Essentials of Computational Chemistry

218 7 INCLUDING ELECTRON CORRELATION IN MO THEORY
To proceed, we first impose intermediate normalization of ; thatis
〈0| (0)
0 〉=1 (7.30)
By use of Eq. (7.22) and normalization of (0)
0 , it must then be true that
〈 (n)
0 |(0)
0 〉=δn0 (7.31)
Now, we multiply on the left by (0)
0 and integrate to solve Eqs. (7.27)–(7.29). In the case
of Eq. (7.27), we have
Using
〈 (0)
0 |A(0) | (1)
0 〉+〈(0)
0 |V|(0)
0 〉=a(0)
0 〈(0)
0 |(1)
0 〉+a(1)
0 〈(0)
〈 (0)
0 |A(0) | (1)
0 〉=〈(1) 0 |A(0) | (0)
0 〉∗
and Eqs. (7.26), (7.30), and (7.31), we can simplify Eq. (7.32) to
〈 (0)
0 |V|(0) 0 〉=a(1)
0
0 |(0)
0
〉 (7.32)
(7.33)
(7.34)
which is the well-known result that the first-order correction to the eigenvalue is the expectation
value of the perturbation operator over the unperturbed wave function.
As for (1)
0 like any function of the electronic coordinates, it can be expressed as a linear
combination of the complete set of eigenfunctions of A (0) , i.e.,
(1)
0
=
i>0
ci (0)
i
(7.35)
To determine the coefficients ci in Eq. (7.35), we can multiple Eq. (7.27) on the left by (0)
j
and integrate to obtain
〈 (0)
j |A (0) | (1)
0 〉+〈(0) j |V| (0)
0 〉=a(0)
0 〈(0)
j | (1)
0 〉+a(1)
0 〈(0)
Using Eq. (7.35), we expand this to
A (0)
(0)
j
a (0)
0
i>0
(0)
j
ci (0)
i
i>0
ci (0)
i
+〈 (0)
j |V| (0)
0 〉=
+ a (1)
0 〈(0) j | (0)
0 〉
which, from the orthonormality of the eigenfunctions, simplifies to
cja (0)
j +〈(0) j |V| (0) (0)
0 〉=cja 0
j | (0)
0
〉 (7.36)
(7.37)
(7.38)