx f ( y( x y ( x F, dx f ( y f dx δy { y [ y] f f f d f d f d f ( y y y dx ( ( ( y / x dx y / x dx y / x δf δy ( x ( x δf δy [ y ] f d f ( x y dx y

5 Basics of functional deivatives Poof : some ules F [ y + δ y] dx f ( y + δy, y + δy δf dx f [ y] ( x f y f d f dx δy x y dx y δf δy f y ( y + δy + δy F[ y] ( + f dx δy + y ( y( x 0 x ± f dx δy y 44 d f dx δy dx y

6 Basics of functional deivatives some ules chain ule [ G( [ y( x ] x ] F, ( x δg( [ y], x ([ y], x y( x δ F δf δg dx δy δ J [ ρ] ρ - ( ρ( d d some examples V H ([ ρ], [ ρ] ( δj d ρ - ( [ ρ] ( ( - δ J

7 Basics of DF Fist Hohenbeg-Kohn-heoem Enegy of goundstate is a mee function of the electon density ρ E ext [ ρ] F[ ρ] + d V (( ρ [ ρ ] [ ρ ] + E [ ρ ] [ ρ ] + J [ ρ ] E [ ρ ] F + ee S XC J [ ρ] S [ ] ρ [ ] ρ E XC ρ - ( ρ( d d classical coulomb inteaction kinetic enegy of a non-inteacting system with density ρ exchange-coelation enegy

8 Basics of DF Second Hohenbeg-Kohn-heoem (Vaiational Pinciple δ ( ( [ ] ( ( E ρ µ d ρ N 0 [ ] ( δe ρ µ Eule-Lagange-Equation µ chemical potential (negative of fist ionization enegy of system

9 Basics of DF What is the kinetic enegy ρ? ϕ ϕ δ ϕ ( ρ( i j ij q i 0 obitals ae solutions of single-paticle Schödinge-equations (Kohn-Sham-equations effective potential has to be detemined self-consistently S i + V [ ] intoduction of a efeence-system of non-inteacting paticles { } ϕ i KS eff ϕ i ε iϕi [ ρ] δe [ ρ] KS δj V eff ([ ρ] + XC, + + Vext VH + VXC Vext S i ϕ ϕ i i

10 Motivation fo OF-DF Intoduction of obitals in DF is not desiable matix diagonalization scales cubically with basis size many data have to be stoed duing computation the goal is to get id of KS-obitals and to expess each enegy contibution in tems of the electon chage density a functional [ ] ρ S has to be found S [ ] high accuacy fo ρ is equied since ρ is of the same ode as E[ ρ] (viial theoem E XC [ ρ] is much smalle than E[ ρ] many schemes fo appoximating E XC [ ρ] might be inappopiate fo ρ S [ ] S [ ]

11 Basics of numeical implementation Once an appopiate appoximation fo S ρ exists, how can the total enegy E ρ of the goundstate be calculated? [ ] [ ] two diffeent appoaches: diect minimization of the total enegy self-consistent calculation

12 Basics of numeical implementation diect minimization enegy has to be minimized unde the constained of constant paticle numbes Π E d ρ ρ µ ρ N ( [ ] [ ] ( vaiable substitution to ensue positivity of electon density ρ minimization of Π with steepest descend, conjugate-gadients, etc. [ ρ] ( δπ δϕ 0 ϕ ( ρ( ϕ n+ ( ϕ ( n [ ρ ] δπ τ δϕ ( ϕ ( n

13 Basics of numeical implementation self-consistent calculation add and subtact kinetic enegy of a bosonic system E [ ρ] [ ρ] + [ ρ] + J[ ρ] + E [ ρ] + d V ( ρ( [ ρ] B S XC B [ ρ] d ρ( ρ( inset total enegy expession in Eule-Lagange-equation ext B integation by pats and chain ule δb [ ] ρ ( L ϕ ϕ ( (

14 Basics of numeical implementation self-consistent calculation Eule-Lagange-equation is tansfeed to a Kohn-Sham-like equation with an additional potential tem δe [ ρ] ( µ ϕ ϕ ( δ δ S VH VXC Vext B ( ( ( µ Ĥϕ µϕ Hˆ + V H + V XC + V ext δ S δb + ( ( same solution methods can be used as in the case of the KS-scheme, but only with one obital no othogonalization

15 KS-scheme OF-scheme solve poisson-equation ( 4 ( VH πρ k solve poisson-equation ( 4 ( VH πρ k calculate exchange-coelation potential ([ ], VˆXC ρ k V ([ ρ ], Gˆ [ ρ ], ˆXC k calculate potentials δˆ ( ( ([ ] ˆ S δb, ( ([ ], k ρk ρk ceate hamiltonian ˆ H + Vˆ ˆ ˆ H + VXC + V ext ceate pseudo-hamiltonian ˆ H + Vˆ ˆ ˆ ˆ H + VXC + G + V ext Hˆ ϕ ε ϕ, i 0, L, q i i solve i solve Ĥϕ µϕ calculate new electon density ~ ρ ρ ( ~ ρ, ρ,, q k + ϕ i i 0 k + f k + 0 L ρk calculate new electon density ~ ρ ϕ ρ ( k + k + f ρk +, ρ0, L, ρk ~ self-consistency achieved? ρ ρ < ε k + k no k k + self-consistency k + ρk ε achieved? ρ < no k k + yes k : inteation index q : numbe of KS-obitals yes

16 he homas-femi appoximation conside a system, consisting of non-inteacting, fee electons H ˆ i ( ( exp k k C i i ϕ ρ ( intoducing peiodic bounday conditions ( C const. k k n π, i L x, y i i, i 8 k kx k y kz ( z π V k F / π ρ

17 he homas-femi appoximation kinetic enegy can be calculated exactly k k k F kf k F V F V k k d k k dk k 8π π k k 0 0 k 4 V π 5 kf 5 V C C, π 0 5/ ρ ( / kinetic enegy density t / V may be used to appoximate the kinetic enegy of a non-homogenous system with sufficiently slowly vaying electon density t V 5/ Cρ d t ρ( [ ]

18 he homas-femi appoximation by constuction, the F appoximation is coect in the limit of a homogenous electon gas F is coect in the limit of infinite nuclea chage ( Z flaws infinite chage density at the nucleus bad total enegies compaed to Kohn-Sham 6 algebaical decay of chage density instead of exponential ( exp( µ no binding of atoms to fom molecules o solids no shell stuctue in atoms (

19 he von-weizsäcke tem oiginally, von-weizsäcke deived intuitively a coection to the homas-femi appoximation to descibe the kinetic enegy of coe paticles to explain mass defects F + vw vw d t ρ ( ρ, the von-weizsäcke tem is exact fo one-obital systems (e.g. bosonic systems, one o two electons

20 he von-weizsäcke tem deivation fo a one-obital system, the kinetic enegy can be calculated exactly vw f d ϕ * ( ϕ( integation by pats d ϕ f g( g g ( ( ( d ρ( vw 8 ( ϕ( 0 d ρ ρ ( ( ρ ( ρ ( ρ (

21 he von-weizsäcke tem impovements compaed to F exponential decay of electonic density finite chage at the nucleus flaws in a homogenous system vw 0, but fo a one-obital system + does not epoduce the two limits S F vw F 0 attempts to impove the bidge between the two extemes: ( vw S G N F + numbe of electons ( ( empiical paametes A A G N δ + N δ N N N expeiences indicated that the pefacto of the vw-em might be too lage

22 Linea-Response heoy Why Linea-Response heoy? elationship between Linea Response and functional deivative of S can be deived coect Linea Response is impotant to descibe chage oscillations in solids definition small change in potential causes a fist-ode change of the chage density 44 ( d χ( - δν ( ( q χ( q δν ( q δν ( ( fouie tansfomation

23 Linea-Response heoy he Linea-Response Function invesion theoem of functional deivatives: δ ( - d δν ( ( δν ( ( d χ ( - δν f ( ( ( - fouie tansfomation: ( ˆ δν ( q ˆ δν χ F F ( ( ( χ( q

24 Linea-Response heoy ˆ H + V Linea-Response in the DF scheme E S + E eff eff δv eff ( ( ( δe V eff ( E S δ S ( ( ( ( δ Eeff δ eff ( δe ( µ const. χ Fˆ δ ( ( ( q goundstate S unknown!

25 ( ( ( ( q F f Lindhad - S ˆ 0 χ δ ρ Linea-Response heoy he Fee-Electon-Gas limit of the LR-function ˆ 0 H F F Lindhad, ln 4 k q k + + η η η η η π χ homogenous system due to symmety...

26 Linea-Response heoy weak logaithmic singulaity fouie tansfomation lim ( cos ( k F Fiedel oscillations singulaity is impotant to descibe the physics of a solid popely

27 Combination of F and vw goal is to put the combination of F and vw on a physical basis by consideing the linea-esponse S F + λ vw conside a nealy homogeneous electon gas with small fluctuations ρ ρ + ρ d ρ ( ( 0 ( 0 kinetic enegy may be expanded aound the aveage density S [ ρ] [ ρ ] S 0 F + d δs ( ρ0 44 const. ρ ( + linea tem vanishes d d S δ ρ ( ( ρ ˆ F χ Lindhad ( q ( ρ(

28 χ Combination of F and vw Lindhad χf λ vw k π F η q L, small η k F K, lage η η Kompaneets, Pavlovskii (957 Kizhnits (957 Le Couteu (964 Stoddat, Beattie, Mach (970 λη, small η λ kf k 9 F π + λη π λ λη, lage η

29 Combination of F and vw λ / 9 valid fo long wavelength petubation e.g. appopiate fo impuity poblems whee long wavelength components of potential ae dominant λ valid fo shot wavelength petubation e.g. appopiate fo pefect lattices flaws bad estimation of total enegy: λ oveestimation, λ / 9 undeestimation (intepolation leads to λ / 5 still no shell stuctues in atoms

30 Conventional Gadient Expansion Basic Idea include highe ode gadient coections to the kinetic enegy functional by expanding the Lindhad function to highe odes χ Lindhad 4 [ ρ] k π F η 4 η 5 L ( ρ 9 ρ( ρ ( ρ ( d ρ π ρ 8ρ ρ F 9 4 vw +L 6[ ρ] L (extemely complicated... (Hodges, 97 (Muphy, 98

31 Conventional Gadient Expansion flaws not suitable fo isolated systems with exponential decay of chage density kinetic enegy potential diveges fo ode fou and highe diveges fo ode six δ in self-consistent calculations, chage density shows wong decay behaviou linea esponse is wong, since expansion does only convege fo η < CGE is of no pactical use

32 Aveage-Density Appoximation Basic idea include non-local effects to the kinetic enegy functional example: (Gacia-Gonzales et al., PRA 54, 897 (996 S 5 [ ρ] [ ρ] + [ ρ] + d ρ( t ( ~ ρ( F F vw [ ρ] d ρ( t ( ρ( coect linea-esponse behaviou is enfoced by appopiate chose of the weight function Ω, ( Fˆ δ S ( ( χ ( q ρ 0 0! ~ ρ ( d ρ( Ω(, Lindhad

33 Aveage-Density Appoximation K Kohn-Sham symmetic ADA non-symmetic ADA FλvW Ω Ω (, Ω(, (, Ω(, non-local functionals can impove numeical esults significantly (Gacia-Gonzales et al., PRA 54, 897 (996

34 Summay goal is to get id of the Kohn-Sham-obitals to educe the computational time tade-off ae less acuate esults due to the necessity of appoximating the kinetic enegy functional state-of-the-at ae non-local functionals which impove the esults of the classical functionals significantly

35 Liteatue Wang, Y.A., Cate, E.A. 000, Obital-fee kinetic enegy density functional theoy, heoetical Methods in Condensed Phase Chemisty, S.D. Schwatz, Ed. Kluwe, 7-84 P. Gacia-Gonzales, J.E. Alvaellos and E. Chacon, Kinetic-enegy density functional: Atoms and shell stuctue, Phys. Rev. A, 54, 897 (996 M. Levy, J.P. Pedew and V. Sahni, Exact diffeential equation fo the density and ionization enegy of a many-paticle system, Phys. Rev. A, 0, 745 (984 C.H. Hodges, Quantum Coections to the homas-femi Appoximation-he Kizhnits Methods, Can. J. Phys., 54, 48 (97 R.G. Pa, S. Liu, A.A. Kugle and A. Nagy, Some identities in density-functional theoy, Phys. Rev. A, 5, 969 (995 Y. Wang and R.G. Pa, Constuction of exact Kohn-Sham obitals fom a given electon density, Phys. Rev. A, 47, R59 (99. Gal and A. Nagy, A method to get an analytical expession fo the non-inteacting kinetic enegy density functional, J. Mol. Stuc , 67 (000

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