I am trying to construct a two body states to calculate the matrix elements of electronic Hamiltonian of Helium atom. $$H=-\frac{1}{2}\nabla^{2}_{1}-\frac{1}{2}\nabla^{2}_{2}-\frac{2}{\vec{r}_{1}}-\frac{2}{\vec{r}_{2}}-\frac{1}{|\vec{r}_{1}-\vec{r}_{2}|}$$

Constructed the following two body (slater determinant) states (spinless fermion so spin states are excluded), $$|\psi_{1s}\psi_{2s}\rangle, |\psi_{1s}\psi_{2p_{x}}\rangle,|\psi_{1s}\psi_{2p_{y}}\rangle,|\psi_{1s}\psi_{2p_{z}}\rangle,|\psi_{2s}\psi_{2p_{x}}\rangle,|\psi_{2s}\psi_{2p_{y}}\rangle,|\psi_{2s}\psi_{2p_{z}}\rangle,|\psi_{2p_{x}}\psi_{2p_{y}}\rangle,|\psi_{2p_{x}}\psi_{2p_{z}}\rangle,|\psi_{2p_{y}}\psi_{2p_{z}}\rangle$$

where each ket is actually slater determinant for example,$$|\psi_{1s}\psi_{2s}\rangle:=\frac{1}{\sqrt{2}}[\psi_{1s}(\vec{r}_{1})\psi_{2s}(\vec{r}_{2})-\psi_{1s}(\vec{r}_{2})\psi_{2s}(\vec{r}_{1})]$$

I am trying to construct the matrix elements for the above hamiltonian with 10 two body slater determinant states (i have used only $1s$,$2s$ and $2p$ single particle states).While constructing the matrix ($10\times10$ matrix), I have a problem in evaluating the coulomb repulsion term. $$\int\frac{1}{\sqrt{2}}[\psi_{1s}(\vec{r}_{1})\psi_{2p_{x}}(\vec{r}_{2})-\psi_{1s}(\vec{r}_{2})\psi_{2p_{x}}(\vec{r}_{1})]\frac{1}{|\vec{r}_{1}-\vec{r}_{2}|}\frac{1}{\sqrt{2}}[\psi_{1s}(\vec{r}_{1})\psi_{2p_{x}}(\vec{r}_{2})-\psi_{1s}(\vec{r}_{2})\psi_{2p_{x}}(\vec{r}_{1})]d^{3}\vec{r}_{1}d^{3}\vec{r}_{2}$$

How do I evaluate those integrals if not analytically, How do i do numerically this six dimensional integration. Can someone suggest a numerical technique to evaluate such an integral or a python code or C code ?

  • $\begingroup$ Is there any reason why not analytically? And what about the 1-electron integrals? The basic principle to solve 1- and 2-electron integrals are the same. So maybe start there. $\endgroup$ – Feodoran Nov 6 '18 at 10:32
  • $\begingroup$ I really don't have any idea about solving it numerically. I am completely new to numerical integration . IF you suggest some already existing codes in python or C it would be nice and be very helpful. I thought there would be no analytical solution.In this case one electron integral is very simple. $\endgroup$ – user135580 Nov 6 '18 at 12:13
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    $\begingroup$ Packages based on analytic expression for GTOs are already linked in one of your previous questions: chemistry.stackexchange.com/questions/103409/… If you want to learn about numerical integration, I would start with simpler things, the harmonic oscillator in 1D to 3D for example. $\endgroup$ – Feodoran Nov 6 '18 at 13:06
  • $\begingroup$ Can you please suggest some book or some reference for that ? $\endgroup$ – user135580 Nov 6 '18 at 14:36
  • $\begingroup$ Again, please refer to the references in your previous question. As far as I understand you, the two questions are identical. Which makes this question either a duplicate or "unclear what you're asking". $\endgroup$ – Feodoran Nov 6 '18 at 14:47