### One-center two-electron integrals

The MNDO, AM1, and MNDO-d one-center two-electron integrals are derived from experimental data on isolated atoms. Most were taken from Oleari's [54] work, but a few were obtained by optimization to fit molecular properties. The values of PM3 one-center two-electron integrals were optimized to reproduce experimental molecular properties.

For each atom there are a maximum of five one-center two-electron integrals. These are <ss|ss>, <ss|pp>, <sp|sp>, <pp|pp>, and <pp|p'p'>, where p and p' are two different p-type atomic orbitals. In the original formulation [55] there was a sixth integral, <pp'|pp'>, but it can be shown that this integral is related to two of the other integrals by:

<pp|p'p'> = 1/2( <pp|pp> - <pp|p'p'>)

Proof: If the molecular frame is rotated by 45 about the z axis the atomic bases mix thus:

R(45o)px = 0.7071(px + py)

R(45o)py = 0.7071(py - px)

R(45o)<pxpy|pxpy> = 1/4<(px + py)(py - px)|(px + py)(py - px)>

= 1/4(<px px|px px> + <py py|py py> - <px px|py py> - <py py|px px>)

or

R(45o)<pxpy|pxpy> = 1/2(<px px|px px>  -  <py py|px px>)

For convenience these five integrals are given the following names:

<ss|ss> = Gss
<pp|pp> = Gpp
<sp|sp> = Hsp
<pp|pp> = Gpp
<pp|p'p'> = Gp2

Using these definitions, the two-electron one-center contributions to the Fock matrix become:
 : : : :

These expressions are common to all methods.