Third-order diagrammatic perturbation

Third-order diagrammatic perturbation theory of nuclei and electrons... 

Prior Y (1984) A complete expression for the third-order susceptibility. Perturbative and diagrammatic approaches. IEEE Journal of Quantum Electronics QE-20 37-42. 

These expressions are identical to our previous results for the second- and third-order energies (Eqs. (6.12) and (6.15)) when i = 1. What if we want the perturbation expansion for some state i, which is not necessarily the lowest What do the diagrams look like One can easily verify that we get the same answers as before if we label our hole lines by the index i and the particle lines by the indices m, n, k,..., which can take on the values 1,2,..., i - 1, I + 1,..., N. Thus we now have a complete diagrammatic representation of RS perturbation theory, which is applicable to any perturbation and any zeroth-order state. 

To illustrate the simplifications introduced by diagrammatic perturbation theory [4], we consider the three-photon processes corresponding to the third-order term in the Dyson expansion of c t). 

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