  The question of which non-interacting Green's function "best" describes an
interacting many-body electronic system is both of fundamental interest as well
as of practical importance in describing electronic properties of materials in
a realistic manner. Here, we study this question within the framework of
Baym-Kadanoff theory, an approach where one locates the stationary point of a
total energy functional of the one-particle Green's function in order to find
the total ground-state energy as well as all one-particle properties such as
the density matrix, chemical potential, or the quasiparticle energy spectrum
and quasiparticle wave functions. For the case of the Klein functional, our
basic finding is that minimizing the length of the gradient of the total energy
functional over non-interacting Green's functions yields a set of
self-consistent equations for quasiparticles that is identical to those of the
Quasiparticle Self-Consistent GW (QSGW) [1] approach, thereby providing an a
priori justification for such an approach to electronic structure calculations.
In fact, this result is general, applies to any self-energy operator, and is
not restricted to any particular approximation, e.g. the GW approximation for
the self-energy. The approach also shows that, when working in the basis of
quasiparticle states, solving the diagonal part of the self-consistent Dyson
equation is of primary importance while the off-diagonals are of secondary
importance, a common observation in the electronic structure literature of
self-energy calculations. Finally, numerical tests and analytical arguments
show that when the Dyson equation produces multiple quasiparticle solutions
corresponding to a single non-interacting state, minimizing the length of the
gradient translates into choosing the solution with largest quasiparticle
weight.
