Generalized Gradient Approximation for Exchange-Correlation Free Energy

Valentin V. Karasiev, J.W. Dufty, and S.B. Trickey

Quantum Theory ProjectPhysics and Chemistry Depts., University of Florida

http://www.qtp.ufl.edu/ofdft

APS March Meeting 2017, New Orleans, LA

© 08 Mar. 2017

Publications, preprints, local pseudopotentials, and codes at http://www.qtp.ufl.edu/ofdft

Funding Acknowledgments: U.S. DoE DE-SC0002139

Univ. Florida Orbital-Free DFT & Free-energy DFT Group

Sam TrickeyJim DuftyLázaro CalderínValentin KarasievKai LuoDaniel MejiaAffiliates: Frank Harris (U. Utah); Keith Runge (U. Arizona)Alumni: Deb Chakraborty, Támas Gál,

Olga Shukruto, Travis Sjostrom

Interior of Saturn (taken from: Fortney J. J., Science 305, 1414 (2004):(1) At an age of ~1.5 billion years(2) The current Saturn according to previous H-

He phase diagram(3) The current Saturn according to new

evolutionary models

Motivation, Physical problem

Schematic temperature-density diagram for Hydrogen (from R. Lee, LLNL).

Warm Dense Matter

Motivation, challenges to Developers

Why does the WDM regime require development of new methods & functionals?

Standard computational methods often cease to work at extremecompressions (high P) and temperatures (high T) –

• Limited transferability of pseudopotentials and PAWsdeveloped for near-ambient thermodynamic conditions.

• Drastic increase of computational cost as T increases: cost ~ (Nband)3

• Strong quantum effects => Usually not possible to go down in Tto WDM regime from the hot plasma regime; classical approaches fail at lower T

• Exchange-correlation effects at finite T are not taken into account by use of ground-state (zero-T) XC functionals

Motivation, challenges to Developers

Need for thermal DFT functionals –

•Thermal DFT is a part of standard treatment (of WDM)

•Choice of the XC free energy Fxc [n] may affect reliability of results

• Common practice is to use a T=0 XC functional:

• First rung XC free-energy functional (VVK, Sjostrom, Dufty, & Trickey, Phys. Rev. Lett. 112, 076403 (2014)) takes into account XC thermal effects in the local density approximation (LDA)

•Next rung GGA XC free-energy is required to take into account XC thermal and non-homogeneity effects which include T-dependent density gradients

xc xc[ , ] [ ( )]n T n T≈F E

0.1 1 10 100rs (bohr)

0.01

0.1

1

10

100

1000

10000

100000

T (

kK)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5XC thermal effects are significant in WDM regime:

fxc = XC free energy per particleεxc= XC energy per particle at T=0fs = non-interacting free energy

Rough WDM region in ellipse.

xc s xc s10

s s xc s

, ( ) ( )( ) ( ),

f r T rlog

f r T rεε

−+

XC thermal effects for the homogeneous electron gas (HEG)

Common practice is to use a T=0 XC functional:

May not be accurate in WDM regimexc xc[ , ] [ ( )]n T n T≈F E

• Identify T-dependent gradient variables for X and C free-energies• Identify relevant finite-T constraints• Use our finite-T LDA XC as an ingredient• Propose appropriate analytical forms, incorporate constraints• Implementation, tests, applications

VVK, Dufty, Trickey, Phys. Rev. Lett. (submitted, 2017)see also arXiv: 1612.06266v1

Framework for GGA XC free-energy functional development

T-dependent density gradient for X

LDA

(2) LDA

LDAx x x

( )22

x 1/2

2x x x

22x x

( , ) ( ) ( ) ; /

( ) ( )2

8( , , ) ( , ) 1 ( , ) ( )81

( , , ) ( , ) ( )

Ff n T n A t t T T

tA t I d

f n n T f n T s n n B t

s n n T s n n B t

βµ

ε

η η−−∞

= =

=

⎡ ⎤∇ = + ∇⎢ ⎥⎣ ⎦∇ ≡ ∇

∫

Start with finite-T gradient expansion for X:

Finite-T GGA X functional: LDAx x x 2xGGA , ] ) F ( )[ ( , F n T n f n T s d= ∫ r

Enhancement factor is defined from several ground-state and finite-T constraints:

x 2xx 2x

2x

F ( ) 11

sss

να

= ++

Constraints:• Reproduce finite-T small-s grad. expansion• Satisfy Lieb-Oxford bound at T=0• Reduce to correct T=0 limit• Reduce to correct high-T limit

T-dependent density gradient for C

(2) 2xc xc(2)

(2) LDA 2 (2) 1/3 2x x x c c

1( , , ) ( , ) | |2

( ) ( , ) ( ) ( , ) ( , )

f n n T g n T n

C n s n n B t C n s n n B n tε

∇ = ∇

= ∇ + ∇

Finite-T gradient expansion for XC:

(2) (2)x c

4/3 ' ''4/3 1/2 1/2

x 1/21/2 1/2

(2)c xc

8 / 81; 0.162125;

( ) ( )3( ) ( ) 32 ( ) ( )

( , ) is defined from equation for (above)

with use of numerical RPIMC-based data for

C C

I IB t II I

B n t f

g

βµ βµβµβµ βµ

− −

− −

= =

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠ ⎣ ⎦

(2)xc ( )n

T-dependent density gradient for C (Contd.)

1/3 2 2c c

c

GGA LDA LDAc c c

( , ) ( , ) ( , )

( , , ) ( , ) ( , )

GGA correlation energy per particle:( , , ) ( , ) ( , )

c

c

n s n n B n t q B n t

q n n T q n n B n t

f n n T f n T H f q

∇ ∝

∇ ≡ ∇

∇ = +

From finite-T gradient expansion for C we identify new T-dependent gradient variable:

where the function H(fcLDA,qc) is defined by the ground-state

PBE functional to guarantee a widely used zero-T limit.

Finite-T GGA C functional:

GGA GGAc c[ , ] ( , , )F n T n f n n T d= ∇∫ r

Constraints:• Reproduce finite-T small-s grad. expansion• Reduce to correct T=0 limit• Reduce to correct high-T limit

where q is a ground-state reduceddensity gradient for correlation.

0.01 0.1 1 10 100 1000t

0

0.5

1

1.5

Bc(r

s,t)

rs=0.5

rs=1

rs=2

rs=4

rs=6

rs=10

0.01 0.1 1 10 100 1000t

-1.5

-1

-0.5

0

0.5

1

1.5

2Ã

x

s2x

/s2

X and C T-dependences

T-dependence of the GGA variable for C

x ( ) - shows -dependence of the GGA reduced gradient for XB t T

x ( ) - shows -dependence of the LDA-XA t T

xB

0 50 100 150 200 250 300T (kK)

-0.16

-0.12

-0.08

-0.04

0

0.04

0.08

∆P (

MB

ar) P

PBE - P

PZ

PKSDT

- PPZ

PKDT16

- PPZ

fcc Al, ρ=3.00 g/cm3

Electronic pressure differences vs. T for the new finite-T GGA (“KSDT16”), KSDT LDA, and ground-state PBE XC functionals, all referenced to PZ ground-state LDA values. Static lattice fcc Aluminum at 3.0 g/cm3.

Thermal GGA XC results on fcc-Al model system

PBE – XC non-homogeneity (T=0) effects

KDT16 – XC thermal GGA and non-homogeneity (explicit-T) effects

KSDT - XC thermal LDA effects

0 20 40 60 80 100 120T (kK)

0

0.4

0.8

1.2

1.6

Pel

(M

bar)

PBEKDT16PIMC

PKDT16

-PPBE

ρD

=0.506 g/cm3

0 20 40 60 80 100 120T (kK)

0

0.4

0.8

1.2

Pel

(M

bar)

PBEKDT16PIMC

PKDT16

-PPBE

ρD

=0.390 g/cm3

Deuterium electronic pressure vs. T for the finite-T GGA (“KDT16”) and ground-state PBE XC functionals, as well as PIMC reference results.

AIMD super-cell 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. Goncharov, and S. Skupsky, Phys. Rev. B 84 224109 (2011).

Thermal GGA XC results on Deuterium EOS

Summary

• Framework for GGA XC free-energy functional development is presented

⇒ virtually any ground-state XC can be extended systematically into an XC free energy

• First GGA XC free-energy (“KDT16”) functional constructed

• Test cases show that KDT16 provides improved accuracy in the description of XC thermal effects

VVK, Dufty, Trickey, Phys. Rev. Lett. (submitted, 2017)see also arXiv: 1612.06266v1