Letting U be the particular subspace of the Hilbert space H such that U is the span of {|I⟩} (in this paper, we take U to be the model space which is spanned by the SA-CASSCF states, as is customary in multi-state perturbation theory and in earlier MS-PDFT methods), then for equal weights, the state-averaged densities maintain the property that they are independent of the reference state basis {|I⟩}, making D 0 and d 0 functionals of the subspace U . Hence ĤL-PDFT , along with its eigenvalues and eigenvectors, are independent of the initial basis {|I⟩} and instead are functionals only of the subspace U that {|I⟩} span. That is, for each subspace U ⊆ H there is a unique quantum operator ĤL-PDFT [U ].