6.2.2. Definition of Effective Properties 3064 6.3. Response Properties to Magnetic Fields 3066 6.3.1. Nuclear Shielding 3066 6.3.2. Indirect Spin−Spin Coupling 3067 6.3.3. EPR Parameters 3068 6.4. Properties of Chiral Systems 3069 6.4.1. Electronic Circular Dichroism (ECD) 3069 6.4.2. Optical Rotation (OR) 3069 6.4.3. VCD and VROA 3070 7. Continuum and Discrete Models 3071 7.1. Continuum Methods within MD and MC Simulations 3072

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Quantum mechanical continuum solvation models.

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.

Optimization of equilibrium geometries and transition structures

An accurate description of the electrical properties of atoms and molecules is critical for quantitative predictions of the nonlinear properties of molecules and of long‐range atomic and molecular interactions between both neutral and charged species. We report a systematic study of the basis sets required to obtain accurate correlated values for the static dipole (α1), quadrupole (α2), and octopole (α3) polarizabilities and the hyperpolarizability (γ) of the rare gas atoms He, Ne, and Ar. Several methods of correlation treatment were examined, including various orders of Moller–Plesset perturbation theory (MP2, MP3, MP4), coupled‐cluster theory with and without perturbative treatment of triple excitations [CCSD, CCSD(T)], and singles and doubles configuration interaction (CISD). All of the basis sets considered here were constructed by adding even‐tempered sets of diffuse functions to the correlation consistent basis sets of Dunning and co‐workers. With multiply‐augmented sets we find that the electrical properties of the rare gas atoms converge smoothly to values that are in excellent agreement with the available experimental data and/or previously computed results. As a further test of the basis sets presented here, the dipole polarizabilities of the F− and Cl− anions and of the HCl and N2 molecules are also reported.

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Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of static electrical response properties

The general expression for the exact forces on the nuclei (negative derivatives of the total energy with respect to the nuclear coordinates) is applied for Hartree-Fock wavefunctions. It is suggested that force constants should be calculated by differentiating the forces numerically. This method, called the force method, is numerically more accurate and requires less computation than the customary one of differentiating the energy numerically twice. It permits the quick determination of the equilibrium geometry by relaxing the nuclear coordinates until the forces vanish. The unreliability of the methods using the Hellmann-Feynman forces is re-emphasized. The question of which force constants can be best calculated ab initio is discussed.

Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules

The Bonn Meson Exchange Model for the Nucleon Nucleon Interaction

Computationally tractable expressions for the evaluation of the linear response function in the multiconfigurational self-consistent field approximation were derived and implemented. The finite lif ...

Near-resonant absorption in the time-dependent self-consistent field and multiconfigurational self-consistent field approximations

Molecular Vibration and Nonlinear Optics

Based on the Ehrenfest theorem, an equation of motion that takes relaxation into account has been presented in wave-function theory, and the resulting response functions are nondivergent in the off-resonant as well as the resonant regions of optical frequencies. The derivation includes single- and multideterminant reference states. When applied to electric dipole properties, the response functions correspond to the phenomenological sum-over-states expressions of Orr and Ward [Mol. Phys. 20, 513 (1971)] for polarizabilities and hyperpolarizabilities of an isolated system. A universal dispersion formula is derived for the complex second-order response function. Response theory calculations are performed on lithium hydride and para-nitroaniline for off-resonant and resonant frequencies in the electro-optical Kerr effect and second-harmonic generation.

Nonlinear response theory with relaxation: The first-order hyperpolarizability

An often overlooked, but nonetheless important, contribution to molecular dipole polarizabilities is that which comes from molecular vibration. This contribution, which was formerly called the atomic polarization, may be related to the intensities of the infrared‐active bands. In this paper we have collected the best available intensity data for some hundred or so molecules and evaluated their vibrational polarizabilities. We have also given estimates of the probable errors of the final numbers.

Vibrational Contributions to Molecular Dipole Polarizabilities

A consistent extension of Onsager’s reaction field model to nonlinear-optical experiments in condensed media is presented. This model is used to derive effective (nonresonant) linear and nonlinear molecular polarizabilities involved in refractive index measurements, electric-field induced second-harmonic generation, hyper-Rayleigh scattering, third-harmonic generation and degenerate four-wave mixing. The commonly applied Lorentz correction, while being a useful approximation in linear optics, is demonstrated to lead to systematic errors of typically 20% in nonlinear optical experiments. This is due to an inherent inconsistency that originates from the implicit assumption of a cavity field factor for the nonlinear molecular response.

David M. Bishop

Papers

Near-resonant absorption in the time-dependent self-consistent field and multiconfigurational self-consistent field approximations

Molecular Vibration and Nonlinear Optics

Nonlinear response theory with relaxation: The first-order hyperpolarizability

Vibrational Contributions to Molecular Dipole Polarizabilities

Effective polarizabilities and local field corrections for nonlinear optical experiments in condensed media