Recursive Computation of Hamiltonian Matrix Elements Using Harmonic Oscillator Eigenfunctions: Application to the Inversion of Ammonia and to the Methyl Torsion + Aldehydic Hydrogen Wagging of Acetaldehyde

A program able to use hybrid free rotor plus harmonic oscillator basis functions for the variational study of large and small amplitude vibrations is developed. The Hamiltonian matrix elements between harmonic oscillator eigenfunctions and polynomial terms are calculated using a recursive algorithm. This technique permits use of only one basic algorithm to compute the kinetic and potential parts of the Hamiltonian. In addition, the program can handle potential functions perturbed with Gaussian barriers and obtain the quantum mechanical average of a magnitude. The program is used to test the efficiency of Taylor series vs polynomial + Gaussian potential functions for the description of the ammonia inversion mode. The data for the construction of the potential functions are obtained by ab initio methodology at the QCISD/6-31 lG+ +(3 dA 3 dp) level. The quantum mechanical average values for the structural parameters of ammonia are computed and compared to the fully optimized ab inifio results. The simultaneous methyl torsion + aldehydic hydrogen wagging motions in the S, state of acetaldehyde are used to illustrate the efficiency of mixed free rotor + harmonic oscillator basis functions. The harmonic approximation is the classical appreach to the study of molecular vibrations. This approximation is applied within the normal coordiINTRODUCT ION nates system with all the vibrational modes being small amplitude and uncoupled to each other. In these conditions, the kinetic terms can be considered constant with the motion, and the potential function shows a simple quadratic variation on the normal mode (Wilson et al., 1980). When the vibrations exhibit some anharmonicity the potential function can be described using higher order polynomials and the kinetic terms can be expanded in a Taylor series on the vibrational coordinate. These anharmonic cases can be solved variationally or perturbationally on the basis of the harmonic oscillator (Harthcock & Laane, 1985). A different approach can be applied to the periodic motions found in organic molecules. These vibrations are usually expressed as angular displacements. Thus, the wavefunction can be expanded in the free rotor basis and the kinetic and potential terms can be described by Fourier expansions. The use of free rotor basis functions and Fourier series for an arbitrary number of vibrations has been previously considered (Mufioz-Caro et al., 1994). An extension of this technique would be the l Author for correspondence. simultaneous use of different types of basis functions, in particular free rotor plus harmonic oscillator hybrid basis functions. This hybrid basis would permit the simultaneous treatment of large and small amplitude vibrational motions. Thus, stretching, bending and torsion could be considered together as well as their interactions. In this paper we present a tool for the variational calculation of vibrational energy levels using free rotor plus harmonic oscillator hybrid basis functions. The program is specially useful for the treatment of several large amplitude vibrations. The computation of the Hamiltonian matrix elements between harmonic oscillator eigenfunctions is implemented recursively. In this form, only one algorithm is needed to describe the potential function as a Taylor series of an arbitrary order. In addition, two algorithms are only necessary to compute the kinetic part when the kinetic elements depend on the vibrational coordinate. The evaluation of matrix elements for potentials with Gaussian perturbations is implemented as well as the computation of quantum mechanical averages in the different vibrational states. The ability of Taylor series and polynomial + Gaussian forms to incorporate the information obtained from ub initio calculations is analyzed through the inversion motion of ammonia. The quantum-mechanical averaged and the equilibrium structural parameters of ammonia are compared. In addition, the use of a mixed free