Elliptical integrals

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 r_A and r_B in terms of l and m.

r_A = Rl – r_B, r_B = r_A – Rm

r_A = Rl – r_A + Rm

2r_A = R(l + m), 2r_B = 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