Quantum Chemistry of Solids / 15-Electron Correlations in Molecules and Crystals

5

Electron Correlations in Molecules and Crystals

5.1 Electron Correlations in Molecules: Post-Hartree–Fock

Methods

5.1.1 What is the Electron Correlation ?

Electrons in molecules and crystals repel each other according to Coulomb’s law, with the repulsion energy depending on the interelectron distance as r12−1. This interaction creates a correlation hole around any electron, i.e. the probability to find any pair of electrons at the same point of spin-coordinate space is zero. From this point of view only the Hartree product Ψ H of molecular or crystalline spin-orbitals ψi(x):

ΨH (x1, x2, . . . , xNe ) = ψ1(x1)ψ2(x2) . . . ψNe (xNe )

(5.1)

is a completely uncorrelated function. The Hartree product (5.1) describes the system of Ne electrons in an independent particle model. This independence means that the probability of simultaneously finding electron 1 at x1, electron 2 at x2, etc. (x means the set of coordinate r and spin σ variables) is given by

|ΨH (x1, x2, . . . , xNe )|2dx1dx2 . . . dxNe
= |ψ1(x1)|2dx1|ψ2(x2)|2dx2 . . . |ψNe (xNe )|2dxNe	(5.2)

which is the probability of finding electron 1 at x1 times the probability of finding electron 2 at x2, etc., i.e. product of probabilities.

The well-known Extended H¨uckel semiempirical method for molecules and the tight-binding approach to crystals are examples of the models with full absence of electron correlation in the wavefunction. The Hamiltonian in these methods does not include explicitly electron–electron interactions (such a Hamiltonian was defined in Chap. 4 as the one-electron Hamiltonian) so that the total many-electron wavefunction is a simple product (5.1) of the one-electron functions and the total electron energy is a sum of one-electron energies. The semiempirical parameters used in these methods allow one to take the correlation into account at least partly. The di erence between the one-electron Hamiltonian and the Hamiltonian of the one-electron approximation (HF method) is the following. The former does not include electron– electron interaction so that the calculation of its eigenvalues and eigenvectors does not

148 5 Electron Correlations in Molecules and Crystals

require a self-consistent procedure. The Hamiltonian of the one-electron approximation includes explicitly the interelectron interactions, the one-electron approximation is made only in the many-electron wavefunction. The one-electron approximation Hamiltonian depends on the one-electron wavefunctions unknown at the beginning of the calculation (for example, the Coulomb and exchange parts of the Hamiltonian in the Hartree–Fock method, see Chap. 4) and the self-consistent calculation is required.

As considered in Chap. 4 the Hartree–Fock (SCF) method replaces the instantaneous electron–electron repulsion with the repulsion of each electron with an average electron charge cloud. The HF method assumes that the many-electron wavefunction can be written as one Slater determinant (4.9). The Hartree–Fock method is usually defined as “uncorrelated”. However, the electron motions are no longer completely independent.

For two electrons with di erent spins, |ϕ1(r1)α(σ1)ϕ2(r2)β(σ2)| the probability of finding electron 1 at r1 and electron 2 at r2 is

		1	P (r1, r2)dr1dr2 = dr1dr2 dσ1 dσ2|Ψ |2
	=		|ϕ1(r1)|2|ϕ2(r2)|2 + |ϕ1(r2)|2|ϕ2(r1)|2 dr1dr2	(5.3)
		2

The electrons are uncorrelated.

For two electrons with the same spin |ϕ1(r1)α(σ1)ϕ2(r2)α(σ2)| the probability of finding electron 1 at r1 and electron 2 at r2 is

P (r1, r2)dr1dr2 =	1	|ϕ1(r1)|2|ϕ2(r2)|2 + |ϕ1(r2)|2|ϕ2(r1)|2
	2
− [ϕ1(r1)ϕ2(r1)ϕ2(r2)ϕ1(r2) + ϕ1(r1)ϕ2(r1)ϕ2(r2)ϕ1(r2)]) dr1dr2			(5.4)

Now, P (r1, r2) = 0. No two electrons with the same spin can be at the same place. This is called the Fermi hole. Thus, same-spin electrons are correlated in Hartree– Fock, di erent-spin electrons are not. Sometimes, it is said that HF methods take into account the so-called spin correlation.

The HF methods are also called by the independent electrons approximation [5] but this independence is restricted by the Pauli principle.

The exact solution of Hartree–Fock–Roothaan equations (4.33) for molecular systems means use of a complete set of basis functions (such a solution corresponds to the Hartree–Fock limit and in practice can be achieved mainly for the simple molecules).

In modern molecular quantum chemistry the correlation energy is defined as the di erence between the exact energy and the HF energy in a complete basis (Hartree– Fock limit). As one does not know the exact energy one uses the experimental total energy (the sum of the experimental cohesive energy and free-atom energies) or calculates the exact energy for a given one-electron basis set and defines the basis set correlation energy as the di erence between the exact and HF energies calculated for the same one-electron basis set. In molecular systems, the correlation energy is about 1 eV per electron pair in a bond or lone pair.

The HF method is usually defined as uncorrelated, however, as we see, the electron motions are no longer completely independent. One of the first attempts to include the electron-correlation in calculations was made by Fock et al. [101], who suggested the incomplete separation of variables for two-valent atoms.

5.1 Electron Correlations in Molecules: Post-Hartree–Fock Methods

149

The key distinction between the Hamiltonian operator and the Fock operator is the following, [102]: the former returns the electronic energy for the many-electron system, the latter is really not a single operator, but the set of all of the interdependent one-electron operators that are used to find the one-electron functions (molecular or crystalline orbitals) from which the HF wavefunction is constructed as a Slater determinant (4.9). The HF wavefunction corresponds to the lowest possible energy for a single-determinant many-electron wavefunction formed from the chosen basis set.

Including electron-correlation in MO theory means an attempt to modify the HF wavefunction to obtain a lower electronic energy when we operate on that modified wavefunction with the Hamiltonian. This is why the name post-Hartree–Fock methods is traditionally used for the methods including the electron-correlation.

As was mentioned in Sect. 4.1.3 in the unrestricted Hartree–Fock approximation (where the coordinate dependence of spin-up and spin-down MOs was allowed to di er) the one-determinant many-electron wavefunction is, in the general case, not an eigenfunction of the total spin operator S2. To repair that deficiency the technique of projection is used [103] so that the resulting wavefunction becomes a sum of several Slater determinants and therefore partly takes into account electron-correlation, i.e. goes beyond the one-derminant HF approximation. However, the coe cients in the sum of Slater determinants are defined only by the projection procedure, i.e. the total spin-symmetry requirements introduced for the many-electron wavefunction.

The sum of Slater determinants

Ψ = C0ΨHF + C1Ψ1 + C2Ψ2 + . . .

(5.5)

is used also in other post-HF approaches: configuartion interaction (CI), multipleconfiguration SCF (MCSCF) and coupled-cluster (CC) methods applied to include the electron-correlation in molecules.

Often, the HF approximation provides an accurate description of the system and the e ects of the inclusion of correlations with CI or MCSCF methods are of secondary importance. In this case, the correlation e ects may be considered as a smaller perturbation and as such treated using the perturbation theory. This is the approach of M¨oller–Plesset [104] or many-body perturbation theory for the inclusion of correlation e ects. In the MP2 approximation only the second-order many-body perturbations are taken into account.

The above-mentioned quantum-chemical approaches to electron-correlations in molecules (also called wavefunction-based correlation methods) are described in detail in monographs [5, 102], recent review articles [105, 106] and are implemented in modern computer codes [35, 107, 108]. The main disadvantage of the wavefunction-based correlation methods is the high scaling of the computational cost with the number of atoms N in a molecule [109], at least when the canonical MOs are used. The scaling of the computational complexity is O(N 5) for the simplest and cheapest method

– second-order perturbation theory MP2. For the CC theory the computational cost scales are O(N 6) and even O(N 7) for the truncated beyond doubles and triples substitutions, respectively. Such a high “scaling wall” [109] restricts the application range of the wavefunction-based correlation methods to molecules of rather modest size. For this reason the density-functional-based correlation methods (see Chap. 7) remain until now the main way to treat large molecular systems. The main disadvantages

150 5 Electron Correlations in Molecules and Crystals

of the latter methods are the principal impossibility for systematic improvements, the underestimation of transition-state energies, and the inability to describe weak interactions (dispersive forces).

The essential progress in the correlation e ects inclusion was achieved in the socalled local correlation methods [109,110] taking into account the short-range nature of the correlation. In these methods, the localized MOs are generated from the occupied canonical MOs using di erent localization criteria, see Sect. 3.3.1. For the virtual space the atomic-orbital basis is projected out of the occupied MO space.

As compared to the molecules the wavefunction-based correlation methods for periodic systems are practically reliable only when the molecular cluster model is used. Unfortunately, the well-known problems of the cluster choice and the influence of the dangling bonds on the numerical results restricts the application range of the molecular cluster model to the essentially ionic systems.

The more sophisticated incremental scheme [111–114] maintains the infinite nature of periodic systems but the correlation e ects are calculated incrementally using standard quantum-chemical codes.

Only recently was the MP2 theory applied to the periodic systems based on the local correlation methods and use of Wannier functions [109, 115].

While for the molecules the local correlation methods are already implemented in the MOLPRO code [116] the implementation of this approach to the periodic systems is the main goal of the new CRYSCOR project [117].

In the next sections we briefly discuss the basic ideas of CI, MCSCF and CC post-HF methods for molecules as they are directly extended to the crystalline solids in the framework of the molecular cluster model. In more detail, the local correlation and MP2 methods are considered both for the molecules and the periodic systems.

5.1.2 Configuration Interaction and Multi-configuration

Self-consistent Field Methods

Methods designed to account for electron correlation in molecules are divided into two classes: wavefunction(WF)-based methods and density-functional (DF)-based methods. The former use, in one or another way, the HF(noncorrelated) orthonormal MOs and therefore are also called post-HF (PHF) methods. In the majority of cases RHF orbitals are used. The latter are based on the density-functional theory (DFT), considered in Chap. 7.

PHF methods can, in turn, be classified as the variational and nonvariational ones. In the former group of methods the coe cients in linear combination of Slater determinants and in some cases LCAO coe cients in HF MOs are optimized in the PHF calculations, in the latter such an optimization is absent. To the former group of PHF methods one refers di erent versions of the configuration interaction (CI) method, the multi-configuration self-consistent field (MCSCF) method, the variational coupled cluster (CC) approach and the rarely used valence bond (VB) and generalized VB methods. The nonvariational PHF methods include the majority of CC realizations and many-body perturbation theory (MBPT), called in its molecular realization the M¨oller-Plessett (MP) method. In MP calculations not only RHF but UHF MOs are also used [107].

In this section, we discuss CI and MCSCF methods, CC and MP approaches are considered in the next sections.

5.1 Electron Correlations in Molecules: Post-Hartree–Fock Methods

151

The concept of the electron configuration has sense only when the one-electron functions of space and spin coordinates are used to approximate the corresponding many-electron wavefunction. The latter is called the configuration-state function (CSF) when the basis of spin-adapted many-electron functions is used. The configuration is specified by fixing the occupation numbers of the molecular spin-orbitals (MSO) and the molecular spin state. For the closed-shell singlets (the ground-state configuration) CSF can always be represented as a single determinant and transforms over identity representation of the molecular point-symmetry group. As the degeneracy of the molecular one-electron states is defined by the dimensions of the irreps of the corresponding point-symmetry group the closed-shell configuration occupation numbers are zero or twice these dimensions (α and β electron-spin projections are allowed). In the majority of open-shell systems (excited-states configurations or ground states of radicals) proper CSFs can only be represented by a combination of two or more Slater determinants. Let the Hartree–Fock–Roothaan (MO LCAO) equations (4.33) be solved and the Slater determinant Φ0 = ΨHF = |ψ1, ψ2, . . . , ψN | in (5.5) is obtained. Solving MO LCAO equations will give M > N orthonormal MSOs (M is the total number of AOs used in the calculation and N is the number of electrons). For the ground state the N energetically lowest MSOs are occupied, which results in the HF determinant. To construct Slater determinants for excited states we may also use energetically higher orbitals, being vacant in the HF solution. It should be added that all MSOs (both those that for the ground state are occupied and those that are empty) are orthonormal as the eigenfunctions to the same Hartree–Fock operator. This represents an important simplification in the calculations.

In the full CI (FCI) method (precise with respect to the basis chosen) M HF MOs

ϕi (i = 1, 2, . . . , M ) generate NCI =C 21 N+MS C 21 N−MS determinants for the system
M	M

with N electrons and fixed total spin projection MS .

Let us rewrite the many-determinant wavefunction (5.5) in the form

	NCI −1
Φ =	I	(5.6)
	CI ΦI
	=0

where index I numbers di erent configurations. The HF approximation corresponds to I = 0 and C0 = 1 and CI = 0 for I > 0 in (5.6).

In the FCI method all MSOs are usually supposed to be fixed as the solutions of HF MO LCAO equations but the parameters CI are varied in the expression

					ˆ			(5.7)
		L = Φ|He|Φ − λ [ Φ|Φ − 1]
We find [8]		I				I
	∂
		" CI ΦI	|Hˆe|		CJ ΦJ − λ	CI ΦI |	CJ ΦJ #
	∂CK
					J		J
				∂
		=	∂CK		I,J CI CJ ΦI |Hˆe|ΦJ − λ ΦI |ΦJ
				= CJ ΦK |Hˆe|ΦJ − λ ΦK |ΦJ = 0				(5.8)

J

152 5 Electron Correlations in Molecules and Crystals

This gives the matrix eigenvalue problem
				HC = λSC			(5.9)
where				ˆ			(5.10)
		HKJ = ΦK |He|ΦJ , SKJ = ΦK |ΦJ
and the eigenvalue in (5.9) is equal to
					ˆ
				λ =	Φ|He|Φ		(5.11)
					Φ|Φ
			ˆ	ˆ		ˆ	N ˆ
The Hamiltonian operator H = h + gˆ consists of the single-electron h =							h(ri) and
1	N	1				ˆ	i
							=1
two-electron gˆ = 2				parts. Calculating for operators h and gˆ the matrix
	=j=1 |ri−rj |
ˆ	i
elements Φ|h|Φ , Φ|gˆ|Φ and taking into account the orthonormality of MSOs it can
be shown [8] that:

1. Di erent configurations are orthonormal;

ˆ

2. For any single-electron operator h there will be nonvanishing matrix elements only between configurations that di er at most in one MSO;

3. For any two-electron operator gˆ there will be nonvanishing matrix elements only between configurations that di er at most in two MSOs. The MOs ϕi are solutions

to the HF equations	ˆ			(5.12)
	F ϕi = iϕi
where	N	Jˆj − Kˆj
Fˆ = hˆ				(5.13)
	+ j=1

The j summation in (5.13) runs over all those orbitals that for the ground-state

				ˆ	ˆ
configuration Φ0 are occupied. The Coulomb Jj					and exchange Kj operators were
defined in (4.16) and (4.17).				ˆ
ν	the matrix elements		Φ
	Let now ν			J |He|ΦI be calculated, where ΦI = Φ0 and ΦJ =
Φn	(in the Φn configuration one electron has been excited from the nth occupied

spin-orbital to the νth empty spin-orbital). It can be shown [8] that both overlap and Hamiltonian matrix elements vanish between the ground-state configuration Φ0 and any single-excited one. Having determined all the relevant matrix elements of (5.9) one may obtain the wavefunction (5.6) as well as the corresponding total electron energy of the FCI method. Notice that in contrast to the HF calculations, the FCI calculations do not require any self-consistency with respect to MOs. Diagonalization of the N -electron Hamiltonian in the basis of Slater determinants gives us energies and WFs of the ground and excited states of the system. It is pertinent to note that for the FCI method the choice of MO basis is not important because any unitary transformation of the MO basis induces unitary transformation of the determinant basis.

The FCI method is the most general of several theories for treating electron correlations. As a variational method it provides upper bounds for the correlation energy. By definition the FCI calculation means that all NCI configurations (possible for a

5.1 Electron Correlations in Molecules: Post-Hartree–Fock Methods

153

given number of electrons N and M AO basis functions) are included in the sum (5.6). FCI calculations can be practically realized only for small molecules and small basis sets as the number of possible configurations drastically increases with increasing number of atoms in a molecule and large-scale CI calculations become very expensive.

As an example, the CH4 molecule with 10 electrons can be taken [5]. Using a minimal basis set of 9 AOs (1s, 2s, 2p for the C atom and 1s for each of four H atoms) one can construct NCI = 43758 Slater determinants (configurations). When the tetrahedral point-symmetry restrictions are taken into account this number considerably reduces (NCI = 5292 for singlet states). Enlarging the basis set to 35 functions (double-zeta plus polarization basis, see Chap. 8) increases the number of singlets to 2 × 1010. When larger molecules are considered and larger basis sets are used, the restriction of the chosen configurations is required. There are several ways of achieving this restriction.

1.Develop some procedure for selection of the most important configurations. In its most general form this method is called GenCI. Frequently used are CIS, CID, CISD, CISDT etc. methods where 1-electron, 2-electron, 1+2-electron, 1+2+3-electron excitations of electrons from occupied HF states to virtual ones are taken into account. These methods can be called restricted CI (RCI) methods. In some computer codes for molecular calculations automatic selection of the most important configurations is performed [119].

2.Subdivide one-electron MOs into several groups. The most primitive subdivision is in three groups: inactive MOs, active MOs, and virtual MOs. Inactive MOs have occupancy 2 in all determinants in CI expansion, virtual MOs have occupancy 0 in all determinants in CI expansion, and occupancies of active MOs are between 0 and 2. It is supposed that in CI wavefunction only excitations within active MOs are taken into account.

3.Combination of 1 and 2, that is subdivision of MOs into groups and use of RCI in the active space.

If RCI expansions are used or orbitals are subdivided into inactive and active groups, or both, then variation of the orbitals themselves may lead to an essential energy decrease (in contrast to the FCI method where it does not happen). Such combined methods that require both optimization of CI coe cients and LCAO co- e cients in MOs are called MCSCF methods. Compared with the CI method, the calculation of the various expansion coe cients is significantly more complicated, and, as for the Hartree–Fock–Roothaan approximation, one has to obtain these using an iterative approach, i.e. the solution has to be self-consistent (this gives the label SCF).

One of the most popular methods of MCSCF class is the complete active space SCF (CASSCF) method by Roos [118] where FCI is performed in active space and optimization of MOs is also done. Note that active–active orbital rotations are irrelevant in the framework of this method.

The most popular version of the RCI method application is to restrict by only singleand double-excited configurations (CISD method). It is well known that the CI single (CIS) method finds no use for ground states as the ground-state HF energy is una ected by inclusion of single excitations (Brillouin theorem, [102]). In the CISD method the singly excited determinants mix with doubles and thus can have some influence on the lowest eigenvalue [102]. What about triple excitations? While there are

154 5 Electron Correlations in Molecules and Crystals

no nonzero matrix elements between the ground state and triply excited states, the triplets do mix with the doubles, and can through them influence the lowest-energy eigenvalue (CISDT method). So, here is some motivation for including them. On the other hand, there are a lot of triples, making their inclusion di cult in a practical sense. As a result, triples and higher-level excitations, are usually not accounted for in the RCI methods so that the CISD approximation dominates in CI calculations. The scaling for CISD with respect to system size, in the large-basis limit, is on the order of N 6. It poses a limit on the sizes of systems that can be practically addressed. Fortunately, the symmetry restrictions allow significant reduction in the computational e ort. Similarly, the core orbitals can be frozen in the generation of the excited state.

5.1.3 Coupled-cluster Methods

The CC method [120] is one of the mathematically elegant techniques for estimating the electron correlation [102,121]. In this method, the FCI wavefunction is represented as

ˆ	(5.14)
ΨCC = exp(T )ΨHF

ˆ

The cluster operator T is defined in terms of standard creation–annihilation operators as

≤N >N			≤N	>N
j	i		j1 2	1 2	† †
ˆ		i	†	i1i2		+ . . .	(5.15)
T =		tj ai aj +		tj1j2 ai1 ai2 aj1 aj1
			<j	i <i

Coe cients t in the last expansion are called the CC amplitudes and they can be defined either variationally or by solving a system of linear equations. The total number of items in (5.15) equals the number of electrons N because no more than N excitations are possible. In most computer codes the nonvariational CC method is implemented since the variational one is technically very complicated. Formally oper-

ˆ

ating on the ΨHF with (1 + T ) gives, in essence, the FCI wavefunction. However, the advantage of the CC representation (5.14) lies in the consequences associated with

ˆ

truncation of T [102]. When, in (5.15), only single or double excitations are involved, the method is called CCS or CCD, respectively, when single and double excitations –

ˆ

then CCSD, etc. Let us consider as an example the CCD approximation when T =

ˆ2 and the expansion (5.15) has the form

T

			ˆ2		ˆ3
ˆ	ˆ		T2		T2
ΨCCD = exp(T2)ΨHF = (1 + T2		+	2!	+	3!	+ · · · )ΨHF	(5.16)

Note that the first two terms in parentheses of (5.16) define the CID method. However,

ˆ

the remaining terms involve products of excitation operators. Each application of T2 generates double excitations, so the product of two applications generates quadruple

ˆ

excitations. Similarly, the cube of T2 generates hextuple substitutions, etc. Such highlevel excitations can not be practically included in CI calculations (in this sense the RCI method is called nonsize consistent).

The computational problem of the CC method is determination of the cluster amplitudes t for all of the operators included in the particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wavefunctions expressed as determinants of the HF

5.1 Electron Correlations in Molecules: Post-Hartree–Fock Methods

155

orbitals. This generates a set of coupled, nonlinear equations in the amplitudes that must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as

ˆ	ˆ		(5.17)
ECC = ΨHF |H| exp(T )ΨHF
	ˆ	in addition to doubles is worth
In practice, the cost of including single excitations T1

the increase in accuracy, and this defines the CCSD model. The scaling behavior of CCSD is on the order of N 6. Inclusion of connected triple excitations (i.e. those

	ˆ	, not the disconnected triples arising
arising with their own unique amplitudes from T3
ˆ	ˆ
as products of T1	and T2) defines CCSDT, but this is very computationally costly

scaling as N 8, and making it intractable for all but the smallest of molecules. Various approaches to estimating the e ects of the connected triples using perturbation theory have been proposed (each with its own acronym). Of these, the most robust, and thus most commonly used, is that in the so-called CCSD(T) method, which also includes a singles/triples coupling term [122]. The (T) approach, in general, slightly overestimates the triples correction, and does so by an amount about equal to the ignored quadruples, i.e. there is a favorable cancelation of errors [123]. This makes the CCSD(T) model extremely e ective in most instances. Analytic gradients [124] and second derivatives [125] are available for CCSD and CCSD(T), which further increases the utility of these methods. Note, however, that truncated coupled-cluster theory is not variational.

5.1.4 Many-electron Perturbation Theory

We follow in this subsection the many-electron perturbation theory description given in [8]. Often, the Hartree–Fock approximation provides an accurate description of the system and the e ects of the inclusion of correlations as, e.g., with the CI or MCSCF methods, may be considered as important but small corrections. Accordingly, the correlation e ects may be considered as a small perturbation and as such treated using the perturbation theory. This is the approach of [126] for the inclusion of correlation e ects.

For the sake of simplification we shall here consider the ground state but mention that the method in principle can be applied for any state, i.e. also for an excited state.

	ˆ
Our starting point is the Hartree–Fock equations (4.13) where the HF operator F ,
	ˆ
(4.14) is a single-electron operator, being a sum of the one-electron operator h, (4.8),
ˆ	ˆ

local Coulomb J, (4.16) and nonlocal exchange K, (4.17) operators.

Solving HF equations (4.13) gives not only the N occupied orbitals, but – in principle – a complete set of M (total number of AO basis functions) orbitals, since

ˆ
F is a Hermitian operator.		ˆ
ˆ
The operator F is a single-electron operator, which we formally wrote as F (i),
where i numbers electrons. We define now first the N -electron operator
	N
Gˆ =	i	(5.18)
	Fˆ(i)
	=1

This is also a Hermitian operator, and the N -electron Slater determinants

156

5 Electron Correlations in Molecules and Crystals

|ψi1 , ψi2 , . . . , ψiN |

(5.19)

define a complete set of eigenfunctions with

ˆ

|ψi1

, ψi2 , . . . , ψiN

| = (εi1 + εi2

+ . . . + εiN )|ψi1 , ψi2 , . . . , ψiN |

(5.20)

G

In particular, for the ground state we have

Gˆ |ψ1, ψ2, . . . , ψN | =

N

" =1 εi# |ψ1, ψ2, . . . , ψN |

(5.21)

i

It follows from (4.38) that

N

1

N

1

1

i=1 εi

= EHF +

i,j=1 ψi

ψj |

|ψiψj − ψj ψi|

|ψiψj

= EHF + E

2

|

r1

−

r2

|

|

r1

−

r2

|

(5.22)

where

N

1

1

1

E

=

i,j=1 ψiψj |

|ψiψj − ψj ψi|

|ψiψj

(5.23)

2

|

r1

−

r2

|

|

r1

−

r2

|

Since the total electronic energy from the Hartree–Fock approximation is the starting point in the perturbation calculation, we see from (5.22) that it is convenient to

consider the operator
ˆ ˆ	− E		(5.24)
G = G

ˆ ˆ

Operator G has the same eigenfunctions as G but the eigenvalues have been shifted

ˆ

by E . The precise form of G is

N

N

N

Jˆj (i) − Kˆj (i)

− E

(5.25)

Gˆ = i=1 Fˆ(i) − E = i=1 hˆ(i) + i,j=1

This can be compared with the true N -electron Hamilton operator,

N

1

1

ˆ

ˆ

He

= i=1 h(i) +

2

i

=j |ri − rj |

(5.26)

In order to apply perturbation theory we write

ˆ

ˆ

ˆ

(5.27)

He = G + ∆H

with

N

1

1

− i,j=1 Jˆj (i) − Kˆj (i)

+ E

(5.28)

∆Hˆ = 2 i =j

|ri − rj |

First-order perturbation theory gives that the ground-state energy changes by

ˆ

(5.29)

Φ0|∆H|Φ0

By using the precise form of the operators in (5.28) one may now show that this term vanishes. The proof of this is very similar to the one showing that the Hamiltonian