combining density-functional theory and many-body methods · 2016-12-14 · combining...

Combining density-functional theory andmany-body methods

Julien ToulouseUniversite Pierre & Marie Curie and CNRS, Paris, France

AlbuquerqueNew Mexico, USA

June 2016

1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states

Outline

1 A brief overview of DFT/many-body hybrids

2 Range-separated hybrids for the ground state

3 Range-separated hybrids for excited states

2/23


Outline




3/23


Overview of DFT/many-body hybrids

1 Kohn-Sham DFT (1965):

E = minΦ

{

〈Φ|T + Vne|Φ〉+ EHxc[nΦ]}

where Φ is a single determinant

Semilocal density-functional approximations (DFAs) for Exc[n]:LDA, GGAs, meta-GGAs

=⇒ better accuracy still needed: self-interaction error, strong/staticcorrelation, van der Waals dispersion interactions

2 Hybrid approximations (Becke 1993):

Exc = axEHFx [Φ] + (1− ax)E

DFAx [n] + E

DFAc [n]

with Hartree-Fock (HF) exchange: EHFx [Φ] = 〈Φ|Wee|Φ〉 − EH[n]

one empirical parameter: ax ≈ 0.25

=⇒ reduces self-interaction error4/23



3 Double-hybrid approximations (Grimme 2006):

Exc = axEHFx [Φ] + (1− ax)E

DFAx [n] + (1− ac)E

DFAc [n] + acE

MP2c

with second-order Møller-Plesset (MP2) perturbative correlation:

EMP2c = −

occ∑

i<j

unocc∑

a<b

|〈Φabij |Wee|Φ〉|2

εa + εb − εi − εj

two empirical parameters: ax ≈ 0.5 and ac ≈ 0.25

=⇒ further reduces self-interaction error, partially account for van derWaals dispersion, but fails for strongly correlated systems

5/23



4 General DFT/many-body hybrid scheme (Sharkas, Toulouse, Savin 2011)

E = minΨ

{

〈Ψ|T + Vne + λWee|Ψ〉+ Eλ

Hxc[nΨ]}

with one empirical parameter λ and the complement density functional:

Eλ

Hx[n] = (1− λ)EHx[n] and Eλ

c [n] ≈ (1− λ2)Ec[n]

Single determinant: Ψ ≈ Φ =⇒ a hybrid approximation:

Exc = λEHFx [Φ] + (1− λ)EDFA

x [n] + (1− λ2)EDFA

c [n]

Second-order perturbation:one-parameter double-hybrid approximation:

Exc = λEHFx [Φ] + (1− λ)EDFA

x [n] + (1− λ2)EDFA

c [n] + λ2EMP2c

Non-perturbative approaches: Ψ =∑

n cnΦn

=⇒ for strong correlation (Sharkas, Savin, Jensen, Toulouse, JCP, 2012)

6/23



5 Range-separated DFT/many-body hybrid scheme (Savin 1996)

E = minΨ

{

〈Ψ|T + Vne + Wlree|Ψ〉+ E

srHxc[nΨ]

}

with a long-range interaction Wlree =

∑

i<j

erf(µrij )rij

a short-range density functional E srHxc[n]

and one empirical parameter µ controlling the range of the separation

Single determinant: Ψ ≈ Φ =⇒ a lrHF+srDFT hybrid:

Exc = Elr,HFx [Φ] + E

sr,DFAx [n] + E

sr,DFAc [n]

similar to the LC scheme (Hirao et al. 2001)

Many-body perturbation theory =⇒ lrMP2/lrRPA+srDFT hybrids:

Exc = Elr,HFx [Φ] + E

sr,DFAx [n] + E

sr,DFAc [n] + E

lr,MP2/RPAc

Angyan, Gerber, Savin, Toulouse, PRA, 2005; Toulouse, Gerber, Jansen, Savin,Angyan, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009

Non-perturbative approaches: Ψ =∑

n cnΦn

Leininger, Stoll, Werner, Savin, CPL, 1997; Fromager, Toulouse, Jensen, JCP, 20077/23


Outline




8/23


Range-separated hybrids: lrHF+srDFT

ElrHF+srDFTxc = E

lr,HFx [Φ] + E

sr,DFAx [n] + E

sr,DFAc [n]

Angyan, Gerber, Savin, Toulouse, PRA, 2005

Short-range exchange and correlation DFAs have been developed, e.g.:

short-range LDA:

EsrLDAx/c [n] =

∫

n(r) ǫsr,unifx/c (n(r)) dr

Toulouse, Savin, Flad, IJQC, 2004; Paziani, Moroni, Gori-Giorgi, Bachelet, PRB, 2006

short-range PBE:

EsrPBEx/c [n] =

∫

n(r) ǫsrPBEx/c (n(r),∇n(r)) dr

Toulouse, Colonna, Savin, JCP, 2005; Goll, Werner, Stoll, Leininger, Gori-Giorgi,

Savin, CP, 2006

9/23


lrHF+srDFT: reduction of the self-interaction error

Dissociation of H+2 molecule: H+

2 −→ H+0.5 + H+0.5

-70

-60

-50

-40

-30

-20

-10

0

10

0 2 4 6 8 10

Bin

din

g e

nerg

y (

kcal/

mo

l)

Internuclear distance (Angstrom)

exact

lrHF+srPBE

PBE0

PBE

=⇒ removal of the large self-interaction error comingfrom the long-range part of the PBE exchange

Mussard, Toulouse, submitted10/23


Adding long-range correlation: lrMP2/lrRPA+srDFT

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

11/23


Fast basis convergence of long-range perturbation theory

Total energy of N2 calculated with Gaussian basis sets(µ = 0.5 bohr−1, srPBE functional, Dunning basis cc-pVXZ):

-109.45

-109.40

-109.35

-109.30

-109.25

VDZ VTZ VQZ V5Z V6Z

To

tal

en

erg

y (

ha

rtre

e)

Basis sets of increasing sizes

MP2lrMP2+srPBE

=⇒ Exponential basis convergence of lrMP2+srPBE

Franck, Mussard, Luppi, Toulouse, JCP, 2015 12/23


Test of lrMP2/lrRPA+srDFT on atomization energies

A G2 subset of 49 atomization energies of small molecules(µ = 0.5 bohr−1, srPBE functional, cc-pVQZ, spin unrestricted):

0

4

8

12

-20 -10 0 10

Sta

nd

ard

devia

tio

n (

kcal/

mo

l)

Mean error (kcal/mol)

MP2dRPARPAx-SO2lrMP2+srPBElrdRPA+srPBElrRPAx-SO2+srPBE

=⇒ Range separation decreases the standard deviation for all methods

=⇒ All range-separated methods give a mean error of ∼ 5 kcal/mol

Mussard, Reinhardt, Angyan, Toulouse, JCP, 201513/23


Test of lrMP2/lrRPA+srDFT on reaction barrier heights

DBH24/08 set: 24 barrier heights of reactions with small molecules(µ = 0.5 bohr−1, srPBE functional, aug-cc-pVQZ, spin unrestricted):

0

2

4

6

8

-1 0 1 2 3 4 5 6

Sta

nd

ard

devia

tio

n (

kcal/

mo

l)

Mean error (kcal/mol)

MP2dRPARPAx-SO2lrMP2+srPBElrdRPA+srPBElrRPAx-SO2+srPBE

=⇒ Range separation improves all methods

=⇒ All range-separated methods give a mean error . 1 kcal/mol

Mussard, Reinhardt, Angyan, Toulouse, JCP, 201514/23


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, 2011 15/23


lrMP2+srDFT for periodic solid-state systems

periodic lrHF+srDFT, followed by local lrMP2 with localized orbitalsTest 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, 201516/23


Outline




17/23


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 exchange kernel is now standard:

fxc = flr,HFx + f

sr,DFAx + f

DFAc

Tawada, Tsuneda, Yanagisawa, Yanai, Hirao, JCP, 2004

Here, range separation for both exchange and correlation kernels:

fxc = flr,HFx + f

sr,DFAx + f

sr,DFAc + f

lr,(2)c (ω)

Rebolini, Savin, Toulouse, MP, 2013; Rebolini, Toulouse, JCP, 2016

Other similar schemes: Pernal, JCP, 2012; Fromager, Knecht, Jensen, JCP, 2013;

Hedegard, Heiden, Knecht, Fromager, Jensen, JCP, 201318/23


Second-order correlation self-energy and kernel

EMP2c

= +

Σ(2)c (2, 1) =

1

2

+

1

2

δ

δG0(1, 2)

Ξ(2)c (2, 4; 1, 3)=

1, 4

2, 3

+

1, 4

2 3

δ

δG0(3, 4)

+

1

2

4

3

+

1

2, 3

4

+

1

2

3

4

+

1

2

3

4

19/23


From the Bethe-Salpeter equation to TDDFT

Bethe-Salpeter equation with a kernel depending on only one frequency:

χ(ω) = χ0(ω) +

∫

dω′dω′′χ0(ω

′, ω)Ξ(2)Hxc(ω

′ ± ω′′)χ(ω′′, ω)

Define an effective second-order kernel:

f(2)Hxc(ω) = χ−1

0 (ω)

[∫

dω′dω′′χ0(ω

′, ω)Ξ(2)Hxc(ω

′ ± ω′′)χ0(ω′′, ω)

]

χ−10 (ω)

which can be used in a TDDFT-like linear-response equation

χ(ω) = χ0(ω) + χ0(ω)f(2)Hxc(ω)χ(ω)

or, equivalently,

χ−1(ω) = χ−10 (ω)− f

(2)Hxc(ω)

Romaniello, Sangalli, Berger, Sottile, Molinari, Reining, Onida, JCP, 2009

Sangalli, Romaniello, Onida, Marini, JCP, 201120/23


In an orbital basis

Effective 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)

=⇒ The correlation kernel brings the effect of the double excitations

Calculation of excitation energies in two steps:

1 lrTDHF+srTDLDA calculation in the TDA: A X0 = ω0 X0

2 perturbative addition of lrBSE2 kernel: ω = ω0 + X†0 f

lr,(2)c (ω0) X0

Zhang, Steinmann, Yang, JCP, 2013

Rebolini, Toulouse, JCP, 2016

21/23


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, 2016 22/23


Summary and Acknowledgments

DFT/many-body hybrid methods based on a decompositionof the e-e interaction into long-range and short-range parts

lrHF+srDFT already helps to reduce the self-interaction error

lrMP2/lrRPA+srDFT has a fast basis convergence

lrMP2/lrRPA+srDFT accounts for van der Waals dispersioninteractions

lrMP2+srDFT has been applied to periodic solid-state systems

lrBSE2+srTDDFT for excitation energies: frequency-dependentsecond-order long-range correlation kernel bringing the effect of thedouble excitations

Acknowledgments

J. Angyan, B. Civalleri, F. Colonna, H.-J. Flad, O. Franck, E. Fromager, I.Gerber, G. Jansen, H. J. Aa. Jensen, E. Luppi, L. Maschio, B. Mussard, E.Rebolini, P. Reinhardt, G. Sansone, A. Savin, K. Sharkas, D. Usvyat, W. Zhu

23/23

combining density-functional theory and many-body methods · 2016-12-14 · combining...

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