## Definitions and notations

Both $$\hat{H}_0$$ and $$\hat{H}_I$$ are invariant under all possible permutations of any two particles and hence commute with $$\hat{A}$$ $$$$[H_0,\hat{A}] = [H_I,\hat{A}] = 0. \tag{5}$$$$ Furthermore, $$\hat{A}$$ satisfies $$$$\hat{A}^2 = \hat{A}, \tag{6}$$$$ since every permutation of the Slater determinant reproduces it.