Generalized Gradient Approximation for Exchange-Correlation Free Energy

Transcription

1 Generalized Gradient Approximation for Exhange-Correlation Free Energy Valentin V. Karasiev, J.W. Dufty, and S.B. Trikey Quantum Theory Projet Physis and Chemistry Depts., University of Florida APS Marh Meeting 2017, New Orleans, LA 08 Mar. 2017

2 Univ. Florida Orbital-Free DFT & Free-energy DFT Group Sam Trikey Jim Dufty Lázaro Calderín Valentin Karasiev Kai Luo Daniel Mejia Affiliates: Frank Harris (U. Utah); Keith Runge (U. Arizona) Alumni: Deb Chakraborty, Támas Gál, Olga Shukruto, Travis Sjostrom Funding Aknowledgments: U.S. DoE DE-SC Publiations, preprints, loal pseudopotentials, and odes at

3 Motivation, Physial problem Warm Dense Matter Shemati temperature-density diagram for Hydrogen (from R. Lee, LLNL). Interior of Saturn (taken from: Fortney J. J., Siene 305, 1414 (2004): (1) At an age of ~1.5 billion years (2) The urrent Saturn aording to previous H- He phase diagram (3) The urrent Saturn aording to new evolutionary models

4 Motivation, hallenges to Developers Why does the WDM regime require development of new methods & funtionals? Standard omputational methods often ease to work at extreme ompressions (high P) and temperatures (high T) Limited transferability of pseudopotentials and PAWs developed for near-ambient thermodynami onditions. Drasti inrease of omputational ost as T inreases: ost ~ (N band ) 3 Strong quantum effets => Usually not possible to go down in T to WDM regime from the hot plasma regime; lassial approahes fail at lower T Exhange-orrelation effets at finite T are not taken into aount by use of ground-state (zero-t) XC funtionals

5 Motivation, hallenges to Developers Need for thermal DFT funtionals Thermal DFT is a part of standard treatment (of WDM) Choie of the XC free energy F x [n] may affet reliability of results Common pratie is to use a T=0 XC funtional: F [ nt, ] E [ nt ( )] First rung XC free-energy funtional (VVK, Sjostrom, Dufty, & Trikey, Phys. Rev. Lett. 112, (2014)) takes into aount XC thermal effets in the loal density approximation (LDA) x x Next rung GGA XC free-energy is required to take into aount XC thermal and non-homogeneity effets whih inlude T-dependent density gradients

6 XC thermal effets for the homogeneous eletron gas (HEG) XC thermal effets are signifiant in WDM regime: log 10 fx( rs, T) ε x( rs ) f ( r, T) + ε ( r) s s x s f x = XC free energy per partile ε x = XC energy per partile at T=0 f s = non-interating free energy Rough WDM region in ellipse. T (kk) r s (bohr) Common pratie is to use a T=0 XC funtional: F [ nt, ] E [ nt ( )] x May not be aurate in WDM regime x

7 Framework for GGA XC free-energy funtional development Identify T-dependent gradient variables for X and C free-energies Identify relevant finite-t onstraints Use our finite-t LDA XC as an ingredient Propose appropriate analytial forms, inorporate onstraints Implementation, tests, appliations VVK, Dufty, Trikey, Phys. Rev. Lett. (submitted, 2017) see also arxiv: v1

8 T-dependent density gradient for X Start with finite-t gradient expansion for X: LDA LDA fx (, nt) = εx () n Ax() t ; t= T/ T F 2 ( βµ ) t 2 x = 1/2 2 A () t I ( η) dη (2) LDA 8 2 fx (, n nt, ) = fx (, nt)1 + s (, n n) B x() t 81 s n nt s n n B t 2 2x(,, ) (, ) x() Finite-T GGA X funtional: Enhanement fator is defined from several ground-state and finite-t onstraints: Constraints: Reprodue finite-t small-s grad. expansion Satisfy Lieb-Oxford bound at T=0 Redue to orret T=0 limit Redue to orret high-t limit = r F GGA [ nt, ] nf ( nt, )F( s ) d LDA x x x 2x νxs2x F( x s2x) = 1+ 1+α s 2x

9 T-dependent density gradient for C Finite-T gradient expansion for XC: (2) 1 (2) 2 fx (, n nt, ) = gx (, nt) n 2 = C ε ( n) s ( n, n) B ( t) + C n s ( n, n) B ( nt, ) C (2) (2) x (2) LDA 2 (2) 1/3 2 x x x = 8 / 81; C = ; 4/3 ' '' 3 4/3 I 1/2 I 1/2 x() = I1/2 ( βµ ) 3 I 1/2 I 1/2 B t ( βµ ) ( βµ ) 2 ( βµ ) ( βµ ) B ( n, t) is defined from equation for f (above) (2) x (2) with use of numerial RPIMC-based data for g ( n) x

10 T-dependent density gradient for C (Contd.) From finite-t gradient expansion for C we identify new T-dependent gradient variable: n s (, n n) B (,) n t q B (,) n t 1/3 2 2 q (, n nt, ) q(, n n) B (,) n t where q is a ground-state redued density gradient for orrelation. GGA orrelation energy per partile: f (, n nt, ) = f (, nt) + H( f, q ) GGA LDA LDA where the funtion H(f LDA,q ) is defined by the ground-state PBE funtional to guarantee a widely used zero-t limit. Finite-T GGA C funtional: F [, nt] nf (, n nt, ) dr GGA GGA = Constraints: Reprodue finite-t small-s grad. expansion Redue to orret T=0 limit Redue to orret high-t limit

11 X and C T-dependenes à x B x s 2x /s t A ( t ) - shows T -dependene of the LDA-X x B x ( t) - shows T-dependene of the GGA redued gradient for X B (r s,t) r s =0.5 r s =1 r s =2 r s =4 r s =6 r s =10 T-dependene of the GGA variable for C t

12 P (MBar) Thermal GGA XC results on f-al model system P PBE - P PZ P KSDT - P PZ P KDT16 - P PZ f Al, ρ=3.00 g/m T (kk) PBE XC non-homogeneity (T=0) effets KSDT - XC thermal LDA effets KDT16 XC thermal GGA and non-homogeneity (expliit-t) effets Eletroni pressure differenes vs. T for the new finite-t GGA ( KSDT16 ), KSDT LDA, and ground-state PBE XC funtionals, all referened to PZ ground-state LDA values. Stati lattie f Aluminum at 3.0 g/m 3.

13 Thermal GGA XC results on Deuterium EOS P el (Mbar) PBE KDT16 PIMC P KDT16 -P PBE ρ D =0.390 g/m T (kk) P el (Mbar) PBE KDT16 PIMC P KDT16 -P PBE ρ D =0.506 g/m T (kk) Deuterium eletroni pressure vs. T for the finite-t GGA ( KDT16 ) and groundstate PBE XC funtionals, as well as PIMC referene results. AIMD super-ell simulations, Γ-point only, for 128 atoms (8500 steps, T 40 kk) or for 64 atoms (4500 steps, T 62 kk PIMC results: S.X. Hu, B. Militzer, V.N. Gonharov, and S. Skupsky, Phys. Rev. B (2011).

14 Summary Framework for GGA XC free-energy funtional development is presented virtually any ground-state XC an be extended systematially into an XC free energy First GGA XC free-energy ( KDT16 ) funtional onstruted Test ases show that KDT16 provides improved auray in the desription of XC thermal effets VVK, Dufty, Trikey, Phys. Rev. Lett. (submitted, 2017) see also arxiv: v1