  Physics-informed neural networks have been widely applied to learn general
parametric solutions of differential equations. Here, we propose a neural
network to discover parametric eigenvalue and eigenfunction surfaces of quantum
systems. We apply our method to solve the hydrogen molecular ion. This is an
ab-initio deep learning method that solves the Schrodinger equation with the
Coulomb potential yielding realistic wavefunctions that include a cusp at the
ion positions. The neural solutions are continuous and differentiable functions
of the interatomic distance and their derivatives are analytically calculated
by applying automatic differentiation. Such a parametric and analytical form of
the solutions is useful for further calculations such as the determination of
force fields.
