Atoms-in-Molecules Discretization

Let us start with a brief comment on the nature of the AIM discretization. It can be viewed as resulting from the phenomenological energy expression  

Continuing this analogy with the local description, one can introduce the modified AIM external potential parameters u, = v, — // = u, which uniquely define the equilibrium AIM electron populations N = N(u). In terms of u (see Eqs. (5), (7), and (12)  

Consider a given molecular system consisting of m atoms. In what follows we adopt the AIM resolution to define the canonical AIM chemical potentials (electron population gradient), p = dE/dN = (fiu fi2. fim), and the corresponding AIM hardness matrix (electron population hessian) tj = d2E/dN dN = dp/dN = here all differentiations are carried out for the fixed external potential v. This canonical charge-sensitivity information will be used to generate a variety of system charge sensitivities (CS) that probe the responses of the system to various populational perturbations at constant v. 

Let us first examine the most important combination problem of determining CS of the system as a whole we call them global or resultant quantities, since they determine the global equilibrium in the system under consideration. The complementary canonical compliant information is obtained by inverting if, 

This softness matrix can be used to generate the local (regional) and global softness quantities  

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