Vibration-rotation wavefunctions and energies for any molecule obtained by a variational method

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Abstract
This paper presents an alternative method to the usual approach via perturbation theory for the determination of vibrational-rotational energy levels of a molecule in a given electronic state. It is assumed that the electronic Born-Oppenheimer equation has been solved, by an ab initio method, to give a potential function which is used in the nuclear Born-Oppenheimer equation. But the method can also be used with any potential obtained by any method. An approximate solution to the nuclear equation is derived in the form of a linear combination of expansion functions, the coefficients being determined by the standard linear variational method. Angular momentum theory is used to show that the nuclear wavefunction for m = 0 can be represented by a linear combination of functions of the form where qi are variables which are closely related to the vibrational normal coordinates, and β, γ are two of the Euler angles. m is the eigenvalue of the Z-component of angular momentum operator in space-fixed axes OXYZ. The Hvi (qi) denote Hermite polynomials while YJ8 (β, γ) are spherical harmonics. It is explicitly shown how all the matrix elements can be evaluated using a (3N–6) dimensional numerical integration technique. The theory in its present form is not suitable for molecules which are linear in the equilibrium configuration. In the following paper the method is used in a calculation on the water molecule.