Quasiparticles in neon using the Faddeev random-phase approximation

The spectral function of the closed-shell neon atom is computed by expanding the electron self-energy through a set of Faddeev equations. This method describes the coupling of single-particle degrees of freedom with correlated two-electron, two-hole, and electron-hole pairs. The excitation spectra are obtained using the random-phase approximation (RPA), rather than the Tamm-Dancoff framework employed in the third-order algebraic diagrammatic construction method. The difference between these two approaches is studied, as well as the interplay between ladder and ring diagrams in the self-energy. Satisfactory results are obtained for the ionization energies as well as the energy of the ground state with the Faddeev RPA scheme, which is also appropriate for the high-density electron gas.

Figure 1

(Color online) (a) Diagrammatic expansion of $R\left(\omega \right)$ in terms of the (antisymmetrized) Coulomb interaction and undressed propagators. (b) $R\left(\omega \right)$ is related to the self-energy according to Eq. (3). (c) By substituting the diagrams (a) in the latter equation, one finds the perturbative expansion of the self-energy.

Figure 2

(Color online) Diagrammatic equations for the polarization (above) and the two-particle (below) propagators in the GRPA approach. Dashed lines are always antisymmetrized Coulomb matrix elements and the full lines represent free (undressed) propagators.

Figure 3

(Color online) Example of one of the diagrams that are summed to all orders by means of the Faddeev equations (7) (left). The corresponding contribution to the self-energy, obtained upon insertion into Eq. (3), is also shown (right).

Figure 4

(Color online) Spectral function for the $s$ states in Ne obtained with various self-energy approximations. From the top down: the second-order $\left({\Sigma }^{\left(2\right)}\right)$, the FRPA (ring), the FRPA (ladder), and the full FRPA self-energies. The strength is given relative to the Hartree-Fock occupation of each shell. Only fragments with strength larger than $Z>0.005$ are shown.

Figure 5

(Color online) Diagrammatic representation of Eq. (A3). Double lines represent fully dressed SP Green’s functions, which, however, are restricted to propagate only in one time direction [i.e., only one of the two terms on the right-hand side of Eq. (1) is retained]. The Faddeev equations (A9, 7) allow for both forward and backward propagation of the phonons ${\Gamma }^{\left(\pi \right)}\left(\omega \right)$ and ${\Gamma }^{\left(II\right)}\left(\omega \right)$ as long as these do not overlap in time. For the propagators, time ordering is assumed with forward propagation in the upward direction.