Using genetic programming to solve the Schrodinger equation

In a recent paper [Makarov, D. E.; Metiu, H. J. Chem. Phys. 1998, 108, 590], we developed a directed genetic programming approach for finding the best functional form that fits the energies provided by ab initio calculations. In this paper, we use this approach to find the analytic solutions of the time-independent Schrodinger equation. This is achieved by inverting the Schrodinger equation such that the potential is a functional depending on the wave function and the energy. A genetic search is then performed for the values of the energy and the analytic form of the wave function that provide the best fit of the given potential on a chosen grid. A procedure for finding excited states is discussed. We test our method for a one-dimensional anharmonic well, a double well, and a two-dimensional anharmonic oscillator.