how to construct rigorous wave-function/density-functional hybrids? · 2018-05-02 · outline 1...

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






3/28

Outline






6/28

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)


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)


PBE

MP2

DH MP2+PBE

10/28

Outline






11/28


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)


PBE

MP2

DH MP2+PBE

RSH MP2+PBE

13/28


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)


PBEMP2

DH MP2+PBE

RSH MP2+PBE

Mussard, Reinhardt, Angyan, Toulouse, JCP, 2015

14/28


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)


PBE

MP2

DH MP2+PBE

RSH MP2+PBE

Mussard, Reinhardt, Angyan, Toulouse, JCP, 2015

15/28

Outline






16/28


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)


PBE

MP2

DH MP2+PBE

RSH MP2+PBE

RSDH MP2+PBE

Kalai, Toulouse, JCP, 2018

18/28


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)


PBEMP2

DH MP2+PBE

RSH MP2+PBE

RSDH MP2+PBE

Kalai, Toulouse, JCP, 2018

19/28


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)


PBE

MP2

DH MP2+PBE

RSH MP2+PBE

RSDH MP2+PBE

Kalai, Toulouse, JCP, 201820/28

Outline






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

how to construct rigorous wave-function/density-functional hybrids? · 2018-05-02 · outline 1...

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