Density matrix embedding theory (DMET) is a quantum embedding theory de-
signed to treat strong correlation e ects in large quantum systems while maintaining
reasonable computation costs. The idea behind DMET is that in complex systems
the region of interest often forms merely one (small) part of a much larger system. It
is therefore natural to think about numerically treating the system with two di erent
approaches|high-level calculations on the active regions of interest, and low-level
calculation on the respective environments|and then `glue' the obtained results to-
gether.
A key component in the DMET formalism is the matching of density matrix blocks
obtained from the high-level and low-level theories; the ability to achieve exact match-
ing is an important issue in the DMET procedure since its inception as, in practical
calculations, this is sometimes not achievable. In such a case, the global band gap of
the low-level theory vanishes, and this can require additional numerical considerations
in order to obtain accurate results. We nd that both the violation of the exact match-
ing 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 where the orbitals are obtained from an auxiliary low-level system.
A potential remedy is to relax the NI-PS-V assumption in DMET and allow for pure
states following arbitrary occupation pro les. This seems to be a daunting problem as
the number of distinct occupation pro les is combinatorially large. We propose to use
an augmented Lagrangian method, coupled with a projected gradient descent method
to solve this modi ed constrained optimization problem. The inclusion of this opti-
mization over all possible occupation pro les into the self-consistent DMET work
ow
is christened alm-DMET. The alm-DMET method relaxes the NI-PS-V assumption,
which allows the pure state to follow any occupation pro le|possibly violating the
Aufbau principle|while yielding an idempotent low-level density matrix. Numer-
ical evidence shows that this relaxation of the Aufbau principle indeed allows the
alm-DMET method to yield exact matching, which improves the numerical accuracy
compared to conventional self-consistent DMET methods.
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