As you no doubt are aware, the standard basis functions used in ab initio MO theory are cartesian gaussian functions, or linear combinations thereof.
This is a basis function of angular momentum 
centered at the
origin, with orbital exponent 
. Standard notation. These atomic
orbital--like basis functions need not be othogonal to one another, but for
later convenience, it would be nice to have them normalized. Thus impose the
condition
Just for fun and to warm up some, evaluate this integral. Assume a
Normalization constant of N for 
, and call 1.2
a self--overlap integral, SO.
This integral over all space is separable when done in cartesian coordinates
(one of the reasons for using gaussian rather than slater orbitals). Using
and 
, we get
Full derivation of these integrals (
, etc) can be found in Appendix I.
The result is that we find
Recall that 
. Thus
Rearranging that to solve for N, the normalization constant,
This result is completly general -- for uncontracted functions. But before we go on to contrations, let's consider the product of two gaussian functions on different centers.
