Accelerating quantum-chemistry wave-function calculations ...

Accelerating quantum-chemistry wave-functioncalculations using density-functional theory

Julien ToulouseLaboratoire de Chimie Theorique

Sorbonne Universite and CNRS, Paris, France

Institut Universitaire de France

Online Nuclear Theory Seminar

Orsay (IJCLab) and Saclay (CEA/IRFU/DPhN)

October 2020

www.lct.jussieu.fr/pagesperso/toulouse/presentations/presentation˙onlineseminar˙20.pdf

www.lct.jussieu.fr/pagesperso/toulouse/presentations/presentation_onlineseminar_20.pdf

Introduction: Quantum chemistry in a nutshell

� The first goal of quantum chemistry is to solve the stationary Schrodinger equation

HΨn = EnΨn

with the Born-Oppenheimer and non-relativistic N-electron Hamiltonian

H = −1

2

N∑

i=1

∆ri −

N∑

i=1

Nnucl∑

α=1

Zα

|ri − Rα|+

N∑

i=1

N∑

j=i+1

1

|ri − rj |+

Nnucl∑

α=1

Nnucl∑

β=α+1

ZαZβ

|Rα − Rβ |

for molecular systems

� We want mostly:

� The ground-state energy for different nuclei positions E0({R})

=⇒ energy (hyper)surfaces to study chemical reactions

� Low-lying excited-state energies En({R})=⇒ excited-state energy (hyper)surfaces to study photochemical reactions

� Energy derivatives with respect to a perturbation∂kEn

∂perturbk

=⇒ spectroscopic properties (e.g., IR, UV-Vis, X-ray, EPR, NMR spectra)

� We aim at about 1% accuracy on energy differences and other properties

2/24

Introduction: Quantum-chemistry electronic-structure approaches

There are two main families of electronic-structure computational approaches in quantum

chemistry:

� Wave-function theory (WFT)

� direct calculation of the correlated many-electron wave function

� accurate and systematically improvable

� computationally costly, in particular due to slow basis-set convergence

� Density-functional theory (DFT)

� avoid the calculation of the correlated wave function thanks to a functional of

the density

� computationally efficient, in particular fast basis-set convergence

� uncontrolled errors due to the use of an approximate density functional

Here, I will give an overview of WFT and DFT, and present a recently developed

hybrid approach for accelerating the basis-set convergence of WFT by using DFT for

correcting for the incompleteness of the basis set

3/24

Outline

1 Overview of wave-function theory and density-functional theory

2 Accelerated wave-function theory using density-functional theory

3 Numerical results on atomic and molecular systems

4/24

Outline




5/24

Wave-function theory

� In quantum chemistry, we usually work with an (incomplete) one-electron basis set

B = {χν(x)}ν=1,...,M

where x = (r, σ) ∈ R3 × {↑, ↓} is a space-spin coordinate

� For a N-electron system, an approximation to the exact ground-state electronic

energy E0 is obtained by full configuration interaction (FCI) in this basis B

EB

FCI = minΨ∈WB

〈Ψ|T + Vne + Wee|Ψ〉 ≥ E0

where the wave function is restricted to the space of N-fold antisymmetric tensor

product of one-electron spaces wB1 = Span(B)

WB = {Ψ ∈∧NwB1 | 〈Ψ|Ψ〉 = 1}

� A wave function Ψ ∈ WB can be written as

Ψ(x1, x2, ..., xN) =∑

IcIΦI(x1, x2, ..., xN)

where ΦI are determinants of orthonormal spin-orbitals

ΦI(x1, x2, ..., xN) = φI1(x1) ∧ φI2(x2) ∧ ... ∧ φIN (xN)

� The total number of determinants is: NFCIdet =(

M

N

)

= O(

MN)

� We must extrapolate to the complete basis set (CBS) limit M →∞6/24

Approximations of FCI for a fixed basis set

We must use approximations avoiding the exponential scaling of FCI with N

� Standard approximations for weakly correlated systems:

1 Hartree-Fock (HF)

2 Single-reference post-HF methods:

� perturbation theory (e.g., MP2)

� configuration interaction (e.g., CISD)

� coupled cluster (e.g., CCSD and CCSD(T))

� Standard approximations for strongly correlated systems:

1 Multiconfiguration self-consistent field (MCSCF) = FCI in a small subspace of

orbitals

2 Multi-reference post-MCSCF methods:

� perturbation theory (e.g., CASPT2)

� configuration interaction (e.g., MRCISD)

� More recent methods to closely approach FCI (in the full space or in a subspace)

� CI with on-the-fly perturbative selection of configurations (CIPSI, SHCI)

� CI with stochastic sampling of configurations (FCI-QMC)

� Tensor-network factorization of the wave function (DMRG)

7/24

Convergence of FCI with respect to the basis set

� We use nucleus-centered Gaussian-type basis functions, with spherical coordinates

r = (r , θ, ϕ) around a nucleus,

χν(x) = rℓν (

∑

i cν,ie−αν,i r

2) Y mνℓν (θ, ϕ) δσ,σν

� Example of He atom (N = 2) with the series of basis sets: BX = “cc-pVXZ”

X (# shells) 2 3 4 5 6

functions 2s,1p 3s,2p,1d 4s,3p,2d,1f 5s,4p,3d,2f,1g 6s,5p,4d,3f,2g,1h

M = O(X 3) 10 28 60 110 182

NFCIdet =

(

M

N

)

45 378 1770 5996 16471

-79.0

-78.9

-78.8

-78.7

-78.6

2 3 4 5 6

exact

En

erg

y (

eV

)

X in cc-pVXZ

FCI

� We have an inverse cubic-law convergence: EBXFCI = E

CBSFCI + c X−3

8/24

Origin of the slow basis-set convergence

� The exact wave function as a cusp as the distance between two opposite-spin electrons

goes to zero, r12 = |r2 − r1| → 0,

Ψ(r12) = Ψ(0)[

1+r12

2+O(r 212)

]


Her1

r2

θ

0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

-180 -120 -60 0 60 120 180

Ψ (

a. u

.)

θ (degree)

X=2X=3X=4X=5X=6

exact

� To accelerate the basis-set convergence, we may use explicitly correlated F12 WFT

methods which construct the wave function out of a two-electron space of the form

wB,f2 = Span{f (r12) χν1(x1) ∧ χν2(x2)}

� Here, we will instead consider DFT to avoid the slow basis-set convergence

9/24

Density-functional theory (DFT)

� DFT is formulated in the complete-basis-set (CBS) limit!

� The exact ground-state electronic energy can be expressed via the

constrained-search approach:

E0 = minΨ∈WCBS

〈Ψ|T + Vne + Wee|Ψ〉

= minn∈DCBS

{

minΨ∈WCBSn

〈Ψ|T + Wee|Ψ〉+

∫

vne(r)n(x)dx}

= minn∈DCBS

{

F [n] +

∫

vne(r)n(x)dx}

� WCBSn is the space of wave functions giving a fixed density n(x)

WCBSn = {Ψ ∈ WCBS | ∀ x, nΨ(x) = n(x)}

with

nΨ(x1)=N

∫

|Ψ(x1, x2, ..., xN)|2dx2...dxN

� DCBS is the space of N-representable densities

DCBS = {n | ∃Ψ ∈ WCBS s.t. ∀ x, nΨ(x) = n(x)}

10/24

Kohn-Sham density-functional theory

� In the Kohn-Sham scheme, we decompose the universal density functional F [n] as

F [n] = minΦ∈SCBSn

〈Φ|T |Φ〉 + EHxc[n]

where SCBSn is the space of single-determinant wave functions giving a fixed density

n(x)SCBSn = {Φ ∈ SCBS | ∀ x, nΦ(x) = n(x)}

and SCBS is the space of all normalized single-determinant wave functions

� The exact ground-state electronic energy can then be expressed as

E0 = minn∈DCBS

{

minΦ∈SCBSn

〈Φ|T |Φ〉 + EHxc[n] +

∫

vne(r)n(x)dx}

= minΦ∈SCBS

{

〈Φ|T + Vne|Φ〉 + EHxc[nΦ]}

� The minimizing single-determinant wave function Φ satisfies an effective

self-consistent one-electron Schrodinger equation(

T + Vne + VHxc[nΦ])

|Φ〉 = E|Φ〉

where VHxc[n] =

∫

δEHxc[n]

δn(x)n(x)dx is an effective one-electron potential operator

11/24

Approximation for the density functional EHxc[n]

� The functional EHxc[n] is decomposed as

EHxc[n] = EH[n] + Exc[n]

where the Hartree energy EH[n] is calculated exactly

EH[n] =1

2

x

n(x1)n(x2)wee(r12)dx1dx2

and the exchange-correlation energy Exc[n] is approximated.

� Many approximations have been proposed for Exc[n]. For example, one widely used is

the Perdew-Burke-Ernzerhof (PBE) approximation

EPBExc [n] =

∫

ePBExc (n(x),∇∇∇n(x))dr

where ePBExc (n,∇∇∇n) is constructed using the uniform electron gas and known exact

conditions (high-density limit, small density gradient expansion, Lieb-Oxford bound)

Importantly, ePBExc implicitly contains the physics of the electron-electron cusp taken

from the uniform electron gas

12/24

Convergence of Kohn-Sham DFT with respect to the basis set

� Kohn-Sham (KS) DFT with the PBE density functional using a basis set B :

EB

KS PBE = minΦ∈SB

{

〈Φ|T + Vne|Φ〉+ EH[nΦ] + EPBExc [nΦ]

}


-79.0

-78.9

-78.8

-78.7

-78.6

-78.5

-78.4

2 3 4 5 6

exact

En

erg

y (

eV

)

X in cc-pVXZ

FCI

KS PBE

� Fast basis-set convergence but uncontrolled error due to the PBE approximation

13/24

Outline




14/24

Accelerated WFT using DFT

� For an incomplete basis set B, we can define an approximation to the ground-state

energy by restriction to densities representable in B

EB

0 = minn∈DB

{

F [n] +

∫

vne(r)n(x)dx}

where DB is the space of densities representable in the basis set B

DB = {n | ∃Ψ ∈ WB s.t. ∀ x, nΨ(x) = n(x)}

� The restriction on densities representable in B is much weaker that the restriction on

wave functions representable in B, so we expect

EB

FCI ≫ EB

0 & E0

� We then decompose the universal density functional F [n] as

F [n] = minΨ∈WBn

〈Ψ|T + Wee|Ψ〉+ EB[n]

where WBn is the space of wave functions representable in B and giving the density n

WB

n = {Ψ ∈ WB | ∀ x, nΨ(x) = n(x)}

and EB[n] is a complementary density functional correcting for the basis-setrestriction on the wave function

Giner, Pradines, Ferte, Assaraf, Savin, Toulouse, JCP, 2018

15/24

Accelerated WFT using DFT

� The energy EB0 can thus be expressed as

EB

0 = minΨ∈WB

{

〈Ψ|T + Vne + Wee|Ψ〉+ EB[nΨ]

}

� The minimizing wave function ΨB is an effective multideterminant wave function

satisfying the effective Schrodinger equation

PB

(

T + Vne + Wee + VB[nΨB ]

)

|ΨB〉 = EB|ΨB〉

where PB is the projector on WB and V B[n] =

∫

δEB[n]

δn(x)n(x)dx is an effective

one-electron potential operator

� We will consider a non-self-consistent approximation using the FCI wave function:

EB

FCI+DFT = 〈ΨBFCI|T + Vne + Wee|ΨB

FCI〉+ EB[nΨB

FCI]

and we have EBFCI ≥ EBFCI+DFT ≥ E

B0

� More generally, we can add the DFT basis correction to any approximate WFT

methodEB

WFT+DFT = EB

WFT + EB[n

ΨBWFT

]

Giner, Pradines, Ferte, Assaraf, Savin, Toulouse, JCP, 201816/24

The complementary basis-correction functional EB[n]

� We need to approximate the complementary basis-correction functional

EB[n] = 〈ΨCBS[n]|T + Wee|Ψ

CBS[n]〉 − 〈ΨB[n]|T + Wee|ΨB[n]〉

where the wave function ΨB[n] is associated with the projected electron-electron

interaction PBWeePB

� We fit this projected interaction by a long-range interaction

〈r1 ↑, r2 ↓ |PBWeePB|r1 ↑, r2 ↓〉 ≈ wlree(r12) =

erf(µ(r1)r12)

r12

with a local range-separation parameter µ(r)

� The obtained µ(r) is a local measure of the incompleteness of the basis set B

� We can then recycle the knowledge developed for DFT variants using this modified

electron-electron interaction to build an approximate PBE-based density functional

EB

PBE[n] =

∫

ePBEc (n(x),∇n(x), µ(r)) dr

� This approximate basis-correction functional contains the physics of the

electron-electron cusp, automatically adapts to each basis set B, and correctly

vanishes in the CBS limit


Loos, Pradines, Scemama, Toulouse, Giner, JPCL, 201917/24

Outline




18/24

Convergence of the total energy with respect to the basis set

Total energy of the He atom (N = 2) with the series of basis sets: BX = “cc-pVXZ”

-79.0

-79.0

-78.9

-78.9

-78.8

-78.8

-78.7

-78.7

2 3 4 5 6

exact

En

erg

y (

eV

)

X in cc-pVXZ

FCI

FCI+PBE

=⇒ Much faster basis-set convergence without altering the CBS limit


19/24

Tests on ionization energies of atoms

Ionization energies of atoms, E0(N − 1)− E0(N), with aug-cc-pVXZ (AVXZ) basis sets:

FCI:

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

B C N O F Ne

Err

or

on

io

niz

ati

on

en

erg

y (

eV

)

X=2X=3X=4X=5

FCI+PBE:

−0.10

−0.05

0.00

0.05

0.10

B C N O F Ne

Err

or

on

io

niz

ati

on

en

erg

y (

eV

)

X=2X=3X=4X=5

=⇒ FCI+PBE reaches “chemical accuracy” (1 kcal/mol ≈ 0.04 eV)with X = 3 basis set


20/24

Test on the atomization energy of the C2 molecule

Atomization energy of the C2 molecule, |E0(C2)− 2E0(C)|, with cc-pVXZ basis sets:

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

2 3 4 5

Err

or

on

ato

miz

ati

on

en

erg

y (

eV

)

X in cc−pVXZ

FCIFCI+PBECCSD(T)

CCSD(T)+PBE

=⇒ FCI+PBE or CCSD(T)+PBE quickly converge to “chemical accuracy”

Loos, Pradines, Scemama, Toulouse, Giner, JPCL, 2019

21/24

Tests on atomization energies of 55 molecules

Atomization energies, |E0(molecule)−∑

E0(atoms)|, with cc-pVXZ basis sets:

FCI:

−1

−0.8

−0.6

−0.4

−0.2

0

LiH

BeH

CH

CH

2 ( 3B

1 )C

H2 ( 1

A1 )

CH

3

CH

4

NH

NH

2

NH

3

OH

H2 O

HF

SiH

2 ( 1A

1 )SiH

2 ( 3B

1 )SiH

3

SiH

4

PH

2

PH

3

H2 S

HC

lLi2

LiF

C2 H

2C

2 H4

C2 H

6C

N

HC

NC

O

HC

OH

2 CO

H3 C

OH

N2

N2 H

4N

O

O2

H2 O

2F

2

CO

2

Na

2

Si2

P2

S2

Cl2

NaC

lSiO

CS

SO

ClO

ClF

Si2 H

6C

H3 C

lH

3 CSH

HO

Cl

SO

2

Err

or

on

ato

miz

ati

on

en

erg

ies

(e

V)

X=2X=3X=4X=5

FCI+PBE:

−1

−0.8

−0.6

−0.4

−0.2

0

LiH

BeH

CH

CH

2 ( 3B

1 )C

H2 ( 1

A1 )

CH

3

CH

4

NH

NH

2

NH

3

OH

H2 O

HF

SiH

2 ( 1A

1 )SiH

2 ( 3B

1 )SiH

3

SiH

4

PH

2

PH

3

H2 S

HC

lLi2

LiF

C2 H

2C

2 H4

C2 H

6C

N

HC

NC

O

HC

OH

2 CO

H3 C

OH

N2

N2 H

4N

O

O2

H2 O

2F

2

CO

2

Na

2

Si2

P2

S2

Cl2

NaC

lSiO

CS

SO

ClO

ClF

Si2 H

6C

H3 C

lH

3 CSH

HO

Cl

SO

2

Err

or

on

ato

miz

ati

on

en

erg

ies

(e

V)

X=2X=3X=4X=5

Mean absolute deviation (in eV):

X === 2 X === 3 X === 4 X === 5FCI 0.90 0.30 0.11 0.05

FCI+PBE 0.28 0.06 0.02 0.02

Yao, Giner, Li, Toulouse, Umrigar, JCP, 2020 22/24

Tests on excitation energies

Excitation energies, En −E0, of the molecule C2H4 with aug-cc-pVXZ (AVXZ) basis sets:

FCI:

−0.15

−0.10

−0.05

0.00

0.05

0.10

3B1u

3B3u

1B3u

1B1u

3B1g

1B1g

Err

or

on

ex

cit

ati

on

en

erg

ies

(e

V)

X=2X=3

FCI+PBE:

−0.15

−0.10

−0.05

0.00

0.05

0.10

3B1u

3B3u

1B3u

1B1u

3B1g

1B1g

Err

or

on

ex

cit

ati

on

en

erg

ies

(e

V)

X=2X=3

=⇒ FCI+PBE reaches near “chemical accuracy” with X === 2 basis set

Giner, Scemama, Toulouse, Loos, JCP, 2019

23/24

Summary and outlook

� Summary:

� We have constructed a DFT correction for the incompleteness of the

one-electron basis set

� It greatly accelerates the basis convergence of WFT calculations

� Outlook:

� Improving the DFT basis correction using new ingredients (pair density)

� Self-consistent implementation and calculation of response properties

� Construct a DFT correction for the error of an approximate WFT method

with respect to FCI in a given basis set?

TAKE-HOME MESSAGE: one can ease the burden of many-body

calculations by informed DFT approximations

Could a similar approach be useful in nuclear physics?

www.lct.jussieu.fr/pagesperso/toulouse/presentations/presentation˙onlineseminar˙20.pdf

24/24

www.lct.jussieu.fr/pagesperso/toulouse/presentations/presentation_onlineseminar_20.pdf

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