Localization energy, definition

We can now consider the theoretical relationships between reactivity indices of the isolated molecule and localization methods. First we quote a generalized definition of the localization energy at atom r given by Fukui et al. (1957a)... 

Figure 1 illustrates the free energy profile of a typical acid-catalyzed chemical reaction that converts a substrate S to a product P. In this case, an intermediate chemical species SH+ is formed on protonation of S. If the activation energy for conversion of SH+ to PH+ is lower than for the conversion of S to P, then the reaction will go faster. It is important at this point to define the difference between an intermediate and a transition state An intermediate is a stable (or semistable) chemical species formed during the reaction and therefore is a local energy minimum, whereas a transition state, by definition, is a local energy maximum. 

A special word on the calculation of the energy is in order. In the last equality in (If) the hamiltonian operator in the numerator can be pushed either to the left or to the right in such a way to operate directly on the trial state. Remembering the definition of the local energy, (4), we can write... 

The pseudo-thermal energy, Epj, associated with deviations of the motion of individual particles from the local average Epj can be represented by the kinetic energy definition of temperature as... 

A natural definition of (z), the local energy density, whidi satisfies the symmetry condition is... 

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