Solution to the schrödinger equation for bound states of polar molecules using shallow neural networks

Abstract

The time independent Schrödinger equation can be solved analytically in very few cases. This is why scientists rely on approximate numerical methods. One that is yet to be a part of the everyday toolbox for physicists are neural networks. With it, it is possible to calculate eigenfunctions and the energy spectra of bound states. A double exponential central potential is used to describe the interaction between two electric dipoles that form through van der Waals interactions. These systems can be important when the electric dipoles are very large and differ from covalent, ionic, and other kinds of bonds, as is the case of the monomers M−C60, where M=Cs,Li or other alkali metals. Based in statistical physics, it has been shown that a dimer can be formed through a spontaneous and exothermic reaction. The ground state and the first two excited states of this system were calculated, also the solutions and their associated energies were plotted. The result for the ground state is compared to the shooting method and the direct variational method. The numerical value of the ground states obtained suggests that the dimers can be dissociated by electromagnetic waves with wavelengths in the far infrared regime, very close to microwaves.