open questions in dft - jussieu · luis sepúlveda historia de una gaviota y del gato que le...
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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”