An effective quasi-one-electron asymptotic technique is developed for treating the electronic part of A+BC molecular interaction. The theory proceeds from a model of the motion of an electron in the field of A(+) and BC. The interaction between the electron and the ion A(+) is handled by a Green function technique while the interaction of the electron with BC is modeled by a short-range approach. The theory provides the electronic wave functions, potential surfaces, and transition dipole moments as functions of three nuclear coordinates specifying the geometrical configuration of the A+BC system. Calculations of potential surfaces are carried out for the case in which A is a hydrogen atom. The main advantage of the theory resides in the fact that the full structure of the potential surfaces and that of transition dipole moments are expressed in terms of a few physical parameters. This feature, along with the ease of computation, turns the theory into a convenient tool for solving the electronic part of molecular dynamics problems for many laser assisted collisions and reactions. The theory significantly extends the method proposed earlier [V.S. Dubov, J. Chem. Phys. 97, 7342 (1992)].