Elliptical integrals for solution of the hydrogen molecule ion

In order to solve the H2+ ion analytically we can use integration in elliptical coordinates. This is shown schematically below.

 

To transform the coordinates we use the following definitions of the elliptical coordinate system.

The volume element is

Within the following limits

Note that there is axial symmetry i.e. there is no f dependence of the 1s-functions in the hydrogen molecule ion.

The overlap integral S becomes

Using the above definitions this integral can be recast as

Let u = -Rl then du = -Rdl and dl = -du/R.

The second term integrates to –e-R, but the first requires integration by parts.

This is evaluated at minus infinity (it is zero there) and at –R where it is

-e-R(R2 +2R+2).

This further simplifies to

 

The Coulomb integral is

 

We can solve for rA and rB in terms of l and m.

rA = Rl – rB, rB = rA – Rm

rA = Rl – rA + Rm

2rA = R(l + m), 2rB = R(l - m)

 

Recognizing that (l2 – m2) = (l + m)(l – m) we can rewrite this integral as.

and then

Using the fact that integration by parts yields

 

The integrals evaluate to

Therefore

The exchange integral can likewise be evaluated.

 

Making substitutions as above

And we can follow the usual steps of expanding the integral

followed by a substitution of variable u = -Rl.

which, when substituted into the initial expression for K gives