Fukui function
The Fukui function \(\require{physics} f( \vb{r} )\) is a DFT-derived concept and is proposed as a tool for deducing the relative reactivity of different positions in a molecule (Parr and Yang 1984; Yang, Parr, and Pucci 1984; Ayers and Levy 2000). The fundamental equation for changes in energy is given by \[\begin{equation} \require{physics} \dd{E} [ N, v( \vb{r} ) ] % = \mu \dd{N} % + \int \rho( \vb{r} ) % \var{ v( \vb{r} ) } % \dd{ \vb{r} } % \thinspace . \end{equation}\] To get more detailed information about the reactivity of the molecule, the second-order changes of the total energy with respect to the electron number \(N\) and external potential \(v( \vb{r} )\) must be considered: \[\begin{align} \dd{\mu} [ N, v( \vb{r} ) ] % &= \qty( % \pdv{\mu}{N} % )_{ v( \vb{r} ) } \dd{N} % + \int \qty( % \fdv{\mu}{ v( \vb{r} ) } % )_N % \var{ v( \vb{r} ) } % \dd{ \vb{r} } % \\ &= 2 \eta \dd{N} % + \int f( \vb{r} ) % \var{ v( \vb{r} ) } % \dd{ \vb{r} } % \thinspace , \end{align}\] where we have defined \[\begin{equation} \eta % = \frac{1}{2} \qty( \pdv{\mu}{N} )_{ v( \vb{r} ) } % = \qty( \pdv[2]{E}{N} )_{ v( \vb{r} ) } \end{equation}\] as the absolute hardness (Parr and Pearson 1983) and the Fukui function \(f( \vb{r} )\) is defined by extending the Legendre-transform formalism from classical thermodynamics to DFT, resulting in a Maxwell relation (Nalewajski and Parr 1982) associated with functional \(E [ N, v(\vb{r}) ]\) \[\begin{equation} f( \vb{r} ) % = \qty( \fdv{\mu}{ v( \vb{r} ) } )_N % = \qty( \pdv{ \rho( \vb{r} ) }{N} )_{ v( \vb{r} ) } \thinspace . \end{equation}\] This Fukui function is normalized (Parr and Yang 1989) \[\begin{equation} \int f(\vb{r}) \dd{\vb{r}} = 1 \thinspace . \end{equation}\]
It has been demonstrated (Perdew et al. 1982) that the derivative of the energy \(E [ N, v(\vb{r}) ]\) w.r.t. \(N\) has a discontinuity of slope at each integer \(N\) so that we must define three distinct chemical potentials for each integer of \(N\): \[\begin{align} \mu^+ % &= \qty( \pdv{E}{N} )^+_{ v( \vb{r} ) } % \qquad \text{(from negative-ion side)} \\ \mu^- % &= \qty( \pdv{E}{N} )^-_{ v( \vb{r} ) } % \qquad \text{(from positive-ion side)} \\ \mu^0 % &= \qty( \pdv{E}{N} )^0_{ v( \vb{r} ) } % = \frac{1}{2} ( \mu^+ + \mu^- ) % \qquad \text{(neutral)} % \thinspace , \end{align}\] meaning the Fukui function must be defined at three reaction indices as well. The first governing nucleophilic attack of reagent \(R\) on substrate \(S\) \[\begin{align} f^+( \vb{r} ) % &= \qty( \fdv{\mu^+}{ v( \vb{r} ) } )_N % \\ &= \qty( \pdv{ \rho( \vb{r} ) }{N} )^+_{ v( \vb{r} ) } % \qquad \mu_S < \mu_R % \\ &= \lim_{\delta \to 0} % \frac{ \rho_{N + \delta} ( \vb{r} ) % - \rho_{N} ( \vb{r} ) }{\delta} % \\ &= \rho_{N + 1} ( \vb{r} ) - \rho_{N} ( \vb{r} ) % \thinspace , \end{align}\] the second one electrophilic attack \[\begin{align} f^-( \vb{r} ) % &= \qty( \fdv{\mu^-}{ v( \vb{r} ) } )_N % \\ &= \qty( \pdv{ \rho( \vb{r} ) }{N} )^-_{ v( \vb{r} ) } % \qquad \mu_S > \mu_R % \\ &= \lim_{\delta \to 0} % \frac{ \rho_{N} ( \vb{r} ) % - \rho_{N - \delta} ( \vb{r} ) }{\delta} % \\ &= \rho_{N} ( \vb{r} ) - \rho_{N-1} ( \vb{r} ) % \end{align}\] and a radical/neutral attack defined as the average of the nucleophilic and electrophilic attack (Ayers and Levy 2000) \[\begin{align} f^0( \vb{r} ) % &= \qty( \fdv{\mu^0}{ v( \vb{r} ) } )_N % \\ &= \qty( \pdv{ \rho( \vb{r} ) }{N} )^0_{ v( \vb{r} ) } % \qquad \mu_S \sim \mu_R \\ &= \frac{1}{2} \qty( f^+( \vb{r} ) + f^-( \vb{r} ) ) \thinspace . \end{align}\]
Frozen-core approximation
One might suppose that the Fukui function is a DFT-analogue of frontier molecular orbital theory. This is not the case because DFT is in principle exact. The Fukui function includes effects such as orbital relaxation and electron corelation, that are a priori neglected in the FMO approach (Chattaraj 2009). When we express the electron density of a \(N\) electron system susceptible to a nucleophilic attack, we get (Yang, Parr, and Pucci 1984; Chattaraj 2009) \[\begin{equation} \rho_{N + 1} % = \sum_i^N % | \phi_i ( \vb{r} ) |^2 % + \delta | \phi_{N+1} ( \vb{r} ) |^2 % \thinspace , \end{equation}\] so that \[\begin{equation} f^+ ( \vb{r} ) % = \lim_{ \delta \to 0 } % \pdv{ \rho_{N + 1} }{N} % = | \phi_{N+1} ( \vb{r} ) |^2 % + \sum_i^N % \pdv{ | \phi_i ( \vb{r} ) |^2 }{N} % \thinspace . \end{equation}\] Similarly, for a \(N\) electron system susceptible to an electrophilic attack \[\begin{equation} f^- ( \vb{r} ) % = \lim_{ \delta \to 0 } % \pdv{ \rho_{N} }{N} % = | \phi_{N} ( \vb{r} ) |^2 % + \sum_i^N % \pdv{ | \phi_i ( \vb{r} ) |^2 }{N} % \thinspace . \end{equation}\]
In 1952, Fukui assumed that only the electrons in the highest \(\pi\)-orbital in the ground state are essential to the reactivity of the system (Fukui, Yonezawa, and Shingu 1952). He called these electrons the frontier electrons, occupying frontier orbitals. This frozen core approximation gives \[\begin{equation} \dd{ \rho( \vb{r} ) } % = \dd{ \rho( \vb{r} )_{\text{valence}} } \end{equation}\] and therefore simplifies the above Fukui functions to \[\begin{align} f^+( \vb{r} ) % &\approx | \phi_{N+1} |^2 % = \rho( \vb{r} )_{\text{LUMO}} \\ f^-( \vb{r} ) % &\approx | \phi_{N} |^2 % = \rho( \vb{r} )_{\text{HOMO}} \\ f^0( \vb{r} ) % &\approx \frac{1}{2} \qty( \rho( \vb{r} )_{\text{LUMO}} % + \rho( \vb{r} )_{\text{HOMO}} ) % \thinspace . \end{align}\] Since the ground-state electron density \(\rho( \vb{r} )\) is a functional of the highest occupied orbital in Kohn-Sham theory (Yang, Parr, and Pucci 1984), the lack of relaxation of the core orbitals is a well-defined first approximation. The significance of this approximation is that the frontier molecular orbital theory results (Fukui, Yonezawa, and Shingu 1952) are recovered as an approximation within the conceptual DFT framework.
Interpretation of the Fukui function
\(f( \vb{r} )\) is a scalar, space-dependent local function that measures the redistribution of additional electron density in a molecule and the extent of the change in chemical potential caused by an external potential. Using DFT, it allows one to predict where the most nucleophilic and electrophilic regions in a molecule are.