Quantum embedding
is an appealing route to fragment a large interacting
quantum system into several smaller auxiliary “cluster”
problems to exploit the locality of the correlated physics. In this
work, we critically review approaches to recombine these fragmented
solutions in order to compute nonlocal expectation values, including
the total energy. Starting from the democratic partitioning of expectation
values used in density matrix embedding theory, we motivate and develop
a number of alternative approaches, numerically demonstrating their
efficiency and improved accuracy as a function of increasing cluster
size for both energetics and nonlocal two-body observables in molecular
and solid state systems. These approaches consider theN-representability of the resulting expectation values via an implicit
global wave function across the clusters, as well as the importance
of including contributions to expectation values spanning multiple
fragments simultaneously, thereby alleviating the fundamental locality
approximation of the embedding. We clearly demonstrate the value of
these introduced functionals for reliable extraction of observables
and robust and systematic convergence as the cluster size increases,
allowing for significantly smaller clusters to be used for a desired
accuracy compared to traditional approaches inab initiowave function quantum embedding.