The Rydberg-Klein-Rees (RKR) method is an inversion procedure used to determine potential energy surfaces from spectroscopic parameters found for a diatomic molecule. The RKR calculations used here were performed using the RKR1.0 program written by LeRoy [80]. This program allows the vibrational energies and rotational constants to be defined by Dunham or near dissociation expansions (not considered here). It offers the advantage that it can prevent unreasonable behaviour of the inner wall (due to inaccuracies in the experimentally derived functions) through a smoothing procedure.
The Klein integrals around which the RKR method is based are:
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(3.16) | ||
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(3.17) |
Where R1 and R2 are the inner and outer turning points of the potential at the energy G(v) associated with the vibrational level v. Bv is the inertial rotational constant and
is
(
is the reduced mass).
The two expressions for f and g can be rearranged to yield:
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(3.18) | ||
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(3.19) |
Integrals for f and g can be evaluated giving the inner and outer turning points of the potential surface, when the vibrational and rotational parameters are known.
The inversion procedure used here is only valid within the WKB approximation (first order semiclassical approach - that energy levels for the vibrating rotor can be quantized). However, this is accurate enough that when the Schrödinger equation is solved for these potentials, the eigenvalues and rotational constants are often seen to agree within the experimental uncertainties. As mentioned earlier, RKR1.0 allows smoothing over irregularities in the inner wall. Due to the steepness of the inner wall, uncertainties in the spectroscopic parameters (at high vibrational energies) lead to non-physical behaviour. RKR1.0 determines non-physical behaviour in the following way: The turning point calculation starts at the potential minimum and calculates at successive energies. After completion of each turning point calculation, the program fits the turning points for that level and the two lying below it, to an equation of the functional form:
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(3.20) |
If the parameter C remains positive and varies slowly from one point to the next, then the inner wall needs no correction. However, if the wall passes through a point of inflexion, C changes sign. Thus the behaviour of C is an indicator of the need to smooth over the inner wall.