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One-particle Greens function methods can model molecular and solid spectra at zero or non-zero temperatures. One-particle Greens functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as $langle{S^2}rangle$ and $langle{N^2}rangle$ can be challenging, because they require a solution of the computationally expensive Bethe--Salpeter equation to find two-particle Greens functions. We demonstrate that the solution of the Bethe--Salpeter equation can be complitely avoided. Applying the thermodynamic Hellmann--Feynman theorem to self-consistent one-particle Greens function methods, we derive expressions for two-particle density matrices in a general case and provide explicit expressions for GF2 and GW methods. Such density matrices can be decomposed into an antisymmetrized product of correlated one-electron density matrices and the two-particle electronic cumulant of the density matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temperature dependence of $langle{S^2}rangle$ and $langle{N^2}rangle$ for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the number of particles for closed-shell systems at the zero-temperature limit.
Due to non-linear structure, iterative Greens function methods can result in multiple different solutions even for simple molecular systems. In contrast to the wave-function methods, a detailed and careful analysis of such molecular solutions was not
We report unphysical irregularities and discontinuities in some key experimentally-measurable quantities computed within the GW approximation of many-body perturbation theory applied to molecular systems. In particular, we show that the solution obta
We present the fundamental techniques and working equations of many-body Greens function theory for calculating ground state properties and the spectral strength. Greens function methods closely relate to other polynomial scaling approaches discussed
The ultrafast hole dynamics triggered by the photoexcitation of molecular targets is a highly correlated process even for those systems, like organic molecules, having a weakly correlated ground state. We here provide a unifying framework and a numer
We review some applications of self-consistent Greens function theory to studies of one- and two-nucleon structure in finite nuclei. Large-scale microscopic calculations that employ realistic nuclear forces are now possible. Effects of long-range c