It can be seen from eq. 2 that for any model space, the L-PDFT Hamiltonian commutes with Ŝ2 and Ŝz and thus the final eigenstates will have definite spin quantum numbers. Similarly, if there is a particular spatial symmetry Ω present in the system such that [ Ω, Ĥel ] = 0, then [ Ω, ĤL-PDFT ] = 0. Just like the real-electronic Hamiltonian, the L-PDFT Hamiltonian can be written as a block-diagonal matrix where only states of the same symmetry are connected (have non-zero coupling terms).