The RPAE method

The RPAE includes the bubble graphs and their exchange counterparts to all orders, but does not include any of the other graphs. This turns out to be sufficient if the atom has a closed shell. It is also possible to 

9 This same series will be discussed again several times in this book for example, it is shown in fig. 6.4 It exhibits a particularly interesting perturbation, which is further discussed in section 8.29. 

Far from the ionisation threshold, where the escaping photoelectron has a large kinetic energy, the normal method of calculation for the continuum states is to compute the orbitals in a field determined by using the frozen orbitals of the neutral atom with one electron removed. However, if one is calculating a resonance which lies close to the threshold, this approach may fail. This happens because the escaping photoelectron moves slowly, so that the residual ion has time to relax as it escapes. It is then better to compute the orbitals in the relaxed field of the ion. This approximation is called the GRPAE or the RPAER, and is referred to as the RPAE with relaxation. 

One can also extend the RPAE by including the self-energy part of the photoelectron s interaction (SEP). In lowest order, this correction is 

Referring back to the derivation leading to equation (4.2), the matrix element for photoionisation may be expressed in the equivalent forms 

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