new$xc$free$energyapproximations from$tddft -...

New XC free energy approximations from TDDFTAurora Pribram-Jones

Lawrence Livermore National LaboratoryUC Berkeley Department of Chemistry

Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

IPAM DFT WorkshopAugust 23, 2016

LLNL-PRES-683261

http://newscenter.lbl.gov/2011/10/19/part-i-energy-stars-earth/

August 23, 2016 1

Inertial Confinement Fusion

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Promotional Materials, SLAC, Stanford University (2015)

August 23, 2016 2

“Warm” Dense Planetary Cores

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LLNL, SNL, LBL websitesAugust 23, 2016 3

Flagship Facilities

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• Magnetic pressure of half a billion refrigerator magnets

• 1 million times atmospheric pressure

• Power >2000 lightning bolts

• 2.5 sticks of dynamite § in 1 cc § over a few ns

Real world equivalents

August 23, 2016 IPAM DFT 4

Basic Research Needs for HEDLP: Report of the Workshop on HEDLP Research, DOE (2009)August 23, 2016 5

The Malfunction Junction

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R.F. Smith et al., Nature 511 (2014) 330-333August 23, 2016 6

Probing Planetary Conditions

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R.F. Smith et al., Nature 511 (2014) 330-333August 23, 2016 7

Probing Planetary Conditions

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Simulations are difficult!

• Quantum effects, strong correlation, partial ionization…

• Approximations affect calculated material properties

Simulations are important!

• Data used in core structure modeling, experimental design

• Experiments hard, expensive, limited

August 23, 2016 8

Quantum and Classical

IPAM DFT

Popular, but...

•no explicit temperature dependence in electrons

•computationally expensive

•need TD electrons for response

•no energy transfer between electrons and ions

IPAM, UCLA

August 23, 2016 9

Quantum Molecular Dynamics

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…some progress:

•building “math toolbox” for approximations

•novel orbital-free method

•derived new linear response proof for thermal ensembles

•no energy transfer between electrons and ions IPAM, UCLA

August 23, 2016 10

Quantum Molecular Dynamics

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Heating Things Up

a highly accurate density approximation in one dimension to illustrate our general result,

and (iv) perform (orbital-free) PFT calculations in the WDM regime. Our method gen-

erates highly accurate density and kentropy approximations, skirts the need for separate

kentropy approximations, provides a roadmap for systematically improved approximations,

and converges more quickly as temperatures increase while maintaining accuracy at low

temperatures. At the same time, it bridges low and high temperature methods, and so is

uniquely suited to WDM.

7.2 Theory

At non-zero temperature, the energy is replaced by the grand canonical potential as the

quantity of interest[56, 169]. The grand canonical Hamiltonian is written

⌦ = H � ⌧ S � µN, (7.1)

where H, S, and N are the Hamiltonian, entropy, and particle-number operators. In elec-

tronic structure theory, we typically deal with non-relativistic electrons, most commonly

within the Born-Oppenheimer approximation. The electronic Hamiltonian (in atomic units

here and thereafter) reads

H = T + Vee + V , (7.2)

where T denotes the kinetic energy operator, Vee the interelectronic repulsion, and v(r) the

static external potential in which the electrons move. (We suppress spin for simplicity of

notation.) The grand canonical potential can be written in terms of potential functionals

165

a highly accurate density approximation in one dimension to illustrate our general result,

and (iv) perform (orbital-free) PFT calculations in the WDM regime. Our method gen-

erates highly accurate density and kentropy approximations, skirts the need for separate

kentropy approximations, provides a roadmap for systematically improved approximations,

and converges more quickly as temperatures increase while maintaining accuracy at low

temperatures. At the same time, it bridges low and high temperature methods, and so is

uniquely suited to WDM.

7.2 Theory

At non-zero temperature, the energy is replaced by the grand canonical potential as the

quantity of interest[56, 169]. The grand canonical Hamiltonian is written

⌦ = H � ⌧ S � µN, (7.1)

where H, S, and N are the Hamiltonian, entropy, and particle-number operators. In elec-

tronic structure theory, we typically deal with non-relativistic electrons, most commonly

within the Born-Oppenheimer approximation. The electronic Hamiltonian (in atomic units

here and thereafter) reads

H = T + Vee + V , (7.2)

where T denotes the kinetic energy operator, Vee the interelectronic repulsion, and v(r) the

static external potential in which the electrons move. (We suppress spin for simplicity of

notation.) The grand canonical potential can be written in terms of potential functionals

165

Mermin, N.D. Phys. Rev. A, 137: 1441 (1965).Pittalis, S. et al. Phys. Rev. Lett., 107: 163001 (2011).

Grand canonical potential operator

Electronic Hamiltonian


Entropy and Statistics

ground-state problem. The former is written

⌦ = H � ⌧ S � µN, (3.83)

where H, S, and N are the Hamiltonian, entropy, and particle-number operators. The crucial

quantity by which the Hamiltonian di↵ers from its grand-canonical version is the entropy

operator:5

S = � kBln� , (3.84)

where

� =X

N,i

wN,i| N,iih N,i| . (3.85)

| N,ii are orthonormal N -particle states (that are not necessarily eigenstates in general) and

wN,i are normalized statistical weights satisfyingP

N,i wN,i = 1. � allows us to describe the

thermal ensembles of interest.

Observables are obtained from the statistical average of Hermitian operators

O[�] = Tr {�O} =X

N

X

i

wN,ih N,i|O| N,ii . (3.86)

These expressions are similar to Eq. (3.53), but here the trace is not restricted to the ground-

state manifold.

In particular, consider the average of the ⌦, ⌦[�], and search for its minimum at a given

temperature, ⌧ , and chemical potential, µ. The quantum version of the Gibbs Principle en-

sures that the minimum exists and is unique (we shall not discuss the possible complications

5Note that, we eventually choose to work in a system of units such that the Boltzmann constant is kB = 1,that is, temperature is measured in energy units.

72


⌦ = H � ⌧ S � µN, (3.83)



operator:5

S = � kBln� , (3.84)

where

� =X

N,i

wN,i| N,iih N,i| . (3.85)






O[�] = Tr {�O} =X

N

X

i

wN,ih N,i|O| N,ii . (3.86)


state manifold.





72


⌦ = H � ⌧ S � µN, (3.83)



operator:5

S = � kBln� , (3.84)

where

� =X

N,i

wN,i| N,iih N,i| . (3.85)






O[�] = Tr {�O} =X

N

X

i

wN,ih N,i|O| N,ii . (3.86)


state manifold.





72

Pittalis, S. et al. Phys. Rev. Lett., 107: 163001 (2011). Pribram-‐Jones et al., “Thermal DFT in Context,” Frontiers and Challenges in Warm Dense Matter, Springer Publishing (2014), p 25-‐60.

Entropy operator:

Statistical operator:

Observables:

August 23, 2016 13

Finite-‐temperature Kohn-‐ShamMap interacting system to non-interacting system with same density.

Kohn and Sham, 1965.

IPAM DFT

Free energies: Helmholtz and XC

Pittalis, S. et al. Phys. Rev. Lett., 107: 163001 (2011). Pribram-‐Jones et al., “Thermal DFT in Context,” Frontiers and Challenges in Warm Dense Matter, Springer Publishing (2014), p 25-‐60.

Temperature-dependent free energy:


Kinetic, potential, entropic exchange-correlation:

Correlation RelationsCorrelation free energy: kentropic, potential, kinetic, entropic


APJ and K. Burke, Phys. Rev. B 93, 205140 (2016)

Correlation free energy: kentropic, potential, kinetic, entropic

Combine with ACF to get a set of relations, such as:

ACF via scaled density/temperature


Change of variables yields thermal connection formula:

Combine finite-temperature ACF (Pittalis, et al., 2011)

with coupling constant-coordinate-temperature scaling (Pittalis, et al., 2011)


Thermal Connection Formula

August 23, 2016

• Relates exact XC free energy to high temperature, high density limit• Need knowledge of XC potential energy at scaled densities, not at scaled interaction strengths

• Reduces to plasma physics coupling-constant relation for uniform systems

• Generalization of plasma physics formula to density functionals and inhomogeneous systems

IPAM DFT 17


An Application

Useful for computation and theory:

•Generates new XC approximations for FT DFT•Provides link between finite-temperature and infinite-temperature limit


Using finite-temperature fluctuation-dissipation theorem for the correlation free energy in terms of the thermal density-density response function:

APJ, P.E. Grabowski, and K. Burke, Phys. Rev. Lett. 116, 233001 (2016)

Approximations to thermal XC kernel:

• = 0 à thermal RPA

•

à Approximate

XC ApproximationsExact expression, as long as exact thermal kernel is used:


APJ, P.E. Grabowski, and K. Burke, Phys. Rev. Lett. 116, 233001 (2016)

AcknowledgmentsMentors and Collaborators

• F. Graziani (LLNL), E. Schwegler (LLNL), M. Head-Gordon (UCB), K. Burke (UCI), A. Cangi (MPI), Z.-Y. Yang (Temple), C.A. Ullrich (UMC), J.C. Smith (UCI), D. Feinblum (UCI), S. Pittalis (CNR-NANO), E.K.U. Gross (MPI), P.E. Grabowski (UCI/LLNL), R.J. Needs (Cambridge), J.R. Trail (Cambridge), D.A. Gross (UCI), M.P. Desjarlais (SNL), K.R. Cochrane (SNL), M.D. Knudson (SNL)

Funding Sources• Lawrence Fellowship, Lawrence Livermore National Laboratory•UC President’s Postdoctoral Fellowship, UC Berkeley Chemistry•Quantum Simulation Group and WCI, LLNL•Krell Institute and the DOE CSGF (DE-FG02-97ER25308)


Summary

August 23, 2016

For further information• Linear response of thermal ensembles: A. Pribram-‐Jones, P.E.

Grabowski, and K. Burke, Phys. Rev. Lett. 116, 233001 (2016).• Thermal connection formula and exact conditions: A. Pribram-‐Jones

and K. Burke, Phys. Rev. B 93, 205140 (2016).• Thermal DFT: A. Pribram-‐Jones, S. Pittalis, E.K.U. Gross, K. Burke,

Frontiers and Challenges in Warm Dense Matter, Springer Publishing (2014), p 25-‐60.

• Quirky overview of DFT: A. Pribram-‐Jones, D.A. Gross, K. Burke, Ann. Rev. Phys. Chem 66 (2015).

IPAM DFT 21

• Exact conditions relating components of correlation free energy• Thermal connection formula: adiabatic connection via

temperature integral• Can generate XC free energy approximations from thermal kernel

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