Quantitative quantum chemistry: Introduction to wave-function methods

Quantitative quantum chemistry:

Introduction to wave-function methods

Julien Toulouse Laboratoire de Chimie Th´eorique

Sorbonne Universit´ e and CNRS, Paris, France Institut Universitaire de France

Course 5C208 of Master 2 Chemistry Sorbonne Universit´e, Paris

www.lct.jussieu.fr/pagesperso/toulouse/enseignement/presentation_wft_19.pdf

Why and how learning wave-function methods?

Understanding the different quantum-chemistry wave-function methods is important for:

improving them and developing new ones;

using them in applications in an appropriate way.

Recommended books:

J.-L. Rivail, El´ ´ ements de chimie quantique ` a l’usage des chimistes, EDP Sciences / CNRS Editions, 1999.

G. Berthier, N´ ecessaire de chimie th´ eorique, Ellipses, 2009.

A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, Dover Publications Inc., NY, 1996.

T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electronic-Structure Theory, Wiley, 2002.

My lecture notes:

www.lct.jussieu.fr/pagesperso/toulouse/enseignement/introduction_pt_cc.pdf

2/58

Outline

1 Overview of wave-function methods The quantum many-electron problem The Hartree-Fock approximation

Electron correlation and post-Hartree-Fock methods

2 Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3 Coupled-cluster theory The exponential ansatz

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4 Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

3/58

Outline

1 Overview of wave-function methods The quantum many-electron problem The Hartree-Fock approximation

Electron correlation and post-Hartree-Fock methods

2

Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3

Coupled-cluster theory

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

4/58

The Hamiltonian and the many-electron wave function

We consider a N-electron system (atom, molecule, solid) in the Born-Oppenheimer and non-relativistic approximations.

The electronic Hamiltonian in the position representation is, in atomic units, H(r

r

r

) =

N

X

i

h(r

) + 1 2

N

X

i N

X

i6=j

1

| r

− r

| where h(r

) = − (1/2) ∇

+ v

(r

) is the one-electron contribution and v

(r

) = −

α

Z

| r

− R

| is the nuclei-electron interaction.

Stationary states satisfy the time-independent Schr¨ odinger equation H(r

r

r

)Ψ(x

x

x

) = EΨ(x

x

x

) where Ψ(x

x

x

) is a wave function written with space-spin coordinates x

= (r

) (with r

∈

and

∈ {↑ , ↓} ) which is antisymmetric with respect to the exchange of two coordinates, and E is the associated energy.

Using Dirac notations (representation-independent formalism):

H ˆ | Ψ i = E | Ψ i

5/58

The ground-state electronic energy and other properties

The ground-state electronic energy E

can be expressed with the variational principle E

= min

Ψ

h Ψ | H ˆ | Ψ i

where the minimization is over all N-electron antisymmetric wave functions Ψ, normalized to unity h Ψ | Ψ i = 1.

Properties (such as dipole moments or optical spectra) can be essentially obtained as derivatives of the ground-state energy with respect to a perturbation.

For systems containing heavy atoms, we need to include relativistic effects, e.g. using the four-component Dirac Hamiltonian.

For photochemistry studies, we sometimes need to go beyond the Born-Oppenheimer approximation, e.g. calculating non-adiabatic couplings (or vibronic couplings) between the electronic and nuclear vibrational states.

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

6/58

Outline

1 Overview of wave-function methods The quantum many-electron problem The Hartree-Fock approximation

Electron correlation and post-Hartree-Fock methods

2

Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3

Coupled-cluster theory

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

7/58

The Hartree-Fock approximation (1/2)

The (unrestricted) Hartree-Fock (HF) method consists in approximating the ground-state wave function as a single Slater determinant, Ψ

≈ Φ

:

Φ

(x

x

x

) = 1

√ N!

χ1

(x

)

(x

) · · ·

(x

)

(x

)

(x

) · · ·

(x

)

.. . .. . . .. .. .

(x

)

(x

) · · ·

(x

)

where

(x) are orthonormal spin orbitals.

The HF electronic energy is written in terms of integrals over these spin orbitals (using Slater’s rules for calculating expectation values over Slater determinants)

E

[ {

} ] = h Φ

| H ˆ | Φ

i =

occ

X

a

h

+ 1 2

occ

X

a,b

h ab || ab i

where a and b refer to occupied spin orbitals, h

are the one-electron integrals h

=

Z

dx

(x)h(r)χ

(x)

and h ab || ab i = h ab | ab i − h ab | ba i are the antisymmetrized two-electron integrals with h ij | kl i =

dx

dx

χ^∗_i

(x

)χ

(x

)χ

(x

)χ

(x

)

| r

− r

| (in physicists’ notation)

8/58

The Hartree-Fock approximation (2/2)

The spin orbitals are determined by minimizing the HF energy subject to the spin-orbital orthonormalization constraints h

|

i =

E

= min

{χa}

E

[ {

} ]

This minimization leads to the (canonical) HF eigenvalue equations f (x)χ

(x) =

(x)

giving both the occupied and virtual spin orbitals

(x) and associated orbital energies

.

The one-electron HF Hamiltonian (or Fock Hamiltonian) is

f (x) = h(r) + v

(x) where v

(x) is the one-electron HF potential operator

v

(x) =

occ

X

a

J

(x) − K

(x)

composed of a Coulomb (or Hartree) operator and an exchange (or Fock) operator J

(x

) =

Z

dx

χ^∗a

(x

)χ

(x

)

| r

− r

| and K

(x

)χ

(x

) =

dx

χ^∗a

(x

)χ

(x

)

| r

− r

|

(x

)

9/58

Hartree-Fock calculations in a basis set

The spin orbitals are written as the product a spatial orbital

(r) and a spin function

χi

(x) =

(r)α(ω) or

(x) =

(r)β(ω) with

↑ ) = 1,

↓ ) = 0 and

↑ ) = 0,

↓ ) = 1.

The spatial orbitals are expanded on a basis set {

(r) }

ψi

(r) =

K

X

µ=1

C

(r)

In the basis, the HF equations take the form a self-consistent generalized eigenvalue matrix equation

F C

=

S C

where F

=

φ^∗µ

(r)f (x)φ

(r)dr are the elements of the Fock matrix and S

=

φ^∗µ

(r)φ

(r)dr are the elements of the overlap matrix.

In practice, we use atom-centered Gaussian-type orbital (GTO) basis sets

(r) =

p

d

g

(r) with g

(r) = N

r

e

Y

p

(θ, φ) using spherical coordinates r = (r, θ, φ) around each nucleus.

In a straightforward implementation, the computational time of HF scales as O (K

).

10/58

Outline

1 Overview of wave-function methods The quantum many-electron problem The Hartree-Fock approximation

Electron correlation and post-Hartree-Fock methods

2

Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3

Coupled-cluster theory

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

11/58

The electron correlation energy

The HF potential v

(x) is an one-electron mean-field potential approximating the effect of the two-electron interaction (1/2)

i

PN

i6=j

1/ | r

− r

| . In other words, the HF approximation only accounts for the electron-electron interaction in an averaged, mean-field way.

The effect of the electron-electron interaction beyond the HF approximation is called electron correlation. The difference between the exact ground-state energy E

and the HF energy E

is called the correlation energy

E

= E

− E

Even though E

is usually a small percentage of the total energy, it very often makes a large and crucial contribution to energy differences (such as reaction energies, reaction barrier heights, ...). It is therefore important to go beyond the HF approximation and calculate the value of the correlation energy, which is the goal of the post-HF methods.

12/58

A straightforward post-HF method: Full configuration interaction (FCI)

In FCI, the wave function is expanded in terms of the HF determinant Φ

, the singly excited determinants Φ

, the doubly excited determinants Φ

, and so on up to N-fold excited determinants:

| Ψ

i = c

| Φ

i +

occ

X

a vir

X

r

c

| Φ

i +

occ

X

a<b vir

X

r<s

c

| Φ

i +

occ

X

a<b<c vir

X

r<s<t

c

| Φ

i + · · ·

and the coefficients c = (c

c

c

c

E

= min

c

h Ψ

| H ˆ | Ψ

i (with the normalization constraint h Ψ

| Ψ

i = 1), which corresponds to diagonalizing ˆ H in the space spanned by all determinants.

In a finite basis set, FCI gives an upper bound to the exact ground-state energy:

E

≥ E

. In the limit of a complete basis set (K → ∞ ), FCI becomes exact:

E

= E

.

The correlation energy E

= E

− E

has a slow convergence with the basis size:

the error on E

typically decreases as O K

.

Combinatorial explosion of number of determinants: N

=

= O (2K)

= ⇒ necessity to find low-power-scaling approximations

To develop approximations to FCI, we identify two regimes of electron correlation:

dynamic (or weak) correlation and static (or strong) correlation.

13/58

Post-HF methods for dynamic/weak correlation: single-reference methods

Correlation is called “dynamic” or “weak” if c

c

≪ c

= ⇒ HF is a good starting point

This corresponds to situations with a large HOMO-LUMO HF gap.

Example: He atom, H

molecule at equilibrium distance.

^B

Even if each coefficient is small, their total contribution can be large.

For dynamic correlation, one considers “single-reference” methods which are approximations to FCI assuming the predominance of the single HF determinant.

Three main families of single-reference post-HF methods:

Truncated configuration interaction (CI)

Møller-Plesset (MP) perturbation theory (PT)

Coupled-cluster (CC) theory

14/58

Single-reference truncated configuration interaction (CI)

The FCI wave function is truncated at a given excitation level, e.g. keeping only single and double excitations (CISD):

| Ψ

i = c

| Φ

i +

occ

X

a vir

X

r

c

| Φ

i +

occ

X

a<b vir

X

r<s

c

| Φ

i

The coefficients c = (c

c

c

) corresponding to the ground-state wave function are found by minimizing the CISD energy (with the constraint of the normalization of the wave function)

E

= min

c

h Ψ

| H ˆ | Ψ

i which, according to the variational principle, gives E

≥ E

.

The CISD computational cost scales as O K

.

Serious shortcoming of truncated CI: it is not size-consistent.

We prefer methods that satisfy the size-consistency property: the energy of a system composed of two non-interacting fragments A and B is equal to the sum of the energies of the separate fragments:

E (A · · · B) = E (A) + E (B )

This property is important in chemistry. There is also the related size-extensity property: E (N) ∝ N for N → ∞ which is important for extended systems.

15/58

Single-reference Møller-Plesset (MP) perturbation theory

Perturbation theory starting from the HF Hamiltonian: ˆ H = ˆ F + ˆ V

The first-order wave function includes only doubly excited determinants:

| Ψ

i =

occ

X

a<b vir

X

r<s

c

| Φ

i with c

= − h rs || ab i

+

−

−

The second-order energy gives the MP2 correlation energy:

E

= h Φ

| V ˆ | Ψ

i = −

occ

X

a<b vir

X

r<s

|h ab || rs i|

+

−

−

MP2 is not variational (E

can be lower than E

) but is size-consistent.

The MP2 computational cost scales as O K

.

MP2 is a simple largely used post-HF method that often reasonably accounts for dynamic/weak correlation.

However, accuracy is limited by missing higher-order terms.

Including higher-order terms (MP3, MP4, etc...) can be computationally costly and the series does not generally converge!

16/58

Single-reference coupled-cluster (CC) theory

In CC, the wave function is taken as an exponential of a truncated excitation expansion, e.g. with single and double excitations (CCSD):

| Ψ

i = e

| Φ

i where ˆ T

=

occ

X

a vir

X

r

t

ˆ a

ˆ a

and ˆ T

=

occ

X

a<b vir

X

r<s

t

ˆ a

ˆ a

ˆ a

ˆ a

are operators generating singly and doubly excited determinants when acting on the HF wave function | Φ

i .

The CCSD wave function contains all excited determinants of the FCI wave

function, but the coefficients of triply, quadruply, etc.. excited determinants are (antisymmetrized) products of the coefficients of singly and doubly excited determinants.

The amplitudes t

and t

are found by projecting the Schr¨ odinger equation H ˆ | Ψ

i = E | Ψ

i onto h Φ

| and h Φ

| .

CCSD is not variational (E

can be lower than E

) but is size-consistent.

The CCSD computational cost scales as O K

.

CCSD is more accurate than MP2 because it contains higher-order terms.

Possibility to perturbatively add the triple-excitation operator ˆ T

in the expansion

= ⇒ CCSD(T) which is often considered as the gold standard for dynamic/weak correlation with computational cost scaling as O K

.

17/58

Post-HF methods for static/strong correlation: multi-reference methods

Correlation is called “static” or “strong” if there are some coefficients in the FCI expansion that are not small compared to c

= ⇒ HF is NOT a good starting point

This corresponds to situations with a small HOMO-LUMO HF gap.

Example: Be atom, H

molecule at dissociation, transition metals (Fe, Cu, etc...).

Single-reference methods give too much importance to the HF determinant and tend to fail for static/strong correlation (e.g., MP2 diverges for zero HOMO-LUMO HF gap).

Instead of HF, we use now the multiconfiguration self-consistent field (MCSCF) method which is a multideterminant extension of HF:

E

= min

{cn,χ_i}

h Ψ

| H ˆ | Ψ

i with | Ψ

i =

n

c

| Φ

i Usually, we include all determinants | Φ

i that can be generated from a small orbital subspace, called complete active space (CAS) = ⇒ accounts for static/strong correlation but combinatorial explosion with the size of the active space!

Starting from MCSCF, remaining dynamic/weak correlation can be added with multi-reference (MR) methods:

MRCI (e.g., MRCISD): not size-consistent but used for very small systems

MRPT (e.g., CASPT2, NEVPT2): fairly used but requires large enough CAS

MRCC: several proposed methods but no consensual method yet

18/58

Outline

1

Overview of wave-function methods

Electron correlation and post-Hartree-Fock methods

2 Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3

Coupled-cluster theory

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

19/58

General Rayleigh-Schr¨odinger perturbation theory (1/3)

Consider a Hamiltonian ˆ H

depending on a coupling constant

H ˆ

= ˆ H

+

V ˆ

where ˆ H

is a zeroth-order Hamiltonian operator and ˆ V is a perturbation operator.

The “physical” Hamiltonian corresponds to

= 1, i.e. ˆ H = ˆ H

.

H ˆ

is chosen such that its eigenstates Φ

and associated eigenvalues E

are known H ˆ

| Φ

i = E

| Φ

i

and the eigenstates are chosen to be orthonormal, i.e. h Φ

| Φ

i =

.

We want to determine the eigenstates Ψ

and associated eigenvalues E

of the

Hamiltonian ˆ H

, e.g. for the ground state

H ˆ

| Ψ

i = E

| Ψ

i

We assume that E

and Ψ

can be expanded in powers of

E

= E

+

+

E

+ · · · and | Ψ

i = | Ψ

i +

| Ψ

i +

| Ψ

i + · · · where Ψ

= Φ

.

It is convenient to choose the so-called intermediate normalization for Ψ

: h Φ

| Ψ

i = 1 for all

= ⇒ h Φ

| Ψ

i = 0 for all i ≥ 1

20/58

General Rayleigh-Schr¨odinger perturbation theory (2/3)

Insertion of the expansion of Ψ

and E

into the eigenvalue equation gives H ˆ

+

V ˆ | Φ

i +

| Ψ

i +

| Ψ

i + · · ·

=

E

+

+

E

+ · · · | Φ

i +

| Ψ

i +

| Ψ

i + · · ·

Looking at this equation order by order in

zeroth order

H ˆ

| Φ

i = E

| Φ

i

At first order, we obtain

H ˆ

| Ψ

i + ˆ V | Φ

i = E

| Ψ

i + E

| Φ

i

Projecting this equation on the bra state h Φ

| gives the first-order energy correction E

= h Φ

| V ˆ | Φ

i

Projecting now on the bra states h Φ

| for n 6 = 0 gives

E

h Φ

| Ψ

i + h Φ

| V ˆ | Φ

i = E

h Φ

| Ψ

i leading to the first-order wave-function correction

h Φ

| Ψ

i = − h Φ

| V ˆ | Φ

i

E

− E

= ⇒ | Ψ

i = −

n6=0

h Φ

| V ˆ | Φ

i E

− E

| Φ

i

21/58

General Rayleigh-Schr¨odinger perturbation theory (3/3)

Similarly, at second order, we obtain

H ˆ

| Ψ

i + ˆ V | Ψ

i = E

| Ψ

i + E

| Ψ

i + E

| Φ

i

Projecting this equation on the bra state h Φ

| gives the second-order energy correction E

= h Φ

| V ˆ | Ψ

i = −

n6=0

|h Φ

| V ˆ | Φ

i|

E

− E

Note that E

diverges if there is a state Φ

(with n 6 = 0) of energy E

equals to E

, i.e. if the zeroth-order Hamiltonian ˆ H

has a degenerate ground state. In this case, one must instead diagonalize the Hamiltonian in the degenerate space before applying perturbation theory, which is known as degenerate perturbation theory.

Similarly, one can show that the third-order energy correction has the following expression

E

= h Φ

| V ˆ | Ψ

i

=

n,m6=0

h Φ

| V ˆ | Φ

ih Φ

| V ˆ | Φ

ih Φ

| V ˆ | Φ

i

(E

− E

)(E

− E

) − E

n6=0

|h Φ

| V ˆ | Φ

i|

(E

− E

)

22/58

Outline

1

Overview of wave-function methods

Electron correlation and post-Hartree-Fock methods

2 Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3

Coupled-cluster theory

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

23/58

Møller-Plesset perturbation theory (1/5)

Møller-Plesset (MP) perturbation theory is a particular case of Rayleigh-Schr¨odinger perturbation theory for which the zeroth-order Hamiltonian is chosen to be the Hartree-Fock (sometimes also simply called Fock) Hamiltonian

H ˆ

= ˆ F

The expression of ˆ F in the position representation is

F (x

x

x

) =

N

X

i

f (x

)

The corresponding perturbation operator ˆ V is thus the difference between the electron-electron Coulomb interaction and the HF potential

V (x

x

x

) = 1 2

N

X

i N

X

i6=j

1

| r

− r

| −

N

X

i

v

(x

)

24/58

Møller-Plesset perturbation theory (2/5)

The zeroth-order ground-state energy is given by the sum of occupied orbital energies E

= h Φ

| F ˆ | Φ

i =

occ

X

a

εa

The first-order energy correction is the expectation value of the HF determinant over the perturbation operator

E

= h Φ

| V ˆ | Φ

i

Therefore, the sum of the zeroth-order energy and first-order energy correction just gives back the HF energy

E

+ E

= h Φ

| F ˆ + ˆ V | Φ

i = E

The second-order energy correction, also called the second-order Møller-Plesset (MP2) correlation energy, is

E

= E

= −

n6=0

|h Φ

| V ˆ | Φ

i|

E

− E

where Φ

can be a priori single, double, triple, ... excited determinants.

25/58

Møller-Plesset perturbation theory (3/5)

In fact, since ˆ V is a two-body operator, according to Slater’s rules, triple and higher excitations with respect to Φ

give vanishing matrix elements h Φ

| V ˆ | Φ

i .

In addition, it turns out that single excitations only give a vanishing contribution h Φ

| V ˆ | Φ

i =

occ

X

b

h ab || rb i − h a | v ˆ

| r i =

occ

X

b

h ab || rb i −

occ

X

b

h ab || rb i = 0

It thus remains only the double excitations, Φ

= Φ

. Only the two-body part of the perturbation operator gives a non-zero matrix element

h Φ

| V ˆ | Φ

i = h ab || rs i

and the zeroth-order energy corresponding to the doubly-excited determinants Φ

is E

= E

= E

+

+

−

−

We arrive at the following expression for the MP2 correlation energy E

= −

occ

X

a<b vir

X

r<s

|h ab || rs i|

+

−

−

= − 1 4

occ

X

a,b vir

X

r,s

|h ab || rs i|

+

−

−

where we have used the antisymmetry property of the integrals, i.e.

h ab || rs i = −h ab || sr i = −h ba || rs i .

26/58

Møller-Plesset perturbation theory (4/5)

The MP2 correlation energy is always negative.

It diverges to −∞ if one energy denominator

+

−

−

is zero. This happens for systems with zero HF HOMO-LUMO gap, in which case MP perturbation theory cannot be applied.

The MP2 energy is

E

= E

+ E

Since it is not a variational theory, E

is not necessarily above E

.

Similarly, it can be shown, after much work, that the third-order Møller-Plesset (MP3) energy correction has the following expression

E

= 1 8

occ

X

a,b,c,d vir

X

r,s

h ab || rs ih rs || cd ih cd || ab i (ε

+

−

−

)(ε

+

−

−

)

+ 1 8

occ

X

a,b vir

X

r,s,t,u

h ab || rs ih rs || tu ih tu || ab i (ε

+

−

−

)(ε

+

−

−

)

+

occ

X

a,b,c vir

X

r,s,t

h ab || rs ih cs || tb ih rt || ac i (ε

+

−

−

)(ε

+

−

−

)

The calculation of the third- or higher-order terms is often considered as not worthwhile

in comparison with coupled-cluster methods for example.

Møller-Plesset perturbation theory (5/5)

The MP2 correlation energy can be rewritten as the sum of a direct and exchange term

E

= − 1 2

occ

X

a,b vir

X

r,s

h ab | rs ih rs | ab i

+

−

−

+ 1 2

occ

X

a,b vir

X

r,s

h ab | rs ih rs | ba i

+

−

−

These terms can be represented by Feynman diagrams (also called Goldstone diagrams in quantum chemistry)

E

= a r s b + a s r

b

When starting from an unrestricted HF calculation, MP perturbation theory is correctly size consistent at each order. In terms of diagrams, this comes from the fact that each diagram contributing to the perturbative expansion of the energy is made of a single linked piece. Unlinked diagrams do not contribute. This is known as the linked-cluster theorem.

28/58

Outline

1

Overview of wave-function methods

Electron correlation and post-Hartree-Fock methods

2

Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3 Coupled-cluster theory The exponential ansatz

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

29/58

The exponential ansatz (1/5)

In coupled-cluster (CC) theory, one starts by an exponential ansatz for the CC wave function

| Ψ

i = e

| Φ

i

where | Φ

i is the HF wave function, and ˆ T is the cluster (excitation) operator which is the sum of cluster operators of different excitation levels

T ˆ = ˆ T

+ ˆ T

+ · · · + ˆ T

T ˆ

is the cluster operator for the single excitations T ˆ

=

occ

X

a vir

X

r

t

ˆ a

ˆ a

where t

are the single-excitation cluster amplitudes to be determined.

We have used the second-quantization formalism:

ˆaais theannihilation operatorof the spin orbitala ˆa^†r is thecreation operatorof the spin orbitalr

When the operator ˆa^†rˆaaacts on the HF single determinant|Φ0i, it generates the single-excited determinant|Φ^r_ai= ˆa^†_rˆaa|Φ0i

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The exponential ansatz (2/5)

Similarly, ˆ T

is the cluster operator for the double excitations T ˆ

=

occ

X

a<b vir

X

r<s

t

ˆ a

ˆ a

ˆ a

ˆ a

where t

are the double-excitation cluster amplitudes to be determined.

When the operator ˆ a

ˆ a

ˆ a

ˆ a

acts on the HF single determinant | Φ

i , it generates the double-excited determinant | Φ

i = ˆ a

ˆ a

ˆ a

ˆ a

| Φ

i .

And so on up to the ˆ T

cluster operator for N-fold excitations.

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The exponential ansatz (3/5)

To understand the action of the operator e

on the HF wave function | Φ

i , one can expand the exponential and rearrange the operators in terms of excitation levels

e

= ˆ 1 + ˆ T + T ˆ

2! + T ˆ

3! + · · · + T ˆ

N! = ˆ 1 + ˆ C

+ ˆ C

+ · · · + ˆ C

where the operator ˆ C

generates single excitations, ˆ C

generates double excitations, etc.

Noting that the cluster operators ˆ T

, ˆ T

, ..., ˆ T

commute with each other, we find for example for the first four excitation operators

C ˆ

= ˆ T

C ˆ

= ˆ T

+ 1 2 T ˆ

C ˆ

= ˆ T

+ ˆ T

T ˆ

+ 1

6 T ˆ

C ˆ

= ˆ T

+ ˆ T

T ˆ

+ 1

2 T ˆ

+ 1

2 T ˆ

T ˆ

+ 1 24 T ˆ

and so on.

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The exponential ansatz (4/5)

The CC wave function has thus the same form as the FCI wave function (with intermediate normalization, i.e. c

= 1)

| Ψ

i = | Φ

i +

occ

X

a vir

X

r

c

| Φ

i +

occ

X

a<b vir

X

r<s

c

| Φ

i

+

occ

X

a<b<c vir

X

r<s<t

c

| Φ

i +

occ

X

a<b<c<d vir

X

r<s<t<u

c

| Φ

i + · · ·

with coefficients related to the cluster amplitudes by c

= t

c

= t

+ t

∗ t

c

= t

+ t

∗ t

+ t

∗ t

∗ t

c

= t

+ t

∗ t

+ t

∗ t

+ t

∗ t

∗ t

+ t

∗ t

∗ t

∗ t

and so on.

Here, ∗ means an antisymmetric product with respect to exchange of indices:

t

∗ t

= t

t

− t

t

t

∗ t

= t

t

− t

t

+ t

t

− t

t

+ t

t

− t

t

+ t

t

− t

t

+ t

t

t

∗ t

∗ t

= t

t

t

− t

t

t

− t

t

t

− t

t

t

+ t

t

t

+ t

t

t

and so on.

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The exponential ansatz (5/5)

Thus, the CC wave function contains all excited determinants, just as the FCI wave function.

If the cluster operator ˆ T is not truncated, the CC wave function is just a nonlinear reparametrization of the FCI wave function with the same number of parameters.

Optimizing the cluster amplitudes t = (t

t

t

would lead to the FCI wave function.

The interest of the CC approach only appears when the cluster operator is truncated.

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Outline

1

Overview of wave-function methods

Electron correlation and post-Hartree-Fock methods

2

Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3 Coupled-cluster theory The exponential ansatz

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

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Truncation of the cluster operator

For example, let us consider coupled-cluster singles doubles (CCSD) T ˆ = ˆ T

+ ˆ T

The expansion of e

gives e

= ˆ 1 + ˆ T

+

T ˆ

+ 1

2 T ˆ

+

T ˆ

T ˆ

+ 1 6 T ˆ

+

1 2 T ˆ

+ 1

2 T ˆ

T ˆ

+ 1 24 T ˆ

+ · · ·

The CCSD wave function has the same form as the FCI wave function, i.e. it contains all excited determinants, but with coefficients now given by

c

= t

c

= t

+ t

∗ t

c

= t

∗ t

+ t

∗ t

∗ t

c

= t

∗ t

+ t

∗ t

∗ t

+ t

∗ t

∗ t

∗ t

and so on.

The coefficients of the triple excitations c

are now only approximated by products of single- and double-excitation amplitudes t

and t

, and similarly for all higher-level excitations.

In comparison with the CISD wave function, the CCSD wave function has many more excited determinants but the same number of free parameters t = (t

t

) to optimize!

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Truncation of the cluster operator: size-consistency

Contrary to truncated CI, truncated CC is size-consistent, which directly stems from the exponential form of the CC wave function.

Proof: consider a system composed of two infinitely separated (and thus non-interacting) fragments A and B :

H ˆ = ˆ H

+ ˆ H

If the HF calculation is size-consistent, the HF wave function is multiplicatively separable:

|Φ^A···B₀ i=|Φ^A₀i ⊗ |Φ^B₀i(where⊗is the antisymmetric tensor product)

Because the orbitals of each fragments do not overlap, the cluster operator of the system is additively separable: ˆT^A···B= ˆT^A+ ˆT^B

We can then show that theCC wave function is multiplicatively separable:

|Ψ^A···B_CC i=e^TˆA···B|Φ^A···B₀ i

=e^{TˆA+ ˆTB}|Φ^A₀i ⊗ |Φ^B₀i

=e^TˆA|Φ^A₀i ⊗e^TˆB|Φ^B₀i

=|Ψ^A_CCi ⊗ |Ψ^B_CCi

This implies that theCC energy is additively separable:

E_CC(A· · ·B) =hΨ^A···B_CC |HˆA+ ˆHB|Ψ^A···B_CC i

=hΨ^A_CC|Hˆ_A|Ψ^A_CCihΨ^B_CC|Ψ^B_CCi+hΨ^A_CC|Ψ^A_CCihΨ^B_CC|Hˆ_B|Ψ^B_CCi

=ECC(A) +E_CC(B)

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Outline

1

Overview of wave-function methods

Electron correlation and post-Hartree-Fock methods

2

Perturbation theory

General Rayleigh-Schr¨odinger perturbation theory Møller-Plesset perturbation theory

3 Coupled-cluster theory The exponential ansatz

Truncation of the cluster operator

The coupled-cluster energy and the coupled-cluster equations Example: coupled-cluster doubles

Coupled-cluster methods in practice

4

Appendix

Slater’s rules for matrix elements The formalism of second quantization Gaussian basis sets

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The coupled-cluster energy and the coupled-cluster equations (1/4)

We consider in all the following an arbitrary truncation level of the cluster operator.

The most natural way to calculate the CC amplitudes t and CC energy would be by using the variational method:

E

= min

t

h Ψ

| H ˆ | Ψ

i with the constraint h Ψ

| Ψ

i = 1.

However, the CC wave function includes all excited determinants up to N-fold excitations which contribute to the expectation value, giving too complex equations to be efficiently solved.

A more convenient approach for obtaining the CC amplitudes and CC energy is the projection method: we require that Ψ

satisfies the Schr¨odinger equation

( ˆ H − E

) | Ψ

i = 0

projected onto the space spanned by h Φ

| , h Φ

| , h Φ

| , h Φ

| ,...

h Φ

| ( ˆ H − E

) | Ψ

i = 0 h Φ

| ( ˆ H − E

) | Ψ

i = 0 h Φ

| ( ˆ H − E

) | Ψ

i = 0

h Φ

| ( ˆ H − E

) | Ψ

i = 0 and so on.

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