Condensed Fukui function
In chemistry, one wishes to identify which atom in a molecule is most likely to react with an attacking nucleophile or electrophile rather than “a point in space”. The form of the Fukui function is not suitable for this purpose, since it only gives us an indication of the reactiveness at a given point in space. This suggests a more convenient way of condensing Fukui functions to an atomic resolution: the condensed Fukui function. The idea is based on the decomposition of a molecular property into atomic contributions, such concept is often called atom in molecules and abbreviated as AIM. The condensed Fukui is obtained from integration of the atomic Fukui function (Bultinck et al. 2007) \[\begin{equation} \require{physics} f^\pm_A % = \int f^\pm_A ( \vb{r}, N ) \dd{ \vb{r} } % \thinspace . \end{equation}\] Contrary to the molecular Fukui function, atomic Fukui functions and condensed values are less well-established. There are several AIM approaches to the Fukui function, resulting in substantially different values of the atomic Fukui functions (Bultinck et al. 2007). However, they all rely on distributing the electron density in every point of space to one or more atoms in the following way: \[\begin{equation} \rho_A( \mathbf{r}, N ) = w_A( \vb{r} ) \rho( \vb{r}, N ) \thinspace , \end{equation}\] where the weight functions for all atoms sum to unity \[\begin{equation} 1 = \sum_A^M w_A( \vb{r} ) % \thinspace . \end{equation}\]
Mulliken’s fragment of molecular response approach (FMR)
In this approach, the atomic Fukui function \(\require{physics} f_A^\pm( \vb{r} )\) for atom \(A\) can be approximated using a finite difference approach (Yang and Mortier 1986; Chattaraj 2009; Bultinck et al. 2007) \[\begin{align} f^\pm_A( \vb{r}, N ) % &= \qty( \pdv{ w_A( \vb{r} ) % \rho( \vb{r}, N ) % }{N} )^\pm_{ v( \vb{r} ) } % \\ &= \qty( \pdv{ \rho_A( \vb{r}, N ) }{N} )^\pm_{ v( \vb{r} ) } \\ &= \rho_A(\vb{r}, N) - \rho_A(\vb{r}, N-1) \thinspace . \end{align}\] Integrating over \(\vb{r}\) allows us to rewrite the condensed Fukui function in terms of the atomic population (Mulliken 1955) \[\begin{align} f^\pm_A(N) &= \int f_A^\pm( \vb{r}, N ) \\ &= q_A(\vb{r}, N) - q_A(\vb{r}, N-1) \thinspace . \end{align}\]