Numerical Potentials and Vibrational Wavefunctions

The Rydberg-Klein-Rees (RKR) procedure is the most widely used method for deriving V(R) from the G(v) and B(v) functions of diatomic molecules. Countless RKR computer programs of independent origin exist, with differences primarily in the way a singularity at the upper limit of integration of Eqs. (5.1.39a or b) and (5.1.40a or c) is handled, all of which give essentially identical results (see Mantz, et al, 1971 for the CO X1E+ potential), except 

Many detailed derivations of the RKR equations have been published (Zare, 1964 Miller, 1971 Elander, et al, 1979). The unknown potential energy for a rotating diatomic molecule, 

Note that the integrals in Eqs. (5.1.39b) and (5.1.40c) become infinite at the upper limit of integration, but this singularity is integrable by various techniques (Mantz, et al., 1971). Note also that, although G(—1/2) = 0, 

The RKR potential may be tested against the input G(v) and B(v) values by exact solution of the nuclear Schrodinger equation [see Wicke and Harris, 1976, review and compare various procedures, e.g., Numerov-Cooley numerical integration (Cooley, 1961), finite difference boundary value matrix diagonaliza-tion (Shore, 1973), and the discrete variable representation (DVR) (Harris, et al., 1965)]. G(v) + y00 typically deviates from EVjj=o by 1 cm-1 except near dissociation. Bv may be computed from Xv,J=o(R) by 

Remember that for every observable quantity, the sign of each wavefunction (electronic and vibrational) always appears twice. This is the reason why the phase convention adopted is irrelevant provided that the same phase convention is used both times each wavefunction appears. It is dangerous to derive an observable quantity, using one matrix element taken from the literature without knowing the phase convention that has been used (see one example in Section 6.2.1). 

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