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The potential energy for the 30 lowest molecular states 2S+1 (+) of the RbCs molecule has been calculated over a wide range of internuclear distances, using an ab initio method based on non-empirical pseudopotentials, parametrized l-dependent polarization potentials and full valence CI calculations. Molecular spectroscopic constants have been derived for the bound states. An accurate description of the few experimentally known states is obtained. Extensive tables of energy values versus internuclear distances are available at the following address: http://lasim.univ-lyon1.fr/allouche/rbcs.htm.

In the diatomic eigenvalue problem, and in related topics, physicists often make use of analytic potential functions to illustrate the numerical application of a new method, or to show the improvement of an old one. Among these analytic potential functions, the most widely used are the Morse and the LennardJones functions. The first has the advantage of having the vibrational eigenvalues E,, and the vibrational wavefunctions I),, both given by analytical expressions. The LennardJones function has no such advantage.Yet a persistent trend makes use of the Lennard-Jones function to illustrate numerical applications (to compare accuracy and efficiency) for many problems, namely the centrifugal distortion constants problem,' and the phase-shift problem in elastic scattering. ' Actually, Lennard-Jones eigenvalues E , are known for the first 22 levels (and not for the two highest levels), having been determined by the Cooley shooting m e t h~d .~ In a recent article, Tellinghuisen* considered these eigenvalues as "exact": he mentioned in a still more recent report lC difficulties (already encountered by Hutsonlb) in obtaining the highest eigenvalues (for u = 22 and u = 23). The aim of this note is to give vibrational eigenvalues for all levels of the Lennard-Jones function, reliable up to 15 significant figures.We make use of the numerical canonical functions m e t h~d ,~ which has been shown to be highly accurate for low and high levels, up to the dissociation.6 The numerical application of this method to the Lennard-Jones potential, U(r) = (1 -r-6)2, gives the vibrational eigenvalues E, shown in Table I, along with those computed by Hutson7 using the Cooley method. The present results were obtained in double precision on the computer HP 10008 with a constant mesh-size h = 0.01 A.We notice that the eigenvalues computed by the two methods are in agreement up to 8 significant figures. In order to test these results, we chose to compare the two programs by using the same c~m p u t e r ,~ for a Morse potential function, where the computed eigenvalues can be compared to the Morse theoretical ones.We list in the third column of Table I The canonical function results are equal to the Morse theoretical eigenvalues up to 15 (or 16) significant figures (the computer precision). By examining the results of the shooting method (second entry) and those of the canonical functions method (first entry), we notice that the agreement for each entry between the two results is practically the

The problem of the determination of the vibration-rotation eigenvalue in diatomic molecules is considered. An eigenvalue equation totally independent from the eigenfunction is written for any potential, analytical or numerical. This equation uses uniquely the vibration-rotation canonical functions; its resolution is reduced to that of a simple and classical numerical problem. Examples of numerical applications for analytical (Morse) and numerical potentials are presented. It is shown that the vibrational eigenvalues deduced from the eigenvalue equation are within 10-6 cm-' of the exact values. Comparison with conventional methods are presented and discussed.