  Partition Density Functional Theory (P-DFT) is a density embedding method
that partitions a molecule into fragments by minimizing the sum of fragment
energies subject to a local density constraint and a global electron-number
constraint. To perform this minimization, we study a two-stage procedure in
which the sum of fragment energies is lowered when electrons flow from
fragments of lower electronegativity to fragments of higher electronegativity.
The global minimum is reached when all electronegativities are equal. The
non-integral fragment populations are dealt with in two different ways: (1) An
ensemble approach (ENS) that involves averaging over calculations with
different numbers of electrons (always integers); and (2) A simpler approach
that involves fractionally occupying orbitals (FOO). We compare and contrast
these two approaches and examine their performance in some of the simplest
systems where one can transparently apply both, including simple models of
heteronuclear diatomic molecules and actual diatomic molecules with 2 and 4
electrons. We find that, although both ENS and FOO methods lead to the same
total energy and density, the ENS fragment densities are less distorted than
those of FOO when compared to their isolated counterparts, and they tend to
retain integer numbers of electrons. We establish the conditions under which
the ENS populations can become fractional and observe that, even in those
cases, the total charge transferred is always lower in ENS than in FOO.
Similarly, the FOO fragment dipole moments provide an upper bound to the ENS
dipoles. We explain why, and discuss implications.
