Supplementary material from "Compressed hydrogen atoms confined within generic boxes∗"

The ‘compressed hydrogen atom’ problem involves solving the Schrödinger equation for a hydrogen atom confined within an impenetrable (often, spherical) box, with the wave function vanishing on the boundary of this box. To better understand the role the structure of the box plays on the compressed hydrogen atom, we study the compressed hydrogen atom in boxes of myriad shapes and sizes, developing a theory for how the structure of a generic box influences the hydrogen ground state energy. Our theory predicts that the ground state energy increases with the first Dirichlet eigenvalue (which encodes information on both the shape and size of a box) of the box and decreases with the distance of the nucleus from the boundary of the box. This theoretical prediction is supported by numerical simulations and variational approximations. Consideration is also given to the problem of a shell-confined hydrogen atom, with the electron confined to a separate ‘cage’ away from the nucleus. The behaviour of the shell-confined hydrogen ground state is well-approximated by a perturbation theory we develop for generic cages. Our results provide a unified framework for understanding confined, high-pressure hydrogen with relevance to applications ranging from crystal lattices of hydrogen to metallic hydrogen.