# Solving the problem using Many Body Perturbation Theory

I am trying to solve the following Hamiltonian using Many body perturbation Theory. $$H=\sum_{i=1}^{N}\Bigg[\frac{P_{i}^{2}}{2m} -\sum_{i,j}\frac{1}{|\vec{r}_{i}-\vec{R}_{j}|}\Bigg]$$. I split this into sum of many hydrogen atoms and thought of using hydrogen atom basis to construct the many body states to solve the Hamiltonian using Many body perturbation theory. $$\vec{r}_{i}-\vec{R}_{i}=\widetilde{\vec{r}_{i}}$$. The Hamiltonian becomes, $$\sum_{i}\bigg[\frac{P_{i}^{2}}{2m}-\frac{1}{|\widetilde{\vec{r}_{i}|}}\bigg]-\sum_{j\neq i}\bigg[\frac{1}{|\widetilde{\vec{r}_{i}}-(\vec{R_{j}}-\vec{R_{i}})|}\bigg]$$ The first part is exactly solvable. However, the other part is not, I thought of solving it perturbatively to get the ground state many body slater determinants. However, I have an issue. I can construct the Many body wave functions from this. But the issue here is that for an arbitrary $$\vec{R}_{i}$$ and $$\vec{R}_{j}$$. As the hydrogen atom energy values are degenerate, Even the ground state sometimes become degenerate depending upon the number of electrons that I take. For instance, If I have $$N=3$$ electrons. we will get degenerate many body energy states from the solvable part $$1s,2s,2p_{z}$$, This can also be $$1s,2s,2p_{x}$$ and $$1s,2s,2p_{y}$$ for the solvable part of the Hamiltonian. To apply the Many body perturbation theory. I need to remove this many body degeneracy for every many body wave functions which I have constructed (slater determinant) from hydrogen atom single particle wave functions.To lift the degeneracy, I need to construct the a matrix for the subspace and to diagonalize the subspace which is degenerate.However, as I the value of R is arbitrary, I cannot diagonalize the Hamiltonian Matrix of more than size 3. So Is it possible to lift the degeneracy for this many body states without using the diagonalizing the degenerate subspace by constructing the matrix.I am looking for first order correction for many body wavefunctions and not energy. Can someone suggest good book/references which deals with Many body perturbation theory with degeneracy for wave functions(apart from szabo) ?