Curl equations as substratum for the derivation of the electronic nonadiabatic coupling terms

The idea to derive the nonadiabatic coupling terms by solving the Curl equations (Avery, J.; Baer, M.; Billing, G. D. Mol Phys 2002, 100, 1011) is extended to a three-state system where the first and second states form one conical intersection, i.e., tau(12) and the second and the third states form another conical intersection, i.e., tau(23). Whereas the two-state Curl equations form a set of linear differential equations, the extension to a three-state system not only increases the number of equations but also leads to nonlinear terms. In the present study, we developed a perturbative scheme, which guarantees convergence if the overlap between the two interacting conical intersections is not too strong. Among other things, we also revealed that the nonadiabatic coupling term between the first and third states, i.e., tau(13) (such interactions do not originate from conical intersection) is formed due to the interaction between tau(12) and tau(23). (C) 2002 Wiley Periodicals, Inc.