ALBERT

Notes: The equilibrium structures, dipole moments, harmonic vibrational frequencies, and infrared intensities of NH3, FON, Be3, BeC2, and BeO2 have been determined using the coupled-cluster and Brueckner electron correlation methods. The singles and doubles coupled-cluster (CCSD) and the Brueckner doubles (BD) methods have been employed and the corresponding methods that include a perturbational estimate of connected triple excitations [i.e., CCSD(T) and BD(T)] have also been investigated. The T1 diagnostic [Int. J. Quantum Chem. Symp. 23, 199 (1989)] is found to provide a good indication of the magnitude of the difference between the results obtained with the coupled-cluster and Brueckner methods. For NH3, the T1 diagnostic is small and so the differences between results obtained from coupled-cluster and Brueckner theories are quite small. For the other four molecules the T1 diagnostic is larger, and so the differences between the coupled-cluster and Brueckner methods become larger. However, it is found for all of the molecules considered in this study that inclusion of the contribution from connected triple excitations is more important than the differences between the Brueckner and coupled-cluster correlation methods.

Notes: This is the second in a series on the ab initio calculation of the second, third, and fourth derivatives of the energy of a molecule with respect to nuclear coordinates. A knowledge of these derivatives yields, in particular, anharmonic spectroscopic constants. Here we discuss our implementation of the formula for the fourth derivative of the self-consistent-field energy and present full quartic force fields in internal coordinates for H2O and CO2.

Notes: The potential energy surfaces of the two lowest-lying singlet electronic states of methylene (CH2) are determined by internally contracted multireference configuration interaction calculations, using a full-valence reference space, with an extended Gaussian basis set. The rotation–vibration levels on these surfaces are calculated by diagonalizing the rovibrational Hamiltonian matrix in a contracted basis. The rovibronic mixing due to the strong Renner–Teller interaction in this system is treated through the Coriolis term in the kinetic energy operator, using geometry-dependent electronic angular momentum matrix elements calculated from ab initio wave functions. The agreement between experiment and this high-quality ab initio calculation is sufficiently good that the calculation can be used to assign the observed vibronic bands in this very complex spectrum, where 90% of the observed lines remain unassigned. Many of the previous vibronic band labels are found to be incorrect. Most of the K〉0 bands previously labeled b˜ 1B1 are actually predominantly a˜ 1A1 in character, and the vibrational numbering of their b˜ 1B1 components are also incorrect. This work demonstrates the importance of supplementing experimental data with good quality ab initio calculations.

Notes: Knowledge of a force field expanded through quartic displacements, together with a dipole field expanded through cubic displacements, yields all the harmonic and anharmonic molecular properties of interest to infrared spectroscopists. Such force fields may also explain much of the mechanism behind intramolecular vibrational energy redistribution. The ab initio quantum chemist can now calculate these fields, either at the self-consistent field level or with the inclusion of electron correlation effects. For accurate predictions, it is important to include electron correlations effects for at least the quadratic part of the force fields. Here we report studies using the second-order Møller–Plesset method for the full quartic fields. We examine the effects of using large basis sets. The quadratic force constants are calculated analytically; cubic and quartic constants are calculated using central differences of second derivatives in reduced normal coordinates. Three molecules are studied. HCCF, for which a large quantity of experimental data has been recently analyzed by Holland, Newnham, and Mills. The calculations are sufficiently accurate that errors in the experimental assignments became apparent. HFCO, where the theoretical anharmonic constants are helpful in understanding the highly excited vibrational states probed by Moore and co-workers. SiH+3, whose high resolution absorption spectra has just recently been detected by Davies and co-workers. The conclusions are that this straightforward way of calculating spectroscopic properties is an extremely valuable tool for the understanding of spectroscopy.

Notes: Existing methods for the calculation of the correlation energy of open shell atoms and molecules using Møller–Plesset perturbation theory are discussed with special emphasis on recent advances in methods based on the restricted Hartree–Fock wave function. The convergence of the Møller–Plesset series is examined to high order for a number of small systems and it is shown that some of the methods offer significant advantage over traditional unrestricted Møller–Plesset theory. The optimized bond length and vibrational frequencies of CN, NO, and O2 are also compared for second and fourth order Møller–Plesset theory, and again it is found that the same methods give much better results than the corresponding unrestricted results. Further supporting calculations are presented on the electron affinity of CN and the dissociation energy of HCN. The new methods involve relatively minor adaptation of existing unrestricted Møller–Plesset (UMP) codes, and in the light of their superior performance are recommended instead of UMP theory for open shell molecules.

Notes: The equilibrium structure of HCN has been determined from the previously published ground state rotational constants of eight isotopomers by using (B0-Be) values obtained from a variational calculation of the vibration–rotation spectrum. The results are re(CH)=1.065 01(8) A(ring), and re(CN)=1.153 24(2) A(ring).

Notes: This is the fourth in a series of papers on the ab initio calculation of the third and fourth derivatives of the energy of a molecule. In this paper we examine anharmonic effects in the infrared and Raman spectra of benzene. The following spectroscopic properties have been calculated; ab initio anharmonic corrections (ω−ν) and estimates of the harmonic frequencies ω for all 30 vibrational modes of C6H6 and C6D6, a complete set of anharmonic constants x and g for C6H6, intensities for the infrared spectrum of C6H6 up to 6148 cm−1, and anharmonic corrections to the Raman scattering factors for the fundamental modes of C6H6. In addition, we have improved on previous calculations of the equilibrium geometry of benzene, using Møller–Plesset perturbation theory and a triple zeta plus double polarization (TZ2P) basis. We have also calculated a zero-point vibrationally averaged geometry which is in good agreement with the experimental R0 value. All these calculations are based on a Hartree–Fock quartic potential, cubic dipole surface, and quadratic polarizability surface, using a double-zeta plus polarization (DZP) basis. This is the first time a complete anharmonic potential has been obtained for a molecule of this size; the computer time required was minimized by the use of analytic derivative programs in favor of finite-difference programs. The quartic potential is presented in three coordinate systems.We discuss efficient algorithms for the nonlinear transformation of the potential from normal coordinates to valence coordinates and for symmetry checking the potential. The approximations used in our calculations have been examined and we find that the use of a Hartree–Fock DZP potential together with a perturbative treatment of the vibrational Hamiltonian is just as accurate for D6h benzene as for smaller molecules. In order to examine correlation effects in the B2u modes 14 and 15, basis-set limit second-order Møller–Plesset TZ2P+f harmonic frequencies have been calculated for these modes. It is suggested that, while these modes are very sensitive to correlation, anharmonicity has only a small effect, so a Hartree–Fock DZP anharmonic potential is adequate. Furthermore, experimental determination of anharmonic corrections to frequencies is very difficult for a molecule of this size so we hope our calculations will fill this gap.

Notes: The Brueckner doubles variant of coupled cluster theory has recently been reintroduced by the authors. The use of Brueckner orbitals means that the governing equations for Tˆ2 take a particularly simple form. Here we give the details for the evaluation of the gradient of the Brueckner doubles energy for (a) the unrestricted spin–orbital formalism and (b) the closed-shell restricted formalism. Applications are presented for H2O, NH3, CH4, H2CO, C2H2, HCN, and CO2 and comparisons are made with the Hartree–Fock, second order Møller–Plesset and quadratic configuration interaction models and with experiment.

Notes: The definition of frequency-dependent polarizabilities α(−ω;ω), β(−2ω;ω,ω), β(−ω;ω,0), and β(0;ω,−ω) is discussed, and it is argued that the most convenient definitions are as energy derivatives, a pseudo-energy being defined as the expectation value of [H−i(∂/∂t)]. This definition outlines a straightforward procedure for obtaining frequency-dependent polarizabilities for all quantum chemistry methods including those which account for the effects of electron correlation. It is demonstrated at the self-consistent field level of theory that αλμ(−ω;ω) cos ωt may be considered as the derivative of the static dipole moment μλ with respect to the strength Eωμ of a frequency-dependent field Eωμ cos ωt (as is usual), or as the derivative of an appropriately defined frequency-dependent dipole moment μμ cos ωt with respect to a static field E0λ. In this way, polarizabilities may be determined from finite static field calculations on lower-order tensors. Therefore, α(−ω;ω) cos ωt is defined within second-order Møller–Plesset perturbation theory (MP2) as the second derivative of the MP2 energy with respect to one static and one frequency-dependent field.An analytic expression is given for αλμ(−ω;ω) at the MP2 level of theory. An MP2 frequency-dependent dipole expression is also defined, which if finite static field calculations are applied, gives the same values for αλμ(−ω;ω). MP2 values are reported for α(−ω;ω) of formaldehyde and ammonia for a range of frequency ω=0.01–0.1 a.u. From comparison of the self-consistent field (SCF) and MP2 values of the frequency-dependent contribution to α¯(−ω;ω), it is concluded that it is appropriate to use an SCF frequency-dependent correction in conjunction with a static polarizability determined at a higher level of theory in order to obtain an accurate value for α¯(−ω;ω) of H2CO in this frequency range. For ammonia, the frequency-dependent contribution to α¯(−ω;ω) is more sensitive to electron correlation. Nevertheless, compared to the total polarizability α¯(−ω;ω), the error in the frequency-dependent contribution determined using the SCF method is small (∼2% at ω=0.1 a.u.)