Essentials of Computational Chemistry

or
7.4 PERTURBATION THEORY 219
cj = 〈(0) j |V| (0)
0 〉
a (0)
0
− a(0)
j
(7.39)
With the first-order eigenvalue and wave function corrections in hand, we can carry out
analogous operations to determine the second-order corrections, then the third-order, etc.
The algebra is tedious, and we simply note the results for the eigenvalue corrections, namely
and
a (3)
0
=
j>0,k>0
a (2)
0
〈 (0)
0 |V|(0) j 〉[〈 (0)
=
j>0
|〈 (0)
j |V| (0)
0 〉|2
a (0)
0
− a(0)
j
j |V| (0)
(0)
k 〉−δjk〈 0 |V|(0) 0 〉]〈(0) k |V| (0)
0 〉
(a (0)
0 − a(0)
j )(a (0)
0 − a(0)
k )
(7.40)
(7.41)
Let us now examine the application of perturbation theory to the particular case of the
Hamiltonian operator and the energy.
7.4.2 Single-reference
We now consider the use of perturbation theory for the case where the complete operator
A is the Hamiltonian, H. Møller and Plesset (1934) proposed choices for A (0) and V with
this goal in mind, and the application of their prescription is now typically referred to by
the acronym MPn where n is the order at which the perturbation theory is truncated, e.g.,
MP2, MP3, etc. Some workers in the field prefer the acronym MBPTn, to emphasize the
more general nature of many-body perturbation theory (Bartlett 1981).
The MP approach takes H (0) to be the sum of the one-electron Fock operators, i.e., the
non-interacting Hamiltonian (see Section 4.5.2)
H (0) =
n
fi
i=1
(7.42)
where n is the number of basis functions and fi is defined in the usual way according to
Eq. (4.52). In addition, (0) is taken to be the HF wave function, which is a Slater determinant
formed from the occupied orbitals. By analogy to Eq. (4.36), it is straightforward to show
that the eigenvalue of H (0) when applied to the HF wave function is the sum of the occupied
orbital energies, i.e.,
H (0) (0) occ.
= εi (0)
(7.43)
where the orbital energies are the usual eigenvalues of the specific one-electron Fock operators.
The sum on the r.h.s. thus defines the eigenvalue a (0) .
i