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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 μA, αAB(−ω;ω), βABC(−ωσ;ω1,ω2), and γABCD(−ωσ;ω1,ω2,ω3), in addition δABCDE(−ωσ;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.

A formulation for calculating frequency-dependent hyperpolarizabilities in the Mo/ller–Plesset perturbation theory is presented as the correlation correction to the TDHF approximation. Our quasienergy derivative (QED) method is applied, and the difference between the QED method and the pseudoenergy derivative (PED) method by Rice and Handy is discussed. The Lagrangian technique is utilized to obtain simple and practical expressions for response properties in which the TDHF orbital rotation parameters satisfy the 2n+1 rule and the Lagrange multipliers satisfy the 2n+2 rule. Explicit expressions for response properties up to third order [μ, α(−ω1;ω1), β(−ωσ;ω1,ω2)] are derived in the second-order Mo/ller-Plesset perturbation theory.

ABSTRACT:We extended the dynamic response theory in the Møller᎐Plesset Ž . Ž . perturbation theory MPPT based on the quasi-energy derivative QED method for closed-shell systems to that for open-shell systems. In this study we perform the Ž . calculations of frequency-dependent polarizabilities ␣ y; for nondegenerate open-Ž shell doublet systems Li, Na, and K atoms and BeH, MgH, CaH, CN, and NH 2 . Ž . molecules in the second-order Møller᎐Plesset perturbation theory MP2 starting with Ž . time-dependent restricted open-shell Hartree᎐Fock TDROHF approximation.

The linear response calculations in the multiconfiguration time-dependent Hartree–Fock (MCTDHF) approximation with a closed-shell-type MCSCF state as the time-independent reference state are discussed. The application to the LiH molecule with a small basis set ([4s2p1d/2s1p]) shows validity of our MCTDHF approach to the singlet ground state. Our MCSCF correlation energy is 97% of the total (=full CI) correlation energy and the MCTDHF excitation energies are in good agreements with the Δ full CI excitation energies. The Born–Oppenheimer potential energy curves for the lowest three singlet states of LiH and the corresponding vibrational level spacings, the transition moments, the oscillator strengths, and the frequency-dependent dipole polarizabilities are reported. All of these results imply the potentiality of our MCTDHF method for the future work with the larger basis set. One of such basis sets ([9s8p4d/8s7p1d]) is referentially used only at the single-configuration TDHF level, and the resultant near-Hartree–Fock polarizability and Thomas–Reiche–Kuhn sum rule is very promising.