Quantum defect theory for the van der Waals plus dipole-dipole interaction

We investigate the scattering dynamics governed by the long-range van der Waals plus dipole-dipole interaction potential, $\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath{-}{C}_{3}/{R}^{3}$, which describes the long-range interaction between two polar molecules in an electric field. In the spirit of quantum defect theory, a set of parameters which are nearly constants in the threshold regime is defined to characterize the scattering process. Using appropriate boundary conditions for the scattering wave functions and relevant parameters, we explore the quantum reflection by and quantum tunneling through the long-range potential. As a sample application, the reactive collision rates of ${}^{40}$K${}^{87}$Rb $+$ ${}^{40}$K${}^{87}$Rb are calculated.