Avoided crossings: A study of the nonadiabatic transition probabilities

摘要

An approximate solution to the problem of constructing a pair of diabatic states exists only if certain requirements are fulfilled, for example, when the nonadiabatic coupling results from an interaction between two electronic configurations which are doubly excited with respect to one another. It is then possible to build up a model in which the series expansion of the elements of the Hamiltonian matrix is truncated after the first nonzero term. This leads to several conclusions concerning the nonadiabatic transition probability which differentiate conical intersections from avoided crossings. For the latter, the nonadiabatic coupling matrix elements (which are Lorentzians with an area equal to pgr;/2) reach their maximum at the nuclear geometry for which Dgr;E(the energy gap between adiabatic surfaces) is a minimum. The loci along which the angle thgr; of the orthogonal transformation which relates adiabatic and diabatic wave functions keeps a constant value are a set of parallel straight lines which coincides with the loci along which Dgr;Eremains constant. This reference direction in the configuration space corresponds to nuclear trajectories which are unable to bring about a nonadiabatic transition. In the case of avoided crossings, there exists only one nuclear degree of freedom which gives rise to surface hopping. Conical intersections, on the other hand, have two such active degrees of freedom. This creates a qualitative difference between the two cases which makes conical intersections more efficient as funnels than avoided crossings. A twohyphen;dimensional extension of the Landaundash;Zener formula is derived for avoided crossings. It contains a factor of anisotropy. It is possible, at least in favorable cases, to extract approximate diabatic quantities fromabinitiocalculations and to compare them with the predictions of these models. This has been done for two2A1electronic states of the CH+2ion. The results are found to agree with the predictions of the model, at least in a restricted range of internuclear distances.