It is often difficult to get accurate electronic states for highly excited adiabatic states. It is thus ideal to consider a Hilbert subspace of the electronic Hamiltonian. Considering a finite subset of electronic states {|ψ α ⟩} (see Eq. 21) where there is a total of N matter states, one can define the following projection operator P = N α=1 |ψ α (R)⟩⟨ψ α (R)|, (26) which defines the truncation of the full electronic Hilbert space 1r = P + Q which has an infinite basis, to a subspace P that contains a total of N states, where 1r is the identity operator in the electronic Hilbert subspace (the subspace containing all of the electron DOF) and Q = 1r -P is the subspace being projected out.