Stuart Carter

Vibrational self-consistent field method for many-mode systems: A new approach and application to the vibrations of CO adsorbed on Cu(100)

We report calculations of the vibrational energies of CO–Cu(100) using a new code to perform vibrational self-consistent field (VSCF) and state-mixing calculations for many-mode systems. The major new feature of the code is the representation of the potential. Unlike recent implementations of the VSCF method, the potential is not expanded in terms of normal coordinates as a multinomial series about a minimum. The full potential, in normal coordinates, is used in the Watson Hamiltonian. This approach, while rigorous, can lead to prohibitively large numerical quadratures, and so we suggest a novel representation of the potential as an expansion in all two-mode, or all three-mode, or all four-mode coupling terms. The new code is tested against previous exact calculations of vibrational states of HCO, and also against previous VSCF calculations that used a fourth-order, normal coordinate force field representation of the global HCO potential. The new code is applied to calculations of the vibrations of CO ads...

MULTIMODE: A code to calculate rovibrational energies of polyatomic molecules

This review focuses on the calculation of rovibrational energies of polyatomic molecules using the code MULTIMODE. This code, which uses normal coordinates and a hierarchical n-mode representation of the potential, aims to be applicable to a wide class of molecules and molecular complexes. The theoretical and computational methods used in this code are described, followed by a review of selected applications. These applications illustrate various features of the code and also point out some limitations of the current version of the code. The review concludes with some ideas about possible future directions in this area of research.

The variational method for the calculation of RO-vibrational energy levels

Abstract In this paper the current status of the variational method for the determination of the rotational-vibrational energy levels of polyatomic systems is reviewed. Special attention is made for the derivation of the kinetic energy operator in various coordinate systems, and several forms are given. Similarly, analytic forms which are in current use for the potentials are given. The calculation of the Hamiltonian matrix elements (expansion functions, numerical integration grid points and weights) is described in detail, and a description of our programs for this problem is given in section 6.

A theoretical determination of the rovibrational energy levels of the water molecule

A redetermination of the quartic force field of the water molecule surface has been recently calculated by the method of Hoy, Mills, and Strey. A novel form of the bending coordinate has been introduced into this force field to ensure that the pure bending potential has the correct form at θ=π, and a few select force constant have been rerefined to the J=0 vibrational spectrum of H2O. The vibrational–rotational energy levels of the resulting surface have been determined by the variational method. Two particular sets have been examined: (i) low vibrational quanta, J≤9 and (ii) all levels below 12 000 cm−1 (relative to zero point), J≤2. The results are compared with the experimental levels in the literature.

An efficient procedure for the calculation of the vibrational energy levels of any triatomic molecule

It is often desirable to determine the parameters of an analytical representation of a molecular potential energy surface by a least squares fit of the vibrational energy levels to experimental values. The variational method is the only accurate method to obtain these levels, and this paper describes our most efficient procedure for the construction of the hamiltonian matrix. Applications of the procedure to CO2 and SO2 are described. Advances made in reducing the size of the variational problem are also discussed.

Potential models and local mode vibrational eigenvalue calculations for acetylene

Two potential models for acetylene are developed and tested by comparison between variational calculations for the stretching vibrational term values and available spectroscopic data. The first model based on local bond potentials with harmonic interbond coupling gives root mean square deviations of 6 cm−1 for C2H2 and 3 cm−1 for C2D2. The second model is more ambitious, being designed to reproduce the dissociation characteristics of the molecules, and the calculated root mean square deviations from the experimental vibrational term values are larger, 32 cm−1 for C2H2 and 24 cm−1 for C2D2. The eigenvalue spectrum of C2H2 is shown to differ from that of C2D2 in showingmarked local mode features and this difference in behaviour is underlined by means of a correlation diagram. Finally it is shown how the known normal mode frequencies and anharmonic constants may be introduced into a simple model in order to predict the excited term values of C2H2, again with a root mean square deviation of 6 cm−1.

A variational method for the calculation of spin-rovibronic levels of Renner-Teller triatomic molecules

This paper reports an extension of our method, introduced several years ago, for the variational treatment of the rovibronic levels of Renner-Teller triatomic molecules. A more complete treatment now introduces the effect of electron spin. Thus new terms enter the Hamiltonian because (a) the total-angular-momentum operator is replaced by - Ľ - S in the kinetic-energy operator, and (b) spin-orbit effects must be introduced. Here we include the latter through a semi-empirical form AĽ · S . Expansion functions having the correct symmetry (Σg +, Σu +, Σg -, Σu -), for D∞h molecules are derived. Hence the present approach takes into account the full dimensionality, anharmonicity and rotation-vibration coupling, as well as the coupling of all relevant angular momenta. This is particularly important for open-shell molecules and/or for electronically degenerate states. For the 2Πg electronic ground state of the CO2 + ion the spin-rotation and spin-orbit split Renner-Teller levels have been calculated using three-...

A variational method for the calculation of vibrational levels of any triatomic molecule

Assuming that the potential energy surface is known, a variational method is presented for the determination of vibrational frequencies which is suitable for any triatomic molecule. The harniltonian is expressed as a function of two bond lengths and the included angle. The expansion functions are products of either Morse oscillator functions or harmonic oscillator functions for the stretching vibrations and spherical harmonics for the bending vibration. Results are presented for linear, quasilinear and bent molecules.

The adiabatic rotation approximation for rovibrational energies of many-mode systems: Description and tests of the method

We extend the vibrational self-consistent field method (VSCF), and two types of state mixing [denoted VSCF-CI and V-CI (configuration interaction)], to include an approximate, adiabatic treatment of overall rotation. In this approach, the asymmetric-top rotational Hamiltonian is diagonalized in an “instantaneous” principal axis system, and the resulting coordinate-dependent rotational energy is added to the exact Hamiltonian of the nonrotating system to form an effective Hamiltonian for the rotation/vibrational energies. The energy eigenvalues of that Hamiltonian are then obtained by the VSCF approach and/or variational, state-mixing methods. In this present formulation for many-mode systems, we use the general Watson Hamiltonian, and also a hierarchical representation of the many-mode potential described previously [S. Carter, S. Culik, and J. M. Bowman, J. Chem. Phys. 107, 10458 (1997)]. This approach, at the VSCF, and VSCF-CI and V-CI levels is tested against recent exact calculations of vibrational/ro...

The vibrational energy levels of ammonia

A variational 6-dimensional method is used to determine the low lying vibrational energy levels of ammonia. The six internal coordinates were chosen to be appropriate for the symmetry and inversion motion of the molecule; they were the three NH bond lengths, r1, r2, r3, the unique angle β which each bond makes with the trisector of them, and two (of the three) angles, θ2 and θ3, between the bonds when projected on to a plane perpendicular to the trisector. The Wilson G matrix was determined for these internal coordinates both by computer algebra and by hand. An appropriate Jacobian for the motion was determined and the full Hermitian kinetic energy operator was obtained using the Podolsky transformation. Expansion functions were in the usual product form. Special attention was given to the θ2, θ3 expansion functions so that appropriate A1, A2 and E symmetry vibrational modes were obtained explicitly. Matrix elements of the kinetic energy operator were expressed in terms of one-dimensional integrals.Variat...