Natural resonance structures and weightings

The mathematical criterion for resonance description of delocalization effects 

The approximation (1.32) of a single Lewis ( resonance ) structure is often inadequate, and the associated model chemistry of Eq. (1.33) cannot accurately describe the system of interest. Because I L) can be expressed in terms of a corresponding density matrix D(L) (cf. Eqs. (1.25) and (1.27)), we can also say that the localized density matrix D(L) does not sufficiently approximate the true delocalized density matrix D(trae) associated with P. 

As suggested by the original resonance theory of Pauling and Wheland,51 such delocalization effects appear to represent some type of average of multiple resonance structures. A general goal of resonance theory is to represent each property (P)true of the true delocalized system in resonance-averaged form 

Equation (1.37) is actually the necessary and sufficient condition that any one-electron property52 (P) (such as electron density, dipole moment, kinetic energy.) could be written in the resonance-averaged form (1.36). The goal of a quantitative resonance theory is to find the resonance weights wa (if any) for which Eq. (1.37) is most accurately satisfied. 

The natural resonance theory (NRT) method53 offers a convenient ab initio means to calculate such resonance weights from the D(true) density matrix for a Fof any 

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