The Gaussian Product Theorem states that the product of two arbitrary angular momentum gaussian functions on different centers can be written as
To show this, we first define the multiplicands as
Here, 
, etc.
For primary analysis, take the angular momentum of 
and 
to be zero, so
These are unnormalized, but normalization can be calculated as in
section 1.1. It would be
convenient if this product could be written as a third gaussian, ie.
, or
Expand equation 1.14 using the definition of 
given above.
Comparing terms,
which leads to the conclusion that
From equation 1.17, we expand 
and use that to get a final
expression for K,
if we define 
.
For two 1--s orbitals,
For more general Cartesian gaussians, ones with arbitrary angular momentum,
where we've used equation 1.22 to take care of the product of the
exponentials. Now, 
and the like need to be considered.
Using a standard binomial expansion,
Likewise,
Using these, we can write 
as a summation of 
to
various powers.
The coefficient of 
in the product
is given by
Perhaps more conveniently for implementing in a computational scheme, 
can
be redefined as
Whence we write the full Gaussian Product Theorem as equation 1.10. A derivation of equation 1.30 might be found in appendix II.
