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In the Hartree scheme for many-electron atoms, the approximated Hamiltonian (in a.u.)

$$\hat{H} = \sum_{i=1}^Z \left(-\frac{1}{2} \nabla_i^2 -\frac{Z}{r_i} + V_\text{H}\left(r_i\right)\right)$$

is separable and commutes with the total angular momentum operators $\left\{\hat{L}^2,\hat{S}^2,\hat{L}_z,\hat{S}_z\right\}$ ($V_\text{H}$ denotes the radial mean-field potential).

In order to combine such commutation relations and the Pauli exclusion principle, one takes linear combinations of Slater-determinants that are eigenfunctions of the total angular momentum operators (i.e., a set of configuration state functions).

What guarantees that this procedure is always possible?

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If two operators $\hat{A}$ and $\hat{B}$ commute then they have a common set of eigenfunctions. In your case, eigenfunctions of the four angular momentum operators are also the eigenfunctions of $\hat{H}$ and so is their linear combination.

You may want to refer to a related post for more details.

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