Accurate and Efficient Solution of the Electronic Schr\"odinger Equation with the Coulomb Singularity by the Distributed Approximating Functional Method

Abstract

We proposed a distributed approximating functional method for efficiently describing the electronic dynamics in atoms and molecules in the presence of the Coulomb singularities, using the kernel of a grid representation derived by using the solutions of the Coulomb differential equation based upon the Schwartz's interpolation formula, and a grid representation using the Lobatto/Radau shape functions. The elements of the resulted Hamiltonian matrix are confined in a narrow diagonal band, which is similar to that using the (higher order) finite difference methods. However, the spectral convergence properties of the original grid representations are retained in the proposed distributed approximating functional method for solving the Schr\"odinger equation involving the Coulomb singularity. Thus the method is effective for solving the electronic Schr\"odinger equation using iterative methods where the action of the Hamiltonian matrix on the wave function need to evaluate many times. The method is investigated by examining its convergence behaviours for calculating the electronic states of the H atom, H$_2^+$ molecule, the H atom in a parallel magnetic and electric fields, as the radial basis functions.