.

$$- \frac{\hbar^2}{2m_e}\nabla^2 - \frac{Z_1e^2}{\sqrt{x^2+y^2+(z-R/2)^2}} -\frac{Z_4e^2}{\sqrt{x^2+y^2+(z+R/2)^2}}$$

Brute force integration usually requires that we first transform to elliptical coordinates, where

$$\lambda = \frac{r_1+r_4}{R}$$

and

$$\mu = \frac{r_1-r_4}{R}$$

(and $\phi$, the common azimuthal angle taken from spherical polar coordinates is a hold over) or, conversely,

$$r_1 = \frac{R(\lambda+\mu)}{2}$$

and

$$r_4 = \frac{R(\lambda-\mu)}{2}$$

This is covered elsewhere in the Computer Guided Reading set of readings.

In order to proceed with the evaluation of the integral, we convert to this new coordinate system, where

$$x = \frac{R}{2} \sqrt{( \lambda^2-1)(1-\mu^2)} cos \phi$$

$$y = \frac{R}{2} \sqrt{( \lambda^2-1)(1-\mu^2)} sin \phi$$

$$z = \frac{R}{2} \lambda \mu$$

and

$$\nabla^2 = {4\over {R^2 \left ( \lambda^2 - \mu^2 \right )}} ... ...}{\partial \phi }\right) }{\partial \phi }\right) \right \}$$

we then form (from Equation 18)

$$\beta^h = \int p_{h_1} \left ( -\frac{\hbar^2}{2 m_e} \nabla^... ...Z_1 e^2 }{r_1} +\frac{-Z_4 e^2 }{r_K} \right ) p_{h_4} d\tau$$