An n-state system

For simplicity, let us begin by considering a two-state system. In this case, we have 

The secular equation for the excited state takes the form 

Equation (2.259) has two roots, of which we take the lowest one. The reaction operators are given by 

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We now generalize our previous development to obtain a diagrammatic representation of RS perturbation theory as applied to an N-state system. Consider the problem of finding the perturbation expansion for the lowest eigenvalue of such a system. Here we still have only one hole state, 1>, but there are now N - 1 particle states n>, n = 2, 3,..., AT. We draw the same set of diagrams as before. However, now we can label the particle lines with any index n. For example, the diagram... 

Exercise 6.2 Use diagrammatic techniques to obtain the fourth-order perturbation energy of a particular state (say, i) of an N-state system. That is, evaluate the diagrams... 

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