Research Article  Open Access
Nucleophilicity Index Based on Atomic Natural Orbitals
Abstract
A simple method of evaluating a semilocal (regional) nucleophilicity is introduced. The concept involves use of the natural orbitals for atomic populations to identify the most “reactive population” of electrons on particular atom in molecule. The results of test calculations considering the regioselectivity problem in electrophilic aromatic substitution to the benzene derivatives are presented and briefly discussed.
1. Introduction
There is a wide variety of reactivity predictors for electrontransfercontrolled reactions among which the Fukui function, originating from the density functional theory (DFT), is one of the most fundamental ones [1, 2]. Within the orbitalbased calculations one of the simplest methods to interpret and implement is the condensed Fukui index [3–6] (a coarsegrained atombyatom representation of the Fukui function) which can be determined from electron population analysis. There are many arguments [7, 8] in favor of the condensed Fukui index based on the Hirshfeld’s populational scheme [9]; however, other population analysis methods, including the Mulliken’s [10–13] and Löwdin’s [14] as well as the NPA schemes [15] are also in common use (e.g., [16–18]).
In this paper we briefly introduce and test a new simple method of evaluating the relative (with respect to particular atom in two “homologous” molecules) reactivity of nucleophiles, based on the natural orbitals for atomic population of electrons [19] and involving their energies as well as occupation numbers. The method is related to the Fukui function concept and its approximation within the framework of Frontier Molecular Orbital (FMO) theory [20–22]. In this work we take into consideration the wellknown problem (somewhat trivial but illustrative) of regioselectivity prediction in the electrophilic aromatic substitution to the following benzene derivatives: fluorobenzene (C_{6}H_{5}F), aniline (C_{6}H_{5}NH_{2}), phenol (C_{6}H_{5}OH), nitrobenzene (C_{6}H_{5}NO_{2}), benzoic acid (C_{6}H_{5}COOH), and benzaldehyde (C_{6}H_{5}CHO).
In general, functional groups –OH and –NH_{2} are classified as electron donating and strongly activating in the electrophilic substitution reactions while functional groups –NO_{2}, –COOH, and CHO remove electron density from the benzene ring and thus strongly deactivate the molecule. Functional groups from the former class tend to be ortho/para directing while those from the latter one direct electrophiles to attack the benzene molecule at the meta position. In fluorobenzene (likewise in other benzene halides) the benzene ring is weakly deactivated due to inductive withdrawal of electrons by electronegative atom F. However, the resonance donation of nonbonding electrons of fluorine atom to the benzene ring causes that the most preferable positions of electrophilic attack are ortho and para.
2. Method Details
Let us assume the closedshell molecular system with electrons doubly occupying lowest molecular orbitals , generated as linear combinations of orthogonalized atomic orbitals (OAO) , or in matrix form, where the rectangular matrix groups the relevant LCAO MO expansion coefficients. It follows directly from the superposition principle of quantum mechanics that the conditional probability of “finding electron” from th occupied molecular orbital on th atomic orbital reads Hence, the LCAO MO expansion coefficient can be regarded as the conditionalprobability amplitude. Alternatively, one can relate to the amplitude of the probability of “finding electron” from th atomic orbital on th canonical molecular orbital. Then Similarly, by replacing the operator with the operator projecting onto the subspace of all OAOs centered on atom , we get the conditional probability of the event that the electron from electron population on atom can be ascribed to th molecular orbital, where normalization constant reads The corresponding matrix of elements representing projections of th occupied MO onto th occupied MO through the subspace of AOs assigned to atom [23], is obviously not diagonal. It follows directly from orthonormality of molecular orbitals that regarding the whole (molecular) electron population ,
The representation of orbitals , in which offdiagonal elements of matrix (7) are zeroes, ; one can straightforwardly obtain from diagonalization of The diagonal matrix of electron probabilities can be used to calculate the electron populations of the corresponding natural orbitals for electron population of atom , , stands for the electron population on atom . The corresponding matrix of the 1electron orbital “energies” (i.e., expectation values of in ) can be determined as follows: Here, is the Fock operator, and diagonal matrix collects orbital energies of occupied canonical MOs. Since natural orbitals are not the eigenvectors of , the matrix of orbital energies (12) is not diagonal. Orbital energies of both frontier canonical molecular orbitals, HOMO and LUMO , are known to be useful qualitative indicators of chemical reactivity. Thus, it is of our special interest to investigate how the expectation values of operator within representation of manage with evaluation of the reactivity of the particular atom . However, due to nonorthogonality of naturalorbital sets for different atoms, only the relative changes of for the same in two homologous species should be compared.
One of the standard frontierorbital treatments of chemical reactivity is the regioselectivity problem, for example, in the electrophilic aromatic substitution to the benzene derivatives. The standard FMO theory analysis of the effect of substituent groups involves the electron populations of the highest occupied molecular orbital (HOMO), whereas within the newly proposed “reverse scenario” we focus on the electron population of particular atom first and then analyze the highest “energies” of occupied atomic NOs.
3. Numerical Results
To examine the presented methodology we have used statefunctions calculated at RHF(ROHF for cations)/STO3G, RHF(ROHF)/631G* as well as DFT/B3LYP/augccpVDZ theory levels, using the standard ab initio quantum chemistry package GAMESS [24], for the following benzene derivatives: fluorobenzene, aniline, phenol, nitrobenzene, benzoic acid, and benzaldehyde. The highest energies of occupied natural orbitals and the corresponding electron populations were compared with two standard MObased atomic descriptors of reactivity: the atomic index of nucleophilicity [25] involving only the highest occupied MO, , and the Fukui nucleophilic reactivity index [26] involving electron populations on atom in cation, , and neutral molecule, , Both indices were calculated only within basis sets STO3G and 631G*, since larger basis sets (especially those including diffuse functions, e.g., augccpVDZ) are well known to dramatically lose their “atomic attributeness” of AOs, and, consequently, many populationtype descriptors usually assume completely unreasonable values. Two alternative procedures of population analysis were used in calculation of indices (13) and (14): the Mulliken’s scheme (superscript ) [10–13] and the Löwdin’s one (superscript ) [14], involving the standard “geometrical” orthogonalization of atomic orbitals.
The results presented in Table 1 clearly indicate that only index calculated within extended basis set allows one to correctly predict the position of electrophilic attack in all molecules under consideration. The Fukui nucleophilic reactivity index properly copes with ortho/para directing, but it completely fails with respect to meta directing. It is worth notice that indicator , involving only electron populations and energies of HOMO, is far less sensitive to population analysis method than index . Furthermore, if one excludes the results for meta directing groups, there is no significant difference between calculated within STO3G and 631G* basis sets.

We performed a similar analysis using the newly proposed method based on natural orbitals . Table 2 collects orbital occupation numbers and the corresponding relative “energetic effects”, , calculated as a difference between the highest energy of occupied natural orbitals of particular carbon atoms in benzene and its derivative: Even a cursory analysis of occupation numbers in Table 2 allows one to correctly predict the preferential ortho/para or meta directing of substituent groups, regardless of basis sets used in calculations. However, evaluation of energetic effects provides more detailed information about activating/deactivating influence of substituent groups on particular carbon atoms. For example, –NH_{2} is properly recognized as the best activator, and the most activated carbon atom is in position para (activated for about 11 kcal·mol^{−1}), while substituent group –NO_{2} strongly deactivates the molecule for about 20–25 kcal·mol^{−1}, and the most active population of electrons is then located on carbon atom in position meta. For the majority of cases orbital energies properly predict the position of electrophilic substitution.

In calculations of natural orbitals from Table 2 we used the standard Löwdin’s orthogonalization procedure to obtain orthogonal AOs. However, it was of our interest to find how and depend on the orthogonalization procedure. Table 3 contains the same data as Table 2 but is calculated within representation of “physically” orthogonalized atomic orbitals (superscript ) [27–29]. At first glance one can observe a small improvement of electron populations from DFT/B3LYP/augccpVDZ. Also, in contrast to calculations involving “geometrical” orthogonalization, for benzaldehyde in STO3G basis set correctly predict meta directing of substituent group. In general, however, calculated energies are similar to those from Table 2 (average distances between the respective energies calculated within “geometrically” and “physically” orthogonalized AO representations are: 0.4 (STO3G), 0.8 (631G*), and 0.9 (augccpVDZ) [kcal·mol^{−1}]) and thus allow one to draw almost the same conclusions about reactivity of particular carbon atom.

The energetic effects from Tables 2 and 3 are relatively small (but comparable with differences between stabilization energies of the corresponding Wheland intermediates [30]). For comparison, below we present several simple species ordered with respect to relative nucleophilicities (from weak to very good nucleophiles) and the corresponding energies [kcal·mol^{−1}] (of orbitals identified as lone pairs in all cases) calculated at DFT/B3LYP/TZP with model core potential [31, 32]: CH_{3}COOH < H_{2}O < NH_{3} < CH_{3}O^{−} < CH_{3}S^{−}, 11.8 61.1 217.9 232.4.
4. Summary
In this work we have briefly introduced a simple method of evaluating the relative nucleophilicity in energy scale. The concept involves the use of natural orbitals for atomic population of electrons and their energies (i.e., expectation values of Fock operator) as well as occupation numbers to identify “the most reactive population of electrons” on particular atom (or molecular fragment). Such scenario is directly related to the standard FMO theory treatment involving atomic populations of electrons of the highest occupied molecular orbital (HOMO); in the newly proposed approach we first focus on the electron population of particular atom and then analyze energies of occupied natural orbital. This strategy has been examined on the regioselectivity problem in the electrophilic aromatic substitution to the benzene derivatives. Analysis of the results allows one to draw the conclusion that evaluation and comparison of relative chemical nucleophilicities of atoms in an energy domain are more reliable and advantageous than analyses involving other popular MObased; this seems to be somehow obvious since, in contradistinction to the majority of condensed atomic indices, energetic descriptors converge systematically to the completebasisset limit. It has to be noticed, however, that electron population of the highest occupied natural orbital of a particular atom in molecule is somewhat insensitive to basis set variations and the corresponding energy seems to exhibit the basis set dependence quite consistent with the variational principle.
The presented methodology is still in need of thorough analysis and examination. Also, it is of our special interest to take advantage of the approach based on natural atomic orbitals involving virtual molecular orbitals to evaluate the energetic descriptor of electrophilicity of atoms and molecular fragments.
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Copyright
Copyright © 2013 Dariusz W. Szczepanik and Janusz Mrozek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.