For large systems with many electrons, the maximum number of electronic states becomes impractically large for standard electronic structure calculations, even considering only the single excitation manifold. Further, for singlereference methods, such as linear response time-dependent density functionally theory (LR-TD-DFT), the efficient and accurate calculation of high-energy excited states is not always trustworthy, so in practice we employ a smaller Hilbert space than that implied by the total molecular Hamiltonian ĤM in Eq. 3, which can be defined through the following projection operator,