New expression for the expectation value integral for a confined helium atom

A new expression for the expectation value integral of Hermitian operators for spherically symmetric states is derived in elliptical coordinates for the Helium atom confined in a cavity with infinite potential barrier walls. The expression is comprised of two components spanning different domains of space. For an atom in infinite space, with a wave function which satisfies Dirichlet boundary conditions, one of these terms is shown to vanish, but for a confined atom that same term cannot be neglected. In a variational calculation of the energy, for a helium atom confined by finite spheres of various radii, it is observed that the principal contribution is due to the first component, with the relative magnitude of the second component becoming smaller as the size of the cavity is increased. The expression derived is shown to be computationally faster than the conventionally employed expression for confined systems derived by ten Seldam-de Groot.