Group theoretical considerations show that an eigenfunction of the Schroedinger equation for a polyatomic molecule may be written as a sum of products of representation coefficients of the group of threehyphen;dimensional rotations and functions which depend only upon the relative configuration of the atoms and hence only upon the normal coordinates. A perturbation solution of the Schroedinger equation is described which is based on zerohyphen;order wave functions which are consistent with this group theoretical result. All of the angular momentum operators appearing in the wave equation are shown to be recursion operators in the representation coefficients. Subsequent utilization of the orthogonality of the representation coefficients then leads to a set of coupled differential equations for the functions which depend only on the normal coordinates. This set of coupled differential equations is a generalization of the matrix equations for the rotational motion of a rigid body.The functions of the normal coordinates are then expanded as sums of products of Hermite functions of the individual normal coordinates. Use of the orthogonality relations of the Hermite functions then yields a set of equations for the expansion coefficients. The eigenvalues of the associated matrix are then, of course, the exact rotationalhyphen;vibrational energy levels of the polyatomic molecule. These energy levels are obtained approximately by a diagonalization in the rotational quantum numbers followed by a perturbation technique in which the coupling between vibrational levels is assumed to be small (except in the case of degeneracy). The lowest approximation to the energy levels contains terms which may be interpreted as due to coupling effects which in previous solutions appear only in higher order approximations.
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