# Condensed Descriptive Tools¶

Local reactivity indicators indicate the susceptibility of a particular point in space to reactions. Chemists usually think, however, in terms of the reactivity of atoms and functional groups. The coarse-graining of pointwise local reactivity indicators into atomic and/or functional group contributions gives condensed reactivity indicators.

In conceptual DFT, the fundamental local reactivity indicators are derivatives of the electron density with respect to either the number of electrons or the chemical potential,

$\lambda \left(\mathbf{r}\right) \equiv \left(\frac{\partial^{k} \rho \left(\mathbf{r}\right)} {\partial N^{k}} \right)_{v\left(\mathbf{r}\right)} \qquad \text{or} \qquad \lambda \left(\mathbf{r}\right) \equiv \left(\frac{\partial^{k} \rho \left(\mathbf{r}\right)} {\partial \mu^{k}} \right)_{v\left(\mathbf{r}\right)}$

Sometimes (e.g., the local electrophilicity ), one will multiply one of the above reactivity indicators by a global reactivity indicator, or consider the sum of two or more local reactivity indicators. To coarse-grain these descriptors, we introduce a method for partitioning the molecule into atoms/functional groups. This partitioning is expressed in terms of atomic weighting function (or, occasionally, atomic weighting operators) in real space,

$\begin{split}w_{A} \left(\mathbf{r}\right) \ge 0 \\ \sum_{A=1}^{{N}_{\text{atoms}}} w_{A} \left(\mathbf{r}\right) = 1\end{split}$

In the fragment-of-molecular-response (FMR) approach, local properties are divided directly,

$\lambda_{A}^{\text{FMR}} = \int w_{A} \left(\mathbf{r}\right) \lambda \left(\mathbf{r}\right) d\mathbf{r}$

In the response-of-molecular-fragment (RMF) approach, the electron density is condensed into atomic populations,

$N_{A} = \int w_{A} \left(\mathbf{r}\right) \rho \left(\mathbf{r}\right) d\mathbf{r}$
and the atomic populations are then differentiated,
$\lambda_{A}^{\text{RMF}} \left(\mathbf{r} \right) \equiv \left(\frac{\partial^{k} N_{A}} {\partial N^{k}} \right)_{v\left(\mathbf{r}\right)} \qquad \text{or} \qquad \lambda_{A}^{\text{RMF}} \left(\mathbf{r} \right) \equiv \left(\frac{\partial^{k} N_{A}} {\partial \mu^{k}} \right)_{v\left(\mathbf{r}\right)}$

The FMR and RMF approaches are just two among many different methods for atom-condensing local reactivity indicators; they give the same results only if the atomic weight functions do not depend on the number of electrons in the molecule, as is true for the (ordinary) Hirshfeld partitioning, the Voronoi/Becke partitioning, and the Mulliken partitioning. For more sophisticated partitioning methods like iterative Hirshfeld charges, the FMR and RMF methods give different results, though we know of no compelling formal or practical reasons to favour one approach over the other. In some contexts, the RMF approach is easier to compute, since it only requires performing population analysis on several different charge states of the molecule being studied. For higher-order reactivity indicators, corresponding to $$k > 2$$ in Eq. (1), the FMR approach seems somewhat simpler.

The fundamental nonlocal reactivity indicators,

$v \left(\mathbf{r},\mathbf{r'}\right) \equiv \left( \frac{ \partial^{k} \chi \left( \mathbf{r},\mathbf{r'} \right)} {\partial N^{k}} \right)_{v\left(\mathbf{r}\right)} \qquad \text{and} \qquad v \left(\mathbf{r},\mathbf{r'} \right)\equiv \left( \frac{ \partial^{k} \chi \left( \mathbf{r},\mathbf{r'} \right)} {\partial \mu^{k}} \right)_{v\left(\mathbf{r}\right)}$

can be condensed in a similar way, producing a matrix that expresses the change in the reactivity of one portion of the molecule in response to a perturbation of a different portion of the molecule. As before, in the fragment-of-molecular-response (FMR) approach one condenses the nonlocal reactivity indicator directly,

$v_{AB}^{\text{FMR}} = \iint w_{A} \left(\mathbf{r}\right) v \left(\mathbf{r},\mathbf{r'}\right) w_{B} \left(\mathbf{r'}\right) d\mathbf{r} d\mathbf{r'}$

while the response-of-molecular-fragment (RMF) approach one condenses the linear response function,

$\chi_{AB} = \iint w_{A} \left(\mathbf{r}\right) \chi \left(\mathbf{r},\mathbf{r'}\right) w_{B} \left(\mathbf{r'}\right) d\mathbf{r} d\mathbf{r'}$

or the softness kernel,

$s_{AB} = \iint w_{A} \left(\mathbf{r}\right) s \left(\mathbf{r},\mathbf{r'}\right) w_{B} \left(\mathbf{r'}\right) d\mathbf{r} d\mathbf{r'}$

and then differentiates these quantities with respect to either the number of electrons or the chemical potential,

$v_{AB}^{\text{FMR}} \equiv \left( \frac{ \partial^{k} \chi_{AB}} {\partial N^{k}} \right)_{v\left(\mathbf{r}\right)} \qquad \text{and} \qquad v_{AB}^{\text{FMR}} \equiv \left( \frac{ \partial^{k} \chi_{AB}} {\partial \mu^{k}} \right)_{v\left(\mathbf{r}\right)}$

Nonlocal reactivity indicators depending on three or more points in space, $$v\left(\mathbf{r},\mathbf{r'},\mathbf{r"},...\right)$$, can be condensed into tensors, $$v_{ABC...}$$, using the same strategy.

Evaluating the FMR and RMF condensed reactivity indicators requires that one select an appropriate model for the dependence of the energy upon the number of electrons. ChemTools can evaluate condensed reactivity indicators for a general energy model using the formulation in ref. , but the most common choices are the linear model and the quadratic model.

In the linear model, the condensed Fukui functions are

$f_{A}^{\text{FMR,+}} \left(\mathbf{r}\right) = \int w_{A} \left(N;\mathbf{r}\right) f^{+} \left(\mathbf{r}\right) d\mathbf{r} = \int w_{A} \left(N;\mathbf{r}\right) \left(\rho \left(N+1;\mathbf{r}\right) - \rho \left(N;\mathbf{r}\right) \right) d\mathbf{r}$
$f_{A}^{\text{FMR,-}} \left(\mathbf{r}\right) = \int w_{A} \left(N;\mathbf{r}\right) f^{-} \left(\mathbf{r}\right) d\mathbf{r} = \int w_{A} \left(N;\mathbf{r}\right) \left(\rho \left(N;\mathbf{r}\right) - \rho \left(N-1;\mathbf{r}\right) \right) d\mathbf{r}$

in the FMR approach and

$\begin{split}f_{A}^{\text{RMF,+}} \left(\mathbf{r}\right) &= N_{A} \left(N+1\right) - N_{A} \left(N\right) = q_{A} \left(N\right) - q_{A} \left(N+1\right) \\ &= \int w_{A} \left(N+1;\mathbf{r}\right) \rho \left(N+1;\mathbf{r}\right) - w_{A} \left(N;\mathbf{r}\right) \rho \left(N;\mathbf{r}\right) d\mathbf{r}\end{split}$
$\begin{split}f_{A}^{\text{RMF,-}} \left(\mathbf{r}\right) &= N_{A} \left(N\right) - N_{A} \left(N-1\right) = q_{A} \left(N-1\right) - q_{A} \left(N\right) \\ &= \int w_{A} \left(N;\mathbf{r}\right) \rho \left(N;\mathbf{r}\right) - w_{A} \left(N-1;\mathbf{r}\right) \rho \left(N-1;\mathbf{r}\right) d\mathbf{r}\end{split}$

in the RMF approach. Here we have used the notation $$\rho\left(N;\mathbf{r}\right)$$ to indicate the $$N-$$ electron ground-state density and $$w_A\left(N;\mathbf{r}\right)$$ to indicate the atomic weighting function for atom $$A$$ in the $$N-$$ electron molecule. Similarly we use $$N_A\left(N\right)$$ and $$q_A\left(N\right)$$ to indicate the population and charge, respectively, of atom $$A$$ in the $$N-$$ electron molecule. In the linear model, the condensed dual descriptor is technically undefined. In the quadratic model, the condensed Fukui function and condensed dual descriptor are defined as,

$\begin{split}f_{A}^{\text{FMR,0}} \left(\mathbf{r}\right) &= \tfrac{1}{2} \int w_{A} \left(N;\mathbf{r}\right) \left( f^{+} \left(\mathbf{r}\right) + f^{-} \left(\mathbf{r}\right) \right) d\mathbf{r} \\ &= \tfrac{1}{2} \int w_{A} \left(N;\mathbf{r}\right) \left( \rho \left(N+1;\mathbf{r}\right) - \rho \left(N-1;\mathbf{r}\right) \right) d\mathbf{r}\end{split}$
$\begin{split}f_{A}^{\text{FMR,(2)}} \left(\mathbf{r}\right) &= \int w_{A} \left(N;\mathbf{r}\right) \left( f^{+} \left(\mathbf{r}\right) + f^{-} \left(\mathbf{r}\right) \right) d\mathbf{r} \\ &= \int w_{A} \left(N;\mathbf{r}\right) \left( \rho \left(N+1;\mathbf{r}\right) - 2 \rho \left(N;\mathbf{r}\right) + \rho \left(N-1;\mathbf{r}\right) \right)d\mathbf{r}\end{split}$

in the FMR approach and

$\begin{split}f_{A}^{\text{RMF,0}} \left(\mathbf{r}\right) &= \tfrac{1}{2} \left( N_{A} \left(N+1\right) - N_{A} \left(N-1\right) \right) \\ &= \tfrac{1}{2} \left( q_{A} \left(N-1\right) - q_{A} \left(N+1\right) \right) \\ &= \tfrac{1}{2} \int w_{A} \left(N+1;\mathbf{r}\right) \rho \left( N+1;\mathbf{r}\right) - w_{A} \left(N-1;\mathbf{r}\right) \rho \left(N-1;\mathbf{r}\right) d\mathbf{r}\end{split}$
$\begin{split}f_{A}^{\text{RMF,(2)}} \left(\mathbf{r}\right) &= \left( N_{A} \left(N+1\right) - 2 N_{A} \left(N\right) + N_{A} \left(N-1\right) \right) \\ &= - \tfrac{1}{2} \left( q_{A} \left(N+1\right) - 2 q_{A} \left(N\right) + q_{A} \left(N-1\right) \right) \\ &= \int w_{A} \left(N+1;\mathbf{r}\right) \rho \left(N+1;\mathbf{r}\right) - 2 w_{A} \left(N;\mathbf{r}\right) \rho \left(N;\mathbf{r}\right) + w_{A}\left(N-1;\mathbf{r}\right) \rho \left(N-1;\mathbf{r}\right) d\mathbf{r}\end{split}$

in the RMF approach.

Condensed reactivity indicators corresponding to derivatives with respect to the chemical potential are computed through the condensed reactivity indicators corresponding to derivatives with respect to electron number. For example, the condensed local softness is defined as

$s_{A} = S f_{A} = \frac{f_{A}}{\eta} = \frac{f_{A}}{I-A}$

and the condensed dual local softness is defined as

$s_{A}^{(2)} = \frac{f_{A}^{(2)}}{\eta^{2}} - \frac{\eta^{(2)}f_{A}^{0}}{\eta^{3}}$

These reactivity indicators can be computed using Fukui functions and/or dual descriptors from either the FMR or RMF approaches.