Correlation energy between two electrons

[eonsv]

Hello. I need to find the expectational value of the correlation energy between two electrons which repel each other via the classical Coulomb interaction. This is given by the 6-dimensional integral:
\left\langle\frac{1}{|\vec{r}_1 - \vec{r}_2|}\right\rangle =\int_{-\infty}^{\infty}d\vec{r}_1d\vec{r}_2e^{-2\alpha(r_1 + r_2)}\frac{1}{|\vec{r_1}-\vec{r_2}|}
r_i = |\vec{r}_i|, \quad \vec{r}_i = x_i \hat{i} + y_i\hat{j} + z_i\hat{k}, \quad i = 1,2
I know that the answer is \frac{5\pi^2}{16^2}, however I can't seem to find a method for solving this type of integrals in any of my books. I have tried to use spherical coordinates, which seemed logical due to the answer having a factor of \pi^2, but with no luck. And it is a couple of years since I've been solving integrals like this one, so a nudge in the right direction is greatly appriciated.

(It is a bonus question in one of the projects in the course: computational physics)

 

[PhysicsGente]

Hello.
Try using the law of cosines to expand the denominator and do one integral at a time.

 

[eonsv]

Thank you for your reply.
This is one of the things which I've been trying to do. However I end up with a result which is a very complicated integral. Expanding the difference of the length of the two vectors gives:
|\vec{r}_1 - \vec{r}_2| = \sqrt{r_1^2 + r_2^2 - 2r_1 r_2 \cos\beta} Where \cos\beta = \cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2 \cos(\phi_1 - \phi_2)
I then need to expand d\vec{r}_1 and d\vec{r}_2, which can be done in spherical coordinates as: d\vec{r}_i = r_i^2 \sin\theta_i dr_id\theta_id\phi_i, i = 1,2 or d\vec{r}_1d\vec{r}_2 = r_1^2r_2^2dr_1dr_2d\cos\theta_1d\cos\theta_2d\phi_1d\phi_2
This is were I end up with the integral
\left\langle\frac{1}{|\vec{r}_1 - \vec{r}_2|}\right\rangle = \int_{-\infty}^{\infty}e^{-2\alpha(r_1 + r_2)} \frac{r_1^2r_2^2dr_1dr_2d\cos\theta_1d\cos\theta_2d\phi_1d\phi_2}{\sqrt{r_1^2 + r_2^2 - 2r_1 r_2\cos\beta}}
I can't seem to find a usefull relation between d\cos\theta_1d\cos\theta_2d\phi_1d\phi_2 and d\beta or \cos\beta. Any ideas on how I can proceed?
Thanks in advance.