Sam Trickey

© 16 June 2015

Finite-temperature Density Functionals –Developments and Computational Consequences

Quantum Theory Project Physics, Chemistry - University of Florida

trickey@qtp.ufl.eduhttp://www.qtp.ufl.edu/ofdft

ES-2015Univ. of WashingtonJune 22-24, 2015

Univ. Florida Finite-Temperature and Orbital-free DFT Group:

Jim Dufty

Frank Harris (also Univ. Utah)

Keith Runge (also Univ. Arizona)

Debajit Chakraborty

Lázaro Calderin

Valentin Karasiev

alumni: T. Gál, Travis Sjostrom, Olga Shukruto

DFT in Magnetic Fields Collaboration:

Wuming Zhu (Hangzhou Normal Univ., China)

XC Functional Collaboration in México:

José Luis Gázquez (UAM- I, México D.F.)

Alberto Vela (Cinvestav, México D.F.)

Jorge Martín del Campo Ramírez (UNAM, México D.F.)

and students and postdocs

Emerging Collaboration at Univ. Washington:

John Rehr

Joshua Kas

Gerald Seidler

Ryan Valenza

Funding Acknowledgments:

U.S. Dept. Energy DE-SC 0002129CONACyT (México)

Publications, preprints, & software at http://www.qtp.ufl.edu/ofdft

Schematic diagram for hydrogen (credit R.W. Lee presentation, LLNL)

Warm Dense Matter: 0.5 eV ≤ T ≤ 100 eV (≈ 1,000,000 K) –

HOT by standards of familiar condensed matter theory

Densities: from gas to ~ 100 × equilibrium density

• Formation of molecules, clusters, and ions.

• High T & P for familiar quantum mechanical

methods (quantum chem, cond. matt. Physics)

Almost prohibitively expensive

• But T is low for classical plasma physics

methods

QM is important.

• Happy but expensive medium: “ab initio

molecular dynamics” (AIMD; also called

“quantum MD”)

A motivating physical problem – Warm Dense Matter

2 / T 1s B

e r kΓ = ≈ T / 1B F

t k E= ≈

Motivating computational problem

Image Credit: W. Lorenzen, V. Karasiev

KS equation

Grand potential

s [n] = Non-interacting (KS) free energy, H[n] = Hartree free energy

xc [n] = eXchange-Correlation (XC) free energy

[ ] [ ] ( ( ) ) ( )extn n d v nµΩ = + −∫ r r rF

[ ] [ ] [ ] [ ]s H xcn n n n= + +F F F F

KS calculational costs scale as

cube of the number of occupied levels.

Scaling worsens with increasing T (non-

integer occupation).

Orbital-free Free Energy DFT –

No explicit KS orbitals.

Scales with system size.

Mermin,

Hohenberg-Kohn

DFT

( ) ( ) ( ) ( ) ( )

( ) ( ) [ ]

1

212 1 1 1 1 1

2

1 1

; ; ; ; ;

; ( ; ) ; ; ; 1 /

r H xc ext j j j

xcj j xc B

j

v v v

n f v n k Tn

ϕ ε ϕ

δε β ϕ β

δ

− ∇ + + + =

= = =∑

r R r R r R r R r R

r R r RF

Electrons Nuclei

•GGA

•LDA

Credit: SCIDAC Review 17

Approaches to complexity, size, and extreme conditions

(0) Bring finite-T DFT up to date relative to much-more-

developed T=0 K version.

(1) Push on orbital-free DFT for AIMD speed, despite

conventional wisdom that OFDFT never has worked.

(2) Work on better functionals at the lower rungs of the

Perdew-Schmidt Jacob’s ladder of XC functionals because

orbital-independent functionals are faster.

[Conventional wisdom – higher rung XC functionals are

required.]

(3) Implement and distribute new functionals and

capabilities

Remark: success for OFDFT, item (1),

requires success with item (2)

Challenges from T-dependence

1] Accurate orbital-free non-interacting

free energy (KS KE & KS entropy).

Why? Speed. OFDFT scales linearly with

system size, independent of T.

KS

2] XC free energy functional –

needed in both conventional

KS-AIMD & orbital-free

AIMD. Why? Non-trivial T-

dependence. At right (for HEG):

10

( ) ( )

( )

,T

,T

xc s xc s

s s

f r rlog

f r

ε−

o DFT theorems provide no constructive

routes to approximations.

o No mechanical recipe (e.g. pert. theory) for adding

complexity (and, presumably, improvements).

T (kK)

Note: no gradient or higher derivative dependence.

Determine fxcHEG from fit to restricted path integral Monte

Carlo (RPIMC) data [Brown et al., Phys. Rev. Lett. 110, 146405

(2013)]

Fit must extrapolate smoothly to correct large-T, T=0, and small

rs limits.

Fit must be augmented with T-dependent interpolation to

intermediate spin polarization.

Procedural issue: Four formally equivalent thermodynamic

relationships between XC internal energy density εxc and XC free

energy density fxc are not computationally equivalent. Detailed

study led to use of

Local spin density approximation (LSDA) xc [n]

( ) ( )HEG

xc xc[ ( ), ] , ( , , )n T T d n T f n T T≈ ∫ r r rF

s

xc sxc s xc s

( , )( , ) ( , ).|r

f r tf r t t r t

tε

∂− =

∂

LSDA xc [n]

Phys. Rev. Lett. 112, 076403 (2014)

1/2

s s

xc s 1/2

s s s

( ) ( ) ( )1( , )

1 ( ) ( )

a t b t r c t rf r t

r d t r e t r

ζ ζ ζζ

ζ ζ

ω + += −

+ +

( ) 1/3

0 1; 1; 2n n n ζ ζζ ω ω= =↑ ↓= − = =

a(t), b(t), c(t), d(t), e(t) are functions of t=T/TF with tabulated coefficients.

Comparison to RPIMC data (red dots) for ζ=0, rs=1 (left) and 40 (right)

for εxc and resulting fxc.

Fitted solution to

thermodynamic

differential

relation

LSDA xc [n] – smooth extrapolation to proper limits

Top: ς=0 ; bottom ς= 1“Fit A” is Phys. Rev. Lett

Also fits rather well to subsequent configuration-PIMC data for rs≤ 1 from Schoof,

Groth, Vorberger, and Bonitz. See arXiv 1502.04616, Fig. 5.

0 5 10 15 20 25 30T (kK)

1.7

1.8

1.9

2

2.1r s

(bohr)

PZ

KSDT

Equilibrium rs for electron gas at temperature T in

external field of H-nuclei fixed in simple-cubic

positions (static, cold sc H ions with hot electrons).

Inhomogeneous Electron Gas (sc-Hydrogen) at finite-T:

Cross-over of

TLDA & ordinary

LDA (with T-dependent

density) is a common

feature of the T-

dependence.

(r) 0 rv nθ θδ δ= ≥ ∀T

Constraints from Pauli KE and Pauli potential. Start at T = 0 K.

An exact decomposition (at all T):

[ ] [ ] [ ], [ ] 0s W

n n n nθ θ= + ≥T T T T

Accurate orbital-free non-interacting free energy functional

( )

( )2 1/3

5/3

2/3

4/3

23

10

1 | |(

[ ] r

) : ;2(3

( ( )

3)

)GGA

s TF t

TF

n c d n F s

cn

sn

ππ

∇=

=

=

∫

r

r rT

2

[ ]

[ ] r [ (r)]

[ ] [ ] [ ]

]

1r

2

[ee xc n

S

e

is i

sE n

n d t n

n E n E n E n

f d ϕ= ∇

= + + +

= ∫∫ ∑

T

T

Global positivity constraint

M. Levy and H. Ou-Yang, Phys. Rev. B 38, 625 (1988); A. Holas and N.H. March,

Phys. Rev. A 44, 5521 (1991); E.V. Ludeña, V.V. Karasiev, R. López-Boada, E. Valderama,

and J. Maldonado, J. Comp. Chem. 20, 155 (1999) and references in these]

Generalized gradient approximation

(GGA)

Local (pointwise) positivity constraint

[ ]( )

2( )1

:8

W

nn d

n

∇= ∫

rr

rT

2 | |

0( ) Z r

ren

−

→∝r

For such a density, the GGA Pauli potential behaves as

The constants A, B, C depend on detailed form of a specific θθθθ approximation

00

( ) . . .r

r

AB C r

rv

nθ

θδδ→

→

≡ = + + +rT

0A≥Pauli potential positivity ⇒⇒⇒⇒ GGA constraint:

GGA Constraints for Non-empirical parameterization of TTTTs

1. Kato cusp condition gives density behavior near nucleus, charge Z,

( ) 2 45( ) 1 ( )27tF s s O s= + +

2. Recover second-order gradient expansion (SGE) for s<<1 :

4. Obey Lieb upper-bound: [ ] [ ] [ ]s TF Wn n n≤ +T T T

3. Recover vW KE for s →∞ : lim ( ) 0s

F sθ→∞=

GGA for TTTTs

2

42 ( )

( )

2 2

( )e 1[ ( )] 1 (1 e ) 1

1 ( ) ( )

asascs

F scs s

θ

−−

= − + − −+

rrr

rr r

2.778; 1.296518c a= =

“VT84F”

Parameters from constraints, not fitting

See Phys. Rev. B 88, 161108R (2013)

Above: Pauli part of enhancement

factors.

Positive slope of VT84F for s2 > s2 (0)≈0.375

guarantees vθ > 0 near nucleus; all others fail.

5 5

3 2 7c a= + −

Kato regions

VT84F gives binding. Other GGAs do not.

At right: Total E (relative to min.) vs. lattice

constant, for simple-cubic H. “APBEK” is L.

Constantin et al. non-empirical GGA.[Phys. Rev. Lett. 106, 186406 (2011)].

Non-empirical ΤΤΤΤ s GGA “VT84F”

Phys. Rev. B 88, 161108R (2013)

TF TF

s 0 0[ ] ( ) ( ) ( ) ( ) ( ) ( )

( ) (d / d )( , ,T) : ( , )

( )

(d / d )( , ,T) : ( , )

( )

/

ftGGA

F

n d n t F s d n t F s

h t t h ts n n s n n

t

t h ts n n s n n

t

t T T

τ τ σ σ

τ

σ

τ ξ τ ζ

ξ

ζ

= −

−∇ = ∇

∇ = ∇

=

∫ ∫r r

% %

%

F

Form of T-dependent

reduced density

derivative variables

motivated by 2nd

order gradient

expansion.

, ζ, ξ are combinations of

Fermi-Dirac integrals. h%

Finite-T GGA via

generalized

reduced density

variables.

Extension to Finite-T GGA for TTTTs[n] and SSSSs [n]

Phys. Rev. B 86, 115101 (2012)

from Perrot’s (1979) analytic fit;

Beware one obviously wrong

coefficient (exponent) in that fit.

h%

Express finite-T KE with non-empirical VT84F written with the finite-T

reduced density variable for KE. Get entropy functional from an

approximate thermodynamic symmetry.

2

42

84

2 2

84 84

( ) :

( ) : 2 ( )

e 11 (1 e ) 1

1

VT F

VT F VT F

asas

F s

F s F s

cs

cs s

ττ

τ τ

σ

τ

τ τ

σσ τ

−−

=

= −

− + − −

+

Non-empirical GGA

“VT84F” with

finite-T gradient

variables.

Non-Empirical finite-T GGA for TTTTs [n] and SSSSs [n]

Phys. Rev. B 88, 161108R (2013)

Deuterium AIMD pressure vs. material density. New VT84F OFDFT functional compared

to KS, both with same T-dependent LDA XC (“TLDA”). OFDFT had 128 atoms in

simulation cell, KS 3x3x3 BZ or Gamma point. APBEF is built analogously from APBEK,

which does not give a bound ground state. [Phys. Rev. B 88, 161108R (2013)]

All together - Warm Dense Deuterium Eq. of State

AIMD (VT84F & TLDA)

FIGS: Electronic pressure as a function of temperature T.

INSET:

METHOD: Kohn-Sham MD at low-T (Quantum-Espresso)

Orbital-free DFT MD at high-T (Profess@Quantum-Espresso)

PIMC: Hu, Militzer, Goncharov, and Skupsky, Phys. Rev. B 84, 224109 (2011)

Warm Dense Deuterium Equation of State

100%LDA TLDA

tot tot tot

TLDA

tot tot

P P P

P P

∆ −≡ ×

50 100 150 200 250T (kK)

70

80

90

100

110

120

130

Pel

(M

Bar

)

KS (LDA XC)KS (TLDA XC)OFDFT (VT84F+LDA)OFDFT (VT84F+TLDA)PIMC

10 100 1000T (kK)

-1

0

1

∆P

tot/P

tot (

%)

KSOFDFT

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

ρD

=10.0 g/cm3

0 20000 40000 60000 80000T (K)

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Pel

(M

Bar

)

KS (LDA)KS (TLDA)OFDFT (VT84F+LDA)OFDFT (VT84F+TLDA)PIMC

1 10 100T (kK)

0

2

4

6

8

∆P

tot/P

tot (

%)

KS OFDFT

ρD

=0.20 g/cm3

3 3.5 4 4.5 5ρ/ρ

0

0

50

100

150

200

P (

GP

a)

PZ

KSDT

PBE

Holst et. al (2008)

Hydrogen principal

Hugoniot; Initial density

ρ0=0.0855 g/cm3

Hugoniots are Insensitive

0 0

0

1 1 1( )

2P P

ρ ρ

− = + −

E E Two issues: (1) Huge error bars on experiment

(not shown). (2) Cancellation between internal

energy difference and PV work difference

terms in Rankine–Hugoniot equation.

Better GGA X enhancement factors

Differing local Lieb-Oxford

bound enforcement

Different large-s

constraints →

subtle low-s changes

Oddity: All the rather different GGAs we (Mexican collaboration) have

constructed (VMT, VT8,4, PBEmol, PBE-LS) have roughly the same MAE

report card. All obey the Lieb-Oxford bound locally.

( )1/ 3

2 1/3 4/3

4/3

3 3

;

:

4

1 | |( ) :

2(3 )

[ ] ( ) ( ( ))

x

GGA

x x x

c

ns

n

E n c n F s d

π

π

= −

∇=

= ∫

r

r r r

Exact limiting behavior of X potential for

finite systems

v

Try the other large-s limit

2 1/3

1[n; ]

(3 )( )

x r

x s

x

vr

F sc

sπ

→∞

→∞

→−

⇒ →−

r

: xx

Ev

nδ

δ=

Remarks – (i) This constraint contradicts the one used in

VT84) and PBE-LS (a GGA X functional can’t do

everything); (ii) It surely is possible to construct a

density for which this kind of Fx will yield a global LO

bound violation. (iii) If such densities are essentially

unphysical, this form may correspond to a very effective

X functional (example of “design choice”).

“Generalized gradient approximation exchange energy functional with correct asymptotic behavior of the

corresponding potential”, Javier Carmona-Espíndola, José L. Gázquez, Alberto Vela, and S. B. Trickey,,

J.Chem. Phys. 142, 054105 (2015)

GGA X enhancement factor oddity

MAJOR ADVANTAGE of CAP:

correct long-range behavior gives

substantially better TDDFT

polarizabilities & hyper-

polarizabilities.

Oddity is confirmed : All the rather different GGAs we have constructed

(VMT, VT8,4, PBEmol, PBE-LS, CAP) have roughly the same MAE report

card on the standard test sets.

J. Carmona-Espíndola, J.L. Gázquez, A. Vela, and S.B. Trickey,, J.Chem. Phys. 142, 045105 (2015)

CAP

Implementation: finite-T LDA in FEFF9

Approaching correct high-T limit?

Under examination.

Huh?

No XC is the same as Finite-T XC?

L. Calderín, May 2015, unpublished

HCP Beryllium Compton profiles at fixed,

T=0 K crystal structure from various XC

approximations.

Software: PROFESS@QuantumEspresso

• Drives QuantumEspresso with

OFDFT forces from modified

PROFESS

• Includes our finite-T functionals

• Provided as patch files and

libraries, plus test cases.

• Download from

www.qtp.ufl.edu/ofdft

and give it a try (GPL).• V. Karasiev, T. Sjostrom and S.B.

Trickey, Comput. Phys. Commnun.

185, 3240 (2014)

Publications, preprints & software -http://www.qtp.ufl.edu/ofdft

New June 2015! The LDA XC free energy module also is downloadable

Summary

1. Explicit T-dependence in XC is important for accurate prediction of

properties of electron gas at finite-T and for accurate equations of state at

elevated T.

2. The KSDT XC functional fitted to Brown et al. data appears to be

consistent with the recent Schoof et al. data as well.

3. There is major progress on a constraint-based, single-point non-interacting

functional (KS KE plus entropy).

4. Caveat: The VT84F functional does not work at low densities – underlying

reasons are under investigation.

5. Caveat: All good OFDFT non-interacting free energy functionals exhibit

odd behavior with respect so some pseudopotentials.

6. Hugoniot calculations of liquid hydrogen are not sensitive to the

LDA(XC)TLDA(XC) replacement. This may change when our finite-T

GGA XC functional (under development) is used.

7. Progress on lower-rung XC functionals still is possible.

8. Introduction of finite-T functionals into spectroscopic and response

function calculations is in its infancy, with some initial surprises.

Publications, preprints, & software at

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