Many-Body Perturbation Theory

4 Electron Correlation Methods

4.8 Many-Body Perturbation Theory

The idea in perturbation methods is that the problem at hand only differs slightly from a problem that has already been solved (exactly or approximately). The solution to the given problem should therefore in some sense be close to the solution to the already known system. This is described mathematically by defining a Hamiltonian operator that consists of two parts, a reference (H₀) and a perturbation (H′). The premise of perturbation methods is that the H′operator in some sense is “small” compared with H₀. Perturbation methods can be used in quantum mechanics for adding corrections to solutions that employ an independent-particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory(MBPT).

Let us assume that the Schrödinger equation for the reference Hamiltonian opera- tor is solved.

(4.28) The solutions for the unperturbed Hamiltonian operator form a complete set (since H₀is Hermitian) which can be chosen to be orthonormal, and lis a (variable) param- eter determining the strength of the perturbation. At present, we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrödinger equation is given by eq. (4.29).

(4.29) If l=0, then H=H₀,Ψ = Φ0and W=E₀. As the perturbation is increased from zero to a finite value, the new energy and wave function must also change continuously, and they can be written as a Taylor expansion in powers of the perturbation parameter l.

HΨ=WΨ

H H H

H

= + ′

= = ∞

0

0 0 1 2

l

Φi EiΦi i , , , . . . ,

(4.30) For l= 0, it is seen that Ψ0= Φ0and W₀=E₀, and this is the unperturbed, or zeroth- order wave function and energy. The Ψ1,Ψ2, . . . and W₁, W₂, . . . are the first-order, second-order,etc.,corrections. The lparameter will eventually be set equal to 1, and the nth-order energy or wave function becomes a sum of all terms up to order n.

It is convenient to choose the perturbed wave function to be intermediately nor- malized, i.e. the overlap with the unperturbed wave function should be 1. This has the consequence that all correction terms are orthogonal to the reference wave function.

(4.31)

Once all the correction terms have been calculated, it is trivial to normalize the total wave function.

With the expansions (eq. (4.30)), the Schrödinger equation (eq. (4.29)) becomes eq.

(4.32).

(4.32) Since this holds for any value of l, we can collect terms with the same power of lto give eq. (4.33).

(4.33)

These are the zero-, first-, second-,nth-order perturbation equations. The zeroth-order equation is just the Schrödinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy,W₁, and the first-order correction to the wave function,Ψ1. The nth-order energy correction can be calculated by multiplying from the left by Φ0and integrating, and using the “turnover rule”〈Φ0|H₀|Ψi〉 = 〈Ψi|H₀|Φ0〉*.

(4.34)

Φ Ψ Φ Ψ Φ Ψ Φ Ψ

Ψ Φ Φ Ψ Φ Ψ

Φ Ψ

0 0 0 1 0

0 1

0 0

0 0 0 1 0 0

0 1

H H

H

H

n n i n i

i n

n

n n n

n n

W W

E W

W

+ ′ = +

+ ′ =

= ′

− −

=

−

−

−

∑

l l l l

0

0 0 0 0

1 0 1 0 0 1 1 0

2 0 2 1 0 2 1 1 2 0

0 1

0

: : : :

H

H H

H H

H H

Ψ Ψ

Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ Ψ

Ψ Ψ Ψ

=

+ ′ = +

+ ′ = + +

+ ′ − = −

∑

=

W

W W

W W W

n W

n n i n i

i n

H₀ H ⁰ ₀ ¹ ₁ ² ₂

0

0 1

1 2

2 0

0 1

1 2

2

+ ′

( )

(

+ + +

)

=

(

+ + +

) (

+ + +

)

l l l l

l l l l l l

Ψ Ψ Ψ

Ψ Ψ Ψ

L

L L

W W W

Ψ Φ

Ψ Ψ Ψ Φ

Φ Φ Ψ Φ Ψ Φ

Ψ Φ

0

0 1 2

2 0

0 0 1 0 2

2 0 0 0

1 1 1 0

=

+ + + =

+ + + =

=

≠

l l

l l

L L

i

W= W + W + W + W +

= + + + +

l l l l

l l l l

0 0 1

1 2

2 3

3

0 0 1

1 2

2 3

3

L L

Ψ Ψ Ψ Ψ Ψ

From this it would appear that the (n −1)th-order wave function is required for cal- culating the nth-order energy. However, by using the turnover rule and the nth- and lower order perturbation equations (4.33), it can be shown that knowledge of the nth- order wave function actually allows a calculation of the (2n+1)th-order energy.

(4.35) Up to this point, we are still dealing with undetermined quantities, energy and wave function corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrödinger equation generate a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh–Schrödinger perturbation theory, and the l¹equation in eq. (4.33) becomes eq. (4.36).

(4.36) Multiplying from the left by Φ* and integrating yields eq. (4.37), where the orthonor-0

mality of the Φis is used (this also follows directly from eq. (4.35)).

(4.37)

The last equation shows that the first-order correction to the energy is an average of the perturbation operator over the unperturbed wave function.

The first-order correction to the wave function can be obtained by multiplying eq.

(4.33) from the left by a function other than Φ0(Φj) and integrating to give eq. (4.38).

(4.38)

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.36)), and they can be calculated from the known unperturbed wave functions and energies. The coefficient in front of Φ0for Ψ1cannot be determined from

c^j E E

j j

= ′

− Φ H Φ0

0

c W c W

c E c E

c E c E

i j i i j i

i i

j j

i i j i

i

j

j j j j

Φ Φ Φ Φ Φ Φ Φ Φ

Φ Φ Φ Φ

Φ Φ

H H

H H

0 0 0 1 0

0 0 0

0 0

0 0 0

− + ′ − =

− + ′ =

− + ′ =

∑

∑

∑

W1= Φ0H′Φ0

c W c W

c E c E W

c E c E W

i i i i

i i

i i i

i

Φ Φ Φ Φ Φ Φ Φ Φ

Φ Φ Φ Φ

Φ Φ

0 0 0 0 0 0 1 0 0

0 0 0 0 0 1

0 0 0 0 0 0 1

0 0 0

H H

H H

− + ′ − =

− + ′ − =

− + ′ − =

∑

∑

∑

Ψ Φ

Φ Φ

1

0 0 1 0 0

=

( − )^_ ^_{ +}( ′ − ) =

∑

∑

c

W c W

i i i i i i

H H

Wn n n Wn k l k l k l

n

2 1 2 1

0

+ + − −

=

= ^{Ψ H}′^Ψ −

∑

^{Ψ Ψ}

,

the above formula, but the assumption of intermediate normalization (eq. (4.31)) makes c₀=0. Starting from the second-order perturbation equation (4.33), analogous formulas can be generated for the second-order corrections. Using intermediate nor- malization (c₀=d₀=0), the second-order energy correction is given by eq. (4.39).

(4.39)

The last equation shows that the second-order energy correction may be written in terms of the first-order wave function (c_i) and matrix elements over unperturbed states.

The second-order wave function correction is given by eq. (4.40).

(4.40)

The formulas for higher order corrections become increasingly complex; the third- order energy correction for example is given in eq. (4.41).

(4.41) The main point, however, is that all corrections can be expressed in terms of matrix elements of the perturbation operator over unperturbed wave functions, and the unperturbed energies.

4.8.1 Møller–Plesset perturbation theory

So far, the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamiltonian operator must

W E E E E

i i j ij j

i j

i j 3

0 0 0 0

0 0

0

= ′ [ ′ − ′ ′ ]

( − )( − )

∑

≠ ^{Φ H Φ Φ H Φ d Φ H Φ Φ H Φ} ,

d W d

c W c W

d E d E c c W

d E d E c c

d

i j i i j i

i i

i j i i j i

i i

j

i i j i

i

j i j i j

i

j j j i j i

i

j

Φ Φ Φ Φ

Φ Φ Φ Φ Φ Φ

Φ Φ Φ Φ

Φ Φ Φ Φ

H H

H

H H

0 0

1 2 0

0 1

0 0 0

0 0 0

− +

′ − − =

− + ′ − =

− + ′ − ′ =

∑

∑

∑

∑

∑ ∑

∑

jj

j i i

j i

i

j

E E E E E Ej

= ′ ′

( − )( − ) ⁻ (^′ ₋ ) ^′

∑

≠ ^{Φ H Φ Φ H Φ0 Φ H Φ Φ H Φ}

0 0

0

0 0 0

0 2

Ψ Φ

Φ Φ Φ

Φ Φ Φ Φ

Φ Φ Φ Φ Φ Φ

Φ Φ

2

0 0 1 2 0

0 0 0 0

0 1 0 2 0 0

0 0

0

0

= ( − )^_ ^_{ +}( ′ − )^_ ^_{ −} =

− +

′ − − =

−

∑

∑ ∑

∑

∑

∑

∑

∑

d

W d W c W

d W d

c W c W

d E d E

i i

i

i i

i i

i i

i i i i

i i

i i i i

i i

i i i

i

H H

H H

0

0 0 0 1 2

0 0 0 0 0 2

2 0

0 0

0 0

0 0

+ ′ − − =

− + ′ − =

= ′ = ′ ′

−

∑

∑

∑ ∑

≠

c c W W

d E d E c W

W c

E E

i i

i

i i

i

i i

i

i i

i i

Φ Φ

Φ Φ

Φ Φ Φ Φ Φ Φ

H H

H H H

be selected. The most common choice is to take this as a sum over Fock operators, leading to Møller–Plesset(MP) perturbation theory.⁹The sum of Fock operators counts the (average) electron–electron repulsion twice (eq. (3.44)), and the perturbation becomes the exact V_eeoperator minus twice the 〈V_ee〉operator. The operator associ- ated with this difference is often referred to as the fluctuation potential. This choice is not really consistent with the basic assumption that the perturbation should be small compared with H₀. However, it does fulfil the other requirement that solutions to the unperturbed Schrödinger equation should be known. Furthermore, this is the only choice that leads to a size extensive method, which is a desirable feature.

(4.42)

The zeroth-order wave function is the HF determinant, and the zeroth-order energy is just a sum of MO energies.

(4.43) Recall that the orbital energy is the energy of an electron in the field of all the nuclei and includes the repulsion to all other electrons, eq. (3.44), and therefore counts the electron–electron repulsion twice. The first-order energy correction is the average of the perturbation operator over the zeroth-order wave function (eq. (4.37)).

(4.44) This yields a correction for the overcounting of the electron–electron repulsion at zeroth order. Comparing eq. (4.44) with the expression for the total energy in eq. (3.32), it is seen that the first-order energy (sum of W0and W1) is exactly the HF energy. Using the notation E(MPn) to indicate the correction at order n, and MPn to indicate the total energy up to order n, we have eq. (4.45).

(4.45)

Electron correlation energy thus starts at order two with this choice of H₀.

In developing perturbation theory, it was assumed that the solutions to the unper- turbed problem formed a complete set. This in general means that there must be an infinite number of functions, which is impossible in actual calculations. The lowest energy solution to the unperturbed problem is the HF wave function, additional higher energy solutions are excited Slater determinants, analogous to the CI method.

When a finite basis set is employed, it is only possible to generate a finite number of excited determinants. The expansion of the many-electron wave function is there- fore truncated.

MP E MP

MP E MP E MP E HF

i i N

0 0

1 0 1

1

= ( )=

= ( )+ (⁼ )= ( )

∑

^elece

W1= Φ0H′Φ0 = Vee −2 Vee = − Vee

W i

i N

i i N

0 0 0 0 0

1 0

1

= = =

= =

∑ ∑

Φ H Φ Φ F Φ

elec elec

e

H F h J K

h g h V

H H H g g V V

0

1 1 1

1 1 1 1

0

1 1

2

2

= = ^_ + ( − )^_

= + = +

′ = − = − = −

= = =

= = = =

=

=

>

∑ ∑ ∑

∑ ∑ ∑ ∑

∑

∑

∑

i i N

i j j

j N i

N

i i N

ij i

i N j

N i N

ij ij

j N i N j i N

elec elec elec

elec elec elec elec

elec elec elec

ee

ee ee

ii N

∑

= 1 elec

Let us look at the expression for the second-order energy correction, eq. (4.39). This involves matrix elements of the perturbation operator between the HF reference and all possible excited states. Since the perturbation is a two-electron operator, all matrix elements involving triple, quadruple, etc., excitations are zero. When canonical HF orbitals are used, matrix elements with singly excited states are also zero, as indicated in eq. (4.46).

(4.46)

The first bracket is zero owing to Brillouin’s theorem (Section 4.2.1), and the second set of brackets is zero owing to the orbitals being eigenfunctions of the Fock opera- tors and orthogonal to each other. The second-order correction to the energy, which is the first contribution to the correlation energy, thus only involves a sum over doubly excited determinants. These can be generated by promoting two electrons from occu- pied orbitals iand jto virtual orbitals aand b. The summation must be restricted such that each excited state is only counted once.

(4.47) The matrix elements between the HF and a doubly excited state are given by two- electron integrals over MOs (eq. (4.8)). The difference in total energy between two Slater determinants becomes a difference in MO energies (essentially Koopmans’

theorem), and the explicit formula for the second-order Møller–Plesset correction is given in eq. (4.48).

(4.48) Once the two-electron integrals over MOs are available, the second-order energy cor- rection can be calculated as a sum over such integrals. There are of the order of M⁴_basis integrals, thus the calculation of the energy (only) increases as M⁴_basiswith the system size. However, the transformation of the integrals from the AO to the MO basis grows as M⁵_basis(Section 4.2.1). MP2 is an M⁵_basismethod, but fairly inexpensive as not all two- electron integrals over MOs are required. Only those corresponding to the combina- tion of two occupied and two virtual MOs are needed. In practical calculations, this means that the MP2 energy for systems with a few hundred basis functions can be cal- culated at a cost similar to or less than what is required for calculating the HF energy.

MP2 typically accounts for 80–90% of the correlation energy, and it is the most economical method for including electron correlation.

The formula for the first-order correction to the wave function (eq. (4.38)) similarly only contains contributions from doubly excited determinants. Since knowledge of the

E MP ^{i j a b i j b a}

i j a b

a b i j

( 2)= ( − )

+ − −

<

<

∑

∑

^{occ vir f f f f}_{e e e}^{f f f f}_e

W E E

ijab ijab ijab a b

i j 2

0 0

0

= ′ ′

< −

<

∑

∑

^{occ vir Φ H Φ Φ H Φ}

Φ Φ Φ Φ

Φ Φ Φ Φ

Φ Φ Φ Φ

0 0 0

0 0 0

0 0 0 0

′ = −

= −

= −  

 =

∑

∑

∑

H H F

H F

H

ia

j j N

ia

ia

j j N

ia

ia

j j N

ia elec

elec

elec

e

first-order wave function allows calculation of the energy up to third order (2n+1 = 3, eq. (4.35)), it is immediately clear that the third-order energy also only contains con- tributions from doubly excited determinants. Qualitative speaking, the MP2 contribu- tion describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is given in eq.

(4.41) and involves a computational effort that formally increases as M⁶_basis. The third- order energy typically accounts for 90–95% of the correlation energy.

The formula for the second-order correction to the wave function (eq. (4.40)) con- tains products of the type 〈Φj|H′|Φi〉〈Φi|H′|Φ0〉. The Φ0is the HF determinant and the lastbracket can only be non-zero if Φiis a doubly excited determinant. This means that the firstbracket only can be non-zero if Φjis either a singly, doubly, triply or quadru- ply excited determinant (since H′is a two-electron operator). The second-order wave function allows calculation of the fourth- and fifth-order energies, and these terms therefore have contributions from determinants that are singly, doubly, triply or quadruply excited. The computational cost of the fourth-order energy without the con- tribution from the triply excited determinants, MP4(SDQ), increases as M⁶_basis, while the triples contribution increases as M⁷_basis. MP4 is still a computationally feasible model for many molecular systems, requiring a time similar to CISD. In typical calculations, the T contribution to MP4 will take roughly the same amount of time as the SDQ con- tributions, but the triples are often the most important at fourth order. The full fourth- order energy typically accounts for 95–98% of the correlation energy.

The fifth-order correction to the energy also involves S, D, T and Q contributions, and the sixth-order term introduces quintuple and sextuple excitations. The working formulas for the MP5 and MP6 contributions are so complex that actual calculations are only possible for small systems. The computational effort for MP5 increases as M⁸_basisand for MP6 as M_basis⁹ . There is very little experience with the performance of MPnbeyond MP4.

As shown in Table 4.2, the most important contribution to the energy in a CI pro- cedure comes from doubly excited determinants. This is also shown by the perturba- tion expansion, the second- and third-order energy corrections only involve doubles.

At fourth order the singles, triples and quadruples enter the expansion for the first time. This is again consistent with Table 4.2, which shows that these types of excita- tions are of similar importance.

CI methods determine the energy by a variational procedure, and the energy is con- sequently an upper bound to the exact energy. There is no such guarantee for pertur- bation methods, and it is possible that the energy will be lower than the exact energy.

This is rarely a problem and may in fact be advantageous. Limitations in the basis set often mean that the error in total energy is several au (thousands of kJ/mol) anyway.

In the large majority of cases, the interest is not in total energies but in energy differ- ences. Having a variational upper bound for two energies does not give any bound for the difference between these two numbers. The main interest is therefore that the error remains relatively constant for different systems, and the absence of a variational bound can allow for error cancellations. The lack of size extensivity of CI methods, on the other hand, is disadvantageous in this respect. The MP perturbation method issize extensive, but other forms of MBPT are not. It is now generally recognized that size extensivity is an important property, and the MP form of MBPT is used almost exclusively.

The main limitation of perturbation methods is the assumption that the zeroth-order wave function is a reasonable approximation to the real wave function, i.e. the per- turbation operator is sufficiently “small”. The more poorly the HF wave function describes the system, the larger are the correction terms, and the more terms must be included to achieve a given level of accuracy. If the reference state is a poor descrip- tion of the system, the convergence may be so slow or erratic that perturbation methods cannot be used. Actually, it is difficult to assess whether the perturbation expansion is convergent or not, although the first few terms for many systems show a behaviour that suggests that it is the case. This may to some extent be deceptive, as it has been demonstrated that the convergence properties depend on the size of the basis set,¹⁰and the majority of studies have employed small- or medium-sized basis sets. A convergent series in for example a DZP type basis may become divergent or oscillat- ing in a larger basis, especially if diffuse functions are present.

The convergence properties for the perturbation series can be analyzed by consid- ering the partitioned Hamiltonian in eq. (4.28), with lbeing a parameter connecting the reference system (l=0) with the real physical system (l=1).¹¹For analyzing the convergence behaviour, we must allow lalso to have complex values.

(4.49) For a given lvalue, the (exact) energy of the ground state can be written as an infi- nite summation of all perturbation terms.

(4.50) The mathematical theory of infinite series states that this summation is only conver- gent within a given radius R, i.e. the infinite series in eq. (4.50) only has a well-defined value if |l| <R. Since we are interested in the situation where l=1, this translates into the condition R>1. The convergence radius is determined by the smallest value of l where another state becomes degenerate with the ground state, i.e. the MP perturba- tion series is only convergent if there are no excited states that become degenerate with the ground state within the circle in the complex plane corresponding to |l| =1.

This includes non-physical situations where lis negative, i.e. where the perturbation corresponds to the electron–electron interaction being attractive. In MP theory the zeroth-order energy is the sum of orbital energies, which includes the average electron–electron interaction twice, and the first-order energy correction W₁ is the negative of the average electron–electron interaction (eq. 4.44). Since W₁is signifi- cantly smaller for excited states than for the ground state (more diffuse orbitals), this means that a negative lvalue will raise the ground state more in energy than an excited state, and this may be sufficient to overcome the energy separation at l =0. This is especially true when diffuse basis functions are included, since they preferentially improve the description of excited states, and such intruder states are the reason for the non-convergent behaviour of the MP perturbation series. A complete search for intruder states within the complex plane corresponding to |l| = 1 is difficult even for simple systems, and a less rigorous search for avoided crossings along the real axis is also demanding. Establishing the convergence or divergence of the MP expansion

E W_i

i

l li

( )=

=

∑

∞ 0

H( )l =H0+lH′

on a case-by-case basis is unmanageable, and one is therefore limited to observing the behaviour for the first few terms.

In the ideal case, the HF, MP2, MP3 and MP4 results show a monotonic convergence towards a limiting value, with the corrections being of the same sign and numerically smaller as the order of perturbation increases. Unfortunately, this is not the typical behaviour. Even in systems where the reference is well described by a single determi- nant, oscillations in a given property as a function of perturbation order are often observed. An analysis by Cremer and He indicates that a smooth convergence (of the total energy) is only expected for systems containing well-separated electron pairs, and that oscillations occur when this is not that case.¹²The latter encompass system con- taining lone pairs and/or multiple bonds, covering the large majority of molecules. It should be noted that one cannot conclude anything about the convergence properties of the whole perturbation series from either the monotonic or oscillating behaviour of the first few terms.

In practice, only low orders of perturbation theory can be carried out, and it is often observed that the HF and MP2 results differ considerably, the MP3 result moves back towards the HF, and MP4 away again. For “well-behaved” systems the correct answer is often somewhere between the MP3 and MP4 results. MP2 typically overshoots the correlation effect, but often gives a better answer than MP3, at least if medium-sized basis sets are used. Just as the first term involving doubles (MP2) tends to overesti- mate the correlation effect, it is often observed that MP4 overestimates the effect of the singles and triples contributions, since they enter the series for the first time at fourth order.

Limitingvalue Property

HF

MP2

MP3

MP4

Figure 4.13 Typical oscillating behaviour of results obtained with the MP method

When the reference wave function contains substantial multi-reference character, a perturbation expansion based on a single determinant will display poor convergence.

If the reference wave function suffers from symmetry breaking (Section 3.8.3), the MP method is almost guaranteed to give absurd results. The questionable convergence of the MP method has caused it to be significantly less popular in recent years, although