In most electronic structure calculations, for a given geometry, the canonical orbitals can be chosen to be orthonormal, i.e. ⟨i|j⟩ = δ ij , ⟨a|b⟩ = δ ab , and ⟨i|a⟩ = 0. However, for different geometries, the canonical orbitals are in general not mutually orthonormal, i.e. ⟨i|j ′ ⟩ and ⟨a|b ′ ⟩ are not diagonal, so that S 0 and S ab ′ ij ′ will be formally dense matrices. As such, when directly constructing the CIS state overlap matrix, one must evaluate det S In the end, the total computational complexity is O(N