where the exp -i ℏ ∆ Ĥ(ξ) ∆t |Ψ (ξ) ⟩ update will be performed using the Chebyshev expansion, and then multiply with a phase exp(-i ℏ ξ ε∆t). With this separation, the accuracy of the Chebyshev propagator is independent of the absolute value of electronic excitation energies. For the Chebyshev expansion with the shifted eigenvalues, the approximate lower eigenvalues used in Eq. 36 is now