The Hartree-Fock approach is based on the assumption that a molecular or atomic wave function may be approximated as an antisymmetrized product of one-electron orbitals in the case of a closed-shell configuration or as a number of antisymme- trized products in other cases. The goal of the Hartree-Fock method or the Self- Consistent Field (SCF) method is the subsequent derivation of the best possible one-electron orbitals by making use of the variational principle.
Rather than vary each orbital at a time, the Hartree-Fock equations have been designed in such a way that all orbitals may be obtained at the same time as the eigenfunctions of an effective one-electron operator, the Hartree-Fock operator.
The latter operator contains the average Coulomb repulsion between the electrons.
The Hartree-Fock equations for most atoms may be solved exactly in numerical form, but in the case of molecules it is usually necessary to introduce additional approximations.
In order to derive the Hartree-Fock equations, we must first derive the expecta- tion values of the Hamiltonian expressed in terms of one-electron orbitals from the antisymmetrized wave function (11-2). The Hartree-Fock equations are then
obtained by varying one of the one-electron orbitals followed by a number of mathematical transformations. We limit ourselves to a discussion of closed-shell ground state configurations. Other configurations lead to similar sets of equations, but their derivations are more complex.
In order to derive the Hartree-Fock equations, we write the Hamiltonian as a sum of one-electron and two-electron terms:
HẳX
j
Gðjị ỵX
j>k
ðj;kị ð11-7ị
Here the one-electron terms Gðjị represent the sum of the kinetic and potential energy of each electron and the two-electron terms are the Coulomb repulsion energies between the electrons. The expectation value E of the energy is then given by
Eẳ 0
X
j
Gðjị ỵX
j>k
ðj;kị
0
* +
ð11-8ị
where0is defined in Eq. (11-2). We assume that the one-electron orbitals are orthonormal,
hfjjfki ẳ0 ifj6ẳk
hfjjfki ẳ1 ifjẳk ð11-9ị It is then easily verified that
h0j0i ẳ1 ð11-10ị
There is no need to consider all possible permutations in both functions0that occur in Eq. (11-8), the permutations in one of them suffices. If in addition we sepa- rate the Hamiltonian into two parts then Eq. (11-8) may be reduced to
Eẳ hf1ð1ịað1ịf1ð2ịbð2ịf2ð3ịað3ịf2ð4ịbð4ị fNð2N1ịað2N1ịfNð2Nịbð2Nị X
j
Gðjị ỵX
k>j
ðj;kị
X
P
PdPẵf1ð1ịað1ịf1ð2ịbð2ịf2ð3ịað3ịf2ð4ịbð4ị
fNð2N1ịað2N1ịfNð2Nịbð2Nịi ð11-11ị
We first consider the term containing the one electron operatorsGðjị, which we denote byE1. It is easily seen that
E1ẳ2X
j
Gjẳ2X
j
hfjjGjfji ð11-12ị
THE HARTREE-FOCK METHOD 147
since any permutation of the function0on the right side of Eq. (1-11) will be zero because of the orthogonality condition (11-9) of the orbitalsfi.
The contribution of the two-electron operators(j;k) to E may be separated into two parts E2 and E3. The part E2 is the contribution of the non-permuted term on the right side of Eq. (11-11). It is given by
E2ẳXN
iẳ1
Ji;iþ4X
j>i
Ji;j ð11-13ị
Here the integralsJi;j are known as Coulomb integral and they are defined as Ji;jẳ hfið1ịfjð2ịjð1;2ịjfið1ịfjð2ịi ð11-14ị It will prove to be convenient to rewrite Eq. (11-14) as
E2 ẳXN
iẳ1
Ji;iþ2XN
iẳ1
X
j6ẳi
Ji;j
ẳ2XN
iẳ1
XN
jẳ1
Ji;jXN
iẳ1
Ji;i
ð11-15ị
The second partE3of the contributions of the operators(j;k) toEis due to all possible permutations in Eq. (11-11) but we should realize that only a fraction of these permutations lead to a nonzero result. Because of the orthogonality of the spin functions we only obtain a nonzero result if we permute either within the set of even or within the set of odd numbered electrons. Subject to this restraint we may only exchange one pair of electrons in order to get a nonzero result because of the ortho- gonality (11-9) of the orbitals. The result is
E3ẳ 2X
j>i
Ki;j ð11-16ị
The exchange integralsKi;jare here defined as
Ki;jẳ hfið1ịfjð2ịjð1;2ịjfjð1ịfið2ịi ð11-17ị We may again rearrange Eq. (11-16) as
E3ẳ XN
iẳ1
XN
jẳ1
Ki;jþXN
iẳ1
Ki;i ð11-18ị
The desired expression for the expectation value E is now obtained by taking the sum of the three contributions (11-12), (11-15) and (11-18) which gives
Eẳ2X
i
Giþ2X
i
X
j
Ji;jX
i
X
j
Ki;j
ẳXN
iẳ1
2GiþXN
jẳ1
ð2Ji;jKi;jị
" # ð11-19ị
since
Ji;iẳKi;i ð11-20ị
The Hartree-Fock equations may now be derived from the energy expression (11-19) by varying one of the orbitals, for example fk, by an amount dfk and by setting the corresponding changedkE in the energy equal to zero,
dkEẳ0 ð11-21ị
It is allowed to vary only the functionsfkon the left of the operators without loss of generality and it follows then that
dkEẳ2hdfkjGjfki ỵ4XN
jẳ1
hdfkð1ịfjð2ịjð1;2ịjfkð1ịfjð2ịi
2XN
jẳ1
hdfkð1ịfjð2ịjð1;2ịjfjð1ịfkð2ịi
ð11-22ị
This expression may be simplified by introducing the operator
Jjðr1ị ẳ ð
fjðr2ịðr1;r2ịfjðr2ịdr2 ð11-23ị
It is possible to transform the exchange integrals in a similar fashion but in the latter case the definition of the operator becomes more complex. We define the operators Kj(r1) by means of the equation
Kjðr1ịcðr1ị ẳ ð
fjðr2ịðr1;r2ịcðr2ịdr2
fjðr1ị ð11-24ị It is easily verified that the Coulomb integral in Eq. (11-22) may now be written as hdfkð1ịfjð2ịjð1;2ịjfkð1ịfjð2ịi ẳ hdfkð1ịjJjð1ịfkjð1ịi ð11-25ị
THE HARTREE-FOCK METHOD 149
while the exchange type integrals may be represented as
hdfkð1ịfjð2ịjð1;2ịjfjð1ịfkð2ịi ẳ hdfkð1ịjKjð1ịfkjð1ịi ð11-26ị By substituting these results into Eq. (11-22) we obtain
dkEẳ2hdfkjGjfki ỵ4XN
jẳ1
hdfkjJjjfki 2XN
jẳ1
hdfkjKjjfki ð11-27ị
By introducing the Hartree-Fock operator
FopẳGỵXN
jẳ1
ð2JjKjị ð11-28ị
We may reduce Eq (11-27) to the simple from
dkEẳ2hdfkjFopjfki ẳ0 ð11-29ị subject to the restraint
hdfkjfki ẳ0 ð11-30ị It is worth noting that the Hartree-Fock operator contains a sum over all occu- pied orbitalsfjso that all occupied orbitals contribute equally to the operator. Even though one of the orbitalsfk was selected as the orbital to be varied this orbital does not have a preferred role in the definition of th Hartree-Fock operator. Conse- quently it follows from Eqs. (11-28) and (11-29) that all occupied orbitals must be eigenfuncitons of the same Hartree-Fock operatorFop. These eigenfunctionsfkand corresponding eigenvalueslkare defined as
Fopfkẳlkfk ð11-31ị We should realize that the Hartree-Fock operatorFopis constructed from a set of approximate orbitals. Even the exact solutions of the eigenvalue problem (11-31) are therefore of an approximate nature. On the other hand, it may be assumed that the set of orbitals which are the solutions of Eq. (11-31) are more accurate than the set of orbitals that were used in constructing the operatorFop. It is therefore advantageous to define an improved operatorFop by substituting the solutions of the previous operator. The solutions of the new and improved operatorFopshould be more accurate than the solutions of the previous operator. In the SCF method this procedure is repeated a number of times until the solutions of the eigenvalue
problem (11-31) become identical with the set of orbitals that were used in con- structing the Hartree-Fock operator. It is said that at this point self-consistency is achieved, hence the name Self Consistent Field or SCF method. It is also customary to refer to the results of the SCF procedure as the solutions of the Hartree-Fock method. It appears that either one of the two names, SCF or HF, is generally accepted for the description of the method.
It may be derived from the definition (11-28) of the Hartree Fock operatorFop that the eigenvaluelk is given by
lkẳ hfkjFopjfki ẳGkỵXN
jẳ1
ð2Jk;jKk;jị ð11-32ị It is important to note that the total energy as defined by Eq. (11-19) is not equal to the sume of the Hartree-Fock parameters but it is instead given by
EẳXN
kẳ1
ðGkỵlkị ð11-33ị
The Hartree-Fock eigenvalue problem (11-31) has in principle an infinite number of eigenvalues and corresponding eigenfunctions. If we assume that self- consistency has been achieved, then the set of eigenfunctions f1, f2,. . . fN corresponding to the lowest N eigenvalues represents the filled orbitals of the system. It may be argued that the additional eigenfunctions fNþ1, fNþ2, and so on, corresponding to higher eigenvalueslNþ1,lNþ2, and so on, have no physical meaning. However, it is generally assumed that these additional Hartree-Fock eigenfunctions may be used for the construction of the wave functions correspond- ing to either singlet or triplet excited molecular configurations. The specific form of these excited state wave functions was presented in Eqs. (11-5) and (11-6). The corresponding excitation energies are given by
1Eðj!nị Eẳ h1<j!nịjHEj1ðj!nịi
ẳlnljJj;nỵ2Kj;n
ð11-34ị
and
3Eðj!nị Eẳ h3ðj!nịjHEj3ðj!nịi
ẳlnljJj;n
ð11-35ị
where the energyE is defined in Eq. (11-8).
There is a much simpler relation between the Hartree-Fock parameterslk and the ionization energies of the system. The energy required to remove an electron from a doubly occupied orbital fk is approximately equal to the corresponding eigenvalue lk. This theorem was proved in 1934 by Tjalling Charles Koopmans
THE HARTREE-FOCK METHOD 151
(1910–1985), who was a graduate student in theoretical physics with Kramers at the time. It is interesting to note that this is the only contribution to theoretical physics by Koopmans because his interest changed to mathematical economics. He received the 1975 Nobel Prize in economics.