Higher‐order response theory based on the quasienergy derivatives: The derivation of the frequency‐dependent polarizabilities and hyperpolarizabilities

The higher‐order response theory to derive frequency‐dependent polarizabilities and hyperpolarizabilities is examined by means of the differentiation of the ‘‘quasienergy’’ with respect to the strengths of the time‐dependent external field, which is referred to as the quasienergy derivative (QED) method. This method is the extension of the energy derivative method to obtain static polarizabilities and hyperpolarizabilities to a time‐dependent perturbation problem. The form of the quasienergy W = 〈Φ‖Ĥ − i(∂/∂t)‖Φ〉 is determined from the time‐dependent Hellmann–Feynman theorem. The QED method is accomplished when the total sum of the signed frequencies of the associated field strengths, with respect to which the quasienergy is differentiated, is equated to 0. The QED method is applied to the single exponential‐transformation (SET) ansatz (up to the fifth‐order QEDs) and the double exponential‐transformation (DET) ansatz (up to the fourth‐order QEDs), where the time‐dependent variational principle (TDVP) is employed to optimize the time development of the system. The SET ansatz covers the full configuration interaction (CI) response and the Hartree–Fock response (i.e., the TDHF approximation), while the DET ansatz covers the multiconfiguration self‐consistent field (MCSCF) response (i.e., the TDMCSCF approximation) and the limited CI response with relaxed orbitals. Since the external field treated in this paper is always ‘‘polychromatic,’’ the response properties explicitly presented for both the SET and DET ansätze are μ_A, α_AB(−ω;ω), β_ABC(−ω_σ;ω₁,ω₂), and γ_ABCD(−ω_σ;ω₁,ω₂,ω₃), in addition δ_ABCDE(−ω_σ;ω₁,ω₂,ω₃,ω₄) is presented for the SET ansatz. All variational formulas for these response properties derived in this study automatically satisfy the (2n+1) rule with respect to the variational parameters.