Chemical transformations proceed through a mechanism consisting of elementary chemical steps representing the reactive collisions of individual molecular species.  Fundamentally, such collisions may be described as a quantum mechanical inelastic scattering process involving the rearrangement of kinetic and internal energy. Important parameters, such as reaction cross-sections and kinetic rate constants, can be derived from the solutions of the Schrödinger equation.  This seminar will review the so-called bipolar counter-propagating wave methodology (CPWM) and its child the bipolar reaction path Hamiltonian (BRPH) approach for calculating stationary scattering state wave functions.  The presentation will illustrate key features of CPWM and BRPH algorithms through pedagogical applications involving 1D and 2D model problems, and will outline recent work and related challenges for applications involving collinear atom plus diatom exchange reactions.