### D.21 Solution of the hydrogen molecular ion

The key to the variational approximation to the hydrogen molecular ion is to be able to accurately evaluate the expectation energy

This can be multiplied out and simplified by noting that and are eigenfunctions of the partial Hamiltonians. For example,

where is the -13.6 eV hydrogen atom ground state energy. The expression can be further simplified by noting that by symmetry

and that and are real, so that the left and right sides of the various inner products can be reversed. Also, and are related by the normalization requirement

Cleaning up the expectation energy in this way, the result is

which includes the proton to proton repulsion energy (the 1). The energy is the 13.6 eV amount of energy when the protons are far apart.

Numerical integration is not needed; the inner product integrals in this expression can be done analytically. To do so, take the origin of a spherical coordinate system at the left proton, and the axis towards the right one, so that

In those terms,

Then integrate angles first using . Do not forget that , not , e.g. 3, not 3. More details are in [25, pp. 305-307].

The overlap integral turns out to be

and provides a measure of how much the regions of the two wave functions overlap. The direct integral is

and gives the classical potential of an electron density of strength in the field of the right proton, except for the factor . The “exchange integral” is

and is somewhat of a twilight term, since suggests that the electron is around the left proton, but suggests it is around the right one.