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.

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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.

The density functional theory was employed to investigate Eu(III) complexes with three beta-diketonates and two phosphine oxides (complex M1: Eu(bdk)3(TPPO)2, complex M2: Eu(bdk)3(TMPO)2, and complex M3: Eu(bdk)3(TPPO)(TMPO)) deemed to be the model complexes of the fluorescence compounds for the ultraviolet LED devices we have recently developed. For each complex, two minimum energy points corresponding to two different optimized geometries (structures A and B) have been found, and the difference of the energy between two minimum energy points is found to be quite small (less than 1 kcal/mol). Vertical excitation energies and oscillator strengths for each complex at two optimized geometries have been obtained by the time-dependent density functional theory, and the character of the excited states has been investigated. For complex M3, the absorption edge is red-shifted, and the oscillator strengths are relatively large. The efficiency of intersystem crossing and energy transfer from the triplet excited state to the Eu(III) ion is considered by calculating DeltaE(ISC) (the energy difference between the first singlet excited state and the first triplet excited state) and DeltaE(ET) (the difference between the excitation energy of the complex for the first triplet excited state and the emission energy of the Eu(III) ion for 5D to 7F).