Perturbed reactivity descriptors in the two parabolas model of fractional electron number

A new procedure based on the two parabolas model of the energy and the electronic density for fractional electron number is used with the assumption that the changes to the isolated values of these two quantities due to the presence of another interacting species can be incorporated through a multiplicative constant in the second order term. The expressions thus obtained for the chemical potential, hardness, Fukui function and dual descriptor reactivity indexes of conceptual density functional theory have the same form of those obtained through a first order perturbation approach within the grand canonical ensemble. The perturbation parameters are then evaluated by imposing the chemical potential and hardness equalization principles for the interaction between species A and B to form AB, and it is applied to show for a group of substituted ethenes that the condensed to atom perturbed local chemical potential and local hardness evaluated at the carbon atom that follows the Markovnikov’s rule lead to better correlation with the activation energy of their reaction with HCl than the unperturbed descriptors. A similar situation is found for the correlation of the condensed to atom local chemical potential evaluated at N in the aniline molecules with the experimental pKa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{p}K}_{\mathrm{a}}$$\end{document} values. The results obtained indicate that through the perturbed descriptors, that introduce information of the electronic structure on each species of the other one with which it interacts allow one to obtain an improved description of their chemical reactivity.


Introduction
The extension of density functional theory to fractional electron number N , considering the three ground states grand canonical ensemble, showed [1][2][3] that at zero temperature the energy is given by a series of straight lines connecting the integer values of N .Thus, because of this behavior, [4] the first derivative of the energy E , with respect to the number of electrons at constant external potential v( ) , the chemical potential , is different at the integer values of N , when evaluated from the left ( − ) or from the right ( +).That is at N = N 0 , where N 0 is an integer, while the second derivative, the hardness is given by where are the vertical first ionization potential and electron affinity, respectively, and N − N 0 is the Dirac delta function.
However, from a conceptual perspective the first and second derivatives of the energy with respect to the number of electrons are fundamental quantities, because through density functional theory, it has been possible to associate them with chemically significative quantities.The first derivative, known as the chemical potential, corresponds to the Lagrange multiplier introduced in the variational procedure to minimize the universal energy functional of Hohenberg-Kohn with respect to the electron density, subject to the constraint that this one must integrate to the number of electrons in the system [5,6].Through this analysis it was possible to link [7] this chemical potential with the intuitive concept of electronegativity ( ) through the generalization proposed by Iczkowski and Margrave [8] to the expression derived by Mulliken [9,10] for this quantity.This relationship implies that the electronegativity equalization principle of Sanderson [11,12] can be seen as a chemical potential equalization.On the other hand, the second derivative was identified [13] with the intuitive concept of hardness introduced by Pearson in the development of the principle of hard and soft acids and bases [14][15][16][17].These associations between the first and second derivatives of the energy with respect to N with the chemical concepts of electronegativity and hardness, together with the first and second derivatives of the electronic density with respect to N , which are closely related to the concepts of frontier orbital theory [18,19], and which were named Fukui function [20][21][22] and dual descriptor [23,24], respectively, led to the development of conceptual density functional theory (CDFT) [6,[25][26][27][28][29][30][31][32][33].Now, in order to avoid the mathematical difficulties associated with the straight lines behavior, Parr and Pearson proposed [13] a smooth quadratic interpolation between the energies of the systems with N 0 − 1 , N 0 , and N 0 + 1 elec- trons, that is with ΔE = E − E N 0 , ΔN = N − N 0 , while the chemical potential and the hardness of the reference system, with N 0 electrons and external potential v 0 ( ) (when it is isolated), are equal to and respectively.
The fact that through this quadratic model the chemical potential becomes equal to the negative of the electronegativity definition suggested by Mulliken [8,9], which is known to closely follow the same tendencies in atoms as those obtained by Pauling [34,35], and the fact that the expression for the hardness reproduces the qualitative scales derived by Pearson for this quantity [14,15], reinforces the (4) identification of these derivatives with the corresponding chemical concepts.Additionally, this parabolic model has been the basis to get a better understanding of several reactivity principles [6,13,[36][37][38][39][40][41][42][43] and charge transfer processes  in terms of the chemical potential (minus electronegativity) and the chemical hardness of the interacting species.However, it should be noted that the calculation of and through Eqs. ( 5) and ( 6) corresponds to the case when the interacting species are isolated from each other, and additionally the calculation of the energy change that arises from charge transference through Eq. ( 4) neglects the effects associated with the change in the external potential of a species, when it is in the presence of another one.Thus, recently [70], it was proposed through a first order perturbation approach that the modifications to the chemical potential of a given species when it is in the presence of another one can be expressed in the form The subindexes Do and Ac distinguish between the chem- ical potential of a species when it acts in the interaction as a donor from the one in which it acts as an acceptor [52].On the other hand, the modifications to the hardness are expressed as [71] The parameters and contain, in principle, the first order perturbation correction caused on a given species.The original derivation of these expressions was done using the three ground states grand canonical ensemble of the systems with N 0 − 1 , N 0 , and N 0 + 1 electrons, to recover through the fractional charge equation evaluated at N = N 0 the expres- sion for the chemical potential given in Eq. ( 5), and introducing then the first order perturbation corrections due to the presence of the other species to the energies E N 0 −1 , E N 0 and E N 0 +1 , to determine the perturbed ionization potential and electron affinity through Eq. ( 3) that must be used to evaluate the perturbed in Eq. ( 5).The parameter is then defined in terms of these quantities so that the perturbed chemical potential adopts the form given in Eq. ( 7), which can be seen as a generalization of the expressions introduced in the two parabolas model of fractional electron number [52], in which the chemical potential for the donating species (base) is −(3I + A)∕4 and for the accepting species (acid) is −(I + 3A)∕4.
An important aspect of this approach is that the expressions given in Eqs.(7) and (8), can be formally used together with Eq. ( 4), even without an explicit evaluation or consideration of the parameters in terms of the perturbed energies, to incorporate into the quadratic charge transfer model, in an approximate manner, the effects of the changes in the external potential that arise in the interaction between two species.This way, the perturbed expressions together with Eq. ( 4) have been successfully used to analyze the maximum hardness and minimum electrophilicity reactivity principles [41] and the |Δ | big is good rule [42].Also, the param- eters have been considered as additional degrees of freedom, whose values can be fixed to impose known conditions associated with the interaction, or simply to improve the description of properties related with the interacting systems [72][73][74][75][76].
Recently, we have shown [77] through a more general procedure than the one adopted in the derivation of the perturbed chemical potential, that one can make use of the grand canonical ensemble formulation to recover the expressions for the perturbed chemical potential and hardness given in Eqs.(7) and (8), and to derive the expressions for the perturbed Fukui function f ( ) and dual descriptor Δf ( ).
However, as mentioned before, since the form adopted for the perturbed chemical potential in Eq. ( 7) in terms of the parameter can be seen as a generalization of the two parabolas model for fractional electron number, the object of the present work is to show that one can directly derive the perturbed expressions for and from this model through a simple assumption about the possible effects of the interaction, and that one can also apply the same procedure to derive the perturbed expressions for f ( ) and Δf ( ) .Addi- tionally, we will present a simple procedure to fix the values of the parameters from known conditions, to show through two specific examples some of the advantages associated with this approach.

Perturbed chemical potential and hardness
The straight lines behavior of the energy as a function of the number of electrons establishes that the response of a system to charge donation is different from the response to charge acceptance, when measured through the derivatives expressed in Eq. ( 1).The former is given by the left derivative while the latter is given by the right derivative.From a chemical viewpoint this seems to be reasonable, because one could expect that the response to each one of these two processes is different.However, at the same time, the straight lines behavior leads to an ill-defined second derivative, which as mentioned it has been identified with the concept of hardness, and it has become, in connection with Eq. ( 4), an important concept that has allowed to explain with a chemically significative language many aspects related with the reactivity of molecules.Additionally, it is important to mention that the straight lines behavior corresponds to the case of zero temperature.It has been shown that for any temperature different from 0 K, treating the chemical species as an open system that may exchange electrons with the bath in which it is immersed, in the grand canonical ensemble, the derivatives of the average energy and the average electronic density of the ensemble as a function of the average number of electrons exist and can be evaluated analytically [78][79][80][81][82].In fact, the smooth quadratic interpolation given in Eq. ( 4) may be thermodynamically justified [83].Thus, at zero temperature, we seem to be at crossroads because, on one hand the straight lines do differentiate the donation from the acceptance processes, but lead to an illdefined hardness, and on the other hand, the smooth quadratic interpolation leads to a well defined hardness, but establishes that the response to donation or acceptance is the same one.
In this context, some time ago, it was proposed [52] that in order to differentiate the response for donation from the response for acceptance and to have a well-defined hardness, one could make the interpolation between the energies of the systems with N 0 − 1 , N 0 , and N 0 + 1 electrons with two parabolas (quadratic functions).However, since in the present work the aim is to describe the situation where a given species is already in presence of the other one, one could consider that the effect of this presence introduces a modification in the second order terms, which is incorporated through a multiplicative parameter.
That is, one may assume that the energy change associated with the donation process can be approximated in the form, and the one associated with the acceptance process may be written as where 0 is defined as and it has been assumed that, in general, the parameter represents the perturbation in the donation and acceptance processes, respectively, on the isolated species, because of the presence of a second one.
By imposing the conditions given in Eq. ( 3), ΔE Do = I for ΔN Do = −1 , and ΔE Ac = −A for ΔN Ac = 1 , and using these two relationships together with Eq. ( 11) one finds that, (9) where one can see that when is equal to one, that corresponds to the case of the isolated species, one recovers the results obtained for the two parabolas original model [52].However, for the case of interacting species one can define so that Eq. ( 12) becomes equal to the perturbed expressions originally derived, which are the ones given in Eq. (7).On the other hand, taking into account that the hardness according to Eqs. ( 9) and ( 10) is given by = 0 , then substituting Eq. ( 12) in Eq. ( 11) one finds that Again, one can see that when = 1 the isolated species hardness of the two parabolas model is recovered, and for the interacting species case one can define so that Eq. ( 14) becomes equal to the perturbed expression originally derived, Eq. ( 8).However, one can see that in this case one can express the parameter in terms of the parameter through the combination of Eqs. ( 13) and (15), that is, Thus, we have seen that indeed, through the two parabolas model, considering that the effect of the perturbation caused on a given species by the presence of a second one can be incorporated through a multiplicative parameter in the second order terms of the quadratic equations related with the charge donation and acceptance processes, one can recover the expressions found through the grand canonical ensemble approach.Additionally, through this procedure one also finds a relationship between the parameter for the chemical potential and the one for the hardness , which may be helpful to reduce the number of parameters.

Perturbed Fukui function and dual descriptor
The extension of density functional theory to fractional electron number based on the three ground states grand canonical ensemble showed also that at zero temperature, the electronic density as a function of N is given by a series of straight lines connecting the integer values of N [1][2][3][4].Thus, the equivalent of Eqs. ( 1) and ( 2) for the electronic density define the Fukui function and the dual descriptor, respectively, and where, if N 0 −1 ( ) , N 0 ( ) and N 0 +1 ( ) represent the elec- tronic densities of the systems with N 0 − 1 , N 0 , and N 0 + 1 electrons obtained at the geometry of the ground state of the N 0 reference system, then are the directional Fukui functions [20,22].As in the case of the energy, one may consider a smooth quadratic interpolation between the electronic densities N 0 −1 ( ) , N 0 ( ) and N 0 +1 ( ) , so that where Δ ( ) = ( ) − N 0 ( ) and However, one may also consider the two parabolas model for fractional electron number, and, as in the case of the energy, one may consider a modification to the second order term through a parameter that takes into account when one species is in the presence of another one, so that the equivalent of Eqs. ( 9) and ( 10) are and where Thus, following the same procedure of the previous section one can find the perturbed expressions for the Fukui function, that is and for the dual descriptor which are the equivalent of Eqs. ( 7) and ( 8).The parameters f and Δf have similar expressions to those given in Eqs. ( 13) and ( 15) with replaced by .
It is important to note that the constant in Eqs. ( 22) and ( 23) allows one, through the present approach, to derive Eqs. ( 25) and ( 26) with f = ( + 2)∕ , and as it has been shown [77] this form contains the local modifications to the electronic density on a given species induced by the presence of another one, through the directional Fukui functions, independently of the global character of the constant .
Thus, it can be seen that the present approach confirms the results obtained for the Fukui function and the dual descriptor through the perturbation procedure in the grand canonical ensemble [77].However, as mentioned in that derivation, if the parameters , , and f , Δf are seen as constants that introduce modifications to the chemical potential, hardness, Fukui function and dual descriptor of a species, because of the presence of a second one, their values could be fixed considering properties that allow one to introduce information on a given species about the electronic structure of the other interacting species.In this context, certainly a plausible approach could be to assume that the parameters and take the same values as the parameters f and Δf , respectively, in order to simplify the process to set their values.In fact, in the two parabolas model presented here, one just needs to assume that the modification to the second order term in the energy in Eqs. ( 9) and ( 10) is equal to the corresponding one in the electronic density in Eqs. ( 22) and (23), or simply that = .

Evaluation of the perturbation parameters
Although in principle one could evaluate the parameters through the first order perturbation expressions derived in previous work [70,77], one can also follow other approaches, like imposing known conditions that are satisfied by the interacting systems, or making use of experimental or theoretical information related with the interaction.Through this procedure, in several works [72][73][74][75][76], it has been shown that the perturbed expressions for the chemical potential and the hardness can be used to incorporate to these fundamental concepts specific information of the interacting systems, that leads to a better description of their chemical reactivity.This last procedure is particularly justified by the present derivation because in order to obtain the form given to the perturbed expressions, as we have seen, one only needs to introduce a multiplicative constant in the second order term of the energy or the electronic density.Since the constant is incorporated on a particular species to model the effect produced by the presence of another species, one can certainly consider to fix the value of this constant through the use of information concerning the electronic structure of the final product.
In this context, if one considers a general interaction of the form then, in principle, one needs to fix the values of A and A for species A, and B and B for species B. Given the present approach of the two parabolas model in which and are related through Eq. ( 16), one needs two more relationships to fix the values of the four parameters.Thus, in the present work the two additional conditions are based in the chemical potential (electronegativity) [7,11,12] and hardness [84][85][86][87][88][89][90][91][92] equalization principles.Through a different approach it had already been established [93][94][95] that these two principles play a very important role in the description of bond and activation energies.
The electronegativity equalization principle was originally proposed by Sanderson [11], who additionally established what he called the geometric mean law for electronegativity equalization.Through the identification of the intuitive concept of electronegativity with the chemical potential of density functional theory [5,7] the equalization of the former follows naturally.Parr and Bartolotti showed [96] that the geometric mean law for chemical potential (electronegativity) equalization can be justified through the observation that, approximately, the energy decays exponentially with the number of electrons.Therefore, in the present work we will assume that the chemical potential of AB may be determined through the chemical potential of A and B, making use of the relationship.
If one considers that A acts as a donor and B as an acceptor, then using Eq. ( 7), one can rewrite the previous equation in the form, so that if AB is determined with Eq. ( 5) through the calculation of I AB and A AB , Eq. ( 29) provides a relationship between A and B .On the other hand, for the hardness equalization it has been shown that a consequence of the approximate exponential decay of the energy as a function of N is that the equilibrium global softness of AB [97], which at zero temperature is the inverse of the global hardness [78,80], can be approximated by the arithmetic average of the softnesses of the isolated species A and B [98], so that the companion of Eq. ( 28) for the hardness equalization is, which combined with Eq. ( 8) leads to where if AB is determined with Eq. ( 6) through the calculated values of I AB and A AB , this expression provides a relationship between A and B .
Since, according to Eq. ( 16), and then Eqs. ( 29), ( 31), ( 32) and ( 33) can be combined to determine the four parameters, A and A for species A, and B and B for species B.
A slightly simpler approach to fix the values of the parameters can be established by assuming that the perturbation parameters for A (donor) and B (acceptor) characterize the specific interaction that leads to AB as a whole in the sense that they are the same for both species, that is A = B , and A = B .Then, with these two conditions plus Eq. ( 32) or (33), because in this case they will be equal, and either Eq. ( 29) or (31), one can fix the values of the four parameters.In order to distinguish it from the previous procedure, which corresponds to a four independent parameters procedure, we will consider this one as a two independent parameters approach.
It is important to note that although the use of Eqs. ( 29) and (31) implies that one needs to perform the calculation of the final product of the interaction between A and B, one may assume that the parameters that characterize the chemical potential and the hardness are equal to those of the Fukui function and the dual descriptor, respectively.This way, one may consider local reactivity indicators [84,86,87,97,[99][100][101][102][103][104][105][106][107][108][109][110][111][112][113][114][115][116][117][118], that usually are composed by products of the global indexes like , , or S (global softness S = 1∕ ) times the local indexes like f ( ) or Δf ( ) .It is through this local descriptors that one can get a better perspective on the site selectivity of a molecule or a family of molecules to obtain a more complete description of their reactivity.

Results and discussion
The object of this section is to present two specific examples that show that the procedure proposed in the previous section may lead to important improvements over the unperturbed reactivity descriptors.The first example refers to the electrophilic addition of hydrogen halides (HX) to several substituted ethenes, see Fig. 1.In this case, according to Markovnikov's rule [119] the hydrogen addition occurs at the carbon atom of the double bond that has more H atoms, which corresponds to C1.
Being an electrophilic addition it implies that the ethene molecule will donate charge around the region of C1, and eventually will form a covalent bond with the hydrogen atom of the interacting species.Thus, since the local chemical potential and the local hardness describe how the global chemical potential and the global hardness are distributed in the molecule, it seems that their values around C1 may contain information about the effects of the substituent R on the charge donation and bond formation in this region, respectively.Thus, one can consider the possible correlation between the activation energy ( E act ) associated to this reaction [104] and these two properties.
The local chemical potential has been defined as [115,116], while the local hardness is given by [102,115,116], As mentioned before, it may be seen that in both cases these local indexes are given by products of the global indexes, chemical potential and hardness, and the local indexes, Fukui function and dual descriptor.Thus, the perturbed expression for these local indices may be determined using the perturbed expressions for the global indexes and and the local indexes f ( ) and Δf ( ) given in Eqs. ( 7), ( 8), ( 25) and ( 26), respectively.Additionally, in order to reduce the number of parameters to be evaluated one may assume that f = and that Δf = .
(34) The calculation of the reactivity indicators requires knowledge of the vertical I and A , and the directional Fukui functions f − ( ) and f + ( ) , which implies that according to Eqs. ( 3) and ( 19) one needs to determine the ground state energy and electronic density of the N 0 electron system, and the energy and electron density of the systems with N 0 − 1 and N 0 + 1 electrons at the geometry of the N 0 elec- tron system.In order to simplify these calculations, within the Kohn-Sham approach [120], one can make use of the frontier orbital approximations which only require computation of the N 0 electron system, since the energy differences in Eq. ( 3) and the density differences in Eq. ( 19) can be expressed in terms of the eigenvalues [6,25,[121][122][123] and the density [6, 20-22, 123, 124] of the highest occupied, H and H ( ) , and lowest unoccupied, L and L ( ) , molecular orbitals, respectively, that is and In the case of these last equations, when one is interested in site selectivity it is more convenient to make use of the condensed to atom Fukui functions [125,126], since these can be determined through a Hirshfeld [127] population analysis of H ( ) and L ( ) , which for the k-th atom in the molecule are denoted as , respectively.Now, since in the case of an electrophilic addition the ethene molecule transfers charge from the region around C1 to the region of the H atom in the hydrogen halide molecule, we need to consider the left derivative for the unperturbed indexes, or the case in which the molecule, ethene (denoted as A) acts as a donor for the perturbed indexes.This implies that for the local chemical potential (Eq. ( 34)) the unperturbed condensed to atom expression becomes equal to [115,116] while the perturbed expression is, where it has been assumed that fA = A .In the case of the local hardness (Eq.( 35)), the unperturbed left derivative is given by [116], and the perturbed expression is, (36) where the assumption ΔfA = A has also been used.
It is important to note that although the perturbed expressions only depend on the properties associated with species A, ethene in the present case, the values of A and A have been obtained through the set of Eqs. ( 29), ( 31), ( 32) and (33), which contain the required electronic properties of species B, the diatomic molecule HCl, so that this way one incorporates in the values of these perturbation parameters for A specific information of the interacting species B, and viceversa.
All the electronic structure calculations required to determine all the quantities involved in the perturbed indexes were done with the PBE0 exchange-correlation energy functional [128][129][130] and the 6-311G** basis set [131,132], in a modified version of the NWChem program [133].To evaluate the condensed Fukui functions f H k and f L k we made use of a developers version of the deMon2k program [134], where each system was determined with the same optimized geometry, functional and basis set employed in the NWChem calculation, together with a GEN-A2* auxiliary basis [135], to perform a Hirshfeld population analysis that led, basically, to the same results obtained in the NWChem code, but that additionally determines the values of f H k and f L k .The results for the unperturbed, and the two and four parameter perturbed local chemical potential are presented in Table 1 and in Fig. 2. (41 Table 1 Condensed to atom ( k = C1 ) local chemical potentials (in hartrees), in decreasing order with respect to those obtained using four parameters (fourth column from left to right).Activation energies in kcal/mol for the reaction given in Fig. 1  For the two parameter model the results correspond to the calculations with the hardness equalization, Eq. ( 31).We found that if one uses Eq. ( 29) instead, one may be led to complex roots in the resulting quadratic equation.Now, since a greater − kA or Do kA indicates that the donation will be facilitated at C1, with respect to a lower value, it can be seen that according to this statement the activation energy decreases as the local chemical potential increases.The three plots show that in the three cases there is a rather good correlation between the chemical potential at C1 and E act .However, the best correlation is found for the two parameter perturbed expression, while the four parameter is slightly better than the unperturbed result, but worse than the two parameter.It is important to note that in the present approach the parameters have not been optimized to get the best possible correlation with the activation energy.In that context, the four parameter would lead to a better description than the two parameter, but in the present work the four parameter results are a consequence of having fixed the values through the satisfaction of Eqs. ( 29), (31), (32) and (33), and the two parameter results introduce a simplifying assumption, which leads to different values of the parameters than the ones obtained with the four parameters approach defined.Thus, for the correlation with the activation energy, it turns out that the four values of the two parameter approach lead to a slightly better correlation with E act than the four values of the four parameter approach.
The results for the unperturbed, two parameter and four parameter perturbed local hardness are presented in Table 2 and in Fig. 2. Since a greater − kA or Do kA indicates that the species will be harder, bond formation will be facilitated at C1, for lower values, and according to this statement the activation energy will increase as the local hardness increases.For this property, one can see that the unperturbed local hardness shows a rather poor correlation with E act , while the two parameter values of Do kA show a much better correlation, and the four parameter values lead to a rather good correlation.The second example refers to the protonation reaction of a group of aniline derivatives shown in Fig. 3, where the protonation occurs at the N atom of the amino group.Thus, this reaction may be seen as an electrophilic addition of the proton to the nitrogen of the amino group.
Being an electrophilic addition it implies that the aniline derivative will donate charge around the region of the N atom of the amino group, and eventually will form a bond with the proton.Therefore, it seems that in this case, the local chemical potential at the N atom of the amino group of the substituted aniline may contain information about the effects of the substituent R on the capacity to bind the proton, indicating that it may correlate with the pK a .Now, since in the case of an electrophilic addition the substituted aniline transfers charge from the region around the N atom of the amino group of the substituted aniline to the region close to the proton, we need to consider the left derivative for the unperturbed indexes, or the case in which the molecule, the aniline derivative (denoted as A) acts as a donor for the perturbed indexes.Therefore, one needs to make use of Eqs.(38) and (39) to determine − kA and Do kA , respectively.However, in the present case, since species B is just a proton, whose presence certainly perturbs the aniline molecule, one could consider a different approach to fix the value of the parameter A .One possibility could be to fix the value of A that makes Do A (Eq. ( 7)) equal to the value of the chemical potential of the protonated species, AB , but because the protonated species has a net positive charge, its chemical potential differs substantially from the chemical potential of the aniline derivative, leading to large values of A that cannot be considered to be of the magnitude of a first order perturbation.Thus, an alternative approach is to determine the value of A that makes A (Eq. ( 8)) equal to the value of the hardness of the protonated species AB , that is, A = AB ∕ I A − A A , and then make use of Eq. ( 32) to determine the value of A to calculate Do A through Eq. ( 7).The results for the unperturbed and the two parameter perturbed local chemical potential are presented in Table 3 and in Fig. 4. The experimental values reported for the pK a were taken from Refs.[136,137].Since a greater − kA or Do kA indicates that the donation will be facilitated at N, with respect to a lower value, it can be seen that according to this statement the pK a values increase when the local chemical potential increases, because the bond between the proton and the aniline molecule is stronger.Additionally, one can also see that − kA , the unperturbed indicator, has a rather poor correlation with the pK a , while Do kA , the perturbed indicator, does show a much better correlation.The two examples presented in this section indicate that the procedure developed to determine the values of the perturbation parameters in the expressions for the perturbed reactivity indicators, leads, in general, to improve the correlation between them and specific properties that characterize a specific interaction.Thus, through this approach one may obtain a more complete description of the chemical reactivity of a family of molecules.

Concluding remarks
The original derivation of the perturbed reactivity descriptors, together with its generalization within the grand canonical ensemble formalism, showed that the parameters and in the expressions for the chemical potential and the hardness, Eqs. ( 7) and ( 8), and the parameters f and Δf in the expressions for the Fukui function and the dual descriptor, Eqs.(25) and (26), can be expressed in terms of first order perturbation theory corrections to the energy and the electronic density, that arise when one chemical species is in the presence of another one.Thus, the parameters in the expressions for the reactivity indexes show how these are modified with respect to their isolated values.
In this context, instead of developing a model for the perturbation, to evaluate the corrections to the energy and to find through this procedure the values of and , these parameters have been used as additional degrees of freedom, whose values have been fixed to impose specific conditions that characterize an interaction, or to improve the correlation with specific properties associated with the interaction [72][73][74][75][76]. Also, the perturbed expressions of the chemical potential and the hardness have been used formally to incorporate in the analysis of important reactivity principles some of the effects on a species that result from an interaction with a second species [41,42].
In this direction the present work provides a strong support to this approach, since we have shown that the effects produced by the presence of the other species can be incorporated through a multiplicative constant in the second order term of the two parabolas model for fractional electron number, that in this case it is not necessarily linked to a perturbative model, justifying this way its presence and its use as an additional degree of freedom.
On the other hand, the fact that the perturbed expressions for the Fukui function and the dual descriptor have been derived through the same assumptions on the electronic density, as those used for the energy, imply that they could be applied in a similar way to study reactivity problems at the local (site) level, and that the assumption about the parameters f and Δf being equal to and seems to be better justified through this approach.
It is also relevant to note that the proposal to fix the values of the perturbation parameters by imposing the chemical potential (electronegativity) and hardness equalization principles, through the use of the chemical potential and the hardness of the final product of the interaction, seems to be an appropriate procedure, in the sense that it leads to parameter values that are close to but different from one, that corresponds to the value of the isolated system, and therefore it seems reasonable from the viewpoint of a perturbation approach.
Finally, one may conclude from the results obtained that the perturbed descriptors, by improving the correlation with specific properties of the interacting systems, may lead to visualize important aspects of their reactivity.

Fig. 1
Fig. 1 Electrophilic addition reaction between hydrogen chloride and substituted ethenes.The product indicated is the one corresponding to Markovnikov's orientation

Fig. 2 Non
Fig.2Plot for the analysis of the correlation between the activation energy for the reaction in Fig.1and (left panel) the condensed to atom local chemical potential, (right panel) the condensed to atom local hardness.The ethene atom considered is C1

Fig. 4
Fig.4 Plot for the analysis of the correlation between the pK a for the reaction in Fig.3and (upper panel) the unperturbed condensed to atom local chemical potential, (lower panel) the two-parameter perturbed condensed to atom local chemical potential.The aniline atom considered is N of the amino group

Table 2
Condensed to atom ( k = C1 ) local hardness (in hartrees), in increasing order with respect to those obtained using four parameters (last column) a Two parameter, A = B and A = B b Four parameter, A ≠ B and A ≠ B