How to construct rigorouswave-function/density-functional hybrids?
Julien ToulouseLaboratoire de Chimie Theorique
Sorbonne Universite and CNRS, Paris, France
LCT, April 2018
Outline
1 Review of Kohn-Sham DFT
2 Multideterminant DFT based on linear separation of e-e interaction
3 Multideterminant DFT based on range separation of e-e interaction
4 Multideterminant DFT based on the combination the above two
5 Extensions: solids, excitation energies, RPA, MCSCF
2/28
Outline
1 Review of Kohn-Sham DFT
2 Multideterminant DFT based on linear separation of e-e interaction
3 Multideterminant DFT based on range separation of e-e interaction
4 Multideterminant DFT based on the combination the above two
5 Extensions: solids, excitation energies, RPA, MCSCF
3/28
4/28
5/28
Outline
1 Review of Kohn-Sham DFT
2 Multideterminant DFT based on linear separation of e-e interaction
3 Multideterminant DFT based on range separation of e-e interaction
4 Multideterminant DFT based on the combination the above two
5 Extensions: solids, excitation energies, RPA, MCSCF
6/28
7/28
Test on atomization energies
A G2 subset of 49 atomization energies of small molecules(DH with λ = 0.7, cc-pVQZ):
0
2
4
6
8
10
-10 -5 0 5 10
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBE
MP2
DH MP2+PBE
8/28
Test on reaction barrier heights
DBH24/08 set: 24 barrier heights of reactions with small molecules(DH with λ = 0.7, aug-cc-pVQZ):
0
2
4
6
8
-10 -8 -6 -4 -2 0 2 4 6 8 10
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBEMP2
DH MP2+PBE
9/28
Test on intermolecular interaction energies
S22 set: 22 equilibrium interaction energies of weakly-interacting molecularsystems from water dimer to DNA base pairs(DH with λ = 0.7, aug-cc-pVTZ):
0
1
2
3
4
5
-4 -3 -2 -1 0 1 2 3 4
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBE
MP2
DH MP2+PBE
10/28
Outline
1 Review of Kohn-Sham DFT
2 Multideterminant DFT based on linear separation of e-e interaction
3 Multideterminant DFT based on range separation of e-e interaction
4 Multideterminant DFT based on the combination the above two
5 Extensions: solids, excitation energies, RPA, MCSCF
11/28
12/28
Test on intermolecular interaction energies
S22 set: 22 equilibrium interaction energies of weakly-interacting molecularsystems from water dimer to DNA base pairs(DH with λ = 0.7, RSH with µ = 0.5, aug-cc-pVTZ):
0
1
2
3
4
5
-4 -3 -2 -1 0 1 2 3 4
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBE
MP2
DH MP2+PBE
RSH MP2+PBE
13/28
Test on reaction barrier heights
DBH24/08 set: 24 barrier heights of reactions with small molecules(DH with λ = 0.7, RSH with µ = 0.58, aug-cc-pVQZ):
0
2
4
6
8
-10 -8 -6 -4 -2 0 2 4 6 8 10
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBEMP2
DH MP2+PBE
RSH MP2+PBE
Mussard, Reinhardt, Angyan, Toulouse, JCP, 2015
14/28
Test on atomization energies
A G2 subset of 49 atomization energies of small molecules(DH with λ = 0.7, RSH with µ = 0.58, cc-pVQZ):
0
2
4
6
8
10
-10 -5 0 5 10
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBE
MP2
DH MP2+PBE
RSH MP2+PBE
Mussard, Reinhardt, Angyan, Toulouse, JCP, 2015
15/28
Outline
1 Review of Kohn-Sham DFT
2 Multideterminant DFT based on linear separation of e-e interaction
3 Multideterminant DFT based on range separation of e-e interaction
4 Multideterminant DFT based on the combination the above two
5 Extensions: solids, excitation energies, RPA, MCSCF
16/28
17/28
Test on atomization energies
A G2 subset of 49 atomization energies of small molecules(DH with λ = 0.7, RSH with µ = 0.58, RSDH with µ = 0.46 and λ = 0.58,cc-pVQZ):
0
2
4
6
8
10
-10 -5 0 5 10
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBE
MP2
DH MP2+PBE
RSH MP2+PBE
RSDH MP2+PBE
Kalai, Toulouse, JCP, 2018
18/28
Test on reaction barrier heights
DBH24/08 set: 24 barrier heights of reactions with small molecules(DH with λ = 0.7, RSH with µ = 0.58, RSDH with µ = 0.46 and λ = 0.58,aug-cc-pVQZ):
0
2
4
6
8
-10 -8 -6 -4 -2 0 2 4 6 8 10
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBEMP2
DH MP2+PBE
RSH MP2+PBE
RSDH MP2+PBE
Kalai, Toulouse, JCP, 2018
19/28
Test on intermolecular interaction energies
S22 set: 22 equilibrium interaction energies of weakly-interacting molecularsystems from water dimer to DNA base pairs(DH with λ = 0.7, RSH with µ = 0.5, RSDH with µ = 0.46 and λ = 0.58,aug-cc-pVTZ):
0
1
2
3
4
5
-4 -3 -2 -1 0 1 2 3 4
Sta
nd
ard
devia
tio
n (
kc
al/m
ol)
Mean error (kcal/mol)
PBE
MP2
DH MP2+PBE
RSH MP2+PBE
RSDH MP2+PBE
Kalai, Toulouse, JCP, 201820/28
Outline
1 Review of Kohn-Sham DFT
2 Multideterminant DFT based on linear separation of e-e interaction
3 Multideterminant DFT based on range separation of e-e interaction
4 Multideterminant DFT based on the combination the above two
5 Extensions: solids, excitation energies, RPA, MCSCF
21/28
lrMP2+srDFT for periodic solid-state systems
periodic lrHF+srDFT, followed by local lrMP2 with localized orbitals
Test on cohesive energies (µ = 0.5 bohr−1, srPBE functional,p-aug-cc-pVDZ)
−20
−10
0
10
20
30
40
50
60
70
Ne
Ar
CO
2
NH
3
HC
N
LiH
LiF
Si
SiC
% o
f err
or
on
co
hesiv
e e
nerg
y
crystal
PBEMP2lrMP2
rare−gas molecular ionic semi−conductor
+srPBE
=⇒ lrMP2+srPBE improves over PBE but is similar to MP2
Sansone, Civalleri, Usvyat, Toulouse, Sharkas, Maschio, JCP, 201522/28
Time-dependent range-separated hybrids
Linear-response TDDFT equation
χ−1(ω) = χ−10 (ω)− fHxc(ω)
=⇒ excitation energies, linear-response properties
Range separation for both exchange and correlation kernels:
fxc = flr,HFx + f
sr,DFAx + f
sr,DFAc + f
lr,(2)c (ω)
Long-range second-order correlation kernel (lrBSE2):
flr,(2)c,ia,jb(ω) =
occ∑
k<l
unocc∑
c<d
〈Φai |W
lree|Φ
cdkl 〉〈Φ
cdkl |W
lree|Φ
bj 〉
ω − (εc + εd − εk − εl)
23/28
Test of lrBSE2+srTDDFT on excitation energies
56 singlet and triplet excitation energies of 4 small molecules N2, CO, H2CO,C2H4 (µ = 0.35 bohr−1, srLDA functional, TDA, Sadlej+ basis):
0
0.1
0.2
0.3
0.4
0.5
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Sta
nd
ard
de
via
tio
n (
eV
)
Mean error (eV)
Valence TDLDAValence lrTDHF+srTDLDAValence lrBSE2+srTDLDARydberg TDLDARydberg lrTDHF+srTDLDARydberg lrBSE2+srTDLDA
=⇒ lrBSE2+srTDLDA provides a slight overall improvement overlrTDHF+srTDLDA
Rebolini, Toulouse, JCP, 201624/28
Long-range correlation by random-phase approximation
Long-range MP2:
Elr,MP2c = −
occ∑
i<j
unocc∑
a<b
|〈Φabij |W
lree|Φ〉|2
εa + εb − εi − εj= +
Angyan, Gerber, Savin, Toulouse, PRA, 2005
Long-range direct RPA (dRPA) = sum of all direct ring diagrams
Elr,dRPAc = + + · · ·
Toulouse, Gerber, Jansen, Savin, Angyan, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009
Long-range RPA with exchange (RPAx-SO2) = sum of all direct+ some exchange ring diagrams
Elr,RPAx-SO2c = + + +
+ + + · · ·Toulouse, Zhu, Savin, Jansen, Angyan, JCP, 2011
25/28
Test of lrMP2/lrRPA+srDFT on weak interactions
S22 set: 22 equilibrium interaction energies of weakly-interacting molecularsystems from water dimer to DNA base pairs(µ = 0.5 bohr−1, srPBE functional, aug-cc-pVDZ):
−60
−40
−20
0
20
40
60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22%
of
err
or
on
in
tera
cti
on
en
erg
y
system in S22 set
lrMP2+srPBElrdRPA+srPBElrRPAx−SO2+srPBE
H bonded dispersion dispersion+multipoles
=⇒ lrRPAx-SO2+srPBE/aVDZ gives a mean absolute error of ∼ 4%
Toulouse, Zhu, Savin, Jansen, Angyan, JCP, 201126/28
Test of MCSCF+DFT on F2 molecule
λ = 0.25, BLYP functional, cc-pVTZ basis:
−199.6
−199.55
−199.5
−199.45
−199.4
−199.35
−199.3
−199.25
2 3 4 5 6 7
To
tal
en
erg
y (
ha
rtre
e)
Internuclear distance R (bohr)
BLYPB3LYP
MCSCF+BLYPAccurate
=⇒ MCSCF+BLYP improves the dissociationbut still a large error on the dissociation energy
Sharkas, Savin, Jensen, Toulouse, JCP, 201227/28
Some ongoing work
Range-separated double hybrids with random-phase approximations(with Cairedine Kalai)
Time-dependent range-separated DFT beyond linear-response theory(with Eleonora Luppi and Felipe Zapata)
Four-component relativistic range-separated DFT(with Julien Paquier)
Multiconfigurational DFT with short-range on-top pair-density functionals(with Emmanuel Giner and Anthony Ferte)
28/28