Introducing a new bond reactivity index: Philicities for natural bond orbitals

Abstract

In the present work, a new methodology defined for obtaining reactivity indices (philicities) is proposed. This is based on reactivity functions such as the Fukui function or the dual descriptor, and makes it possible to project the information from reactivity functions onto molecular orbitals, instead of onto the atoms of the molecule (atomic reactivity indices). The methodology focuses on the molecules’ natural bond orbitals (bond reactivity indices) because these orbitals have the advantage of being localized, allowing the reaction site of an electrophile or nucleophile to be determined within a very precise molecular region. This methodology provides a “philicity” index for every NBO, and a representative set of molecules has been used to test the new definition. A new methodology has also been developed to compare the “finite difference” and the “frontier molecular orbital” approximations. To facilitate their use, the proposed methodology as well as the possibility of calculating the new indices have been implemented in a new version of UCA-FUKUI software. In addition, condensation schemes based on atomic populations of the “atoms in molecules” theory, the Hirshfeld population analysis, the approximation of Mulliken (with a minimal basis set) and electrostatic potential-derived charges have also been implemented, including the calculation of “bond reactivity indices” defined in previous studies.

Introduction

To understand some detailed reaction mechanisms such as regio-selectivity, besides global descriptors [1,2,3,4,5,6,7,8,9,10,11], local reactivity parameters to differentiate the reactive behavior of the atoms forming molecules are necessary. The Fukui function [12, 13] (f(r)) and the softness [14,15,16] (s(r)) are two of the most commonly used reactivity descriptors Eq.1

$$ {\displaystyle \begin{array}{l}f(r)={\left(\frac{\partial \rho (r)}{\partial N}\right)}_{\nu };S={\left(\frac{\partial N}{\partial \mu}\right)}_{\nu}\\ {}s(r)={\left(\frac{\partial \rho (r)}{\partial \mu}\right)}_{\nu }={\left(\frac{\partial \rho (r)}{\partial N}\right)}_{\nu}\cdot {\left(\frac{\partial N}{\partial \mu}\right)}_{\nu }=f(r)\cdot S\end{array}} $$

(1)

The Fukui function is primarily associated with the response of the density function of a system when changing the number of electrons (N) under the constraint of a constant external potential [v(r)]. The Fukui function can also represent the response of the electronic chemical potential [17] of a system when changing its external potential. Since the chemical potential is a measure of the intrinsic acidic or base strength, and softness incorporates global reactivity, they provide two indices that make it possible to determine, for example, the specific sites of interaction between two reagents.

Due to the discontinuity of the electron density with respect to N, finite difference approximation (FD) leads to three types of Fukui function for a system, namely: f ⁺(r) Eq. (2), f ⁻(r) Eq. (3), and f ⁰(r) Eq. (4) for nucleophilic, electrophilic and radical attack, respectively. f ⁺(r) is measured by the electron density change following the addition of an electron, and f ⁻(r) by the electron density change upon removal of an electron. f ⁰(r) is approximated to the average of the both previous terms, which are defined as follows:

$$ {f}^{+}(r)={\rho}_{N_0+1}(r)-{\rho}_{N_0}(r),\mathrm{for}\ \mathrm{nucleophilic}\ \mathrm{attack}, $$

(2)

$$ {f}^{-}(r)={\rho}_{N_0}(r)-{\rho}_{N_0-1}(r),\mathrm{for}\ \mathrm{electrophilic}\ \mathrm{attack}, $$

(3)

$$ {f}^0(r)=\frac{1}{2}\left({\rho}_{N_0+1}(r)-{\rho}_{N_0-1}(r)\right),\mathrm{for}\ \mathrm{neutral}\ \left(\mathrm{or}\ \mathrm{radical}\right)\ \mathrm{attack}. $$

(4)

The condensation of reactivity descriptors to atoms can be carried out in several ways, such as: the famous Yang–Mortier scheme [18] or other “population strategies”, as can be seen in references [19, 20].

Frontier molecular orbital approximation

Considering the frozen orbital approximation (FOA) of Fukui [21,22,23,24,25], and neglecting the orbital relaxation effects, the Fukui function can be approximated as follows:

$$ {f}^{\alpha }(r)\approx {\left|{\phi}^{\alpha }(r)\right|}^2 $$

(5)

where φ ^α(r) is a particular frontier molecular orbital (FMO) chosen depending upon the value of α = + or α = −. Eq. (5) can be used to develop an approximated definition of the Fukui function:

$$ {f}^{-}(r)={\rho}_{HOMO}(r),\mathrm{for}\ \mathrm{electrophilic}\ \mathrm{attack}, $$

(6)

$$ {f}^{+}(r)={\rho}_{LUMO}(r),\mathrm{for}\ \mathrm{nucleophilic}\ \mathrm{attack}, $$

(7)

$$ {f}^0(r)=\frac{1}{2}\left({\rho}_{LUMO}(r)+{\rho}_{HOMO}(r)\right),\mathrm{for}\ \mathrm{neutral}\ \left(\mathrm{or}\ \mathrm{radical}\right)\ \mathrm{attack}. $$

(8)

Expanding the FMO in terms of the atomic basis functions, the condensed Fukui function at the atom k is:

$$ {f}_k^{\alpha }=\sum \limits_{\nu \in k}\left[{\left|{C}_{\nu\;\alpha}\right|}^2+\sum \limits_{\chi \notin \nu }{C_{\chi\;\alpha}}^{\ast }{C}_{\nu\;\alpha}\kern0.36em {S}_{\chi \nu}\right] $$

(9)

$$ {f}_k^{-}=\sum \limits_{\nu \in k}\left[{\left|{C}_{\nu\;H}\right|}^2+\sum \limits_{\chi \notin \nu }{C_{\chi H}}^{\ast }{C}_{\nu\;H}\kern0.36em {S}_{\chi \nu}\right]\kern0.37em \left(\mathrm{electrophilic}\ \mathrm{attack}\right) $$

(10)

$$ {f}_k^{+}=\sum \limits_{\nu \in k}\left[{\left|{C}_{\nu\;L}\right|}^2+\sum \limits_{\chi \notin \nu }{C_{\chi\;L}}^{\ast }{C}_{\nu\;L}\kern0.36em {S}_{\chi \nu}\right]\;\left(\mathrm{nucleophilic}\ \mathrm{attack}\right) $$

(11)

$$ {f}_k^0=\frac{1}{2}\left({f}_k^{+}+{f}_k^{-}\right)\;\left(\mathrm{radical}\ \mathrm{attack}\right) $$

(12)

where C _να are the molecular frontier orbital coefficients, and S _χν are the elements of the atomic orbital overlap matrix. The subscripts “H” and “L” in Eqs. (10, 11) are referenced to the HOMO and LUMO orbitals, respectively. This definition of the condensed Fukui function was pioneered by Pérez and Chamorro [26, 27] and has been used in a variety of studies yielding reliable results [28,29,30].

Chemical reactions are mainly adjustment of valence electrons among the reactant orbitals. Fukui proposed his frontier orbital theory [31], which allows a chemical reaction to be understood in terms of HOMO and LUMO only. Fukui functions capture this concept of frontier orbital theory. The largest value of f(r) at the reaction site will be preferable [32] since that will imply a large electronic-potential change. Since the hard species [33] are generally of small size and high charge and the soft species are large in size with a low charge, it is expected that in the hard–hard reactions ionic bonding would predominate [34, 35] and in the soft–soft reactions covalent bonding would predominate. For the soft species, the nuclear charge is adequately screened by the core electrons and the two soft species will mainly interact via frontier orbitals but the core orbitals are not just “spectators” for the hard-hard reactions, implying that soft-soft interactions are frontier-controlled (follow “through bond” interactions) while hard–hard interactions are charge-controlled (follow “through space” interactions) [34]. One should not expect FOT to work in the case of hard–hard interactions. While soft–soft interactions are controlled by f(r), for hard–hard interactions the charges on each atom will decide the actual reaction site [36, 37]. It has also been shown [34] that for the interaction between a hard and a soft species the reactivity is generally very low and it cannot be identified as a charge/frontier-controlled reaction, vindicating the HSAB principle [36, 38,39,40,41,42,43].

Reactivity indices of natural bond orbitals

In a previous study [44], we proposed the ${FF}_i^{NBO}$ reactivity index (Eq. 13) for NBO_i (ST1 in the Supplementary Material shows a summary of the nomenclature and definitions),

$$ {FF}_i^{NBO}=\sum \limits_k{\left|{C}_{i\;\alpha}\right|}^2{f}_{k\;i}^{(NBO)} $$

(13)

which is based on the approximation:

$$ {f}_k^{\alpha }=\sum \limits_{\nu \in k}\left[{\left|{A}_{\nu\;\alpha}\right|}^2+\sum \limits_{\chi \notin \mu }{A_{\chi\;\alpha}}^{\ast }{A}_{\nu\;\alpha}\kern0.36em {S}_{\chi \nu}\right]\approx \sum \limits_i{\left|{C}_{i\;\alpha}\right|}^2\;{f}_{k\;i}^{(NBO)};\left(\mathrm{a}=+\mathrm{or}-\right) $$

(14)

where

$$ {f}_{k\;i}^{(NBO)}=\sum \limits_{\nu \in k}\left[{\left|{B}_{\nu\;i}^{(NBO)}\right|}^2+\sum \limits_{\chi \notin \nu }{B_{\chi\;i}^{\left(\; NBO\right)}}^{\ast }{B}_{\nu\;i}^{(NBO)}\kern0.36em {S}_{\chi \nu}\right] $$

(15)

A _iα and B _ij represent the molecular orbital coefficients and χ _i the basis functions employed to develop ${\varphi}^{HOMO}=\sum \limits_i{A}_i{\chi}_i$ and ${\varphi}_j^{NBO}=\sum \limits_i{B}_{ij}\;{\chi}_i$. The subscripts “i” and “k” in Eqs. (13–15) are referenced to orbitals and atoms, respectively. The C _i coefficients of Eq. (16) were obtained by the least-squares method by applying Eq. (17), which leads to the linear system of Eq. (18).

$$ {\varphi}^{(HOMO)}\approx \sum \limits_i{C}_i{\varphi}_i^{(NBO)} $$

(16)

$$ \int {\left[{\varphi}^{(HOMO)}-\sum \limits_i{C}_i{\varphi}_i^{(NBO)}\right]}^2 d\tau = MINIMUM $$

(17)

$$ \sum \limits_i\sum \limits_l{A}_i{B}_{l\;j}{S}_{i\;l}\approx \sum \limits_k\sum \limits_i\sum \limits_l{C}_k{B}_{ik}{B}_{l\;j}{S}_{i\;l} $$

(18)

Bond reactivity indices

NBO-based Fukui functions and related quantities have been previously studied in other works performed by different authors [45, 46], and the bond reactivity indices: ${f}_i^{+(NBO)}$ and ${f}_i^{-(NBO)}$ were defined in a previous own study [47] (table ST1 in the Supplementary Material shows a summary of the nomenclature and definitions). The main difference in the work performed in reference [45] regarding the proposed one in this paper is that it is the first one based on the partial occupations of the NBOs of the neutral molecule, the cation, and the anion (without considering possible changes in the NBOS set), while in the proposed work, the Fukui’s functions are projected on the NBOs of the neutral molecule, and it is based on the approximation that states that NBOs do not change when the molecule loses or gains an electron, only their effective occupancies. $\left\{{f}_i^{-(NBO)}\right\}$ and $\left\{{f}_i^{+(NBO)}\right\}$ sets, were estimated by using least-squares method, fitting Eq. (19).

$$ {\displaystyle \begin{array}{l}\int {\left({f}^{-}\left(\overrightarrow{r}\right)-\sum \limits_{i=1}^{all\kern0.24em orbitals}{f}_i^{-(NBO)}\;{\left|{\varphi}_i^{(NBO)}\left(\overrightarrow{r}\right)\right|}^2\right)}^2d\overrightarrow{r}= MINIMUM\\ {}\int {\left({f}^{+}\left(\overrightarrow{r}\right)-\sum \limits_{i=1}^{all\kern0.24em orbitals}{f}_i^{+(NBO)}\;{\left|{\varphi}_i^{(NBO)}\left(\overrightarrow{r}\right)\right|}^2\right)}^2d\overrightarrow{r}= MINIMUM\end{array}} $$

(19)

The Lagrange multipliers [48] method was used to obtain the reactivity indices of Eq. (19), and finally, the following relations were obtained:

$$ \int {\left({f}^{-}\left(\overrightarrow{r}\right)-\sum \limits_{i=1}^{all\kern0.24em orbitals}{f}_i^{-(NBO)}\;{\left|{\varphi}_i^{(NBO)}\left(\overrightarrow{r}\right)\right|}^2\right)}^2d\overrightarrow{r}+\lambda \left[\sum \limits_{i=1}^{all\kern0.24em orbitals}{f}_i^{-(NBO)}-1\right]= MINIMUM $$

(20)

$$ \int {\left({f}^{+}\left(\overrightarrow{r}\right)-\sum \limits_{i=1}^{all\kern0.24em orbitals}{f}_i^{+(NBO)}\;{\left|{\varphi}_i^{(NBO)}\left(\overrightarrow{r}\right)\right|}^2\right)}^2d\overrightarrow{r}+\lambda \left[\sum \limits_{i=1}^{all\kern0.24em orbitals}{f}_i^{+(NBO)}-1\right]= MINIMUM $$

(21)

$$ \int {\left(\Delta f\left(\overrightarrow{r}\right)-\sum \limits_{i=1}^{all\kern0.24em orbitals}\Delta {f}_i^{(NBO)}\;{\left|{\varphi}_i^{(NBO)}\left(\overrightarrow{r}\right)\right|}^2\right)}^2d\overrightarrow{r}+\lambda \left[\sum \limits_{i=1}^{all\kern0.24em orbitals}\Delta {f}_i^{(NBO)}\right]= MINIMUM $$

(22)

Computational details

All the structures included in this study were optimized at B3LYP/6-31G(d) [49, 50] theory level by using the Gaussian 09 package [51]. For the FMO approximation, the electrophilic Fukui function was evaluated from a single point calculation in terms of molecular orbital coefficients and overlap integrals. For the FD approximation, the densities were calculated for the neutral molecule, the cation, and anion from a single-point calculation (the wave functions were obtained through Gaussian 09 software).

The calculation of the new indices and the new methodology defined in this study (see below) have been implemented in the new version of the UCA-FUKUI software (http://www2.uca.es/dept/quimica_fisica/software/UCA-FUKUI_v2.exe). A detailed description of all the improvements implemented in the new version of the program has been included in Appendix I of the Supplementary Material as well as several examples showing how to use them.

Results and discussion

In a previous paper [47] formula (23) was obtained, which allows us to relate the atomic reactivity indices, ${f}_k^{-}$, and those of the bond reactivity, ${f}_i^{-(NBO)}$, where the “k” subscript refers to atoms and the “i” subscript to orbitals.

$$ {f}_k^{-}\approx \sum \limits_i{f}_i^{-(NBO)}\cdot {f}_{k\;i}^{\left( NBO\;\right)} $$

(23)

Furthermore, considering the relationship between the atomic philicities (${w}_k^{-}$) and the global one (w) [52], by means of the condensed indices of Fukui ${w}_k^{-}=w\cdot {f}_k^{-}$ [30] and relating to the previous expression (23):

$$ {w}_k^{-}\approx w\cdot \sum \limits_i\left({f}_i^{-(NBO)}\cdot {f}_{k\;i}^{\left( NBO\;\right)}\right) $$

(24)

Since $w=\sum \limits_k{w}_k^{-}$, then:

$$ w\approx \sum \limits_k\sum \limits_i\left(w\cdot {f}_i^{-(NBO)}\cdot {f}_{k\;i}^{\left( NBO\;\right)}\right) $$

(25)

or in other words:

$$ w\approx \sum \limits_i\left(w\cdot {f}_i^{-(NBO)}\cdot \sum \limits_k{f}_{ki}^{(NBO)}\right) $$

(26)

and since $\sum \limits_k{f}_{ki}^{(NBO)}=1;\forall i$, the following expression can be achieved:

$$ w\approx \sum \limits_i\left(w\cdot {f}_i^{-(NBO)}\right)=\sum \limits_i{w}_i^{-(NBO)} $$

(27)

where ${w}_i^{-(NBO)}$ is defined as:

$$ {w}_i^{-(NBO)}=w\cdot {f}_i^{-(NBO)} $$

(28)

Likewise, we can define (29) for the ${f}_i^{+(NBO)}$ index

$$ {w}_i^{+(NBO)}=w\cdot {f}_i^{+(NBO)} $$

(29)

In Eqs. (28, 29), we have obtained definitions of philicity in terms of orbitals instead of atoms. A multiphilic descriptor for chemical reactivity [53,54,55,56,57] can also be defined based on Eqs. (28, 29), as indicated below:

$$ \Delta {w}_i^{(NBO)}={w}_i^{+(NBO)}-{w}_i^{-(NBO)}=w\cdot \left({f}_i^{+(NBO)}-{f}_i^{-(NBO)}\right) $$

(30)

Testing the new philicities in a sample set of representative molecules

The new definitions (28 and 29) have been implemented in software UCA-FUKUI v.2 [58] and the new indices were calculated for a representative molecules set. The molecules used in the test were: CH₂CHCl, CH₂CHCHO, CH₂CHNO₂, CH₂CHCN, CH₂CHCH₃, and CH₂CHOCH₃ in order to get a representative sample with diversity of functional groups (double bonds, halogen (Cl), nitro, ether and aldehyde). Also, the very simple molecules specially chosen for the example, allow achieving results easy to interpret.

The ${w}_i^{-(NBO)}$ and ${w}_i^{+(NBO)}$ indices are the maximum stabilizing energies (with opposite signs) when an NBO yields or takes a fraction of charge respectively. Hence, the ${w}_i^{-(NBO)}$ index can be used as an estimate of the nucleophile character of the orbital, and the ${w}_i^{+(NBO)}$ index, of the electrophilic character. On the other hand, the $\Delta {w}_i^{(NBO)}$ index (based on dual descriptor) takes positive and negative values. Positive corresponds to electrophilic character, and negative to nucleophilic one.

To obtain the indices presented in Tables 1, 2, 3, 4, 5 and 6, all NBOs have been used (as can be seen in Figs. 1 and 2). Tables 1, 2, 3, 4, 5 and 6 are an extract of the calculated indices. They include those with important values and some ones that are interesting for comparative purposes (for example, all of the Cl atom LPs of the CH₂CHCl reagent).

Table 1 ${w}_i^{-(NBO)}$, ${w}_i^{+(NBO)}$and $\Delta {w}_i^{(NBO)}$ (Eqs. 28, 29, and 30) under FD approximation (electrophile and nucleophile attack) and FMO approximation, (HOMO and LUMO) for CH₂CHCl reagent

Table 2 ${w}_i^{-(NBO)}$ ${w}_i^{+(NBO)}$ and $\Delta {w}_i^{(NBO)}$ (Eqs. 28, 29, and 30) under FD approximation (electrophile and nucleophile attack) and FMO approximation (HOMO and LUMO) for CH₂CHCHO reagent

Table 3 ${w}_i^{-(NBO)}$, ${w}_i^{+(NBO)}$ and $\Delta {w}_i^{(NBO)}$ (Eqs. 28, 29, and 30) under FD approximation (electrophile and nucleophile attack) and FMO approximation (HOMO and LUMO) for CH₂CHNO₂ reagent

Table 4 ${w}_i^{-(NBO)}$, ${w}_i^{+(NBO)}$ and $\Delta {w}_i^{(NBO)}$ (Eqs. 28, 29, and 30) under FD approximation (electrophile and nucleophile attack) and FMO approximation (HOMO and LUMO) for CH₂CHCN reagent

Table 5 ${w}_i^{-(NBO)}$, ${w}_i^{+(NBO)}$ and $\Delta {w}_i^{(NBO)}$ (Eqs. 28, 29, and 30) under FD approximation (electrophile and nucleophile attack) and FMO approximation (HOMO and LUMO) for CH₂CHCH₃ reagent

Table 6 ${w}_i^{+(NBO)}$, ${w}_i^{+(NBO)}$ and $\Delta {w}_i^{(NBO)}$ (Eqs. 28, 29, and 30) under FD approximation (electrophile and nucleophile attack) and FMO approximation (HOMO and LUMO) for CH₂CHOCH₃ reagent

Fig. 1

Graphical representation of a ${w}_i^{-(NBO)}$ and b ${w}_i^{+(NBO)}$ parameters (28 and 29) calculated under frontier molecular orbital approximation for CH₂CHCl reagent. The NBOs are represented on the x-axis (sorted by energy); the red color means that the partial occupancy is greater than 1.92 and the blue color that the partial occupancy is less than 0.08

Fig. 2

Graphical representation of bond reactivity indices $\Delta {w}_i^{(NBO)}$ based on the dual descriptor function for CH₂CHCl reagent: a Frontier molecular orbital approximation and b finite difference approximation. The NBOs are represented on the x-axis (sorted by energy); the red color means that the partial occupancy is greater than 1.92 and the blue color that the partial occupancy is less than 0.08

As can be seen in Figs. 1 and 2 and Tables 1, 2, 3, 4, 5 and 6, the most nucleophilic NBOs are those corresponding to double bonds and the non-shared pairs of the highest energy but with a high partial occupation of ≈ 2.0 (SF1 in the Supplementary Material shows some NBOs of CH₂CHCl reagent and their schematic representations). The most electrophilic NBOs correspond to double bonds (anti-bonding) with the lowest energy but with a low partial occupation (≈ 0.0). These conclusions coincide with those expected and, in addition, the two approaches used (FD and FMO), provide numerically different, but qualitatively equivalent results, leading to the same conclusions, with the exception of Table 3 (see the $\Delta {w}_i^{(NBO)}$ parameter) where very different results are shown. This is due to the orbital relaxation effects [58], which is treated in the following section. On the other hand, the conclusions obtained when analyzing the results of the $\Delta {w}_i^{(NBO)}$ index agree with those of the other indices. Examples of how to calculate this type of indices have been included in the Supplementary Material (EXAMPLE IV).

The proposed methodology (obtaining the indices based in the variation of the orbitals energy) can be employed as a tool for analyzing the atomic electrophilicities, that is, this approach allows determining the energetic contribution of each orbital of a concrete atom.

From the combination of Eqs. (24) and (28), the next expression is achieved:

$$ {w}_k^{-}\approx \sum \limits_i\left({w}_i^{-(NBO)}\cdot {f}_{k\;i}^{\left( NBO\;\right)}\right) $$

(31)

where the term ${f}_{k\;i}^{\left( NBO\;\right)}$ is the philicity fraction of the bonding orbital $\left({w}_i^{-(NBO)}\right)$ which make its contribution to the philicity of atom “k” (${w}_k^{-}$), keeping in mind that: $\sum \limits_k{f}_{ki}^{(NBO)}=1;\forall i$ and $0\le {f}_{ki}^{(NBO)}\le 1;\forall i,\forall k$. Tables 7 and 8 show the ${f}_{ki}^{(NBO)}$ values for the reagents: CH₂CHCl and CH₂CHCHO (Tables S2-S5 in the Supplementary Material show the ${f}_{ki}^{(NBO)}$ values for the reagents: CH₂CHNO₂, CH₂CHCN, CH₂CHCH₃ and CH₂CHOCH₃). Tables 9 and 10 show the atomic philicities calculated per the relation: ${w}_k^{\alpha }=w\cdot {f}_k^{\alpha };\left(\alpha =- or+\right)$ for the example molecules disclosed therein. The Fukui (${f}_k^{-}$ and ${f}_k^{+}$) indices were obtained from the Hirshfeld population analysis. These atomic philicities are consistent with those obtained by using the new methodology proposed for the NBOs (shown in Tables 1, 2, 3, 4, 5 and 6). As an example, in the case of the reagent: CH₂CHCl, the highest value for ${w}_k^{-}$ is the one corresponding to the Cl6 atom, which can be justified by the contributions of the NBOs: 14 (LP Cl6; ${f}_{Cl\;6\kern0.48em NBO14}^{\left( NBO\;\right)}$= 0.9554, value obtained from Table 7), 15 (LP Cl6; ${f}_{Cl\kern0.24em 6\kern0.48em NBO15}^{\left( NBO\;\right)}$= 0.9737) and 16 (LP Cl6; ${f}_{Cl\kern0.24em 6\kern0.48em NBO16}^{\left( NBO\;\right)}$= 0.9855), since they have important ${w}_i^{-(NBO)}$ values. On the other hand, the NBO 2 (BD C1- C2) possesses the highest ${w}_i^{-(NBO)}$ value; however, it is distributed between C1 (${f}_{C1\kern0.48em NBO2}^{\left( NBO\;\right)}$= 0.4791) and C2 (${f}_{C2\kern0.48em NBO2}^{\left( NBO\;\right)}$= 0.5165) obtaining significant ${w}_k^{-}$ values, but not as high as for the Cl6 atom. The higher ${w}_k^{+}$ values correspond to the C1 and C2 atoms, which can be justified by the contribution of the NBO 51 (BD C1-C2; ${f}_{C1\kern0.36em NBO51}^{\left( NBO\;\right)}$= 0.5014; ${f}_{C2\kern0.48em NBO51}^{\left( NBO\;\right)}$= 0.4952) that has the highest ${w}_i^{-(NBO)}$ value. For the case of the reagent: CH₂CHCHO, the most remarkable ${w}_k^{-}$ value corresponds to the O7 atom, due to the contribution of the NBO 14 (LP O7; ${f}_{O7\kern0.36em NBO14}^{\left( NBO\;\right)}$= 0.9078) that has the most important ${w}_i^{-(NBO)}$ value. On the other hand, the highest values of the atoms: C1, C2, and O7 can be justified by the ${w}_i^{+(NBO)}$ contribution of the NBOs: 61 (BD C1-C2; ${f}_{C1\kern0.48em NBO61}^{\left( NBO\;\right)}$= 0.4958; ${f}_{C2\kern0.48em NBO61}^{\left( NBO\;\right)}$= 0.4977) and 67 (BD C6-O7; ${f}_{C6\kern0.48em NBO67}^{\left( NBO\;\right)}$= 0.6116; ${f}_{O7\kern0.48em NBO67}^{\left( NBO\;\right)}$= 0.3786). Similar conclusions can be reached for the rest of the reagents. Finally, Tables S6-S13 of the Supplementary Material show the atomic philicities ${w}_k^{-}$ and ${w}_k^{+}$, calculated with the NPA, MBS, AIM populations and potential-derived charges for the same sample of molecules. The values obtained with these methodologies lead, qualitatively, to the same conclusions as those achieved by the Hirshfeld population analysis.

Table 7 ${f}_{k\;i}^{\left( NBO\;\right)}$ values for the considered bond orbitals of the reagent: CH₂CHCl

Table 8 ${f}_{k\;i}^{\left( NBO\;\right)}$ values for the considered bond orbitals of the reagent: CH₂CHCHO

Table 9 Atomic philicities ${w}_k^{-}$ (a.u.) calculated by using Hirshfeld charges

Table 10 Atomic philicities ${w}_k^{+}$ (a.u.) calculated by using Hirshfeld charges

A new methodology to compare the frontier molecular orbital (FMO) and the finite difference (FD) approximations

A similar definition to Eq. (19) can be used to analyze the orbital relaxation effects [59] and how they affect the comparison between the frontier molecular orbital (FMO) and the finite difference approximations (FD). This analysis of the Fukui function and the treatment of relaxation effects pursue a similar aim as the Fukui matrix decomposition proposed by Bultinck et al. [60, 61] but following a different methodology. The FMO approximation is in agreement with Eq. (6), which is based on assuming that MOs change very little when they lose or gain an electron. In other words:

$$ {\displaystyle \begin{array}{l}{\rho}_{N-1}(r)\approx {\left|{\varphi}^{HOMO}(r)\right|}^2+\sum \limits_{i=1}^{\frac{N}{2}-1}2\cdot {\left|{\varphi}_i^{CMO}(r)\right|}^2\\ {}{\rho}_{N+1}(r)\approx {\left|{\varphi}^{LUMO}(r)\right|}^2+\sum \limits_{i=1}^{\frac{N}{2}}2\cdot {\left|{\varphi}_i^{CMO}(r)\right|}^2\end{array}} $$

(32)

where N is the number of electrons in the neutral molecule, and, by substituting (32) in f ⁻(r) = ρ _N (r) − ρ _N − 1(r) and f ⁺(r) = ρ _N + 1(r) − ρ _N (r), the expressions: f ⁻(r) ≈ |φ ^HOMO(r)|²and f ⁺(r) ≈ |φ ^LUMO(r)|² are achieved. However, this relation is not fulfilled for all cases due to the orbital relaxation effects. To study this effect, we have used the least-squares method and applied the condition (Eq. 33):

$$ \int {\left[{f}^{\alpha }(r)-\sum \limits_{i=0}^a{E}_i\cdot {\left|{\varphi}^{CMO_i}(r)\right|}^2\right]}^2 dr= MINIMUM\left(\mathrm{with}\alpha =-\mathrm{or}+\right) $$

(33)

which leads to the Eq. (34):

$$ {\displaystyle \begin{array}{l}\int {f}^{-}(r)\cdot {\left|{\varphi}^{HOMO-j}(r)\right|}^2\kern0.36em dr=\sum \limits_{i=0}^a{E}_i\int {\left|{\varphi}_i^{HOMO-i}(r)\right|}^2\cdot {\left|{\varphi}^{HOMO-j}(r)\right|}^2\kern0.36em dr;j= 0, 1, 2,\dots, a\\ {}\int {f}^{+}(r)\cdot {\left|{\varphi}^{LUMO+j}(r)\right|}^2\kern0.36em dr=\sum \limits_{i=0}^a{E}_i\int {\left|{\varphi}_i^{LUMO+i}(r)\right|}^2\cdot {\left|{\varphi}^{LUMO+j}(r)\right|}^2\kern0.36em dr;j= 0, 1, 2,\dots, a\end{array}} $$

(34)

where “a” is the number of orbitals to be considered in the analysis (e.g., all occupied orbitals) and “E _i ” is a parameter that can help to study the orbital relaxation effects.

$$ {\displaystyle \begin{array}{l}{f}^{-}(r)={\rho}_N(r)-{\rho}_{N-1}(r)\approx \sum \limits_{i=0}^a{E}_i{\left|{\varphi}^{HOMO-i}(r)\right|}^2\\ {}{f}^{+}(r)={\rho}_{N+1}(r)-{\rho}_N(r)\approx \sum \limits_{i=0}^a{E}_i{\left|{\varphi}^{LUMO+i}(r)\right|}^2\end{array}} $$

(35)

The systems of Eq. (34) provide the “E _i ” coefficients of Eq. (35) that establish a relationship between the f ^α(r) function (with α = − or +) and the canonical orbitals. This provides a very useful tool for comparing the FMO and FD approximations, so it has been added to the UCA-FUKUI v.2 package. It is selectable from the “Additional Tools/FMO-FD Test” option in the menu bar of the main screen. To show the meaning of the E _i coefficients, the following example, considering the two cases of Fig. 3, is presented, where N represents the number of electrons of the neutral molecule.

Fig. 3

Left: the lost electron belongs to the HOMO. Right: the lost electron belongs to the HOMO-1

In case 1, the FMO approximation is fulfilled, (f ⁻(r) ≈ |φ ^HOMO(r)|)². However, in case 2, this approximation is not properly fulfilled, having to be replaced, in this case, by f ⁻(r) ≈ |φ ^{HOMO − 1}(r)|². If the fit shown in Eq. (36) is performed for each one of those cases, E ₁ will be the largest coefficient for case 1 while E ₂ will be the largest one for case 2.

$$ {f}^{-}(r)\approx {E}_1{\left|{\varphi}^{HOMO}(r)\right|}^2+{E}_2{\left|{\varphi}^{HOMO-1}(r)\right|}^2 $$

(36)

In any case, the E _i coefficient of largest magnitude should determine the orbital that has lost (or gained) the electron when the molecule is ionized.

To test this methodology, we have used, as a sample, a set of simple molecules: N₂, CO, CN, H2O, CH₂O, NH₃, C₂H₄, C₂H₃F, C₂H₃OH, CH₃NO₂, CHOCH₃, HCOOH. For each molecule, the fit (37) has been applied by using Eq. (34).

$$ {f}^{-}(r)={\rho}_N(r)-{\rho}_{N-1}(r)\approx {E}_1{\left|{\varphi}^{HOMO}(r)\right|}^2+{E}_2{\left|{\varphi}^{HOMO-1}(r)\right|}^2+{E}_3{\left|{\varphi}^{HOMO-2}(r)\right|}^2 $$

(37)

Figure 4 shows the E _i coefficients (Eq. 34) calculated for the molecules of the sample. It is relevant to point out that, as expected, in most cases the E ₁ parameter, corresponding to HOMO, is the largest of the three, while for the CN and FO molecules, this parameter is smaller than the rest (E ₃ being the largest parameter for the CN molecule and E ₂ the largest one for FO).

Fig. 4

E _i coefficients (Eq. 34) for the molecules of the sample

Figure 5 (for N₂ molecule) shows that the HOMO of the neutral molecule is CMO7 (FMO approximation), which matches the BETA-CMO7 orbital that has lost one electron in the cation (FD approximation). For the CO molecule (Fig. 6), the HOMO of the neutral molecule is CMO7, which matches with the BETA-CMO7 orbital.

Fig. 5

Canonical-orbital energy levels for the N₂ molecule. Left: neutral-molecule energy levels. Right: those of the cation

Fig. 6

Canonical-orbital energy levels for the CO molecule. Left: neutral-molecule energy levels. Right: those of the cation

However, a different situation is presented for the CN and FO molecules, as shown in Fig. 4. In those cases, FMO and FD approximations provide equivalent results by changing the HOMOs, where ALPHA-CMO5 and ALPHA-CMO8 should be the new HOMOs of the FMO approximation (probably, due to an orbital relaxation effect in the cation), as shown in Fig. 7 (for CN molecule) and Fig. 8 (for FO molecule). Finally, in the Supplementary Material, an example (EXAMPLE V) has been added showing how to perform this type of calculation.

Fig. 7

Canonical-orbital energy levels for the CN molecule. Left: neutral-molecule energy levels. Right: those of the cation

Fig. 8

Canonical-orbital energy levels for the FO molecule. Left: neutral-molecule energy levels. Right: those of the cation

Conclusions

A new methodology for obtaining reactivity indices (philicities) has been defined and subsequently used on a representative sample of organic molecules selected as an example. This methodology has been included in the new update of the UCA-FUKUI software (version 2.0). Also, condensation schemes based on atomic populations of the atoms in molecules theory, the Mulliken approximation (with a minimum basis) and electrostatic potential-derived charges have been implemented. In addition, the calculation of “bond reactivity indices”, defined in previous studies, has been implemented. New “utilities” have also been included in this second version of the program, including a tool that allows the comparisons of finite difference and frontier molecular orbital approximations (described in detail in this paper) or a utility that automatically builds “* .cub” files with the ALIE (average local ionization energy) function. This could be very useful since the construction of this kind of functions can be very laborious. Finally, a set of commented examples that show the use of the program are also available in this new version.