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Potential Energy Surfaces Using Algebraic Methods Based on Unitary Groups

DOI: 10.1155/2011/593872

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This contribution reviews the recent advances to estimate the potential energy surfaces through algebraic methods based on the unitary groups used to describe the molecular vibrational degrees of freedom. The basic idea is to introduce the unitary group approach in the context of the traditional approach, where the Hamiltonian is expanded in terms of coordinates and momenta. In the presentation of this paper, several representative molecular systems that permit to illustrate both the different algebraic approaches as well as the usual problems encountered in the vibrational description in terms of internal coordinates are presented. Methods based on coherent states are also discussed. 1. Introduction The description of molecular systems involves the solution of the corresponding Schr?dinger equation. This task is so difficult that an approach involving just numerical methods needs powerful computers even for three-or four-particle systems. An alternative approach is based on choosing the basis functions in such a way that they resemble the exact eigenfunctions as much as possible. The suitable basis are obtained by making approximations that simplify the Hamiltonian of the molecule. The advantage of this method is that the functions reproduce correctly the gross features of the spectrum, and consequently they provide a better physical insight in understanding the solutions. The first step in simplifying the molecular problem consists in taking advantage of the large difference between the nucleus and electron masses, a fact that leads to the Born-Oppenheimer approximation [1, 2]. As a result of this approximation the original Schr?dinger equation is split into two coupled equations, one corresponding to the electronic degrees of freedom which is solved for many nuclear geometries and the other one associated with the rotation-vibration Schr?dinger equation for the nuclei whose potential is basically provided by the electronic energy [3, 4]. On the other hand, the rotation-vibration Schr?dinger equation is usually solved making the rigid-rotor approximation together with the harmonic oscillator approximation. The total wave function is then approximated as the direct product of three contributions: electronic, rotational, and vibrational wave functions. Corrections to this description are allowed by introducing the braking of the Born-Oppenheimer approximation, distortion effects, anharmonicity, centrifugal distortion, and Coriolis coupling [3–5]. Within the Born-Oppenheimer approximation the potential energy surface (PES) is provided by the solution of

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