The HIG of tdDFT - UCLAhelper.ipam.ucla.edu/publications/pltut/pltut_10579.pdf · IPAM HEDP Long...

Sam Trickey

Quantum Theory Project Physics, Chemistry - University of Florida

IPAM HEDP Long Program Tutorial 16 March 2012

http://www.qtp.ufl.edu/ofdft [email protected]

©March 2012

The HIG of tdDFT

Sam Trickey

The Highly Important Games of

Temperature-dependent DFT and Hartree-Fock

Quantum Theory Project Physics, Chemistry - University of Florida

IPAM HEDP Long Program Tutorial 16 March 2012

http://www.qtp.ufl.edu/ofdft [email protected]

©March 2012

Funding: U.S. DoE DE-SC 0002139; CONACyT (México)

Collaborators (Orbital-free KE & FE functionals):

Jim Dufty (Univ. Florida)

Frank Harris (Univ. Utah and Univ. Florida)

Keith Runge (BWD Associates)

Tamás Gál (Univ. Florida)

Vivek Kapila (Univ. Arizona)

Valentin Karasiev (Univ. Florida)

Travis Sjostrom (Univ. Florida)

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Collaborators (Materials and molecular calculations):

Jon Boettger (Los Alamos National Laboratory)

PUPIL (Joan Torras, Erik Deumens, Ben Roberts, Gustavo Seabra, Oscar Bertran)

deMon2k (Andreas Köster, Gerald Geudtner, Patrizia Calaminici, Carlos Quintanar, …)

Notker Rösch and company, TU München

Collaborators (XC functionals):

Klaus Capelle (IFSC, Univ. do ABC, Brazil)

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

Alberto Vela (Cinvestav, México D.F.)

Victór Medel (Virginia Commonwealth Univ.)

Mariana Odashima (MPI Mikrostrukturphysik, Halle Germany)

Juan Pacheco Kato (Univ. Guanajuato, México)

Jorge Martín del Campo Ramirez (UAM- I, México D.F.)

Challenge: Orbital-free functionals

Progress: New functionals, new reference calculations

Topics

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Motivation: Physics of Warm Dense Matter: see M. Desjarlais on 12 Mar.

and F. Graziani on 13 Mar.

Survey

Problems of naïve variational theory (Zero T, finite T)

Implications for molecular dynamics on complicated materials

Density functional theory basics – Zero T, finite T

Finite-T Hartree-Fock

Open questions for WDM applications

1 2( , , , )I I I Nm V R R R R

ion ionV E R R R

Molecular dynamics is a major simulation tool; See M. Murillo 14, 15 March.

MD implements Newton’s 2nd Principle:

Bottleneck: the potential. For electrons & ions, that

should be the Born-Oppenheimer free- energy

surface

WDM Systems and Simulation Bottleneck

QTP

Adapted from R. Redmer, LBNL WDM, 2011

H-He (8.6%) @ 1 Mbar, 4 kK

F({R}) is the electronic

free energy. Quantum Stat Mech

$ $ $ $ or € € € € € Quantum Stat Mech Box

WDM: NO small

parameters to do the QM!



Topics

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Survey


Implications for molecular dynamics on complicated systems




• The time-independent Schrödinger equation & Hamiltonian for the

Ne-electron problem with N fixed nuclei (Hartree atomic units):

Many-electron Quantum Mechanics

1 0 1 1 0 1 10;

,21 1

2 21

1 ,

12

1

ˆ , , , , ; , , , ;

1ˆ ,

: ,

e e e e e

e e e

e

e e

N N N N N

N N N N

IN i

i i I i ji I i j

N N

i i j

i i j

Z

h g

R R

R

r r r r R r r R

r rr R r r

r r r

The many-electron wave function is anti-symmetric on interchange of a pair:

0 1 1 0 1 1, , , , , ; , , , , , ;e e e ek k j j N N j j k k N N r r r r R r r r r R

Ne = number of electrons

N = number of nuclei

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Sum of 1- & 2-body

Hamiltonians; symmetric

on interchange of a pair

Nuclear (ion) positions (Born-Oppenheimer approx.);

suppressed hereafter unless directly relevant.

2

1

1 E 27.2116 eV 1 au.=0.5292 Å

1One-electron KE: r2

electron electron

Hartree

m q

d

Hartree & traditional units:

2

2 / 2 1

1 E 13.6058 eV 1 au.=0.5292 Å

One-electron KE: r

electron electron

Rydberg

m q

d

Rydberg & traditional units:

Why many-electron problem is tough; T=0 K Variational Principle

The variational principle

So why not just guess a trial wave function with a bunch of physically and

chemically plausible parameters and do the minimization calculation by brute

force numerical quadrature?

*

1

0 *

1

e

e

trial trial N

trial

trial trial N

d d

d d

r r

r r

*

*

1

0 0 0

* 0

( ) 0e

trial trial

trial trial N

trial

trial

H dr dr

H

Lagrange Multiplier to

preserve normalization

is equivalent to the Schrödinger equation. That is, the Schrödinger eqn. arises from requiring the first-order variation of Etrial to be zero:

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Approximate vs. exact wave functions:

HF is approx

0

1

2

ψ 1,2, , 1 2

1

1

2

2

trial e a b z e

A

B

b z e

a z e

N c N

c N

c N

All “excited” Slater

determinants

Full CI

Question 1: is this expansion exact? YES

Question 2: is this expansion rapidly converging? NO

Question 3: can the expansion be truncated and used as

a variational expression? Well YES; scheme is called MCSCF

(multi-configuration self-consistent field). But still have issues:

Selection of orbitals φ, parameters in them, … . Did we keep

the variation principle? Are we capturing the physics and

chemistry? Can we do the integrals? Credit: So Hirata

J.C. Slater (by SBT)

Examples, T=0 K Variational Principle

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ψ 1,2, , 1 2trial e a b z eN N ↑Anti-symmetrizer

ˆ1,2, , exp 1 2trial e a b z eN N T

Jastrow correlated wave function; much

used in Monte Carlo. Note: nodes are

fixed by the determinant. Explicit “r12”

dependence makes integrals difficult.

1,2, , ( ) 1 2trial e ij a b z e

i j

N f r N

1

1 1 1

† †

1

ˆe

j

j j j

Na a

i i a i a i

j

t a a a a

T

All possible orbital

substitutions

in the determinant

More examples, T=0 K Variational Principle

Coupled Cluster (CC) Ansatz ( single

reference). t- amplitudes are not

determined variationally in practice.

Keeping all substitutions through

doubles scales computationally as Ne6


The variational principle

So why not just guess a trial wave function with a bunch of physically and

chemically plausible parameters and do the minimization calculation by brute

force numerical quadrature?

*

1

0 *

1

e

e

trial trial N

trial

trial trial N

d d

d d

r r

r r

*

*

1

0 0 0

* 0

( ) 0e

trial trial

trial trial N

trial

trial

H dr dr

H

Lagrange Multiplier to

preserve normalization

is equivalent to the Schrödinger equation. That is, the Schrödinger eqn. arises from requiring the first-order variation of Etrial to be zero:

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Too many arbitrary forms, too many diverse parameterizations to guess,

too many parameters to vary, too many variables of integration!

Computational cost OK for one or a few molecules, NOT for Born-

Oppenheimer Molecular Dynamics

For a normalized ground state, the energy is

*

0 0 1 0 1 1, , , ,e e eN N NH d d r r r r r r

2-particle reduced density matrix “2-RDM”

State function: anti-symmetric under particle exchange (Fermions!)

Hamiltonian: symmetric sum of 1- and 2-body interactions.

We end up doing integrals over all but one or two of the coordinates just to

eliminate them (change of variables), thus (introducing spin here):

1 1;0 01 1

2 2

0

0

(2)

0 1 1 1 1 1 2 1 2 1 2 1 2

(2) *

1 2 1 2 0 1 1 2 2 3 3 0 1 1 2 2 3 3 3

*

1 1 0 1 1 2 2 3 3

( ) | ( , ) |

1| : , , , , , , , ,

2

| : , , , ,

N e N e ee e

N

x xx x

x x

e e

N N N

e

h x x dx g x x x x dx dx

N Nx x x x dx dx

x x N

r r r

r r r r r r r r

r r r r 0 1 1 2 2 3 3, 2 3, , ,e e e ee

N N N Ndx dx dx r r r r

1-particle reduced density matrix “1-RDM”

Sum over spins

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Guessing a 2-RDM and using it in the variational principle doesn’t work either.

This is the notorious N-representability problem: How can we know (NASC) that

a trial 2-RDM comes from a Ne -Fermion wave function? The answer is known

but using it is essentially as intractable as using the many-electron wave function.

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( ) ( )

( )

( ) ( )( )1

0

( ):

oneelementof complete set fermio

( ) ( ) l

n stat

n

( ) : ( (

s

) ; 1/ )

e

e e

e

e

e

e

d nN

N

ext B

N N

i i

i

N

i

N

e

p V Tr e

v

A A

k

Tr

N

T

r r r

r r

∣ ∣

Why many-electron problem is tough; T≠0 K

Grand canonical ensemble



Topics

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Survey






Hohenberg-Kohn Theorems – consider the Ne -electron Hamiltonian, which includes

an external potential (for us, the nuclear-electron attraction)

,

21 12 21

1 ,

1ˆ ,

ˆ ˆˆ ˆ ˆ ˆ:

e e e

e

N N N N

IN i

i i j i I i Ii j

ee ext ee Nuc electr

Z

Rr r

r Rr r

HK-I: “A given ground state density n0 (r) determines the ground state

wave function and hence all the ground state properties of an Ne -electron system.”

Original Proof: by contradiction [P. Hohenberg and W. Kohn, Phys. Rev. 136,

B864 (1964)

Modern Proof: Levy-Lieb constrained search (sequential application of the

Variational Principle, density by density)

[M. Levy, Proc. Natl. Acad. Sci. USA 76, 6062 (1979); L. Lieb, Internat. J. Quantum Chem.

24, 243 (1983)]

Density Functional Theory Rudiments

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DFT Rudiments: HK-I Proof by Contradiction

0ˆ ˆ,ext ext n

1. Consider a non-degenerate ground state. Suppose that there are two external

potentials that yield the same ground-state density:

2. Then by the variational principle

0 0 0 0 0 0 0

0 0 0

0 0

ˆ ˆ

ˆ ˆ

ext ext

ext ext

ext extd n v v

r r r r

3. But the argument can be done reversing primed and unprimed quantities.

This gives

0 0 ext extd n v v r r r r

4. Adding these last two results gives the contradiction 0 0 0 0

5. Therefore the ground state density determines the external potential and

hence, the ground state wave function.

Remark: this proof does not address the possibility of a density n which is

not associated with any vext (the v-representability problem)

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DFT Rudiments: HK Theorems & Equivalence Classes (Constrained Search)

Mel Levy’s idea: take all the wave-functions that give the same density and vary

over them first. Then vary the density. The variational expression becomes

*

1

*

0 1

*

1

1

min min ( )

min min ( ) ( )

e

etrial

etrial

trial trial N

trial trial ee trial Nn n

trial ee tri

Ne

Nal Nn n

e

d d

d d

d d v n d

r r

r r

r r r r r

Just a rewriting but establishes the Hohenberg-Kohn theorems

by constrained search.

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HK-I: “A given ground state density n0 (r) determines the

ground state wave function and hence all the ground state

properties of an Ne -electron system.”

HK-II: “For such a system in an external potential vext(r), there

is a universal (i.e., independent of vext ) functional F[n] with

the following properties:

0

0 0( )

min

ext

ext ext

v

v

e

r

x

vn

tE n F n d n E

E n E E n

v

r r r

Mel Levy, Perdew Fest, Mar. 2008

DFT Rudiments: HK-I Proof by Constrained Search ← blue heading means supplemental

2. Minimize this functional over all the Ne electron states that give n0

0

0

ˆ ˆmin

|

ee

n

r r r

E

3. Then at most 0 is a normalized linear combination of ground states, since

the external potential contribution to the total energy depends only on the

density:

0ext extd n v r r r

4. Therefore the ground state density determines the ground state wave function.

1. Consider the positive operator ˆ ˆee and form

ˆ ˆee E

Remarks: No v-representability problem, no restriction to non-degenerate

ground states. We have not shown that, subject to mild conditions, every density

is associated with at least one Ne electron state. In fact, there are infinitely many

such states for each density (Harriman, Phys. Rev. A 24, 680 (1981)).

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| 1-particle reduced

density matrix

r r

HK-II: “For an Ne –electron system with an external potential vext(r), there

exists a universal (i.e., independent of vext ) functional F[n] with the

following properties:

Original Proof: Essentially by announcement; the paper assumes that

HK-I holds for non-ground-state densities.

Modern Proof: Levy-Lieb Constrained Search

Form

DFT Rudiments : HK Theorems from Constrained Search (continued)

; ;ˆ ˆˆ ˆ: min ee min n ee min n

nF n

0 0 0 ; ;ˆ ˆˆ ˆ ˆ ˆ

:ext

ee ext min n ee ext min n

ext v

E

F n d n v E n

r r r

0

0 0( )

min physically acceptable ( )

ext

ext ext

v ext

v vn r

E n F n d n v E

E n E E n n

r r r

r

Then

which is the first piece of the theorem.

QTP

Remark: This F[n] , the Levy functional, is NOT the same mathematical object as the

one originally defined by HK. In particular, there is no v-representability issue with

regard to the variation over n nor any restriction to non-degenerate ground state. This

functional does fulfill the role of the one defined by HK. It has its own problems

however. It is not convex and has some unpleasant functional derivative problems. For

most purposes, the Levy functional is good enough. See Lieb (1983).

Now apply the variational principle again

DFT Rudiments: HK-II proof (continued)

0 0

0 0

0 0 0 ; ;

0 0 ; ;

ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆˆ ˆ

ee ext min n ee ext min n

ee min n ee min n

E

But the definition of min;n means that 0 0; ; 0 0

ˆ ˆˆ ˆmin n ee min n ee

Convexity: 1 2 1 2(1 ) [ ] (1 ) [ ]A an a n aA n a A n

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0 0 0

0

0 0 ; ; 0 0

0 0 0 0

ˆ ˆ ˆˆ ˆ ˆ

ˆ ˆ [ ]ext

ee min n ee min n ee

ee v

F n

F n E n E

Taken together, these give bracketing inequalities

0

0 0( )

min physically acceptable ( )

ext

ext ext

v ext

v vn r

e

E n F n d n v E

E n E E n n

d n N

r r r

r

r r

Recall HK-II

Do the variation with a Lagrange multiplier for fixed particle number

0ext e

ext

F n d n v d n N

Fv

n

r r r r r

rr

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DFT Rudiments: Existence Thms and Proceeding Constructively (Kohn-Sham)

DFT Rudiments: Existence Thms and Proceeding Constructively (Kohn-Sham)

The big challenge: HK-II says that F[n] exists but does not give a form that can

be used to do the variational problem just displayed.

But we do know how to form F[n] explicitly for a system of non-interacting

fermions with the same density set by some vKS as follows:

; ;

non-inter

non-inter ; ;

ˆ ˆˆ ˆ: min

ˆ ˆˆ 0 : min

KS

ee min n ee min nn

v KS

ee S min n min nn

F n

E n F n d n v

F n n

r r r

Non-interacting fermions means

min;n is a Slater determinant. ; 1

11, , det

! emin n e Ne

NN

*

1

1 1 1 2(2)

1 2 1 2

2 1 2 2

|

| |1| det

| |2

eN

SD j

j

SD SD

SD

SD SD

x x x x

x x x xx x x x

x x x x

The Slater determinant 1- and 2-RDMs are

So the KS kinetic energy is * 2

1

1[ ]

2

eN

s s j j

j

n n d x x

r

How is this useful?

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DFT Rudiments: Existence Theorems and Proceeding Constructively (cont’d.)

By definition, the non-interacting (independent particle) system must have energy

[ ]s S KSE n n d n v r r r

Again, do the variation with a Lagrange multiplier for fixed particle number

0S KS e

SKS

n d n v d n N

vn

r r r r r

rr

Choose the Lagrange multiplier to be the same as in the original system

by setting the zero of the potential vKS (adding a constant, if necessary).

So we may be able to map the problem into that of constructing vKS.

How can that be done?

Remark: v-representability of the ground-state density n0 has been

reintroduced as an assumption.

QTP

21

2

1 21 2

1

. .

1 2

2

(depending on spins)

( ) ( )1

2

;

0 1,2

ext

F

v coul ext

S ee ext

S ee ext

S

quant chem

x c

j j

j

orrel S

xc

ee

N

j j

j

j

E n n E n E n

n E n E n

n E n E n

n n d

n nE

E n E n n n

E

n d d

n n

n

n

r r

r rr r

r r

r r

The non-interacting system becomes useful as follows. Rearrange the

interacting system functional as follows:

DFT Rudiments: Kohn-Sham Rearrangement

QTP

Lu Jeu Sham, Sanibel, 2010.

What we have done is

0

xc

ext e

See ext

F n d n v d

E

n N

v vn n

r r r r r

r rr r

:extv ext S ee ext

S ee

xc

xc

EE n F n E n n E n E n

F n n E

n

E nn

Thus, the variation principle and Euler equation for the physical system become

S

KSvn

r

rComparison with the Euler Equation for the KS system

shows that

KS ee exx

tcE

v v vn

r r r

r

New task: construct Exc

QTP

DFT Rudiments: Details of the Kohn-Sham Construction -

The Kohn-Sham equation (non-spin-polarized for simplicity) is:

221

2 2 1 1 1

1 1 2

( ); ;

R KSIxc j j j

I I

nZd v

rr r r R r R

r R r r

xcxc

Ev n

n

with the eXchange-Correlation potential given by

Remark: Exact exchange in DFT is defined by the KS determinant, NOT the

Hartree-Fock determinant.

Rudiments of K-S DFT

QTP

2

12

ˆ: min

the Kohn-Sham determinant

Sn

j j

j

n

n d

r r

Everything is known except for the XC functional EXC . It is known exactly

only in a few special cases.

Remark:

xcxc

Ev n

n

1/3

constant original LDA, also

dimensionless functional of density

only later LDAs (Hedin-Lundqvist,

Perdew-Zunger, Vosko-Wilk-Nu

( ) [ ( )]

( ) ( ) [ ( )]

Rxc xcLDA

LDA

LDA

LDA

X

E n dr n U n

d n n F n

F

F

R

R

R R

r r

r r r r

ssair)

There are several classes of approximate functionals that

are demonstrably successful at different levels of prediction:

◊ Local Density Approximation

Rudiments of K-S DFT

QTP

◊ Generalized Gradient approximations add

terms dependent on density gradients:

1/3

2 1/3 4/3

( ) [ ( ), ( )]

( ) ( ) [ ( ), ( )]

( ) = 2(3 )

GGA

xc xcGGA

GGA

E n d n U n s

d n n F n s

s n n

R

R

R

R R

r r r r

r r r r r

r

a) FGGA determined by constraints, scaling inequalities, other exact

results: Perdew-Wang 91, Perdew-Burke- Ernzerhof, VMT, VT{84}

PBEmol, (Mexican DFT collaboration), etc.

b) FGGA determined in part by fitting to atomization energies: Becke etc.

QTP

Free Energy Density Functional Theory

Eion-ion omitted for brevity.

1 2 21 1

.

2 2

1 22

1 2

.

1

[ ]

[ ]

: [ ]

: 1 ex

[ ] [ ]

p

( ) ( )1;

2

ionv coul ion

S s ee ion

S

S s ee ion

S j j j j

j j

ee

quant chem

x correl s

xc

n n E n E n T n

n T n E

E n E n n n T n

n E n

n T n E n E n

n d f d

n nE n d

n

n

d

r r r r

r rr r

r r

F

F

1 2 2

( ) ( )

: 1 exp

1/

ion ion

j j j j

j j

B

E n d n v

n f

k T

r r r

r r r

Mermin-Hohenberg-Kohn finite-temperature DFT

Basic equations are:

1 2 2

: 1 exp

xcxc

j j j j

j j

v nn

n f

r r r

Otherwise the KS equations look the same as in the ground state.

K-S Free-energy DFT

QTP

QTP

Thermal Hartree-Fock Approximation

The trace “Ne,SD” is over all Ne-electron Slater determinants Φ.

ˆ( ( ) ( ) )( , )1

0

21

2

( | ) ( ) ( )

( | ) ln Tr

[ ]

: ; [ ] : ln (1 ) ln(1 )

ee ione

e

HF

HF HF ion

H d v nN SD

HF

N

HF HF ee ex ion

HF j j HF B j j j j

j

d v n

e

T E E E n

f d k f f f f

r r r

r r r

r r

F

F T

T

* *

1 2 1 2

1 2

1 2

* *

1 1 2 2

1 2

1 2

1

( ) ( ) ( ) ( )1

2

( ) ( ) ( ) ( )1

2

: 1 exp ; : 1/

i j

j

i j i j

ee i j

ij

i j i j

ex i j

ij

j j B

E f f d d

E f f d d

f k T

r r r rr r

r r

r r r rr r

r r

Mermin [Ann. Phys. (NY) 21, 99 (1963)] proved that the Finite Temperature

Hartree-Fock approximation is the “obvious” generalization of ground-state

Hartree-Fock theory. Basic equations are:

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Thermal Hartree-Fock Approximation - continued

*

2 22

2

2

*

2 2

2

2

1

2

( ) ( )( ) ( ) ( ) ( )

( ) ( )( )

i j

j j

i i ion i j i

j

i j

j j

j

v f d

f d

r rr r r r r

r r

r rr r

r r

=

Variational minimization leads to the obvious generalization of the

ground-state HF equation:



Topics

QTP



Survey






DFT for WDM – Major Open Questions

QTP

• Observation: OF-DFT may be necessary, not merely an attractive option for

WDM. Large numbers of small occupation numbers make the number of

MD steps comparatively small for conventional KS at finite T .

• Technical issue: pseudopotentials? (mean ionization state?)

[ , ] ?

[ , ] ln (1 ) ln(1 ) ?

[ , ] ?

s

s B j j j j

j

xc

n T

n T k f f f f

n T

T

• Temperature and density dependence of Fxc ?

• Implicit temperature dependence of ground-state Exc[n] used at finite T. Is Fxc [n, T]≈ Exc[n(T)] good enough?

• Same question for ground state OF-KE functionals: Ts[n,T] ≈ Ts[n(T)]?

• Orbital-free free-energy functionals in WDM regime -

Functional forms for tS [n, T] , sS [n,T] , uxc [n,T]?

Accurate ground-state limiting forms?



Topics

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Survey






• The KS eigenvalue problem is the high-cost computational step in B-O

MD. At every step:

2

; ; ;

2

; ;

for each configuration( ) ( ) | ( ) |

ˆ ( ) ( )

1ˆ ( ; )2

( ; ) ( ; ) ( ) ( )

k kk

KS

k k k

KS

KS

KS ion H xc

n f T

h

h v

v v v v

R R

R R R R

R

R R

r r R

r r

r R

r R r R r r

Too slow! Order NeM with 2.6 ≤ M ≤ 3 at best. Requires heroic calculations

(some of which have been done – need for the data is HIGH!)

There are “order-N” approximate methods but they are not general

(additional assumptions, e.g. about basis locality, etc., limit applicability).

B-O Forces from DFT

→ So there’s both urgency and opportunity for better methods. Or do we just

wait on bigger, faster machines?

QTP

“Everyone knows Amdahl’s Law but quickly forgets it”

T.Puzak, IBM (2007)

QTP

1

(1 )

speedup; fraction parallel; number of processors

PP

N

P N

S

S =

99.9 % parallel, 2048 processors ,

speedup ≈ 675

IEE Computer, 41, 33 (2008)

Also see www.cs.wisc.edu/multifacet/papers/amdahl_multicore.ppt

21

2 ( )

( )[ ( )]

[ , ] ln (1 ) ln(1 )

[ , ] [ ( ), ]

[ ( ), ]

k k

k

B k k k k

k

s

K

B

sS

s S

S

f T d

v

k f f f

nn

n T k f

n T d t n T

d s n T

r

r rr r

r r

T

T

The Ideal - Orbital-free DFT

With the K-S density kernels ts, sS and a good density-dependent (NOT orbital-dependent) FXC , DFT B-O forces would be simple:

Even at T=0, there are many well-known barriers to

getting ts , e.g., Teller non-binding theorem for Thomas-

Fermi-Dirac, failures of TF-von Weizsäcker, ….

L.H. Thomas,

Sanibel, 1975.

Photo by SBT

QTP

5/3 2 2/3

2

3(3 )

10

[ ] : , 0 1

[ ] : ( ) ,

( )1:

8

TF W TF W W W

TF TF TF

W

n

n c d n c

nn d

n

r r

rr

r

Scaled Thomas-

Fermi-von

Weiszäcker

The Other KS Orbital Problem (Jacob’s Ladder)

Biblical (Genesis 28: 11-19)

William Blake, 1800; Also African-

American Spiritual (folk hymn)

• DFT – XC approximations

(John Perdew)

n LSDA

Hartree world

Heaven of chemical accuracy

n GGA

ts and/or 2n meta-GGA

x hyper-GGA

unocc. {φi} generalized RPA, etc.

Red = explicit orbital dependence

J. P. Perdew and K. Schmidt, in Density Functional

Theory and its Application to Materials,

V. Van Doren, C. Van Alsenoy, and P. Geerlings,

Eds., AIP, Melville, New York, 2001

Dilemma: “Everybody” is working on orbital-dependent XC functionals!

See Mexican Group work on “lower rung” XC functionals

QTP

Here, in somewhat sloppy translationally invariant notation, is the basic

theme of the response function approach

2

(r) r (r - r ) (r ) (q) (q) (q)

(r )(r)(r - r ) r (q - q ) (q) (q,q )

(r ) (r )

[ (r)] (q - q ) (q) (q,q )

KS KS

KS KS

KS

KSs s

n d v n v

v vnd

v n n

v nn n n

Orbital-free KE Approaches – Response Function

Result is a set of non-local (two-point) approximations based on Average

Density Approximation, Weighted Density approximation, etc.

Various versions work moderately well for metals.

Another recent version works moderately well for insulators.

The non-locality is too complicated (computationally costly, introduces local

pseudo-potentials) for our purposes

QTP

Orbital-free KE Approaches – Response Function

Density-dependent kernel for semiconductors C. Huang and E.A. Carter, Phys. Rev. B 81, 045206 (2010)

,

,

0

2

2 1/3

4/3

r r (r) (r ) r,r

(r,r ) : (r)(1 ) r-r

(3 )

80.65 0.89

3

s TF W

F

F

n

c d d n n w

w w k x

nk n x

n

QTP

Introducing density-dependence in the kernel “… comes at the expense of

greatly complicating the derivation and making a straightforward numerical

implementation computationally expensive…” Y.A. Wang, N.Govind, and

E.A.Carter, Phys. Rev. B 60, 16350 (1999)

Two-point OFKE Functional Problems

QTP

Density-independent kernels suffer from nonlinear instability “ … in the sense

that the corresponding kinetic-energy functionals are not bounded from below.” X.

Blanc and E. Cancès, J. Chem. Phys. 122, 214106 (2005) N-representability issue?

Multiple parameters without physical constraints.

Different kernels for metals and semi-conductors: but what about pressure-

induced insulator-metal transitions?

Reference density is non-unique for bulk systems and undefined for finite ones.

* 2

i

,

1r (r) (

[ ]

[ ] r [ (r)

[ ] [ ] [ ] [ ]

[ ] r [ (r)] (r) r (

)2

r

] r

)

ee xc ne

sI electronic I KS I I

i

s

e

is

n

S

E n E n E n E n

F E n d v n n d n

n

n

v

d t dn n

n

• Goal: a workable recipe for a one-point ts[n] purely for driving MD. Remember

the ground state formulation:

Orbital-free Density Functional Theory: Single-Point Strategy

• Obviously need analogous progress for T > 0 K: tS [n, T] , sS [n,T] , uxc [n,T]

• Observe: we do NOT seek a KE density kernel that will do everything that is in

the basic DFT theorems

• Assumption: continued progress on pure EXC approximations, i.e., not

hybrids or OEP (optimized effective potential).

QTP

• Desired: ts[n(r)] be no more complicated than GGA EXC (depends on gradient of

the density) or meta-GGA (depends on Laplacian of the density also).

• Key Ingredient - Pauli KE, Pauli potential, and square root of density [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] An exact expression is

[ ] [ ] [ ], [ ] 0s Wn n n n

21(r) (r) (r) (r)

2

(r) 0 r

KSv v n n

v T n

Stationarity of variation with respect to density n yields

Single-point OF-KE Functional Construction

• Useful (but incorrect) clue: “Conjointness conjecture” [H. Lee, C. Lee, and R.G. Parr,

Phys. Rev. A 44, 768 (1991)]. The GGA EX is

4/3

2 1/3 4/3

[ ] r (r) (r)

(r) = 2(3 )

GGA

x x xE n c d n F s

s n n

• Conjointness:

5/3[ ] r (r) (r)

(r) (r)

GGA

s t

t x

n d n F s

F s F s

QTP

SiO Stretch; Total energy vs.

bond length

All six Ts approximations fail to

bind! V violates positivity.

OF-DFT: Test of Existing Single-point (GGA) KE functionals

• Tests of 6 existing functionals, three

conjoint. KS density as input to each.

PW91: Lacks and Gordon, J.Chem.

Phys. 100, 4446 (1994) [conjoint]

PBE-TW: Tran and Wesolowski,

Internat. J. Quantum Chem. 89, 441

(2002) [conjoint]

GGA-Perdew: Perdew, Phys. Lett. A

165, 79 (1992) [conjoint]

DPK: DePristo and Kress, Phys. Rev. A

35, 438 (1987)

Thakkar: Thakkar, Phys. Rev. A 46,

6920 (1992)

SGA: Second order Gradient Approx. TS = TTF + (1/9) TW

J. Comput. Aided Matl. Design 13, 111 (2006)

QTP

Parameterizations

• Tried two simple forms for modified enhancement factors. These forms would

be convenient for use in MD. No guarantee that any of these is optimal.

21

21

( ) 11

jN

PBE N

t j

j

sF s c

a s

2 21 2exp4

1 2(1 ) (1 )a s a s

tF C e C e

N=2 is typical PBE form as also used by Tran & Weslowski

N=3 is the form used by Adamo and Barone [J. Chem. Phys. 116, 5933 (2002)]

N=4 highest tried

• Constrain parameters to 0v

Initial parameterization used

(a) single SiO or

(b) SiO, H4SiO4, and H6Si2O7.

Stretched single Si-O bond in all cases with self-consistent KS densities.

OF-DFT: Modified conjoint KE functionals

QTP

Single bond

stretching

gradient in

H2O.

OF-KE

parameters

from 3-member

training set

(SiO, H4SiO4,

and H6Si2O7)

except PBE2.

NO information

about H2O in

the set.

OF-DFT - Performance of Modified Conjoint, Positive-definite Functionals

All of our functionals give too large an equilibrium bond length.

QTP

OF-DFT - Pauli Potentials Compared

Phys. Rev. B 80, 245120 (2009)

Comput. Phys. Commun. (submitted)

QTP

Pauli potential vθ , vicinity of the

Si site in SiO at R = 1.926 Å.

“KS” is the exact inversion from

the KS solution.

“GGA” is the Tran -Wesolowski

potential.

“mcGGA” is our PBE2 modified

conjoint-form (positivity

enforced).

TF+vW and scaled TF+vW/9 for

comparison.

PROBLEM:

mcGGA Pauli potential vθ singularities at the nuclei. TOO positive!

GGA goes negative. TF+vW/9 does too but TF+vW doesn’t ⇒ any

answer you want with TF+λvW!

Beware of scaled Thomas-Fermi-von Weiszäcker used variationally in

OF-DFT, or even with the actual KS n0 (for a given XC) as

input, can lead to weirdness:

Paul Ayers (Sanibel 2007): “An ab initio quantum chemist will

wonder- “Is an N-representability constraint missing?”

Orbital-free KE – Problems with Simple Approaches

“If so, should we surrender? N-representability problems are very

difficult ….”

Functional

(all simple LDA XC) Etotal (Hartree)

TFλW, λW = 1 -85.734

TFλW, λW = 1/5 -128.801

TFλW, λW = 1/9 -139.887

KS (13s8p GTO) -127.484

TFλW from G. Chan, A. Cohen, and

N. Handy, J. Chem. Phys. 114, 631

(2001)

QTP

[ ]

[ ] [ ] [

[ ]

]

TFW

ee xc

TF W

ne

E n

E n E n E

n

n

0, 0,[ ] [ ]TF W TF W KS KSE n E n

[ ]:TF W TF W Wn

Singularities are not fatal.

Left: Bcc Li lattice constant; Right fcc Al lattice

constant. mcGGA is our PBE-2,

parameterized to SiO implemented in modified

PROFESS code. GGA=Tran-Wesołowski

QTP

Are Positive Singularities Fatal to mcGGA?


Solving the OFKE Euler equation with a KS code?

QTP

[ ] [ ] [ ], [ ] 0s Wn n n n

Why did we switch to the PROFESS code?

21(r) (r) (r) (r)

2

(r) 0 r

KSv v n n

v n

Chan, Cohen, and Handy

J. Chem. Phys. 114, 631

(2001)

Levy, Perdew, and

Sahni, Phys. Rev. A 30,

2745 (1984)

Stationarity of variation with respect to density n yields

QTP

TFDvW, λW = 1

Etotal (Hartree)

Chan, Cohen

& Handy

Numerical KS

code

GTO KS

code

H atom -0.2618 -0.26183 -0.25997

Li atom -4.1054 -4.10542 -4.09635

Ne atom -85.7343 -85.73445 -85.73004

Solving the OFKE Euler equation with a KS code?

However, with simple linear mixing of densities and

starting from the pure von Weizsäcker KE, the

iterative convergence is very slow and unpredictable

Even a numerical grid diatomic molecule code is

slow to converge and unpredictable.


Current work is to remove mcGGA singularities by partial resummation of

gradient expansion to respect Kato cusp condition. One functional has

been explored a little.

OF-DFT - Removing Pauli Potential Singularities

Phys. Rev. B 80, 245120 (2009)

H2O molecular total energy

as function of one bond

KS = reference calc (source of

input density; middle curve)

DPK = DePristo-Kress OFKE

functional reparameterized

for this case (2nd from top)

MGGA = meta-GGA functional of

Perdew and Constantin (top) Phys. Rev. B 75, 155109 (2007).

Thakkar functional (lowest curve)

RDA = This work (second lowest)

QTP

• The model problem is a hard-walled rectangular parallelepiped

containing a few (1-8 for now) hydrogen atoms.

• Initial exploration with cubic box, edge-length L.

• A few fixed atomic positions are sampled.

• Box size is from 1 au3 (L = 1 au) to free-system limit (L → ∞).

• Temperature range: 0 ≤ T ≤ 200,000 K.

QTP

Confined System – finite-T Hartree-Fock

Requirements:

• Match boundary conditions.

• Represent ground state and sufficient number of excited states at

different box sizes.

• Allow for efficient calculation of 2-electron integrals.

Basis:

Cartesian Gaussians truncated to

match BCs.

Coefficients a0 , aL set by requiring

each basis function to be continuous.

QTP

Basis Set – Confined System

0 0( ) : ( ; ) 0

: ( ; )x

n n

box c c

n

L c L c x

g x a g x x x x

a g x x x x L

0 (0) ; : ( ):x

n n

L xg g L 2( )

( ; ) ( ) cx xn n

c cg x x x x e

Technical issues and resolution:

• Continuity of first derivative at matching point and corrections to

piecewise evaluation of KE matrix elements – works with a non-zero

piece-wise correction for p-type functions

• Efficient calculation of 2- electron integrals – finite-range integrals of

Gaussians and error functions done analytically as much as possible,

rest via Gauss-Legendre quadrature.

QTP

Basis Set - continued

The ion configuration. Each ion is at the center of its own octant.

(An initial choice for exploration. In general, the ion configuration

is arbitrary.)

QTP

Results – 8 Atom Array of Cubical Symmetry in a Cubical Box

Comparison of Exchange Functionals

QTP

Ex from finite-temperature Hartree-Fock (FTHF)

Ex from ground state LDA with FTHF density

Ex from temperature-dependent LDA (Perrot & Dharma-Wardana 1984

parameterization) with FTHF density.

All for 8 atoms, cubic symmetry, in cubic box, L = 6 bohr

Phys. Rev. B 85, 045125 (2012)

3 2 4/3( ) 3 3 4 ( , )/xE T d n T r r

QTP

8 Atom Cubical Array – Low-T plateaus?

Fermi distribution for a single spin for four temperatures at L = 6 bohr.

Note that the T=0 K occupied orbitals induced by cubic symmetry are a

single a1g and a triply degenerate t1u.

Finite-T Hartree-Fock, LDAx, LDAx(T) Exchange free energy

Periodic bcc Lithium at ambient and 2x density. Modified AbInit code.

(orbital-dependent, nor orbital free).

∆FX(T) = FX(T) - FX(T =100 K) per atom.

LDAx(T) is from Perrot and Dharma-Wardana, Phys. Rev. A 30, 2619 (1984).

QTP

Finite-temperature Hartree-Fock, LDAx, LDAx(T) Total free energy

∆Ftotal(T) = Ftotal(T) - Ftotal(T= 100 K) per atom.

LDAx(T) is from Perrot and Dharma-Wardana, Phys. Rev. A 30, 2619 (1984).

QTP

Periodic bcc Lithium at ambient and 2x density.

Modified AbInit code (orbital dependent, not orbital-free).

Finite-temperature Hartree-Fock, LDAx, LDAx(T) Eq. of State: ρ=0.6 g/cm3

Pressure vs T: HF, LDAx, LDAx(T)

∆P(T) = PHF(T) - PLDAx(T);

∆P(T) = PLDAx(T) (T) - PLDAx(T)

QTP

Bcc Lithium at ambient density. Modified AbInit

code (orbital dependent, not orbital-free).

Pressure vs T: HF, LDAx, LDAx(T)

∆P(T) = PHF(T) - PLDAx(T);

∆P(T) = PLDAx(T) (T) - PLDAx(T)

Finite-temperature Hartree-Fock, LDAx, LDAx(T) Eq. of State: ρ=1.2 g/cm3

Bcc Lithium at 2 x ambient density. Modified AbInit code

(orbital-dependent, not orbital-free.

QTP

• “Finite Temperature Scaling, Bounds, and Inequalities for the Noninteracting Density Functionals”,

J.W. Dufty and S. B. Trickey, Phys. Rev. B 84, 125118 (2011)

• “Positivity Constraints and Information-theoretical Kinetic Energy Functionals”, S.B. Trickey, V.V.

Karasiev, and A. Vela, Phys. Rev. B 84, 075146 [7 pp] (2011).

• “Temperature-Dependent Behavior of Confined Many-electron Systems in the Hartree-Fock

Approximation”, T. Sjostrom, F.E. Harris, and S.B. Trickey, Phys. Rev. B 85, 045125 (2012)

• “Issues and Challenges in Orbital-free Density Functional Calculations”, V.V. Karasiev and S.B.

Trickey, Comput. Phys. Commun. [submitted]

• “Comparison of Density Functional Approximations and the Finite-temperature Hartree-Fock

Approximation for Warm Dense Lithium”, V.V. Karasiev, T. Sjostrom, and S.B. Trickey, in

preparation

• “Constraint-based, Single-point Approximate Kinetic Energy Functionals”, V.V. Karasiev, R.S.

Jones, S.B. Trickey, and F.E. Harris, Phys. Rev. B 80 245120 (2009)

• “Conditions on the Kohn-Sham Kinetic Energy and Associated Density”, S.B. Trickey, V.V. Karasiev,

and R.S. Jones, Internat. J. Quantum Chem.109, 2943 (2009)

• “Recent Advances in Developing Orbital-free Kinetic Energy Functionals”, V.V. Karasiev, R.S. Jones,

S.B. Trickey, and F.E. Harris, in New Developments in Quantum Chemistry, J.L. Paz and A.J.

Hernández eds. [Research Signposts, 2009] 25-54

• “Born-Oppenheimer Interatomic Forces from Simple, Local Kinetic Energy Density Functionals”,

V.V. Karasiev, S.B. Trickey, and F.E. Harris, J. Computer-Aided Mat. Design, 13, 111-129 (2006)

References – OF-KE and thermal DFT

QTP

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

“Improved Constraint Satisfaction in a Simple GGA Exchange Functional ”, A. Vela, J. Pacheco Kato,

J.L. Gázquez, J. Martín del Campo, and S.B. Trickey, J. Chem. Phys. [accepted]

“Non-empirical Improvement of PBE and Its Hybrid PBE0 for General Description of Molecular

Properties”

J.M. del Campo, J.L. Gázquez, S.B. Trickey, and A. Vela, J. Chem. Phys. [accepted]

“A new meta-GGA Exchange Functional Based on an Improved Constraint-based GGA”

J.M. del Campo, J.L. Gázquez, S.B. Trickey, and A. Vela, Chem. Phys. Lett. [submitted]

“Structure –dependence of the Magnetic Moment in Small Paladium Clusters: Surprising Results

from the M06-L Meta-GGA Functional”, R. Koitz, T.M. Soini, A. Genest, S.B. Trickey, and N. Rösch,

Internat. J. Quantum Chem. 112, 113-120 (2012)

“Variable Lieb-Oxford Bound Satisfaction in a Generalized Gradient Exchange-Correlation

Functional”, A. Vela, V. Medel, and S.B. Trickey, J. Chem. Phys. 130, 244103 (2009)

“Tightened Lieb-Oxford Bound for Systems of Fixed Particle Number”,

M. M. Odashima, K. Capelle, and S.B. Trickey, J. Chem. Theory and Comput. 5, 798 (2009).

References – better XC functionals

QTP

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

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