Algorithm for Solving the Two-Dimensional Boundary Value Problem for Model of Quantum Tunneling of a Diatomic Molecule Through Repulsive Barriers

Algorithm for solving the boundary value problems that describe the model of quantum tunneling of a diatomic molecule through repulsive barriers in s-wave approximation is presented. The boundary value problems are formulated and reduced to the one-dimensional ones for systems of coupled second-order differential equations by means of the Galerkin and Kantorovich methods. The description of elaborated algorithms and the calculated asymptotes of parametric basis functions, matrices of variable coefficients, and fundamental solutions of the systems of the coupled second-order differential equations needed for solving the boundary problems on a finite interval are given. The BVPs were solved by the elaborated set of programs implementing the finite element method. Analysis of benchmark calculations of quantum tunneling of a diatomic molecule model with the nuclei coupled by the Morse potential through Gaussian barriers and quantum transparency effect induced by metastable states embedded in continuous spectrum below dissociation threshold are presented.

quantum tunneling problem, diatomic molecule, repulsive barriers, boundary-value problems, Galerkin method, Kantorovich method, asymptotic solutions, finite element method