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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