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Numerical Treatment of Schrödinger’s Equation for One-Particle and Two-Particle Systems Using Matrix Method

With an increased interest in accurately predicting aerothermal environments for planetary entry, recent researches have concentrated on developing high fidelity computational models using quantum-mechanical first principles. These calculations are dependent on solving the Schrödinger equation using molecular orbital theory techniques. They are either approximate solutions to actual equations or exact solutions to approximate equations. Exact solutions have not found favor due to the computational expense of the problem. Hence, a novel numerical approach is developed and tested here using linear algebraic matrix methods to enable precise solutions. The finite-difference technique is used to cast the Time-Independent Schrödinger equation (TISE) in the form of a matrix eigenvalue problem. The numerical singularity in the Coulomb potential term is handled using Taylor series extrapolation, Least Squares polynomial fit, soft-core potential, and Coulomb potential approximation methods. A C++ code is developed to extract the eigenvalues and eigenfunctions with the help of PETSc and SLEPc open-source packages. The eigenvalues represent the energies, and eigenfunctions represent the orbitals of atoms. The test cases include both one-particle and two-particle systems. The singularity poses an issue in the computation by affecting the accuracy of energies and wavefunctions obtained. Higher-order differences and non-uniform meshes are implemented to resolve the issue, but they provide counterintuitive results and fail to achieve the desired level of accuracy.