  Density matrix embedding theory (DMET) formally requires the matching of
density matrix blocks obtained from high-level and low-level theories, but this
is sometimes not achievable in practical calculations. In such a case, the
global band gap of the low-level theory vanishes, and this can require
additional numerical considerations. We find that both the violation of the
exact matching condition and the vanishing low-level gap are related to the
assumption that the high-level density matrix blocks are non-interacting
pure-state $v$-representable (NI-PS-V), which assumes that the low-level
density matrix is constructed following the Aufbau principle. In order to relax
the NI-PS-V condition, we develop an augmented Lagrangian method to match the
density matrix blocks without referring to the Aufbau principle. Numerical
results for 2D Hubbard and hydrogen model systems indicate that in some
challenging scenarios, the relaxation of the Aufbau principle directly leads to
exact matching of the density matrix blocks, which also yields improved
accuracy.
