Failures of the Feynman-Dyson diagrammatic perturbation expansion of propagators

Using a general-order many-body Green's-function method for molecules, we illustrate numerically three pathological behaviors of the Feynman-Dyson diagrammatic perturbation expansion of one-particle many-body Green's functions as electron propagators. First, the perturbation expansion of the frequency-dependent self-energy is nonconvergent at the exact self-energy in wide domains of frequency. Second, the Dyson equation with an odd-order self-energy has a qualitatively wrong shape and, as a result, most of their satellite roots are complex and nonphysical. Third, the Dyson equation with an even-order self-energy has an exponentially increasing number of roots as the perturbation order is raised, which quickly exceeds the correct number of roots. Infinite partial summation of diagrams by vertex or edge modification exacerbates these problems. Not only does the nonconvergence render higher-order perturbation theories useless for satellite roots, but it also calls into question the validity of their combined use with the ans\"{a}tze requiring the knowledge of all poles and residues. Such ans\"{a}tze include the Galitskii-Migdal formula, self-consistent Green's-function methods, Luttinger-Ward functional, and some models of the algebraic diagrammatic construction.