However, it is important to note that the zero-order densities of L-PDFT are taken to be averaged over the entire model-space, which may include averaging over states of different spatial and spin symmetries. This means that whenever there is more than one state in the state-average manifold, regardless of their symmetries, the final diagonal elements of the L-PDFT Hamiltonian will generally differ from the MC-PDFT energies. For example, if we have one singlet (|S⟩) and one triplet state (|T ⟩), then the model space is 2-dimensional subspace defined as the span of |S⟩ and |T ⟩. The zero-order densities for which we construct our ĤL-PDFT in this case are taken as the average of singlet and triplet densities: