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Sam Trickey
© 16 June 2015
Finite-temperature Density Functionals –Developments and Computational Consequences
Quantum Theory Project Physics, Chemistry - University of Florida
[email protected]://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