bastien mussard · developments in electronic structure theory. bastien mussard laboratoire de...

Developments in Electronic Structure Theory.

Bastien Mussard

Laboratoire de Chimie TheoriqueInstitut du Calcul et de la Simulation

Sorbonne Universites, Universite Pierre et Marie Curie

[email protected]/pagesperso/mussard/

www.lct.jussieu.fr/pagesperso/mussard/

“Developments in Electronic Structure Theory”

Get the accurate answer for the right reason at a reasonable cost .

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DFT (the density n(r) is the key quantity)

4 cheap methods ; applicable to big systems

6 unable to systematically improve the results

RangeSeparation

WFT (the wavefunction Ψ(xN) is the key quantity)

6 rather costly

4 improvement of the description is systematic

QMC ( stochastic method)

4 good method in terms of accuracy and scaling

6 costly ; ideal for parallelisation

WFT asinputfor QMC

DFT+WFT

I post-DFT calculations in the context of Range Separation.

I In the WFT part, for the treatment of London dispersion forces,we use the Random Phase Approximation.

Range Separation [Savin, Rec.dev. (1996)]

WFT suffers at short range from the description of the e-e coalescence.DFAs are essentially (semi)local approximation and are best at short-range.

The idea is to split the e-e interaction into long- and short- range parts:

1

r= v lr

ee(r) + v sree(r)

1/r

vlree(r)

vsree(r)versus

. . . and as a result to rigourously mix WFT and DFT methods :

I in a variational way:

E = minΨ

{〈Ψ| T + Vne + W lr

ee |Ψ〉+ E srHxc[nΨ]

}e.g. MCSCF+DFT, CI+DFT,. . . -> for the static correlation.

I with a perturbative treatment, from a mono-determinental reference (RSH):

E = minΦ

{〈Φ|T + Vne + W lr

ee|Φ〉+ E srHxc[nΦ]

}+ E lr

c

e.g. with RPA -> for long range dynamical correlation.

Random Phase Approximation: Origins [Bohm,Pines PR 82 (1951)]

The original intention is to decouple by a canonical transformation the particle andfield variables in the description of the UEG.

Go from H = Hpart. + Hfield + Hpart./field to HRPA = Hpart. + Hoscillations + Hsrpart./part.

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The physics: mix of long-range organized behavior (collective plasma oscillations)and of short-range screened interactions.

Hence, in RPA, the energy is the sum of oscillator energies and of a short-rangecorrection. This seems ideal in a range separation context.

Only the particles in phase with the oscillating field contribute to the oscillationsthe other particles, having a random phase , are neglected.

Random Phase Approximation: formalisms and flavors

ACFDT equation [Langreth,Perdew PRB 15 (1977)]

EACFDTc =

1

2

∫ 1

0

dα

∫ ∞−∞

dω

2πTr(Wee

[�RPAα (iω)− �0(iω)

]) ∣∣∣∣ (�RPAα )−1 = �−1

0 − fHx,α

Flavors (include/exclude exchange) [Angyan,Liu,Toulouse,Jansen JCTC 7 (2011)]I direct or exchange RPAI single-bar or double-bar

This yields the flavors of RPA called: dRPA-I, dRPA-II, RPAx-I, RPAx-II

Formulations (analytical or numerical integrals)

I density-matrix formulation E = 12

∫dα tr (Pα W)

I dielectric-matrix formulation E = 12

∫dω

2πtr (log(ε(iω)) + 1− ε(iω))

I plasmon formula E = 12

∑ωRPA − ωTDA

I Ricatti equations and rCCD E = 12 tr(BT)

+Approximations (in each formulations, for each flavors...) SOSEX, RPAx-SO2

Full-range DFT + RPA

DFT calculation followed by an RPA calculation with KS orbitals and energies.

EDFT = minΦ

{〈Φ|T + Vne|Φ〉+ EHxc[nΦ]

}E = 〈Φ|T + Vne|Φ〉+ EHF

Hx [Φ] + ERPAc

I DFT (LDA/GGA/...) do not describe correctly thelong-range dispersion forces, while RPA does .

I Still problems with: short-range correlation energies (far too negative)strong dependence on basis sizesimple van der Waals dimers

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Internuclear distance (bohr)

AccuratedRPA-ISOSEX

RPAx-IIRPAx-SO2

Ne2

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ener

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AccuratedRPA-ISOSEX

RPAx-IIRPAx-SO2

Ar2

>PBE, aug-cc-pV6Z [Toulouse,Zhu,Savin,Jansen,Angyan JCP 135 (2011)]

Range-Separated Hybrid + RPA

Hybrid DFT with exact HF exchange and RPA correlation, both at long-range.

E = minΦ



}+ E lr,RPA

c

I Range separation greatly improves RPA (see srPBE+RPAx-SO2).

I Basis dependence is reduced.

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ract

ion

ener

gy(m

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ree)


AccuratedRPA-ISOSEX

RPAx-IIRPAx-SO2

Ne2

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0.0

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0.4

6 7 8 9 10 11 12 13

Inte

ract

ion

ener

gy(m

hart

ree)


AccuratedRPA-ISOSEX

RPAx-IIRPAx-SO2

Ar2

>srPBE, aug-cc-pV6Z, µ = 0.5 [Toulouse,Zhu,Savin,Jansen,Angyan JCP 135 (2011)]

Range-Separated Hybrid + RPA

Hybrid DFT with exact HF exchange and RPA correlation, both at long-range.

E = minΦ



}+ E lr,RPA

c

I Range separation greatly improves RPA (see srPBE+RPAx-SO2).

I Basis dependence is reduced.

I for ex.: performance on the S22 dataset is very good for srPBE+RPAx-SO2.

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60

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system in S22 set

MP2dRPA-ISOSEXRPAx-IIRPAx-SO2

H bonded dispersion dispersion+multipoles

>srPBE, aug-cc-pVDZ, µ = 0.5 [Toulouse,Zhu,Savin,Jansen,Angyan JCP 135 (2011)]

New developments

DFT+WFT

I Basis Set Convergence studyand Three-point extrapolation scheme.

I Spin-Unrestricted generalizationand Calculations on DBH24/08 and AE49.

I Development and Implementation ofthe gradients of RSH+RPA methods.

In general

I Development of Localized Virtual Orbitalsand Application for RPA calculations.

WFT+QMC

I Implementation of the calculationof weights of VB structures in QMC.

Basis Set Convergence [Franck,BM,Luppi,Toulouse JCP 142 (2015)]

Wavefunction basis convergence [Gori-Giorgi,Savin PRA 73 (2006)]

Both partial wave and principal number expansionslead to exponential convergence of the wavefunction.

Her1

r2θ

0.21

0.23

0.25

0.27

0.29

-180 -120 -60 0 60 120 180

Ψ(a

.u.)

θ (degree)

HFVDZVTZVQZV5ZV6Z

Hylleraas0.21

0.23

0.25

0.27

0.29

-180 -120 -60 0 60 120 180

Ψlr,µ

(a.u

.)

θ (degree)

0.263

0.264

0.265

-120 -60 0 60 120

RSHVDZVTZVQZV5ZV6Z

Extrapolation formula for total energiesWe proposed a three-point extrapolationscheme for total RSH+correlation energies(fitting EX = E∞ + B exp(−βX )).

E∞ = EXYZ =E 2Y − EXEZ

2EY − EX − EZ

>errors (mHartree) wrt. cc-pV6Z

He Ne N2 H2O

∆ED 8.488 74.523 51.581 55.850∆ET 0.781 20.337 13.406 14.736∆EQ 0.245 5.763 4.090 4.499∆E5 0.078 0.751 0.810 0.726

∆EDTQ 0.205 0.401 1.083 1.105∆ETQ5 0.002 -1.876 -0.972 -1.475

Spin-Unrestricted RPA [BM,Reinhardt,Angyan,Toulouse JCP 142 (2015)]

Implemented most of the variants with anospinflip/spinflip block structure.

Normal distributions of errors(kcal/mol) of calculations on AE49.

I Mean errors of post-RSH calculationare better.

I Post-RSH calculations dist. of errorshave sharper distributions .

Similar results were obtained on theDBH24/08 dataset.

We argue that RPAx-SO2 is a goodmethod for a wide range of

applications.

0

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HF+MP2

0

0.04

0.08

0.12

-40 -20 0 20

RSH+MP2

0

0.04

0.08

0.12

-40 -20 0 20

HF+dRPA-I

0

0.04

0.08

0.12

-40 -20 0 20

RSH+dRPA-I

0

0.04

0.08

0.12

-40 -20 0 20

HF+RPAx-SO2

0

0.04

0.08

0.12

-40 -20 0 20

RSH+RPAx-SO2

>cc-pVQZ (post-RSH: srPBE, µ = 0.5)

Analytical Gradients: Lagrangian formalism [BM,Szalay,Angyan JCTC 10 (2014)]

Given an energy E [κ,V(κ),T(κ)]

you have rules for V: ∂E/∂V = 0

you have rules for T: R[T] = 0

energie

energie

(c)(b)(a)

parametreparametreparametre

energie

lagrangien

T TV

R[T] = 0E[V]

E[T] E[T]

L[T]

energie

energie

(c)(b)(a)


energie

lagrangien

T TV

R[T] = 0E[V]

E[T] E[T]

L[T]

dE

dκ=∂E

∂κ+∂E

∂V

∂V

∂κ×+∂E

∂T

∂T

∂κ

(in∂E

∂κthere is h(κ), (µν|σρ)(κ) and S(κ)

)For non-variational methods

The solution is to work with an alternative object that is variational :

L[κ,V,T,λ] = E [κ,V,T] + tr(λR[T])

∂L∂V

=∂E

∂V= 0 4

∂L∂λ

= R[T] = 0 4

∂L∂T

=∂E

∂T+ tr

(λ∂R

∂T

)= 0

energie

energie

(c)(b)(a)


energie

lagrangien

T TV

R[T] = 0E[V]

E[T] E[T]

L[T]

For RSH+RPA the energy is: E = minΦ



}+E lr,RPA

c

The equations were derived (sr and lr terms) and implemented in MOLPRO.(based on RSH+MP2 of [Chabbal, Stoll, Werner, Leininger, MP 108 (2010)])

Analytical Gradients of RSH+RPA [BM,Szalay,Angyan JCTC 10 (2014)]

Bond Lengths of simple molecules> aug-cc-pVQZ

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0.00

H 2/H

−HHF/

F−H

H 2O/

O−H

HOF

/O−H

HNC/

N−H

NH 3

/N−H

N 2H 2

/N−H

HNO/

N−H

C 2H 2

/C−H

HCN

/C−H

C 2H 4

/C−H

CH4/C−H

N 2/N

−NCH

2O/

C−H

CH2/C−H

CO/C−O

HCN

/C−N

CO2/C−O

HNC/

C−N

C 2H 2

/C−C

CH2O/

C−O

HNO/

O−N

N 2H 2

/N−N

C 2H 4

/C−C

F 2/F−F

HOF

/O−F

∆r

(Å)

RHF+dRPA-ILDA+dRPA-IRHF+SOSEXLDA+SOSEX

Mean Absolute Errors

Bond Lengths of simple molecules

RHF+dRPA-I 0.016

LDA+dRPA-I 0.013

RHF+SOSEX 0.021

LDA+SOSEX 0.014

Interaction Energies Intermonomer distances

MP2 0.76 MP2 0.075

LDA+MP2 0.63 LDA+MP2 0.053

LDA+dRPA-I 0.35 LDA+dRPA-I 0.023

Interaction Energies> aug-cc-pVTZ

-1.0-0.50.00.51.01.52.02.5

C 2H 4

. .. F

2NH 3

. .. F

2C 2

H 2. .

. ClF

HCN

. .. C

lFNH 3

. .. C

l 2H 2

O.. .

ClF

NH 3

. .. C

lF

∆(∆

E)

(kca

l.mol

−1) MP2

LDA+MP2LDA+dRPA-I

Intermonomer distances> aug-cc-pVTZ

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C 2H 4

. .. F

2NH 3

. .. F

2C 2

H 2. .

. ClF

HCN

. .. C

lFNH 3

. .. C

l 2H 2

O.. .

ClF

NH 3

. .. C

lF

∆R

(Å)

MP2LDA+MP2LDA+dRPA-I

Projected Oscillator Orbitals [BM,Angyan, TCA (submitted)]

The objective is to construct a set of localized virtual orbitals by multiplying theset of occupied Localized Molecular Orbitals (LMOs) with solid spherical harmonicfunctions. The orthogonality to the occupied space is ensured by projection.

Dipolar oscillator orbital [Foster, Boys RMP 32 (1960)]

|iα〉= (1− P)︸︷︷︸projector

(rα − D iα)︸︷︷︸

harmonic

|i〉︸︷︷︸LMO

where D iα = 〈i |rα|i〉

C-H bond in CH2O: Lone pair in CH2O:

Projected Oscillator Orbitals: Key features [BM,Angyan, TCA (submitted)]

Dipolar oscillator orbital

|iα〉= (1− P)(rα − D iα) |i〉 where D i

α = 〈i |rα|i〉

This formulation is best employed with Boys’ LMOs, see:

|iα〉= rα |i〉−∑|m〉〈m|rα|i〉

The POOs are non-orthogonal, with overlap:

S ijαβ = 〈i |rαrβ |j〉 −

∑〈i |rα|m〉〈m|rβ |j〉

The fock matrix element are found to be:

f iiαβ = 12δαβ + 1

2

∑(fim〈m|rαrβ |i〉+ 〈i |rαrβ |m〉fmi )−

∑〈i |rα|m〉fmn〈n|rβ |i〉

The two-elec integrals are written with a multipole expansion of the lr interaction:

〈iαj |klβ〉 =∑〈iα|rγ |k〉Lijγδ〈j |rδ|lβ〉+ . . .

A nice result simplifies the equations:

〈iα|rβ |j〉 = 〈i |rαrβ |j〉 −∑〈i |rα|m〉〈m|rβ |j〉 = S ij

αβ

Projected Oscillator Orbitals: RPA and C6 [BM,Angyan, TCA (submitted)]

The working equations are the local RPA Ricatti equations (i.e. local rCCD)

with the local excitation approximation (Mijkαlβ→Mij

iαjβ

.= Mij

αβ)

and spherical average approximation (Miiαβ → δαβ

13

∑Miiαα

.= δαβ

13m

i ).

RPA correlation energy

ERPA,lrc =

occ∑ij

4

9s i s j tr

(LijTij

)where Rij = · · · = 0

C6 coefficients

E (2),lrc =

occ∑ij

4

9

s i s j

∆i + ∆jtr(LijLij

)=

occ∑ij

8

3

s i s j

∆i + ∆j

Fµdamp(D ij)

D ij 6

where ∆i = fii − f i/s i

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0

50

100

150

H 2HF

H 2O N 2

CONH 3

CH4

CO2

H 2CON 2O

C 2H 2

CH3OH

C 2H 4

CH3NH 2

C 2H 6

CH3CH

OCH

3OC

H 3C 3

H 6HCl

HBr H 2S

SO2

SiH 4Cl2

COS

CS2

CCl 4

Err

orin

the

mol

ecul

arC

6co

effici

ents

(%) LDA

PBERHFRSHLDA

> aug-cc-pVTZ (RSH: µ = 0.5)

MA%E

LDA 59.8PBE 56.7RHF 15.2RSHLDA 11.8

VB-QMC: Motivation

VB-SCF (VB structures = Lewis structures)

4 Good dissociation energies

4 Interpretative tools (weight of structures, . . . )

6 Non-orthogonal orbitals -> costly

6 Introduce dynamical correlation by “breathing”(L-BO-, S-BO-, SD-BOVB) -> costly

VB-QMC in the hope of treating big systems

Ψ = eJ( )

I Static correlation taken care of by VB.

I Non-orthogonality of orbitals is not an issue for QMC.

I Dynamical correlation hopefully taken care of by Jastrow.

I But: we wish to conserve the interpretative tools,i.e. the weights (and the calculation of separated structure).

WFT asinputfor QMC

VB-QMC: weights of VB structures

We wish to transfert the weights of structures to VB-QMC.

Chirgwin-Coulson weights wi =

∑jcicjSij∑

ijcicjSij

and Lowdin weights wi =(∑

jcj (S

12 )ji )

2∑i(∑

jcj (S

12 )ji )2

In both cases, Sij needs to be computed, via Sij =

∫dR

Ψi (R)

Ψ0(R)

Ψj(R)

Ψ0(R)Ψ2

0(R)

0

0.2

0.4

0.6

0.8

1

1 2 3

VB-SCFVB-QMCS-BOVB(jc)(jco)

Li2

0

0.2

0.4

0.6

0.8

1

1 2 3


LiH

0

0.2

0.4

0.6

0.8

1

1 2 3


F2

0

0.2

0.4

0.6

0.8

1

1 2 3


FH

4 VB-QMC = VB-SCF

Li2: mainly covalent characterionic too important in BOVB(structures are mixing)

F2: Jastrow very similar to BOno mixing of structures

It is unclear if theVB weights are the referent ones(when there is a too preponder-ant structure, mixing occurs).

Other Works

All (and more) is implemented in the Quantum Chemistry Package MOLPRO

Fractional occupation number calculationsand Instabilities in the RPA problem.

Efficient calculations of determinants in QMC usingthe Sherman-Morrison-Woodbury formula.

Collaborators at the UPMC. . .Odile FranckEleanora LuppiPeter ReinhardtJulien Toulouse

. . . and at the Universite de Lorraine:Janos AngyanDario Rocca



bastien mussard · developments in electronic structure theory. bastien mussard laboratoire de...

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