The structure of the matrix in Eq. 14 reflects the Kronecker delta functions that appear in the respective block equations. The structure of the matrix of the projected version of Eq. 2 is comparable to that in Eq. 14 with the only substantive difference being that the matrix elements of P ˆd ˆP ˆare offset by ⟨d ˆ⟩, i.e. d αβ -⟨d ˆ⟩δ αβ . We have a choice in what state to compute ⟨d ˆ⟩, and in this work, we choose the ground state of the uncoupled system unless otherwise specified.