  Energy functionals of the Green's function can simultaneously provide
spectral and thermodynamic properties of interacting electrons' systems. Though
powerful in principle, these formulations need to deal with dynamical
(frequency-dependent) quantities, increasing the algorithmic and numerical
complexity and limiting applications. We first show that, when representing all
frequency-dependent propagators as sums over poles, the typical operations of
dynamical formulations become closed (i.e., all quantities are expressed as
sums over poles) and analytical. Importantly, we map the Dyson equation into a
nonlinear eigenvalue problem that can be solved exactly; this is achieved by
introducing a fictitious non-interacting system with additional degrees of
freedom which shares, upon projection, the same Green's function of the real
system. Last, we introduce an approximation to the exchange-correlation part of
the Klein functional adopting a localized $GW$ approach; this is a
generalization of the static Hubbard extension of density-functional theory
with a dynamical screened potential $U(\omega)$. We showcase the algorithmic
efficiency of the methods, and the physical accuracy of the functional, by
computing the spectral, thermodynamic, and vibrational properties of SrVO$_3$,
finding results in close agreement with experiments and state-of-the-art
methods, at highly reduced computational costs and with a transparent physical
interpretation.
