Hello. I need to find the expectational value of the correlation energy between two electrons which repel eachother via the classical Coulomb interaction. This is given by the 6-dimensional integral:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\left\langle\frac{1}{|\vec{r}_1 - \vec{r}_2|}\right\rangle =[/itex][itex]\int_{-\infty}^{\infty}[/itex][itex]d\vec{r}_1d\vec{r}_2[/itex][itex]e^{-2\alpha(r_1 + r_2)}[/itex][itex]\frac{1}{|\vec{r_1}-\vec{r_2}|}[/itex][itex][/itex]

[itex]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[/itex]

I know that the answer is [itex]\frac{5\pi^2}{16^2}[/itex], 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 [itex]\pi^2[/itex], 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)

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# Homework Help: Correlation energy between two electrons

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