The AO approximation

The only uncharged molecule with two electrons is H2, and we will consider this molecule for a while. The ESE allows us to do something that cannot be done in the laboratory. It assumes the nuclei are stationary, so for the moment we consider a very stretched out H2 molecule. If the atoms are distant enough we expect each one 

Perhaps a small digression is in order on the use of the term centered in the last paragraph. When we write the ESE and its solutions, we use a single coordinate system, which, of course, has one origin. Then the position of each of the particles, r, for electrons and for nuclei, is given by a vector from this common origin. When determining the state of H (with an infinitely massive proton), one obtains the result (in au) 

In actuality it will be useful later to generalize the function of Eq. (2.10) by changing its size. We do this by introducing a scale factor in the exponent and write 

When we work out integrals for VB functions, we will normally do them in terms of this version of the H-atom function. We may reclaim the real H-atom function any time by setting a = 1. 

Let us now investigate the normalization constant in Eq. (2.9). Direct substitution yields 

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