ANALYTICAL TREATMENT OF THE QUADRATIC AND NONRELATIVISTIC RENNER-TELLER EFFECT FOR A TRIATOMIC MOLECULE IN A PI-ELECTRONIC STATE .3. INTERMEDIATE LIMIT CASE (EPSILON-LESS-THAN-OR-EQUAL-TO-1-MUCH-LESS-THAN-M-SIMILAR-TO-E(2/3)) AND COMPARISON WITH SEMICLASSICAL RESULTS

This work completes our analytical treatment of the quadratic and nonrelativistic Renner-Teller effect and deals with a situation (intermediate limit case, ILC) placed between the recently studied first and second limit cases (FLC, high energy limit, and SLC, high energy and high angular momentum limit, respectively). In the ILC the second order ordinary differential equation in the complex plane that results from a recently reported matrix-integral transformation is also of singular perturbation type, as also occurred in the FLC and SLC, but the method of matched asymptotic expansions cannot be applied in the same extent. However, thanks to the calculation of the non-adiabatic transition probability based on results of an earlier semiclassical study, it has been possible to achieve a fairly deep insight into this problem. To the best of our knowledge this is the first time that the semiclassical non-adiabatic transition probability has been determined, the results being valid for a wide interval of energy and angular momentum values. Moreover, we have shown that the choice of a proper matrix connection between the coefficients of the direct asymptotic expansion for two relevant regions of the direct domain leads to complete agreement between the quantal and semiclassical analyses. This result gives us additional confidence about the correctness of both approaches. Although the ILC is more difficult to study than the FLC and SLC, probably it is the most attractive one from both the experimental and theoretical points of view, as from the semiclassical calculations it becomes evident that it corresponds to a situation where the energy splitting and transition probability caused by the non-adiabatic coupling reach maximal values.