Effective potentials in molecular collisions

Procedures are developed for obtaining effective potentials V^eff between two linear molecules. The potential V_eff couples the internal rotational states of the molecules, regardless of the spatial effects associated with the projection quantum numbers M. Since just V^eff is not sufficient for treating the dynamics, an effective Hamiltonian H ^eff = H₀^eff+V^eff is obtained from the original Hamiltonian H = H₀+V, where Ha describes the unperturbed motion of the molecules. A consistent procedure is presented for finding both effective operators H₀^eff and V^eff, where the matrix elements of an effective operator Q^eff are written as 〈J₁J₂|Q^eff|J₁′J ₂′〉. The effective states 1/1/2) are orthonormal eigenstates of the rotational portion of H₀^eff with the same level spacings as for H₀. The effective potential is shown to have the form of a radial function times an internal state coupling matrix, or in general a sum of such terms. The effective close-coupled equations are developed from H^eff, and it is demonstrated that there is a decoupling of the orbital and internal rotational angular momenta. The significantly reduced dimensionality of the effective equations is shown by comparison with the corresponding equations derived from H. A numerical example using V^eff in semiclassical time-dependent perturbation theory is presented.