  Quantum mechanical calculations require the repeated solution of a
Schr\"odinger equation for the wavefunctions of the system. Recent work has
shown that enriched finite element methods significantly reduce the degrees of
freedom required to obtain accurate solutions. However, time to solution has
been adversely affected by the need to solve a generalized eigenvalue problem
and the ill-conditioning of associated systems matrices. In this work, we
address both issues by proposing a stable and efficient orbital-enriched
partition-of-unity method to solve the Schr\"odinger boundary-value problem in
a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In
our proposed PUM, the three-dimensional domain is covered by overlapping
patches, with a compactly-supported, non-negative weight function, that is
identically equal to unity over some finite subset of its support associated
with each patch. This so-called flat-top property provides a pathway to devise
a stable approximation over the whole domain. On each patch, we use $p$-th
degree orthogonal polynomials that ensure $p$-th order completeness, and in
addition include eigenfunctions of the radial solution of the Schr\"odinger
equation. Furthermore, we adopt a variational lumping approach to construct a
block-diagonal overlap matrix that yields a standard eigenvalue problem and
demonstrate accuracy, stability and efficiency of the method.
