Local and nonlocal counterparts of global descriptors: the cases of chemical softness and hardness

  • Marco Franco-PérezEmail author
  • Carlos A. Polanco-Ramírez
  • José L. GázquezEmail author
  • Paul W. Ayers
Original Paper
Part of the following topical collections:
  1. International Conference on Systems and Processes in Physics, Chemistry and Biology (ICSPPCB-2018) in honor of Professor Pratim K. Chattaraj on his sixtieth birthday


A new strategy, recently reported by us to develop local and linear (nonlocal) counterparts of global response functions, is applied to study the local behavior of the global softness and hardness reactivity descriptors. Within this approach a local counterpart is designed to identify the most important molecular fragments for a given chemical response. The local counterpart of the global softness obtained through our methodology corresponds to the well-known definition of local softness and, in agreement with what standard conceptual chemical reactivity in density functional theory dictates, it simply reveals the softest sites in a molecule. For the case of the local hardness, we obtain two expressions that lead to different information regarding the values of the hardness at the different sites within a chemical species. The performance of these two proposal were tested by comparing their corresponding atom-condensed values to experimentally observed reactivity trends for electrophilic attack on benzene and ethene derivatives.


Global response functions Local and nonlocal reactivity indices Thermodynamic softness Chemical hardness Local softness Local hardness 


Chemical reactivity indices constitute the main tools in the density functional theory of chemical reactivity (CR-DFT) approach to the description, prediction, and quantification of the reactivity features of chemical species [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. A chemical reactivity index is commonly defined as a response coefficient of a zero-temperature state function, like the partial derivative of the electronic energy E (which is equivalent to the zero temperature Helmholtz potential due to the vanishing entropic term at T = 0) or the grand canonical potential (GCP), Ω, with respect to its natural variables, the number of electrons N and the external potential υ(r), in the case of E, or the chemical potential of the reservoir μBath and the external potential, in the case of Ω [1]. Partial derivatives with respect to N or μBath provide global reactivity descriptors that depict the reactivity of a chemical species as a whole, while functional derivatives with respect to υ(r) provide local response functions that are useful to reveal regio-selectivity features of chemical species. Second-order functional derivatives with respect to the external potential provide linear response kernels (nonlocal response functions), which can be used to discern the degree of coupling between two molecular reactive sites [13, 14, 15]. Table 1 displays the definition of the widely known first- and second-order global, local, and linear reactivity response functions coming from both the electronic energy and the GCP.
Table 1

Zero temperature electronic energy and grand potential response functions up to second-order


Descriptors from the electronic energy

Descriptors from the grand potential






μ e

Chemical potential [16]

Global descriptor

\( {\left(\frac{\partial E}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)} \)


Electron Number [1]

Global descriptor

\( -{\left(\frac{\partial\;\varOmega }{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)} \)


Electron density [1]

Local descriptor

\( {\left(\frac{\delta E}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_N \)


Electron density [1]

Local descriptor

\( {\left(\frac{\delta \kern0.1em \varOmega }{\delta \kern0.1em \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{Bath}} \)


η e

Chemical hardness [17]

Global descriptor

\( {\left(\frac{\partial {\mu}_e}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)} \)


Thermodynamic softness [18]

Global descriptor

\( {\left(\frac{\partial N}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)} \)


Fukui Function [19]

Local descriptor

\( \Big\{{\displaystyle \begin{array}{l}\kern1em {\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)}\\ {}={\left(\frac{\delta \mu}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_N\end{array}} \)


Local softness [18]

Local descriptor

\( \Big\{{\displaystyle \begin{array}{l}\kern1.5em {\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}\\ {}=-{\left(\frac{\delta N}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{Bath}}\end{array}} \)

χe(r, r)

Linear density kernel [1]

Linear descriptor

\( {\left(\frac{\delta \rho \left(\mathbf{r}\right)}{\delta \upsilon \left({\mathbf{r}}^{\prime}\right)}\right)}_N \)

s(r, r)

Softness kernel [13]

Linear descriptor

\( -{\left(\frac{\delta \rho \left(\mathbf{r}\right)}{\delta \upsilon \left({\mathbf{r}}^{\prime}\right)}\right)}_{\mu_{Bath}} \)

In addition to the response functions contained in Table 1, a local hardness [20, 21] and a hardness kernel [13] were also proposed. The local hardness is defined as:
$$ {\eta}_e\left(\mathbf{r}\right)={\left(\frac{\delta \kern0.1em {\mu}_e}{\delta \kern0.1em \rho \left(\mathbf{r}\right)}\right)}_{v\left(\mathbf{r}\right)}, $$
while the hardness kernel is given by:
$$ {\eta}_e\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)=-\frac{\delta \kern0.1em u\left(\mathbf{r}\right)}{\delta \kern0.1em \rho \left({\mathbf{r}}^{\prime}\right)}, $$
where u(r) = v(r) − μe is a modified potential.
The softness and hardness quantities are inverses of each other at zero-temperature, that is:
$$ {\eta}_e\;S=1, $$
$$ \int {\eta}_e\left(\mathbf{r}\right)\;s\left(\mathbf{r}\right)\kern0.24em d\kern0.1em \mathbf{r}=1 $$
$$ \int {\eta}_e\left({\mathbf{r}}^{\hbox{'}},{\mathbf{r}}^{"}\right)\;s\left(\mathbf{r},{\mathbf{r}}^{\hbox{'}}\right)\kern0.24em d{\mathbf{r}}^{\hbox{'}}=\delta \left(\mathbf{r}-{\mathbf{r}}^{"}\right). $$
It is interesting to note that the global, local, and nonlocal softness in addition to corresponding to response functions in the sense that they are expressed as derivatives, also fulfill the relationship [18].
$$ S=\int s\left(\mathbf{r}\right)d\mathbf{r}=\iint s\left(\mathbf{r},{\mathbf{r}}^{\hbox{'}}\right)\;d\mathbf{r}\;d{\mathbf{r}}^{"}, $$
where one can see that the integral over the whole space of one of the variables of the softness kernel leads to the local softness, for which the integral over the whole space leads to the global softness.
On the other hand, the hardness kernel and the local hardness do not satisfy a relationship like the one established in Eq. 6, since in the case of these quantities one has that
$$ {\eta}_e\left(\mathbf{r}\right)=\int {\eta}_e\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)\;\alpha \left({\mathbf{r}}^{\prime}\right)\;d{\mathbf{r}}^{\prime } $$
$$ {\eta}_e=\int {\eta}_e\left(\mathbf{r}\right)\;f\left(\mathbf{r}\right)\;d\mathbf{r}, $$
where the function α(r) must be such that its integral over the whole space must be equal to one. The two options that are most frequently considered are [20] α(r) = σ(r) = ρ(r)/N or [21] α(r) = fe(r). In the first case, because it depends on the density as a whole it has been named the total local hardness ηt(r), while in the second case it has been named the frontier local hardness, ηf(r) = ηe [22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

In the last few years there has been interest in establishing a connection between a global reactivity index and the corresponding local descriptor. In this context, it has been suggested [32, 33] that in the cases of the softness and the hardness, the local reactivity indices provide the description of the distribution of the global index in the molecule. Other proposals take a different approach, like the one established by Heidar-Zadeh et al. [34] in which the local response of a global descriptor is defined as the functional derivative of the global property with respect to the external potential at constant number of electrons or chemical potential.

A relevant aspect to consider in this analysis is related to the value of the integral of the local property over all the space. As we have seen, according to the first equality in Eq. 6, this integral for the local softness is equal to the global softness. However, according to Eq. 8, this is not the case for the local hardness and the global hardness. Also, in the proposal made by Heidar-Zadeh et al., the local response does not integrate to the global property, as a matter of fact, in this case the units per volume of the local property are not equal to the units of the global property.

Recently, based on the work of Gál et al. [28] and in some temperature dependent derivations [31], we proposed [30] a procedure based on the chain rule for functional derivatives that led to a local chemical potential that integrates to the global chemical potential, a local hardness that integrates to the global hardness defined in Table 1, and a hardness kernel that when integrated over one of the variables leads to the local hardness. That is, in this case the hardnesses fulfill the same integration rules given in Eq. 6 for the softnesses. This procedure was generalized to develop new local descriptors [35] and to derive [36] the local and linear (nonlocal) counterparts of a global property, which is given by a first or second derivative of a state function with respect to a global variable. Unlike the electron density, the Fukui function or any local reactivity index shown in Table 1, the local counterpart defined through the chain rule for functional derivatives is not strictly conceived as a response coefficient, instead, it is designed to understand how a molecular global response is distributed over the whole molecular space.

In the following sections we will use this approach to define and analyze expressions for the local softness and the local hardness. Because the softness and hardness are second-order derivatives, our approach provides two possible expressions for their local counterparts. We show, however, that there is only one possible local counterpart for the thermodynamic softness, the local softness presented in Table 1. Regarding the chemical hardness, two different expressions for the local hardness are found and analyzed as indicators of chemical reactivity within a molecule in several systems.

Theoretical framework

The local R(i)(r) and linear (nonlocal) R(i)(r, r) counterparts belonging to an ith order global response function R(i) defined as [36]:
$$ {R}^{(i)}={\left(\frac{\partial^{(i)}F}{\partial {X}^{(i)}}\right)}_Y, $$
where F is a zero temperature state function, while X and Y are the corresponding free variables, which must be chosen so as to satisfy the following integration conditions,
$$ {R}^{(i)}=\int {R}^{(i)}\left(\mathbf{r}\right)d\mathbf{r}=\int {R}^{(i)}\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)d\mathbf{r}\;d{\mathbf{r}}^{\prime }. $$
In this context R(i)(r) indicates how the global property R(i) is distributed over the whole molecular space, while R(i)(r, r) specifies how a particular point in the scalar field described by R(i)(r) is correlated to any other point in the same field. As a second requirement, any R(i)(r) must be developed from the R(1)(r) local index (the local counterpart of the first-order global response), which by construction is connected to the electron density, the fundamental variable in density functional theory, through the following chain rule,
$$ {R}^{(1)}\left(\mathbf{r}\right)=\frac{\delta F}{\delta \rho \left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial X}\right)}_Y, $$
$$ {R}^{(1)}={\left(\frac{\partial F}{\partial X}\right)}_Y=\int \frac{\delta F}{\delta \rho \left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial X}\right)}_Yd\mathbf{r}=\int {R}^{(1)}\left(\mathbf{r}\right)d\mathbf{r} $$
This additional requirement introduces the Fukui function as the link between a global response and its corresponding local counterpart. This is an important aspect because the Fukui function is the density-change that minimizes the energy change associated with the addition or the removal of an electron [23, 24]. With this additional constraint there are only two possibilities for the development of the R(i)(r) functions. One of them corresponds to the case when one takes the derivative of the integrand in Eq. 12 with respect to one of the natural variables corresponding to the selected state function F, to get the local counterpart of the R(i) global index,
$$ {R}_a^{(i)}\left(\mathbf{r}\right)={\left(\frac{\partial^{i-1}{R}^{(1)}\left(\mathbf{r}\right)}{\partial {X}^{i-1}}\right)}_Y, $$
where R(1)(r) is given in Eq. 11. The other one corresponds to the case when one applies the chain rule in Eq. 11, using the R(i − 1) global indicator instead of the state function F,
$$ {R}_b^{(i)}\left(\mathbf{r}\right)={\left(\frac{\delta {R}^{\left(i-1\right)}}{\delta \rho \left(\mathbf{r}\right)}\right)}_Y{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial X}\right)}_Y. $$

There are no mathematical or conceptual reasons to prefer local descriptors defined by Eq. 13 more, or less, than those defined by Eq. 14. Both approaches, in fact, may be useful in a particular chemical reactivity study.

The two procedures explained above can be extended to develop the linear (nonlocal) counterpart R(i)(r, r) of the chemical reactivity index R(i). Linear descriptors developed in this way can be used to reveal possible couplings between molecular fragments when a molecule is perturbed as the index R(i) indicates. The equivalent of Eq. 13 is
$$ {R}_a^{(i)}\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)=\frac{\delta {R}^{\left(i-1\right)}\left(\mathbf{r}\right)}{\delta \rho \left({\mathbf{r}}^{\prime}\right)}{\left(\frac{\partial \rho \left({\mathbf{r}}^{\prime}\right)}{\partial X}\right)}_Y. $$
As in the case of Eq. 14, one has two possibilities. The first one leads to
$$ {R}_b^{(i)}\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)={\left(\frac{\delta {R}^{\left(i-1\right)}}{\delta \rho \left({\mathbf{r}}^{\prime}\right)}\right)}_X{\left(\frac{\delta \rho \left({\mathbf{r}}^{\prime}\right)}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_X, $$
where the relationship
$$ {\left(\frac{\partial {R}^{\left(i-1\right)}}{\partial \upsilon \left(\mathbf{r}\right)}\right)}_X={R}_{a,b}^{(i)}\left(\mathbf{r}\right) $$
must be fulfilled (see ref. [36] for a more complete explanation). If this relationship is not fulfilled, then one cannot generate the linear kernel given in Eq. 16. The subindices a and b in Eq. 17 indicate that \( {R}_{a,b}^{(i)}\left(\mathbf{r}\right) \) may refer to any of the local counterparts defined by Eq. 13 or Eq. 14. Note that the two-point dependence is introduced in a different manner if one uses Eq. 15 or Eq. 16. In the former case, the linear (nonlocal) descriptor is given by taking the functional derivative with respect to the electron density, which must be evaluated in a different position in the space, while in Eq. 16 one must consider the functional derivative of the electron density with respect to the external potential. The linear kernel in Eq. 15 is restricted so that it integrates to Eq. 13, while the linear response in Eq. 16 can, in principle, integrate to Eq. 13 or Eq. 14 if they are connected to the partial derivative given in Eq. 17.

From the above definitions, one can confirm that any local and linear counterparts indices obtained from the formalism here presented will satisfy the integration hierarchy given in Eq. 10 for all i > 1. It is pertinent to remark that within this formulation, it is not possible to define a linear response kernel for first-order response functions, and only the first equality in Eq. 10 holds in this case. Note, furthermore, that the whole framework could be extended to three-, four-, and more-point descriptors using the same general strategy.

Local and linear counterpart models for the chemical hardness and softness

Electron number and softness family of descriptors

Setting F ≡ Ω (μBath, υ(r)), the local counterpart of the first-order descriptor, the number of electrons, is obtained by applying Eq. 11, that is:
$$ -{\left(\frac{\partial \varOmega }{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}=N=-\int \frac{\delta\;\varOmega }{\delta \kern0.1em \rho \left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}d\mathbf{r} $$
For any zero-temperature stationary state, the free functional derivative at the right hand side of Eq. 18 vanishes,
$$ \frac{\delta\;\varOmega }{\delta \kern0em \rho \left(\mathbf{r}\right)}=\frac{\delta \left[E-{\mu}_eN\right]}{\delta \rho \left(\mathbf{r}\right)}=\frac{\delta E}{\delta \kern0em \rho \left(\mathbf{r}\right)}-{\mu}_e\frac{\delta N}{\delta \kern0em \rho \left(\mathbf{r}\right)}=0, $$
where we used the normalization condition,
$$ N=\int \rho \left(\mathbf{r}\right)d\mathbf{r}. $$

Equations 18 and 19 indicate that the local counterpart of the number of electrons obtained through this approach is equal to zero. Nevertheless, one may consider the integrand in Eq. 20, the electron density, to be the local counterpart of the electrons number. Note that Eq. 19 is, actually, a direct consequence of Eq. 20 (the variational constraint in density functional theory).

Applying Eq. 13 to the grand potential case, with the consideration that R(1)(r) = ρ(r) and taking into account Eq. 20, we get the original local softness definition [18].
$$ S={\left(\frac{\partial N}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}={\left(\frac{\partial }{\partial {\mu}_{Bath}}\left[\int \rho \left(\mathbf{r}\right)d\mathbf{r}\right]\right)}_{\upsilon \left(\mathbf{r}\right)}=\int {\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}d\mathbf{r}, $$
$$ {S}_a\left(\mathbf{r}\right)={\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}. $$
The local softness Sa(r) thus obtained is connected to the Fukui function by the following chain rule,
$$ {S}_a\left(\mathbf{r}\right)={\left(\frac{\partial N}{\partial {\mu}_{Bath}}\right)}_{\upsilon (r)}{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial N}\right)}_{\upsilon (r)}=S\kern0.1em {f}_e\left(\mathbf{r}\right), $$
where the zero temperature equivalency μe ≡ μBath and the zero temperature reciprocity relation given by 3 have been used [13, 18, 37].
Now, applying the operation indicated in Eq. 14, one finds that the local softness counterpart obtained through this approach is given by:
$$ S={\left(\frac{\partial N}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}=\int {\left(\frac{\delta N}{\delta \rho \left(\mathbf{r}\right)}\right)}_{\upsilon \left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}d\mathbf{r} $$
One can verify that the result in Eq. 24 is equivalent to Eq. 22 since:
$$ {S}_b\left(\mathbf{r}\right)={\left(\frac{\delta N}{\delta \rho \left(\mathbf{r}\right)}\right)}_{\upsilon \left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}={\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}={S}_a\left(\mathbf{r}\right), $$
where we made use of the fact that (δN/δ ρ(r))υ(r) = 1.
Because of Eqs. 18 and 19 one cannot make use of Eq. 15 to obtain the softness kernel. Thus, the only alternative is the one provided by Eqs. 16 and 17. Since the latter corresponds to a Maxwell relationship, using the one that appears in Table 1 for the local softness, one finds that:
$$ \kern0.36em s\left(\mathbf{r}\right)=-\int {\left(\frac{\delta \kern0.1em N}{\delta \kern0.1em \rho \left({\mathbf{r}}^{\prime}\right)}\right)}_{\mu_{Bath}}{\left(\frac{\delta \kern0.1em \rho \left({\mathbf{r}}^{\prime}\right)}{\delta \kern0.1em \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{Bath}}d\kern0em {\mathbf{r}}^{\prime }=-\int {\left(\frac{\delta \kern0.1em \rho \left({\mathbf{r}}^{\prime}\right)}{\delta \kern0.1em \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{Bath}}d\kern0em {\mathbf{r}}^{\prime }, $$
which implies that the softness kernel is given by:
$$ s\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)=-{\left(\frac{\delta \rho \left({\mathbf{r}}^{\prime}\right)}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{bath}}, $$
which corresponds to the original softness kernel definition [13], and fulfills the condition that when integrated over the whole space of one of the variables it leads to the local softness, as Eq. 26 shows.

In conclusion, our procedure to derive the local and linear counterparts of a global index recovers, in the case of the softness, the well-established original proposals [13, 18]. It is important to note that even though there are two possibilities in our methodology, the results for the local softness are the same whether one makes use of Eq. 13 or Eq. 14, while the derivation of the softness kernel is only possible through Eq. 16.

Chemical potential and hardness family of descriptors

If we now set F ≡ E (N, υ(r)), Eq. 13 or Eq. 14 provide the following local counterpart of the first-order global descriptor, the electronic chemical potential, μe,
$$ {\mu}_e={\left(\frac{\partial E}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)}=\int {\left(\frac{\delta E}{\delta \rho \left(\mathbf{r}\right)}\right)}_{\upsilon \left(\mathbf{r}\right)}{\left(\frac{\rho \left(\mathbf{r}\right)}{\delta N}\right)}_{\upsilon \left(\mathbf{r}\right)}d\mathbf{r}=\int {\mu}_e\;{f}_e\left(\mathbf{r}\right)d\mathbf{r}, $$
where we used the chemical potential expression from the Euler-Lagrange equation of density functional theory, [1, 16, 38].
$$ {\mu}_e={\left(\frac{\delta E}{\delta \kern0em \rho \left(\mathbf{r}\right)}\right)}_{\upsilon \left(\mathbf{r}\right)}. $$
Thus, the integrand in Eq. 28 is composed of two quantities, the electronic chemical potential and the Fukui function as the local scalar field. Together they provide the definition of the local chemical potential,
$$ {\mu}_e\left(\mathbf{r}\right)={\mu}_e\kern0.2em {f}_e\left(\mathbf{r}\right). $$
At second-order, there are two possible ways to define the local counterpart of the chemical hardness. If one applies Eq. 13 one obtains:
$$ {\eta}_e^a\left(\mathbf{r}\right)={\left(\frac{\partial {\mu}_e}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)}{f}_e\left(\mathbf{r}\right)+{\mu}_e\kern0.1em {\left(\frac{\partial {f}_e\left(\mathbf{r}\right)}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)}={\eta}_e\kern0.1em {f}_e\left(\mathbf{r}\right)+{\mu}_e\kern0.1em \varDelta \kern0em {f}_e\left(\mathbf{r}\right), $$
where for the second equality we used the expression for the global hardness given in Table 1 and the definition of the dual descriptor, [39, 40, 41].
$$ \varDelta \kern0em {f}_e\left(\mathbf{r}\right)={\left(\frac{\partial {f}_e\left(\mathbf{r}\right)}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)}={\left(\frac{\partial^2\rho \left(\mathbf{r}\right)}{\partial {N}^2}\right)}_{\upsilon \left(\mathbf{r}\right)}. $$

Thus, in this first proposal the local hardness expression, Eq. 31, which we already reported [30], contains the contribution of the local fields generated by the Fukui function and the dual descriptor, weighted by the global hardness and the global chemical potential, respectively.

The second local hardness candidate is obtained through Eq. 14. In this case one has:
$$ {\eta}_e^b\left(\mathbf{r}\right)={\left(\frac{\delta \kern0.1em {\mu}_e}{\delta \kern0.1em \rho \left(\mathbf{r}\right)}\right)}_{\upsilon \left(\mathbf{r}\right)}{f}_e\left(\mathbf{r}\right) $$
It is important to note that the functional derivative on the right-hand side of Eq. 33 corresponds to the original definition of local hardness, [13, 20, 21], which is known to be ambiguous because, due to the relationship between the electron density and the external potential, one cannot vary the density freely at fixed external potential. Therefore, one might prefer \( {\eta}_e^a\left(\mathbf{r}\right) \) over \( {\eta}_e^b\left(\mathbf{r}\right) \) since the former does not exhibit ambiguities. However, taking into account that the only possible variations of the electron density at constant external potential are the variations associated with the number of electrons, one may consider that the functional derivative of Eq. 33 describes a response that is equivalent to the one coming from the global chemical hardness ηe. In this context this functional derivative could be expressed in the form:
$$ {\left(\frac{\delta {\mu}_e}{\delta \rho \left(\mathbf{r}\right)}\right)}_{\upsilon \left(\mathbf{r}\right)}={\eta}_e. $$
This relationship is the frontier local hardness definition proposed by Ayers and Parr [22], and it is a consequence of the hardness equalization [22, 42, 43]. Substituting Eq. 34 in Eq. 33, one obtains that:
$$ {\eta}_e^b\left(\mathbf{r}\right)={\eta}_e\;{f}_e\left(\mathbf{r}\right). $$

Unlike the local hardness definition given in Eq. 34, in Eq. 35 the local behavior comes from the presence of the Fukui function, which distributes the global hardness over the molecular space.

Following a similar procedure, it is also possible to develop a formula for the hardness kernel using Eq. 15, that is [30]:
$$ {\eta}_e^a\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)=\frac{\delta^2E}{\delta \kern0em \rho \left(\mathbf{r}\right)\delta \kern0em \rho \left({\mathbf{r}}^{\prime}\right)}{f}_e\left(\mathbf{r}\right){f}_e\left({\mathbf{r}}^{\prime}\right)+\frac{\delta E}{\delta \kern0em \rho \left(\mathbf{r}\right)}{\left(\frac{\delta {f}_e\left(\mathbf{r}\right)}{\delta \kern0em \rho \left({\mathbf{r}}^{\prime}\right)}\right)}_{\upsilon \left({\mathbf{r}}^{\prime}\right)}{f}_e\left({\mathbf{r}}^{\prime}\right). $$
For the hardness kernel associated with the local hardness defined in Eq. 35, one cannot make use of Eq. 16 because in this case Eq. 17 is not fulfilled. However, an alternative procedure may be established through the use of Eq. 12, with F replaced by μe, that is:
$$ {\left(\frac{\partial {\mu}_e}{\partial N}\right)}_{v\left(\mathbf{r}\right)}=\int {\left(\frac{\delta }{\delta \kern0.1em \rho \left({\mathbf{r}}^{\prime}\right)}{\left(\frac{\delta E}{\delta \kern0.1em \rho \left(\mathbf{r}\right)}\right)}_{v\left(\mathbf{r}\right)}\right)}_{v\left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left({\mathbf{r}}^{\prime}\right)}{\partial \kern0.1em N}\right)}_{v\left(\mathbf{r}\right)}\;d{\mathbf{r}}^{\prime }, $$
because this way one can express the sequence:
$$ {\eta}_e={\left(\frac{\partial {\mu}_e}{\partial N}\right)}_{v\left(\mathbf{r}\right)}=\int {\eta}_e\;{f}_e\left(\mathbf{r}\right)\;d\mathbf{r}=\iint {\left(\frac{\delta }{\delta \kern0.1em \rho \left({\mathbf{r}}^{\prime}\right)}{\left(\frac{\delta E}{\delta \kern0.1em \rho \left(\mathbf{r}\right)}\right)}_{v\left(\mathbf{r}\right)}\right)}_{v\left(\mathbf{r}\right)}{\left(\frac{\partial \rho \left({\mathbf{r}}^{\prime}\right)}{\partial \kern0.1em N}\right)}_{v\left(\mathbf{r}\right)}{f}_e\left(\mathbf{r}\right)\;d\mathbf{r}\;d{\mathbf{r}}^{\prime }, $$
which indicates that the hardness kernel is given by:
$$ {\eta}_e^b\left(\mathbf{r},{\mathbf{r}}^{\prime}\right)=\frac{\delta^2E}{\delta \kern0em \rho \left(\mathbf{r}\right)\delta \kern0em \rho \left({\mathbf{r}}^{\prime}\right)}\;{f}_e\left({\mathbf{r}}^{\prime}\right)\;{f}_e\left(\mathbf{r}\right). $$

One can verify that the integral over one of the variables in Eq. 36 leads to Eq. 31, which when integrated over the whole space lead to the global hardness ηe. On the other hand, the integral over one of the variables in Eq. 39 leads to the local hardness defined in Eq. 35, and the integral of the latter over the whole space leads to the global hardness ηe.

Numerical performance of the local hardness proposals

Through the strategy followed we obtained only one definition for the local softness, Eqs. 22 or 25, which is consistent with the original local softness [13, 18], while for the local hardness we obtained two different expressions given by Eq. 31 and Eq. 35. In this section we test the performance of the two local hardness expressions for the prediction of the reactivity trends associated with the electrophilic attack on benzene and ethene derivatives.

In order to determine the local hardness we first make use of the approximations for the chemical potential and the global hardness that result from making a smooth quadratic interpolation for the energy of the systems with N0 − 1, N0, and N0 + 1 electrons (N0 is an integer), as a function of the number of electrons [17] that leads to μe = (I + A)/2 and ηe = I − A, where I and A are the vertical first ionization potential and electron affinity, respectively, which are, in turn, approximated in the case of I by the negative of the eigenvalue of the highest occupied molecular orbital (HOMO) I ≈ − εH, and in the case of A by the negative of the eigenvalue of the lowest unoccupied molecular orbital (LUMO) A ≈ − εL. For the Fukui function and the dual descriptor one may also use a smooth quadratic interpolation which implies that \( {f}_e\left(\mathbf{r}\right)=\left({f}_e^{-}\left(\mathbf{r}\right)+{f}_e^{+}\left(\mathbf{r}\right)\right)/2 \) and \( \varDelta \kern0em {f}_e\left(\mathbf{r}\right)={f}_e^{+}\left(\mathbf{r}\right)-{f}_e^{-}\left(\mathbf{r}\right) \), where \( {f}_e^{-}\left(\mathbf{r}\right)={\rho}_{N_0}\left(\mathbf{r}\right)-{\rho}_{N_0-1}\left(\mathbf{r}\right) \) and \( {f}_e^{+}\left(\mathbf{r}\right)={\rho}_{N_0+1}\left(\mathbf{r}\right)-{\rho}_{N_0}\left(\mathbf{r}\right) \). These two quantities may also be approximated in terms of the density of the HOMO, \( {f}_e^{-}\left(\mathbf{r}\right)\approx {\rho}_H\left(\mathbf{r}\right) \), and the density of the LUMO, \( {f}_e^{+}\left(\mathbf{r}\right)\approx {\rho}_L\left(\mathbf{r}\right) \). Finally, in order to simplify the study of the results, we used the condensed-to-atom values of the Fukui function for each atom in the molecule [44, 45, 46, 47], based on the Hirshfeld (or stockholder) population analysis [48, 49, 50], applied to the HOMO and LUMO, so that: \( {f}_k^{-}\approx {\rho}_k^H \) and \( {f}_k^{+}\approx {\rho}_k^L \), where \( {\rho}_k^H \) and \( {\rho}_k^L \) are the fraction of the HOMO and LUMO, respectively, associated with the kth atom in the molecule. With these considerations, the local hardness proposal given in Eq. 31 becomes:
$$ {\eta}_{a,k}=I\kern0.1em {f}_k^{-}-A\kern0.1em {f}_k^{+}, $$
which may be approximated by:
$$ {\eta}_{a,k}\approx {\varepsilon}_L{\rho}_k^L-{\varepsilon}_H{\rho}_k^H, $$
while the local hardness expressed in Eq. 35 is equal to:
$$ {\eta}_{b,k}=\left(I-A\right)\kern0.22em \left({f}_k^{+}+{f}_k^{-}\right)/2, $$
which may be approximated by:
$$ {\eta}_{b,k}\approx \left[{\varepsilon}_L-{\varepsilon}_H\right]\left({\rho}_k^H+{\rho}_k^L\right)/2 $$

For each molecule, an initial geometry optimization for the N0-electron system was performed with the Gaussian program [51], using the PBE0 functional [52, 53, 54], and the 6-311G** basis set [55, 56]. Then the condensed Fukui function values were calculated using a developmental version of deMon2k 4.3. The deMon2k calculations used the same optimized geometry, functional, and basis set of the corresponding Gaussian calculations.

In Table 2 we present the results for the electrophilic attack of some substituted benzenes (see Fig. 1 for atom numbering). Experimental evidence indicates that, for the first five benzene derivatives, the electrophilic attack is most likely to occur at the para position (C-4 atom), followed by the ortho one (C-2 atom), while the meta position is the least reactive one, carbon (C-3). The situation is inverted in the last two derivatives, in which the meta position (C-3 atom) is the most susceptible site for an electrophilic attack. As can be observed, the whole trend is perfectly recovered by the condensed to atoms ηa, k proposal, while the ηb, k local counterpart wrongly predicts the C-2 atom to be the most reactive one in practically all cases.
Table 2

Condensed to atoms local hardness values for the set of substituted benzenes under consideration (in atomic units)


ηa, k (Eq. 41)

ηb, k (Eq. 43)










0.038 √

0.032 ×






0.044 √

0.033 ×






0.053 √

0.034 ×






0.051 √

0.036 ×






0.064 √



0.039 √



0.058 √


0.029 ×



Benzoic Acid


0.040 √




0.035 ×

Fig. 1

Atom numbering for substituted benzenes (left) and ethenes (right)

The electrophilic attack of a substituted ethene by a protic acid (H-X) is usually described in terms of Markovnikov’s rule. The carbon atom with more hydrogen bonds in the ethene double bond will bind the acidic proton (H), while the halogen (X) will bind to the most substituted carbon. Following our atom numbering (Fig. 1), the C-1 will attach the acidic proton, and thus, accordingly to the HSAB principle, it must be the hardest atom in the bond. It can be seen in Table 3 that for the set considered, \( {\eta}_b^k \) is in complete agreement with the experimental observation, while \( {\eta}_a^k \) incorrectly predicts that the acidic proton binds to the C-2 atom in the case of the NO2, the COF, and the NH2 derivatives.
Table 3

Condensed to atoms and local hardness values for the set of substituted ethenes under consideration (in atomic units)


ηa, k (Eq. 41)

ηb, k (Eq. 43)






0.094 √


0.084 √




−0.004 ×

0.026 √



0.117 √


0.108 √




0.103 ×

0.086 √



0.122 √


0.115 √



0.100 √


0.095 √





0.086 √



0.090 √


0.036 √




0.014 ×

0.049 √



0.117 √


0.112 √


Thus, one can see that for the cases presented here, the results obtained with Eqs. 40 or 41 are, in general, better than the ones obtained with Eqs. 42 or 43. This situation seems to indicate that it is better to scale the local components \( \kern0.1em {f}_k^{-} \) and \( {f}_k^{+} \) by the global components I and A, respectively, first, and make the subtraction afterwards.


We have shown that the chain rule for functional derivatives and the integration rules expressed in Eq. 10 can be used to define local and nonlocal reactivity descriptors associated with a global property; these descriptors show how the global properties are distributed over the space occupied by the molecule. These descriptors, in turn, can be used to reveal the most important molecular moieties for a specified type of chemical interaction. A relevant feature of the local counterparts developed in this way is their connection with the Fukui function, which is a very important site reactivity indicator.

In the present work, we applied this procedure to analyze specifically the consequences it has for the global softness and the global hardness. In the case of the local softness, the original definition of the local and nonlocal counterparts already satisfied Eq. 10, and we, unsurprisingly, found that our procedure leads to exactly the same local softness descriptor. In fact, both possible derivations for the local softness—one in which we use Eq. 13 to derive Eq. 22, and the other one in which we use Eq. 14 to derive Eq. 25—lead to the same result, which is equal to the original local softness proposed by Yang and Parr, and from this local function the original softness kernel proposed by Berkowitz and Parr is obtained. This means that our procedure, which is based on the fulfillment of Eq. 10, is consistent with previous approaches that satisfy Eq. 10 without imposing it. In the case of the hardness, two different expressions for the local counterpart and, consequently, for the nonlocal counterpart, were found. With respect to the local expressions, the numerical evidence presented seems to indicate that ηa, k, Eqs. 40 or 41, lead to a better description in the case of the substituted benzenes, while ηb, k, Eqs. 42 or 43, provide a better description in the case of the substituted ethenes. Therefore more studies should be carried out to assess the performance of both in a greater variety of chemical systems. It should be noted that neither definition is the inverse of the local softness and that a similar situation applies for the two hardness kernels obtained: neither is the inverse of the softness kernel. This is an interesting result because it indicates that the local softness and the softness kernel provide chemical information that is different from the one obtained through either of these expressions for the local hardness and the hardness kernel, respectively.



PWA thanks NSERC and support from the Canada Research Chairs and Compute Canada. We thank the Laboratorio de Supercómputo y Visualización of Universidad Autónoma Metropolitana-Iztapalapa and the Laboratorio Nacional de Cómputo de Alto Desempeño (LANCAD) for the use of their facilities. CPR was supported in part by Conacyt through a doctoral fellowship. JLG thanks Conacyt for grant 237045. We dedicate this work to Prof. Pratim Chattaraj on the occasion of his 60th anniversary for his great contributions to the density functional theory of chemical reactivity.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Facultad de QuímicaUniversidad Nacional Autónoma de MéxicoCiudad de MéxicoMexico
  2. 2.Departamento de QuímicaUniversidad Autónoma Metropolitana-IztapalapaCiudad de MéxicoMexico
  3. 3.Department of ChemistryMcMaster UniversityHamiltonCanada

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