Open questions in DFT

A. Savin

Santiago, 2011

Luis SepúlvedaHistoria de una gaviota y del gato que le enseñó a volar

Story of a Seagull and of the Cat who Taught Her to Fly

What can an independent particle Hamiltoniantell about an interacting particle Hamiltoinan?

Introduction

Introduction

Impact of DFTWeb of Science : "density functional theory"

2050?

Introduction

Why this talk?

1989, 10th Canadian Symposium on Theoretical Chemistry

John Pople: Why is H2+ not working ?

Axel Becke: Take another method for H2+!

A. Becke showed with systematic tests that DFAs work well.

J. Pople became an adept of DFT

Introduction

H2+ problem

0 2 4 6 8 10

-0.60

-0.58

-0.56

-0.54

-0.52

-0.50

-0.48

-0.46

R

E

exact, LDA

Introduction

Some papers on H2+, et al.

1992R. Merkle et al., JCP 97, 9216 (1992) :

X2+ molecules

1998B. Braïda, et al., JPC A 104, 7872 (1998):

3-e bonds

Introduction

Some people still interested in this problemCitations of paper by Braïda et al. : ¶

Negative results not published or publication irrelevant?

Overview

£ Is accurate DFT accessible?

£ Are systematic constructions of DFAs accessible?

£ What should be approximated?

£ Is degeneracy correctly described?

£ Are statistics relevant ?

Is accurate DFT accessible?

DFT is exact. Can we approach it numerically ?

Accurate DFT is difficult to reach and

this is done in an uncotrolled way.

Accurate DFT?

DFT and DFAs

£ D: density

£ F: functional

£ T: theory

£ A: approximations

X: "DFT is not working"

M. Levy: "Is DFT not working or the approximation?"

Accurate DFT?

T in DFT• Wave function theory:

E0@vD = minY XY T + V + Vee Y\ V = Úi=1,N vHriL• DFT (Hohenberg-Kohn / E. Lieb):

F@nD = maxvIE0@vD - Ù v nM E0@vD = minnIF@nD + Ù v nM

Accurate DFT?

"F @nD is unfortunately unknown"

F@nD is known, = maxvIE0@vD - Ù v nMExample: n = N 1

Πã-2 r

N 1 2 4F@nD 0.5 1.588 6.827

F. Colonna, AS, JCP110, 2828 (1999)

See also: Teale, Coriani, Helgaker

More difficult to calculate than minY XY H Y\

Accurate DFT?

Is v accessible?

F@nD = maxvIE0@vD - Ù v nMJ.E. Harriman (~1980’s):

HK theorem not valid in a finite basis

Accurate DFT?

Is v accessible for the H atom in a finite basis?

Basis: 1s, 2s

v = -1 � r + Dv

Dv = 12 256 213 732 454 395 36872 539 740 899 199 421 ã-4 r +

4 658 295 494 941 200 00072 539 740 899 199 421 ã-3 r H2 - 2 rL +

1 179 894 519 818 100 00072 539 740 899 199 421 ã-2 r H2 - 2 rL2

makes 1s and 2s degenerate (in this space):

1s is an acceptable solution.

Accurate DFT?

Is v accessible for the H atom in a finite basis?

H atom, corrections to v = -1 � r,making the orbitals degenerate, for small basis sets

1 2 3 4 5 6r

-3

-2

-1

1

2

r Dv1s,2s

1 2 3 4 5 6r

-40

-30

-20

-10

10

20

30

r Dv1s,2s,3s

Accurate DFT?

Is v accessible for Kohn-Sham?

Kohn-Sham defines:

F@nD ® Ts@nD = maxvIE0,Vee ®0@vD - Ù v nMvKS = vmax

Success of KS ¬ fast calculation for non-interacting particles

HT + VKSL YKS = EKS YKS

Accurate DFT?

Dv(r) making YFCI = YKS = YOEP

... for He atom, VDZ basis (via orbital degeneracy)

D. Rohr, AS (2010)

Accurate DFT?

Conclusion

• F@nD : second-order errors

• vKSHnL, ...: ?

..., ..., E. Lieb

Are systematic constructions of DFAs accessible?

DFAs - semi-local.

How to overcome this?

Systematic constructions of DFAs are not known.

DFA construction?

Construction of approximations

F @nD » Ù f H@ nHrL, !nHrL , ...D D d3 r

DFA construction?

Available approximations

• LDA

• GGAs, m-GGAs, ...

• Hybrids: B3LYP, ...

• ...

Quite limited ansatz

DFA construction?

Quite limited ansatz. Example

Eh@nD = 12 Ù Ù nHr1L nHr2L � r1 - r2

»? Ù f @nHrL, !nHrL , ...D d3 r

Eh@nD is known.

Is a (semi-)local approximation working?

E. Cancès’ answer

DFA construction?

Behavior of U at ¥ separation

A B

n

n = nA ... nB

Eh = ... + 12 Ù nAHrL nBHr 'L � r - r '

Eh ® ... + 12 NA NB � RA - RB

DFA construction?

Conclusion

• From n Þ explicit dependence on RAB?

(Eh@nD, but also van der Waals interaction,...)

What should be approximated?

Not F@nD is approximated, but a part of it.

There is no ideal choice

for the part of F@nD to approximate.

Approximate what?

Hohenberg-Kohn

F@nD = Eh@nD + Ekxc@nDEh@nD º 1

2 Ù Ù d3 r1 d3 r2 nHr1L nHr2L � r12

Ekxc@nD defined above, by choice of Eh

Motivation: electrostatics right (as are Ù vne n, vnn)

cf. Madelung potential, semi-empirical methods, W. Kutzelnigg on dipole moments...,

Approximate what?

Kohn-Sham

F@nD = Eh@nD + Ts@nD + Exc@nDTs@nD = minY®n XY T Y\Exc@nD defined above, by F@nD,Eh@nD, Ts@nD Motivation: Pauli principle (repulsion) mainly in XT\

Approximate what?

Kohn-Sham (b)

F@nD = Eh@nD + Ts@nD + Ex@nD + Ec@nDEx@nD = XYmin,®n Vee Ymin,®n\ - Eh@nDEc@nD defined above

Motivation: a smaller part to approximate

Approximate what?

A term remains to be approximated

Hope : approximations work better for remaining term

Approximate what?

Example: one-electron systems

no e-e repulsion

Eh@nD + Exc@nD = 0, or Exc@nD = -Eh@nDIf we cannot approximate Eh@nD, how can we approximate Exc@nD?

cf. H2+ problem

R. Merkle et al., JCP 97, 9216 (1992)

Approximate what?

Conclusion

• Problems avoidable?

- electrostatics : separate Eh !

- self-interaction : Eh + Exc !

Is degeneracy correctly described?

Degeneracies are commonly observed.

With DFAs

we do not known how to deal with degeneracies.

Degeneracy?

Degeneracy in DFAs

F@niD = const, " ni Î degenerate set

• “N-conserving”

• “non N-conserving”

Using spin-density amplifies the problem

Degeneracy?

Example of N -conserving degeneracy

EIB, ... + 12 I px

2 + py2MM = EHB, ... + px

2L

Degeneracy?

Example of non N -conserving degeneracy

• Dissociated H2+

EHH2+, R ® ¥L = 2 EI"H+1�2"M = EHHL + EHH+L

nA ... nB = Σg2 ® 1

2 nA + 12 nB

Degeneracy?

How important are density changes?N2 HCCH

-3 -2 -1 0 1 2 3

0

50

100

150

200

-4 -2 0 2 4

0

20

40

60

80

100

120

X ray ,e .g., SrSi

- 3 - 2 - 1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

- 4 - 2 0 2 40.0

0.2

0.4

0.6

0.8

1.0

Chemical Bonding

Degeneracy?

Spin-density in DFAs

For DFAs

F@nD ® FAn = nΑ + nΒ, m = nΑ - nΒDTheory, parallels F@nD, adding “magnetic field”.

m ¹ 0 Þ degenerate states, when B = 0.

Thus, F@n, mD should be independent of m.

DFAs for F@n, mD deliberately m-dependent.

Degeneracy?

Why care about degeneracy?

Can we choose among degenerate states,e.g., choose maximal m ?

Not always free: degeneracy can be lifted (approached) in a specific way,e.g., for potential energy surfaces(chemical reactions, adsorbtion, ...).

Degeneracy?

Potential energy curve for H2, LDA

0 2 4 6 8 10-1.15-1.10-1.05-1.00-0.95-0.90-0.85

R

E

Restricted and Unrestricted Kohn-Sham (RKS, UKSL

Choice:

• right m (RKS)

• »right E (UKS), right MS = 12 Ù m = 1

2 HNΑ - NΒL

Degeneracy?

O2 H3SL R ® ¥ : MS Ior NΑ, NΒM not conserved

O2: triplet, MS = 1Size-consistency requires:

OHC CL ... OHC CL Þ MS = 2, not 1

OHC CL ... OHE EL Þ MS = 0, not 1

NΑ, NΒ = ? What Slater determinant?

Degeneracy?

O2 H3SL R ® ¥ : "phase transition"

For Re, ground state has MS = 1.

For large R, in DFAs

EHMS = 0 or 2L < EHMS = 1L R

E

• "phase transition"

Degeneracy?

Conclusion

DF is

• sensitive to small Dn , to describe binding

• insensitive to large Dn, to describe degeneracy

How can this be realized with a DFA?

Degeneracy ¬ external potential, not n

Problem for parts of F @nD even more complicated (supplement)

Are statistics relevant?

Functionals trained, judged by test sets.

Statistics on test sets need care.

Statistics are not applicable for particular cases.

Statistics relevant?

Strategy

Test sets

• statistics ® general idea of the quality of the DFA

• grouping of data ® when to use it, and when not

DFAs: 'good'

Statistics relevant?

For those who are in a hurry to get good results

Scientifically indisputable: no hiding of bad results

No need for new• parametrizations• calculations

Example: better three electron bonding energies HHe2

+, ...L

Statistics relevant?

Errors in dissociation energies for X2+

Data: B. Braïda, Ph. C. Hiberty, A.S., J. Phys. Chem. A 104, 7872 (1998).

Statistics relevant?

Explanation: how n o t to group results

set 1, set 2

Statistics relevant?

Explanation: how to group results

set 1, set 2

Statistics relevant?

Conclusion

How to judge the relevance of test sets?

Do statistics help in a particular case?

Summary

I know what must be, but I do not know what may be

Idries Shas, The Pleasantries of Mulla Nasrudin

£ Accurate DFT: accessible, but difficult to reach and in an uncotrolled way

£ Systematic constructions of DFAs: not known

£ Part of F@nD to approximate: no ideal choice

£ DFAs when degeneracies occur: not known

£ Statistics on test sets: need care

What could be done

Luis SepúlvedaHistoria de una gaviota y del gato que le enseñó a volar

Story of a Seagull and of the Cat who Taught Her to Fly

“... el mas importante... Que sólo vuela el que se atreve al hacerlo”

“... the most important... That only flies the one who dares to do it”