Page 63 - Mechanism and Theory in Organic Chemistry

somewhere must be unity, Equation A2.5 must hold, where d~ indicates a multiple

integration throughout all space over all the coordinates of the electron. It is an
easy matter'to normalize an orbital function; if t+b is not normalized, t+b/W~
will be. In practice, normalized atomic orbital functions vj are chosen initially;
then the 4's are normalized when Equation A2.6 is satisfied.

THE SECULAR EQUATION
We now return to Equation A2.3 and substitute into it Equation A2.4. We
then obtain A2.7, where N is the total number of basis orbitals being used. The
variation principle now has to be applied to A2.7 to find the values of the c's
which will give the best t+bYs possible with the chosen basis. The energy is mini-)?
mized simultaneously with respect to all the c's by carrying out a partial differen-
tiation with respect to each c and making the derivatives of the energy satisfy i

A2.8. The result, after some manipulations, is a set of N equations of the form
A2.9, where the index i takes a different value for each equation.

aE
- = 0 for eachj = 1,2,. . ., N (A2.8)
acj

We now introduce the following new notation.

St, is the overlap integral, and Hij is called the Hamiltonian matrix element between
basis functions i and j. We can now rewrite A2.9 as A2.12:

Rearranging, we have :

Because orbital functions can be complex (that is, they can contain the quantity dq), one must
actually use +h*+h instead of +ha. The functions one ordinarily encounters in approximate molecular
orbital theories are real; therefore this distinction makes no practical difference for our purposes.