Solution of the Schrödinger equation for two electrons in axially symmetric cavities

Abstract

We present a method of solving the Schr\"odinger equation for two electrons confined in an almost arbitrary infinite-potential well of ${D}_{\ensuremath{\infty}h}$ symmetry. The boundary conditions for the wave function are imposed by devising a one-electron basis set that incorporates detailed information about the shape of the cavity. Two-electron integrals are evaluated using a fully separable expansion of the Coulomb operator similar to that developed earlier by the authors for a cylindrical cavity. To illustrate the capabilities of our approach, we present full configuration-interaction solutions for the lowest ${}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and ${}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ states of two electrons in four axially symmetric cavities: a sphere, a cylinder, a bulged cylinder, and a capsule. By comparing the ground-state electronic energies for different cavities of equal volume $V$, we find that spherical confinement is optimal for $V\ensuremath{\lesssim}1500{a}_{0}^{3}$; for larger volumes, the capsule-shaped system has a significantly lower energy. This switchover is driven by the same physical mechanism as Wigner crystallization and may serve as an indication of that process. The magnitude of the singlet-triplet gap strongly depends on the shape of the cavity and, for a fixed volume, decreases in the following order: bulged cylinder, sphere, cylinder, capsule.