Fundamental model equations general formulation

This chapter aims to present the fundamental formal and exact relations between polarizabilities and other DFT descriptors and is organized as follows. For pedagogical reasons, we present first the polarizability responses for simple models in Section 24.2. In particular, we introduce a new concept the dipole atomic hardnesses (Equation 24.20). The relationship between polarizability and chemical reactivity is described in Section 24.3. In this section, we clarify the relationship between the different Fukui functions and the polarizabilities, we introduce new concepts as, for instance, the polarization Fukui function, and the interacting Fukui function and their corresponding hardnesses. The formulation of the local softness for a fragment in a molecule and its relation to polarization is also reviewed in detail. Generalization of the polarizability and chemical responses to an arbitrary perturbation order is summarized in Section 24.4. 

However, the conventional reactor model formulations are normally strictly not precise on the microscopic scales (yet usually sufficient in engineering practice) as the fundamental or rigorous version of the governing transport equations were considered too complex to be conveniently solved for practical applications. There are generally two ways to obtain simplified models,... 

Formulation of a proper reactor model and incorporation of kinetic expressions for rs and rx- Using the fundamental law of conservation of mass (cf. Equs. 4.76 and 4.101), the general partial differential equation of S concentration in a control volume is... 

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