Finite-Temperature Hartree-Fock Exchange and Exchange- Correlation Free Energy Functionals. Travis Sjostrom. IPAM 2012 Workshop IV

Transcription

1 1 of 45 Finite-Temperature Hartree-Fock Exchange and Exchange- Correlation Free Energy Functionals Travis Sjostrom Quantum Theory Project Depts. of Physics and Chemistry IPAM 2012 Workshop IV 2012

2 2 of 45 Acknowledgments: UFL OFDFT Group Sam Trickey Jim Dufty Frank Harris Keith Runge Travis Sjostrom Valentin Karasiev Tamas Gal Vivek Kapila Funding US DOE - TMS Grant DE-SC

3 3 of 45 Overview Exchange and Correlation (XC) in DFT Introduction to finite temperature XC functionals Confined and extended system results Future work for finite temperature XC functionals

4 4 of 45 Exchange and Correlation in DFT

5 5 of 45 DFT Basics 1 2 H = 2i + i i<j 1 r i r j i,i ZI r i R I, H Ψ = EΨ Hohenberg-Kohn (1964) E[n] = T + U ee + U ion Kohn-Sham (1965) occ n(r) = ϕ i (r)ϕi (r) dr i T [n] Ts [ϕi ], KS single particle equations EH = n(r)n(r ) r r dr dr, U ion = I n(r)z I r i R I E = Ts + E H + U ion + E xc E xc = (T Ts ) + (U ee E H ) vxc (r) = δe xc δn(r) W. Kohn

6 6 of 45 E xc approximations LDA (local density approximation) Prior to 1980 approximate electron gas calculations: RPA, high-low density interpolation, GW, local field corrections (i.e. STLS) Post 1980 Ceperley-Alder QMC calculation parameterizations: Vosko-Wilk-Nusair (1980) VWN Perdew-Zunger (1981) PZ81 Perdew-Wang (1992) PW92 D.M. Ceperley

7 7 of 45 E xc approximations Gradient expansions as proposed by Kohn-Sham implemented by several others. Often not better or worse than LDA results. GGA (generalized gradient approximation) Can include corrections beyond second order expansion and enforce various constraints on XC. Becke (1988) B88 Perdew-Wang (1991) PW91 Perdew-Burke-Ernzerhof (1996) PBE Perdew's Jacob's Ladder... J.P. Perdew

8 8 of 45 Extension to Finite Temperature

9 9 of 45 Finite Temperature DFT Mermin (1965) extension of Hohenberg-Kohn to finite temperature. Development through the grand canonical ensemble. F[n, T] = T + U ee + U ion TS KS scheme also applies n(r) = f i ϕ i (r) ϕ i (r) dr, = i f i 1 e β( ε i μ) +1 T = f i ϕ i (r) 2 ϕ i (r) dr 1 T s 2 i S S s = k B f i ln( f i ) + (1 f i ) ln(1 f i ) i N.D. Mermin F = T s + E H + U ion T S s + F xc F xc = (T T s ) + ( U ee E H ) + (S S s ) (r) = v xc δf xc δn(r)

10 10 of 45 F xc approximations - LDA No finite temperature benchmarks equivalent to ground state QMC results. Again interpolation and approximate theoretic methods are required. T = 0 Known and High limits QMC and Debye-Hückel, respectively. Useful thermodynamics for the UEG Ω = pv = F μn p xc /n = f xc μ xc μ xc = v xc, f xc = F xc /N T t = T/ T F = 1/βε F 4π 3 r3 s r s 1 n = = 3π 2 k 3 F = 1

11 11 of 45 Exchange only F x F HEG x V k 4π (2π) 6 k k = dk d f(k)f( ) k βμ 0 1 I 2 2π 3 β 2 = 1/2 (x) dx, I α (x) = dz 3 x (n) = ε x (n) = 4 ( 3n π ) (n, T) = (n)j(n, T) f LDA f LDA(T) x ε x 1/3 0 e z x z α +1

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13 13 of 45 F x beyond LDA Geldart (1993) has provided a closed form expression (in Fermi integrals) for the first term in the finite temperature gradient expansion. Developed in grand canonical ensemble, analgous to KS original proposal for gradient expansion. B x πβ 2 I 1/2 (η) 1/2 (n, T) = [( ) 3 ( ) ] 72 I 1/2 (η) 2 I I 1/2 (η) (η) 2 F g x = B x (n, T) n(r) dr 2 η = βμ 0 2 n = I (β ) πβ 3/2 1/2 μ 0 B x (t) = ( 18 k 4 f /π) B x

14 14 of 45 Finite temperature Correlation

15 15 of 45 RPA Originally calculated by Gupta and Rajagopal (1980), revised by Perrot and Dharma-wardana (1984). Ω(μ, T) = Ω 0 + Ω x + Ω c Ω x βμ V I 2 2π 3 β 2 1/2 = (x) dx V 1 Ω c Ω r = {ln[1 u q π 0 ( v n, q)] + u q π 0 ( v n, q)} dq (2π) 3 n 2β 0 Linearized inversion for μ xc, direct inversion will differ. (n, T) = (, T) F xc Ω xc μ 0 Ω xc ) μ=μ 0 ( / μ μ xc (n, T) = ( n/ μ) μ=μ 0

16 16 of 45 STLS type Dielectric Finite temperature version of Singwi-Tosi-Land-Sjölander local field corrections. Calculated and fit by Ichimaru et. al. (1985) S(x) = 3 t 2 l= 1+(2Γt/πα x 2 Φ(x,l) )[1 G(x)]Φ(x,l) 3 G(x) = y 2 x [S(y) 1](1 + 2 y 2 ln x+y x y ) dy 4 0 2xy Γ = β/ r s, α = (4/9π) 1/3, x = q/ q F After self-consistent solutions fit, ε int Γ παβ 0 = [S(x) 1] dx, Γ 1 f xc (n, T) = β dγ Γ (β ε int ) t 0

17 17 of 45 Classical Mapping Idea is to map quantum variables to classical variables, for which classical calculation results are equivalent/correct for the quantum system. Perrot and Dharma-wardana realization through CHNC. T = 0 n T cf At, given density, the CHNC equations are solved with which is adjusted until T cf = T q is found which produces the same ε c (n) as the benchmark QMC. The CHNC calculation also produce pair distribution functions in good agreement with the benchmarks, and are always non-negative. T q / E F = 1/(a + b r s + c r s ) For T > 0 T cf = T 2 + T 2, q interpolates between 0 and high. T

18 18 of 45 Exchange-correlation energy per electron for r s = 1. Approximately 5% difference at T = K from ground state.

19 19 of 45 Exchange-correlation energy per electron for r s = 0.5. Approximately 2% difference at T = K from ground state.

20 20 of 45 Exchange-correlation energy per electron for r s = 5. Approximately 50% difference at T = K from ground state.

21 21 of 45 (n f v = xc ) xc n Not necessarily related to temperature dependence.

22 22 of 45 Finite temperature Hartree-Fock Mermin (1963) examined finite temperature Hartree-Fock, showing in the grand ensemble that there is an ensemble of Slater determinants that minimizes the Hartree-Fock free energy and provides an upper bound as at zero temperature. F HF = T HF T S HF + U ion + E H + F HF x 1 T HF = T s, S HF = S s, with respective orbitals and f i = e β( ε i μ) +1 F x HF 1 ϕ ( ) ( ) ( ) ( ) = f i f j δ( σ i, σ j ) dr 1 dr i r 1 ϕ j r 1 ϕ i r 2 ϕ j r i,j r 1 r 2 Single particle equations 2 ϕ j ( r 2 ) ϕ j ( r 2 ) ε i ϕ i (r) = ( + v ion (r)) ϕ (r) + d (r) 2 i f j r 2 ϕ r i ϕ i ( r 2 ) ϕ j ( r 2 ) f j δ( σ i, σ j ) d r 2 ϕ (r) r j j j r 2 r 2

23 23 of 45 Confined hydrogen

24 24 of 45 Model Problem Hard-wall rectangular parallelepiped containing a few (1-32) hydrogen atoms, where fixed atomic positions are sampled. Box size is from 1 au 3 (Bohr 3 ) to free limit (arbitrarily large). Temperatures range from 0 to 300,000 K (0 26 ev) Methods: Finite Temperature Hartree-Fock (FTHF) and Kohn-Sham density functional theory (DFT). Unique truncated Gaussian type orbital basis is employed. 8 Hydrogen atoms in cubic arrangement of spacing a confined within a hardwall cube of side length. L

25 25 of 45 Truncated Gaussian Type Orbitals Basis Requirements Large number of orbitals for thermal occupation. Matching of hardwall boundary condition. Efficient calculation of electron repulsion integrals. Cartesian Gaussians truncated to match BCs. Coefficients a 0 and a L set by requiring continuity of basis functions. g n box = a 0 [ g n (x) g n (0)] 0 x x c a L [ g n (x) g n ( L x )] x c x L x g n (x) = (x x c ) n e α(x x c) 2 x c is nucleus posistion.

26 26 of 45 Results - Free Energy T = to K. a = 2.0 au; for simple cubic lattice this corresponds to r s = HF free energy is better approximated by temperature dependent LDA exchange functional.

27 27 of 45 Comparison of previous with Geldart finite temperature exchange gradient correction included.

28 28 of 45 Free Energy Components Entropic and kinetic terms contribute most to overall change in free energy, but change in exchange free energy is not negligible. At 300 kk ΔF x is 25% of ΔE and 14% of ΔF.

29 29 of 45 fthf for solid Li

30 30 of 45 Pseudopotentials PPs are required, but must be specialized for WDM systems. Issues are transferability to higher densities and temperatures. Inclusion of more core orbitals in calculation is necessary. All electron zero temperature calculations for clusters can benchmark PP calculations.

31 31 of 45 PP for Hartree-Fock fthf calculation will be for Li densities = g/cc. and temperatures up to 100 kk This allows use of standard 1 electron PP ρ

32 32 of 45 Li fthf free energy results

33 33 of 45 Li fthf pressure results

34 34 of 45 Finite temperature XC results

35 35 of 45 Li F xc - LDA free energy Free energy for ground state LDA and 3 ftxc LDA functionals.

36 36 of 45 Li LDA pressure Pressure results for bcc Li at 1 g/cc up to 100 kk. All temperature dependent functionals show about 5% lowering of the pressure at 100 kk.

37 37 of 45 ftxc beyond LDA Consider the PBE functional for exchange and correlation. Replace the uniform energy densities by the finite temperature versions. ε x, ε c f x, f c The constraint based parameters of the ehancement factors are taken to be temperature independent.

38 38 of 45 Pressure in MD P = P ID F int V Nk B T P ID = V

39 39 of 45 Simple cubic hydrogen ρ H = 1 g/cc. Pressure and percent difference, for LDA and PBE. Green is minimum percent difference and red is maximum, for functional.

40 40 of 45 Simple cubic hydrogen ρ H = 2 g/cc. Pressure and percent difference, for LDA and PBE. Green is minimum percent difference and red is maximum, for functional.

41 41 of 45 Simple cubic hydrogen ρ H = 4 g/cc. Pressure and percent difference, for LDA and PBE. Green is minimum percent difference and red is maximum, for functional.

42 42 of 45 Temperature dependent XC results for eigenvalues. Simple cubic hydrogen ρ H = g/cc, T = 95250K. 64 atom Gamma point calculation with 500 bands.

43 43 of 45 Temperature dependent XC QMD results. Deuterium ρ D = g/cc, at T = 95250, K. 64 atom Gamma point calculation.

44 44 of 45 Summary and Outlook Finite temperature exchange-correlation functionals are in infancy with respect to ground state functions. This is in part due to lack of high quality LDA data, and lack of benchmark systems. Finite temperature Hartree-Fock provides one means of benchmarks for exchange, quantum chemistry post-hf procedures may provide benchmarks for correlation. Finite temperature functionals add no real additional cost to DFT methods. High accuracy DFT calculations in warm dense matter will require temperature dependent exchange-correlation functionals. Thank You!

45 45 of 45 References HEDLP Basic Research Needs Workshop Report, Nov , (2009) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964) W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965) D.M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980) J.P. Perdew and Y. Wang, Phys. Rev. B 45, (1992) J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996) N.D. Mermin, Phys. Rev. 137, A1441 (1965) M.A. Pokrant, Phys. Rev. A 16, 413 (1977) U. Gupta and A.K. Rajagopal, Phys. Rev. A 22, 2792 (1980) E. Dunlap and D.J.W. Geldart, Can. J. Phys. 72, 1-6 (1994) M.L. Glasser, D.J.W. Geldart, and E. Dunlap, Can. J. Phys. 72, 7-13 (1994) F. Perrot and M.W.C. Dharma-wardana, Phys. Rev. A 30, 2619 (1984) S. Tanaka and S. Ichimaru, J. Phys. Soc. Jpn. 55, 2278 (1986) F. Perrot and M.W.C. Dharma-wardana, Phys. Rev. B 62, (2000) N.D. Mermin, Ann. Phys. (NY) 21, 99 (1963) T. Sjostrom, F.E. Harris, and S.B. Trickey, Phys. Rev. B 85, (2012) S.X. Hu, B. Militzer, V.N. Goncharov, and S. Skupsky, Phys. Rev. B 84, (2011)