Local and nonlocal counterparts of global descriptors: the cases of chemical softness and hardness
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Abstract
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.
Keywords
Global response functions Local and nonlocal reactivity indices Thermodynamic softness Chemical hardness Local softness Local hardnessIntroduction
Zero temperature electronic energy and grand potential response functions up to second-order
Descriptors from the electronic energy | Descriptors from the grand potential | |||
---|---|---|---|---|
Symbol | Definition | Symbol | Definition | |
First-order | μ _{ e} Chemical potential [16] Global descriptor | \( {\left(\frac{\partial E}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)} \) | N Electron Number [1] Global descriptor | \( -{\left(\frac{\partial\;\varOmega }{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)} \) |
ρ(r) Electron density [1] Local descriptor | \( {\left(\frac{\delta E}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_N \) | ρ(r) Electron density [1] Local descriptor | \( {\left(\frac{\delta \kern0.1em \varOmega }{\delta \kern0.1em \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{Bath}} \) | |
Second-order | η _{ e} Chemical hardness [17] Global descriptor | \( {\left(\frac{\partial {\mu}_e}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)} \) | S Thermodynamic softness [18] Global descriptor | \( {\left(\frac{\partial N}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)} \) |
f_{e}(r) Fukui Function [19] Local descriptor | \( \Big\{{\displaystyle \begin{array}{l}\kern1em {\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial N}\right)}_{\upsilon \left(\mathbf{r}\right)}\\ {}={\left(\frac{\delta \mu}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_N\end{array}} \) | s(r) Local softness [18] Local descriptor | \( \Big\{{\displaystyle \begin{array}{l}\kern1.5em {\left(\frac{\partial \rho \left(\mathbf{r}\right)}{\partial {\mu}_{Bath}}\right)}_{\upsilon \left(\mathbf{r}\right)}\\ {}=-{\left(\frac{\delta N}{\delta \upsilon \left(\mathbf{r}\right)}\right)}_{\mu_{Bath}}\end{array}} \) | |
χ_{e}(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 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
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.
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
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).
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
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.
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.
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 η_{e}. 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 η_{e}.
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.
For each molecule, an initial geometry optimization for the N_{0}-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.
Condensed to atoms local hardness values for the set of substituted benzenes under consideration (in atomic units)
η_{a, k} (Eq. 41) | η_{b, k} (Eq. 43) | |||||
---|---|---|---|---|---|---|
C2 | C3 | C4 | C2 | C3 | C4 | |
Aniline | 0.026 | 0.012 | 0.038 √ | 0.032 × | 0.026 | 0.026 |
Bromobenzene | 0.018 | 0.011 | 0.044 √ | 0.033 × | 0.030 | 0.029 |
Chlorobenzene | 0.021 | 0.015 | 0.053 √ | 0.034 × | 0.032 | 0.033 |
Phenol | 0.028 | 0.013 | 0.051 √ | 0.036 × | 0.029 | 0.033 |
Fluorobenzene | 0.025 | 0.019 | 0.064 √ | 0.036 | 0.034 | 0.039 √ |
Nitrobenzene | 0.055 | 0.058 √ | 0.004 | 0.029 × | 0.025 | 0.014 |
Benzoic Acid | 0.004 | 0.040 √ | 0.030 | 0.017 | 0.024 | 0.035 × |
Condensed to atoms and local hardness values for the set of substituted ethenes under consideration (in atomic units)
R | η_{a, k} (Eq. 41) | η_{b, k} (Eq. 43) | ||
---|---|---|---|---|
C-1 | C-2 | C-1 | C-2 | |
NH_{2} | 0.094 √ | 0.054 | 0.084 √ | 0.062 |
NO_{2} | −0.016 | −0.004 × | 0.026 √ | 0.015 |
OH | 0.117 √ | 0.074 | 0.108 √ | 0.090 |
COF | 0.094 | 0.103 × | 0.086 √ | 0.070 |
CH_{3} | 0.122 √ | 0.101 | 0.115 √ | 0.102 |
OCH_{3} | 0.100 √ | 0.062 | 0.095 √ | 0.075 |
NHCH_{3} | 0.090√ | 0.051 | 0.086 √ | 0.066 |
CH_{2}NO_{2} | 0.090 √ | 0.073 | 0.036 √ | 0.031 |
CH_{2}NH_{2} | 0.013 | 0.014 × | 0.049 √ | 0.045 |
CH_{2}F | 0.117 √ | 0.104 | 0.112 √ | 0.099 |
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.
Conclusions
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 η_{a, k}, Eqs. 40 or 41, lead to a better description in the case of the substituted benzenes, while η_{b, 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.
Notes
Acknowledgments
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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