Conceptual Density Functional Theory

Within the context of Density Functional Theory (DFT), Conceptual DFT (CDFT) has proven its merit for the analysis and interpretation of both experimental and theoretical studies of chemical phenomena, especially the thermodynamics and kinetics of chemical reactions. In the first part of this chapter, CDFT is introduced from a perturbative perspective. This approach exposes the machinery necessary to approximate reactivity indicators, computationally. Starting from an (approximate) energy model, E = E [ v ( r ) ; N ] , a reference state is chosen which is usually the reactant molecule of interest. Its change in energy upon interaction with a second reactant, caused by changes in the number of electrons, N , and external potential, v ( r ) , can then be written as a Taylor series where the expansion coefficients are partial derivatives with respect to N and/or functional derivatives with respect to v ( r ) . These coefficients are only dependent on the (first) reactant, whereas the characteristics of the second reactant are reflected in the changes in N and v ( r ) . The coefficients or response functions therefore characterize the intrinsic reactivity of the reactant molecule and can be considered reactivity indicators. These indicators are introduced up to third order in the expansion, and identified with chemical concepts, both in the canonical ensemble, characterized by the E = E [ v ( r ) ; N ] state function and in other Legendre-transformed ensembles. The Grand Canonical Ensemble with its associated Grand Potential Ω = Ω [ v ( r ) ; µ ] turns out to be more suitable for systems where the number of electrons is not easily controlled, paving the way to the description of open systems. The second part highlights well-known rules of thumb, explains the limitations of CDFT, and discusses some pitfalls in its application. The focus on the onset of a chemical reaction or, in a comparative context, on different reactions, as advocated in the perturbative approach, is discussed via general considerations on reaction paths. Klopman’s non-crossing rule is highlighted, leading to the avoidance of late transition states or multi-step reactions. The limitations of several principles where CDFT is often used, e.g., the Hard and Soft Acids and Bases principle, are highlighted scrutinizing the proviso in Pearson’s formulation “all other things being equal, hard acids prefer binding to hard bases and soft acids prefer to bind to soft bases”. The selection of adequate reactivity descriptors among the Taylor expansion coefficients is discussed within the context of the one-reactant approach. The influence of the nature of the second reactant on this choice is paramount, leading to more general-purpose reactivity indicators, containing for example a mix of hard and soft components. This part closes by addressing the conundrum of how negative electron affinities should be regarded when evaluating both electronegativity and hardness. The final part addresses important issues regarding the dissemination of CDFT-tools to help both specialists and non-specialists (including experimentalists) analyze their results and even, perchance, predict chemical reactivity. This requires user-friendly and well-documented software packages, like ChemTools, offering a unified approach to the evaluation of as many response functions as possible. ChemTools can post-process outputs of essentially any Gaussian basis-set-based quantum-chemistry code. A further advantage as compared to the variety of in-house research codes is the consistent use of various energy models. One thereby avoids the piecemeal approach to calculating response functions in existing codes. Starting from a chosen energy-model, a straightforward evaluation of the functional derivatives of the energy with respect to the external potential is then possible; all other reactivity indicators can readily be obtained by differentiating these descriptors with respect to the number of electrons.