Self-energy second-order

All second-order terms are retained in the P3 self-energy formulae for ionization energies and electron affinities. There are no differences between the expressions used for ionization energies and electron affinities in the second-order self-energy, which reads... 

Fig. 3.15. Exact second order self-energy operator...

The renormalized second-order self-energy takes the form... 

Second-Order Self-Energy Calculations for Tightly Bound Electrons in Hydrogen-Like Ions... 

Second-Order Self-Energy 631 For the integration over u.>,u> a following transformation was made in Eq. 

Loop-After-Loop Contribution to the Second-Order Self-Energy in Hydrogen... 

SEVP-mixed second order self energy-vacuum polarization correction. 

It is interesting to note that the second-order energy E can be written as the diagonal matrix element of the second-order self-energy operator defined by... 

FIG. 6.1. All Hugenholtz and Brandow second-order self-energy diagrams. 

In particular, the matrix elements of the second-order self-energy (P E)) are... 

We will now apply GF theory to this system and compare the results obtained with the exact answers. The second-order self-energy for this model is easily found from Eq. (7.39) by setting all the hole and particle indices equal to one and two, respectively (i.e., a = b = 1, r = s = 2 ... 

This behaviour of GF theory at the level of the second-order self-energy is more striking for the lowest ionization potentials of the ten-electron series shown in Table 7.4. At the level of 6-31G both the Koopmans and GF results appear to have converged. For all four molecules, Koopmans theorem overestimates the IP s, and in all cases the GF treatment lowers the IP too far, so that the Koopmans result actually agrees better with experiment. 

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