Post on 16-Mar-2022
transcript
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
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
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
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)
]
� Example of He atom (N = 2) with the series of basis sets: BX = “cc-pVXZ”
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]
}
� Example of He atom (N = 2) with the series of basis sets: BX = “cc-pVXZ”
-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
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
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
Giner, Pradines, Ferte, Assaraf, Savin, Toulouse, JCP, 2018
Loos, Pradines, Scemama, Toulouse, Giner, JPCL, 201917/24
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
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
Giner, Pradines, Ferte, Assaraf, Savin, Toulouse, JCP, 2018
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
Giner, Pradines, Ferte, Assaraf, Savin, Toulouse, JCP, 2018
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