An Economical Approximation

Perturbative expressions for the self-energy operator can achieve this goal for large, closed-shell molecules. In this review, we will concentrate on an approximation developed for this purpose, the partial third-order, or P3, approximation. P3 calculations have been carried out for a variety of molecules. A tabulation of these calculations is given in Table 5.1. 

The original derivation of the P3 method was accompanied by test calculations on challenging, but small, closed-shell molecules with various basis sets [12], The average absolute error was approximately 0.2 eV 

This procedure requires analytical expressions for EPP(E) and its derivative with respect to E it usually converges in three iterations. Neglect of off-diagonal elements of the self-energy matrix also implies that the corresponding Dyson orbital is given by  

In the latter expression, the derivative is evaluated at the converged energy. Diagonal self-energy approximations therefore subject a frozen Hartree-Fock orbital F(x) to an energy-dependent correlation potential Epp(E). 

Diagonal matrix elements of the P3 self-energy approximation may be expressed in terms of canonical Hartree-Fock orbital energies and electron repulsion integrals in this basis. For ionization energies, where the index p pertains to an occupied spinorbital in the Hartree-Fock determinant, 

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