In the literature, the eigenstates of P μ P, {|ϕ ν ⟩}, are referred to as the Mulliken-Hush (MH) diabatic states, which are commonly used as approximate diabatic states that are defined based on their characters. They are approximate diabatic states in the sense that ⟨ϕ ν |∇ R |ϕ ϵ ⟩ ≈ 0; (32) hence, we drop the R-dependence in |ϕ ν ⟩. Constructing rigorous diabatic states (where the derivative coupling is rigorously zero for all possible nuclear configurations) in a finite set of electronic Hilbert spaces is generally impossible, except for diatomic molecules. Recent theoretical progress on diabatization can be found in Ref. In the electronic subspace defined within the MH diabatic subspace using P (Eq. 31), Ĥel (Eq. 20) has off-diagonal (or "diabatic") coupling terms 33) We can explicitly express the matter state projected