Can molecular and atomic descriptors predict the electrophilicity of Michael acceptors?

  • Guillaume Hoffmann
  • Vincent TognettiEmail author
  • Laurent JoubertEmail author
Original Paper
Part of the following topical collections:
  1. International Conference on Systems and Processes in Physics, Chemistry and Biology (ICSPPCB-2018) in honor of Professor Pratim K. Chattaraj on his sixtieth birthday


In this paper, we assess the ability of various intrinsic molecular and atomic descriptors, grounded in the conceptual density functional theory and the quantum theory of atoms-in-molecules frameworks, to predict the electrophilicity of Michael acceptors, which are fundamental organic reagents involved in the formation of carbon–carbon bonds. To this aim, linear and multilinear regressions between these theoretical properties and the experimental values gathered in Mayr-Patz’ scale were performed. The relevance of quantum chemical descriptors are then discussed.


Mayr’s electrophilicity scale Michael additions Molecular descriptors Atomic descriptors Reactivity indices Conceptual DFT Quantum theory of atoms-in-molecules (QTAIM) 


Being able to differentiate and classify molecules by their propensity to accept or donate electrons, from either a theoretical or experimental point of view, is of particular importance in organic chemistry for a better understanding of reactivity. For this purpose, several quantitative scales for electrophilicity and nucleophilicity have been established throughout the years [1, 2, 3]. The scale proposed by Mayr et al. [4] is certainly among the most comprehensive and useful available. It basically relies on the following equation, linking the rate constant measured at 20 °C (k20 °C) for the reaction between a given nucleophile and electrophile with the intrinsic reactivity properties of the reactants:
$$ \log {k}_{20{}^{\circ}C}={s}_N\left({N}_{Mayr}+{E}_{Mayr}\right) $$

In Eq. 1EMayr denotes the electrophilicity parameter; NMayr the nucleophilicity parameter; and sN, the so-called nucleophile-specific susceptibility parameter. It is worth mentioning that EMayr is related almost exclusively to electronic effects and is independent of solvent. In practice, these molecular descriptors can then easily be used to predict rates and selectivity in organic chemistry [5, 6], and to guide synthetic chemists to selecting the most relevant reagents. From a theoretical aspect, such an extensive scale with an online freely available database of over 1400 molecules [7] fosters the design of new theoretical tools capable of recovering and extending (for instance on not yet synthetized molecules) these experimental values.

With this aim in mind, a large number of theoretical reactivity indices, mainly within the framework of conceptual density functional theory (CDFT) [8, 9], have been proposed in recent years and tested with respect to Mayr’s scale. One can, for instance, cite the recent papers by Pereira et al. [10] and González et al. [11]. One should notice that, in the literature, efforts have been concentrated mainly on retrieving the experimental electrophilic reactivities of benzhydrylium ions [12, 13, 14], which constitute a subclass of particular importance because they were established as reference electrophiles for building Mayr-Patz’ scale [6, 15]. The use of frontier molecular orbitals [16, 17], or of the intrinsic reactivity index [18], or of descriptors derived from information theory, have been recently discussed [19]. From our side, we showed that the constant sign domains of the dual descriptor [20], as well as the atomic electronegativities [13] and atomic energies [21] were also successful.

This prominence of benzhylidrium ions in electrophilicity studies is notable due to their ability to correctly mimic the behavior of the carbocations that are generated as typical intermediates in organic synthesis [22]. However, despite their predating omnipresence in theoretical studies devoted to Mayr-Patz’ scale, many other electrophiles, such as aldehydes, ketones and quinones are of considerable interest and should not be overlooked.

For such reasons, we will focus in this work on the electrophilic reactivity of α-β-unsaturated compounds known as Michael acceptors, already investigated by Domingo and co-workers [23]. The generic Michael addition [24] is epitomized by the reaction depicted in Scheme 1: reagent 1 features an acidic proton in the α position with respect to two electron-withdrawing groups (EWGs) that can be abstracted by a base B. The generated carbanion can attack an enone molecule in the β position of the C=C double bond, forming an enolate intermediate that can be subsequently protonated to afford the final product.
Scheme 1

A prototypical Michael reaction

Such 1,4-nucleophilic addition reactions (which can be extended to compounds involving heteroatoms) are often key processes in total synthesis since they result in the formation of a C–C bond between the two initial reactants. From a physicochemical point of view, notably thanks to the very recent addition of 15 new empirical electrophilicity (EMayr) parameters by Mayr’s group [25], the amount of available experimental data has become large enough to assess the performances of quantum theoretical descriptors for this particular reaction.

In this paper, we will focus mainly on chemical descriptors stemming from CDFT and from the quantum theory of atoms-in-molecules (QTAIM) [26, 27]. Both frameworks are indeed based on the same ingredient—the electron density—which is the primary observable in quantum chemistry. They are thus complementary from our point of view. In the last decade, we have notably exploited the synergy of CDFT and QTAIM in organic chemistry and organometallics [28, 29, 30, 31, 32, 33, 34, 35, 36, 37].

All CDFT and QTAIM descriptors will be described in detail in the next section. They will be then applied (using the protocol explained in the computational section) to the experimental dataset (taken from Mayr’s Database of Reactivity Parameters [7]) in order to design a predicting tool for the potential reactivity of Michael acceptors.

It should be stressed that this approach is centered on the intrinsic properties of the reactants, independently of the way they interact along the reaction, and relies on a strong assumption that we explain now. First, we presume that the reaction is under kinetic (rather than thermodynamical) control. Then, according to Arrhenius’ law, the rate constant depends on the activation barrier and thus on the property of the transition state, which is a point on the potential energy surface where the reagents may strongly interact and might be significantly distorted. However, in case the reaction is very exothermic, one can then conjecture that the transition state (TS) is early. The application of the Hammond postulate implies that the physicochemical TS properties are expected to be enough close to the reactants, or at least can be safely extrapolated from them.

The studied reactivity descriptors

The first category of descriptors gathers those from CDFT. They can be classified into two categories: basic descriptors (BD) and composite descriptors (CD). We define BDs as those corresponding to direct derivatives of the electronic energy Ee with respect to the natural variables within the chosen ensemble. For instance, in the canonical representation, Ee is expressed as a function of the number of electrons N and as a functional of the external potential generated by the nuclei, \( v\left(\overrightarrow{r}\right) \). The primary aim of CDFT is to provide a chemical gist to these derivatives [38, 39] . They can be either global (one value for the whole molecule), local (depending on each real space position \( \overrightarrow{r} \)), or non-local [depending or two (\( \overrightarrow{r},{\overrightarrow{r}}^{\prime } \)) or more positions].

The three global BDs (GBDs) we will consider are the electronic chemical potential μ [40], the molecular hardness η [41], and the hyperhardness γ [42, 43] according to:
$$ \upmu ={\left(\frac{{\partial E}_e\left[N,v\left(\overrightarrow{r}\right)\right]}{\partial N}\right)}_{v\left(\overrightarrow{r}\right)};\eta ={\left(\frac{\partial^2{E}_e\left[N,v\left(\overrightarrow{r}\right)\right]}{\partial {N}^2}\right)}_{v\left(\overrightarrow{r}\right)};\gamma ={\left(\frac{\partial^3{E}_e\left[N,v\left(\overrightarrow{r}\right)\right]}{\partial {N}^3}\right)}_{v\left(\overrightarrow{r}\right)} $$
The three local BSs (LBDs) under study are the electron density ρ, the Fukui function f [44], and the standard dual descriptor f(2) [45]:
$$ \rho \left(\overrightarrow{r}\right)={\left(\frac{\delta {E}_e\left[N,v\left(\overrightarrow{r}\right)\right]}{\delta v\left(\overrightarrow{r}\right)}\right)}_N;f\ \left(\overrightarrow{r}\right)=\frac{\delta^2{E}_e\left[N,v\left(\overrightarrow{r}\right)\right]}{\partial N\delta v\left(\overrightarrow{r}\right)};{f}^{(2)}\ \left(\overrightarrow{r}\right)=\frac{\delta^3{E}_e\left[N,v\left(\overrightarrow{r}\right)\right]}{\partial^2 N\delta v\left(\overrightarrow{r}\right)} $$

Note that global descriptors may characterize the whole reactivity of the molecule, but they do not give insight into regioselectivity, unlike local functions.

In practice, there exist two main models to evaluate these derivatives: the so-called parabolic (P) and straight-lines (S) models. In the P model, the value for a non-integer number of electrons between N − 1 and N + 1 is obtained through the unique continuously differentiable parabola that gives the correct energy for the N − 1, N, and N + 1 points. In the S model, there is a derivative discontinuity [46] for the energy when crossing an integer value for the electron number: the line connecting Ee(N − 1) to Ee(N) has a different slope than that linking Ee(N) to Ee(N − 1). In this model, one thus distinguishes between the right and left derivatives, giving rise, for instance, to the μ+, μ global descriptors, and to the f+ and f local functions.

Once the energy model is chosen, one can use two different approaches for the numerical evaluation: the finite difference linearization (FDL) and the frontier molecular orbital (FMO) one (in the following, εHOMO and εLUMO will denote the Kohn-Sham orbital energy of the highest occupied MO and of the lowest unoccupied one, respectively). Table 1 collects the working equations for both approaches.
Table 1

Basic descriptors from conceptual density functional theory (DFT) considered in this study. P Parabolic model, S straight-lines model, GBD global basic descriptors, LBD local basic descriptors, FMO frontier molecular orbital, FDL finite difference linearization, HOMO highest occupied molecular orbital, LUMO lowest unoccupied molecular orbital




Electronic chemical potential


\( \left\{\begin{array}{l}{\upmu}_{\mathrm{FDL}}=\frac{E_e\left(N+1\right)-{E}_e\left(N-1\right)}{2}\\ {}{\upmu}_{\mathrm{FMO}}=\frac{\upvarepsilon_{\mathrm{HOMO}}+{\upvarepsilon}_{\mathrm{LUMO}}}{2}\end{array}\right. \)


\( \left\{\kern0.5em \begin{array}{c}{\upmu}_{\mathrm{FDL}}^{+}={\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}+1\right)-{\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}\right)\\ {}{\upmu}_{\mathrm{FMO}}^{+}={\upvarepsilon}_{\mathrm{LUMO}}\kern6em \end{array}\kern1.5em \right.\left\{\kern0.5em \begin{array}{c}{\upmu}_{\mathrm{FDL}}^{-}={\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}\right)-{\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}-1\right)\\ {}{\upmu}_{\mathrm{FMO}}^{-}={\upvarepsilon}_{\mathrm{HOMO}}\kern5.75em \end{array}\right. \)

Molecular hardness


\( \left\{\kern0.5em \begin{array}{c}{\upeta}_{\mathrm{FDL}}={\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}+1\right)-2{\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}\right)+{\mathrm{E}}_{\mathrm{e}}\left(\mathrm{N}-1\right)\\ {}{\upeta}_{\mathrm{FMO}={\upvarepsilon}_{\mathrm{LUMO}}-{\upvarepsilon}_{\mathrm{HOMO}}}\kern11.25em \end{array}\right. \)



γFMO = εLUMO − 2εHOMO + εHOMO − 1

Fukui functions


\( \left\{\kern0.5em \begin{array}{c}{\mathrm{f}}_{\mathrm{FDL}}^{+}\left(\overrightarrow{\mathrm{r}}\right)={\uprho}_{\mathrm{N}+1}\left(\overrightarrow{\mathrm{r}}\right)-{\uprho}_{\mathrm{N}}\left(\overrightarrow{\mathrm{r}}\right)\\ {}{\mathrm{f}}_{\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)={\uprho}_{\mathrm{LUMO}}\left(\overrightarrow{\mathrm{r}}\right)\kern3em \end{array}\kern1.5em \right.\left\{\kern0.5em \begin{array}{c}{\mathrm{f}}_{\mathrm{FDL}}^{-}\left(\overrightarrow{\mathrm{r}}\right)={\uprho}_{\mathrm{N}}\left(\overrightarrow{\mathrm{r}}\right)-{\uprho}_{\mathrm{N}-1}\left(\overrightarrow{\mathrm{r}}\right)\\ {}{\mathrm{f}}_{\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right)={\uprho}_{\mathrm{HOMO}}\left(\overrightarrow{\mathrm{r}}\right)\kern3em \end{array}\kern1.5em \right. \)

Dual descriptor


\( \left\{\kern0.5em \begin{array}{c}{\mathrm{f}}_{\mathrm{FDL}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right)={\uprho}_{\mathrm{N}+1}\left(\overrightarrow{\mathrm{r}}\right)-2{\uprho}_{\mathrm{N}}\left(\overrightarrow{\mathrm{r}}\right)+{\uprho}_{\mathrm{N}-1}\left(\overrightarrow{\mathrm{r}}\right)\\ {}{\mathrm{f}}_{\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right)={\uprho}_{\mathrm{LUMO}}\left(\overrightarrow{\mathrm{r}}\right)-{\uprho}_{\mathrm{HOMO}}\left(\overrightarrow{\mathrm{r}}\right)\kern3em \end{array}\kern1.5em \right. \)

As previously mentioned, these basic descriptors can be mixed to design CDs, which can be evaluated within either the FDL or FMO approaches, and which are presented in Table 2. One of the most popular CD is probably the global electrophilicity index, whose scope has been extensively reviewed by Chattaraj [47, 48]:
$$ \omega =\frac{\upmu^2}{2\eta } $$
Table 2

Composite descriptors from conceptual DFT considered in this study. GCD General composite descriptors, LCD local composite descriptors




Molecular softness





\( {\Delta \mathrm{E}}_{\mathrm{FDL},\mathrm{FMO}}^{\mathrm{nucleo}}=\frac{{\left({\upmu}_{\mathrm{FDL},\mathrm{FMO}}+{\upeta}_{\mathrm{FDL},\mathrm{FMO}}\right)}^2\ }{2{\upeta}_{\mathrm{FDL},\mathrm{FMO}}};\kern0.75em {\uplambda}_{\mathrm{FDL},\mathrm{FMO}}^{\mathrm{nucleo}}={\mathrm{e}}^{-{\upbeta}_{\mathrm{N}}{\Delta \mathrm{E}}_{\mathrm{FDL},\mathrm{FMO}}^{\mathrm{nucleo}}} \)



\( {\Delta \mathrm{E}}_{\mathrm{FDL},\mathrm{FMO}}^{\mathrm{e}\mathrm{lectro}}=\frac{{\left({\upmu}_{\mathrm{FDL},\mathrm{FMO}}-{\upeta}_{\mathrm{FDL},\mathrm{FMO}}\right)}^2\ }{2{\upeta}_{\mathrm{FDL},\mathrm{FMO}}};\kern0.75em {\uplambda}_{\mathrm{FDL},\mathrm{FMO}}^{\mathrm{e}\mathrm{lectro}}={\mathrm{e}}^{-{\upbeta}_{\mathrm{E}}{\Delta \mathrm{E}}_{\mathrm{FDL},\mathrm{FMO}}^{\mathrm{e}\mathrm{lectro}}} \)

Electrophilicity index


\( {\upomega}_{\mathrm{FDL},\mathrm{FMO}}=\frac{\upmu_{\mathrm{FDL},\mathrm{FMO}}^2}{2{\upeta}_{\mathrm{FDL},\mathrm{FMO}}} \)

Electrodonating power


\( {\upomega}_{\mathrm{FDL},\mathrm{FMO}}^{-}=\frac{{\left(3{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}+{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}\right)}^2}{16\left({\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}-{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}\right)} \)

Electroaccepting power


\( {\upomega}_{\mathrm{FDL},\mathrm{FMO}}^{+}=\frac{{\left({\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}+3{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}\right)}^2}{16\left({\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}-{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}\right)} \)

Local softness


\( \left\{\begin{array}{l}{\mathrm{s}}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)={\mathrm{Sf}}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)\\ {}{\mathrm{s}}_{\mathrm{FDL},\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right)={\mathrm{Sf}}_{\mathrm{FDL},\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right)\end{array}\right. \)

Composite Fukui functions


\( {\displaystyle \begin{array}{l}{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right);{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right);\\ {}{\upomega}_{\mathrm{FDL},\mathrm{FMO}}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)\end{array}} \)

Local chemical potential


\( {\upmu}_{\mathrm{FDL},\mathrm{FMO}}\left(\overrightarrow{\mathrm{r}}\right)=\frac{1}{2}\left({\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right)+{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)\right) \)

Local hardness


\( {\upeta}_{\mathrm{FDL},\mathrm{FMO}}\left(\overrightarrow{\mathrm{r}}\right)={\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{+}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)-{\upmu}_{\mathrm{FDL},\mathrm{FMO}}^{-}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right) \)

Multiphilic descriptor


\( {\mathrm{f}}_{\upomega}^{(2)}\left(\overrightarrow{\mathrm{r}}\right)={\upomega}_{\mathrm{FDL},\mathrm{FMO}}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right) \)

Composite dual descriptors


\( {\mathrm{S}}_{\mathrm{FDL},\mathrm{FMO}}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right);{\mathrm{s}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right)={\mathrm{S}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right) \)

Local electrophilicity


\( {\Delta \uprho}_{\mathrm{FDL},\mathrm{FMO}}^{+}\left(\overrightarrow{\mathrm{r}}\right)=-\frac{\upmu_{\mathrm{FDL},\mathrm{FMO}}}{\upeta_{\mathrm{FDL},\mathrm{FMO}}}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{+}+\frac{1}{2}{\left(\frac{\upmu_{\mathrm{FDL},\mathrm{FMO}}}{\upeta_{\mathrm{FDL},\mathrm{FMO}}}\right)}^2{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right) \)

Local nucleophilicity


\( {\Delta \uprho}_{\mathrm{FDL},\mathrm{FMO}}^{-}\left(\overrightarrow{\mathrm{r}}\right)=\frac{\upmu_{\mathrm{FDL},\mathrm{FMO}}}{\upeta_{\mathrm{FDL},\mathrm{FMO}}}{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{-}+\frac{1}{2}{\left(\frac{\upmu_{\mathrm{FDL},\mathrm{FMO}}}{\upeta_{\mathrm{FDL},\mathrm{FMO}}}\right)}^2{\mathrm{f}}_{\mathrm{FDL},\mathrm{FMO}}^{(2)}\left(\overrightarrow{\mathrm{r}}\right) \)

Similarly, Fukui functions can be multiplied by selected global descriptors to afford new local descriptors, as notably proposed by Chattaraj and co-workers [49, 50, 51]. They can be theoretically justified either by shifting towards other ensembles (for instance, the global softness S [52], and the local softness \( s\left(\overrightarrow{r}\right)=f\ \left(\overrightarrow{r}\right)/\eta \)[53] are basic descriptors in the so-called grand-canonical ensemble, recommended to compare species with different electron numbers) or by scrutinizing relevant Taylor expansions (this is the case for ω, ∆ρ+and ∆ρ) [54], in either the P or S models.

However, experimentalists may be not well accustomed with local functions that convey infinite information (at any real space point). Indeed, organic chemistry is often explained in terms of reactive sites, which can be viewed as a coarse-grain representation of the molecule. One thus needs a “translation” of local functions into this usual language of reactive centers. This can be conveniently achieved by a procedure coined “condensation” [36]. In this paper, we will mainly consider atomic condensation, but the same approach can be used for whole substituents (several atoms grouped together) of interest. Several non-equivalent ways for partitioning a molecule into its atoms have been proposed over the last decades: Hirshfeld partition [55], fuzzy atoms [56], Parr’s atoms [57], etc.

Here, as already stated in Introduction, we will rely mainly on QTAIM that divides real space into non-overlapping basins ΩA. For any local function \( \boldsymbol{a}\left(\overrightarrow{\boldsymbol{r}}\right) \), one can derive the corresponding atomic value by summing all contributions of points pertaining to the chosen atom:
$$ a(A)=\underset{\varOmega_A}{\int }a\left(\overrightarrow{r}\right){d}^3r $$
For instance, in the FMO approximation, the atomic electrophilic Fukui function will be evaluated simply by:
$$ {f}_{FMO}^{+}(A)=\underset{\varOmega_A}{\int }{\rho}_{LUMO}\left(\overrightarrow{r}\right){d}^3r $$
Within the FDL approach, the situation is a little bit trickier. Two main schemes actually exist [58]: the response of molecular fragment (RMF) and the fragment of molecular response (FMR) schemes. In the latter, the partition of the species with N electrons is also used for the condensation of the functions involving N + 1 or N − 1 electrons. In the first approach, the atomic partition is specifically recalculated from the electron densities of the oxidized or reduced molecules. One then gets:
$$ {f}_{FDL}^{+}(A)=\underset{\varOmega_A\left(N+1\right)}{\int }{\rho}_{N+1}\left(\overrightarrow{r}\right){d}^3r-\underset{\varOmega_A(N)}{\int }{\rho}_N\left(\overrightarrow{r}\right){d}^3r={q}_N(A)-{q}_{N+1}(A) $$
where qN(A) denotes the atomic charge in the molecule with N electrons. This is the famous Yang-Mortier scheme [59]. It can be applied on any population analysis. Thus, in order to enrich the number of descriptors, we decided to use it (besides QTAIM) with the natural population analysis (NPA) [60], Mulliken [61], Hirshfeld, and electrostatic-fitted ChelpG charges [62] for all local functions derived from Fukui functions and the dual descriptor. Note that these atomic values can also be combined, as for instance in the relative electrophilicity defined by the sk(A) = s+(A)/s(A) ratio [53] that we also computed.

Furthermore, atomic descriptors specific to QTAIM have been included, for instance the atomic localization index, LI(A). Some others are energies, such as the atomic Kohn-Sham kinetic energy TS(A), the atomic Thomas-Fermi [TTF(A)] and von Weizsäcker kinetic energies TW(A) (part of Liu’s steric energy [63]), the atomic virial energy [TW(A)], and the interacting quantum atoms additive atomic energy, EIQA(A). We refer the interested reader to some of our recent publications for the corresponding formulas [21, 64, 65].

To be more comprehensive about atomic descriptors, it should be stressed that another method for condensing Fukui functions was proposed by Chamorro, Pérez and coworkers (CP) [66, 67], and was used in particular by Mayr and coworkers [25, 68]. The CP method is no longer based atomic charges, but on discriminating localized Gaussian primitives. This approach will also be considered in our paper.

Lastly, three descriptors recently proposed by Demir [69] were added: the electronic part of the molecular electrostatic potential evaluated at a given nucleus, \( h\left({\overrightarrow{R}}_A\right) \), and the two finite difference combinations (\( {\varDelta}^{+}h\left({\overrightarrow{R}}_A\right)={h}_{N+1}\left({\overrightarrow{R}}_A\right)-{h}_N\left({\overrightarrow{R}}_A\right) \) and \( {\varDelta}^{-}h\left({\overrightarrow{R}}_A\right)={h}_N\left({\overrightarrow{R}}_A\right)-{h}_{N-1}\left({\overrightarrow{R}}_A\right) \)).

Computational details

The molecular structures and the values for the different chemical reactivity descriptors presented in the previous section were obtained using the following steps. First, the structures of the most stable conformers were generated with the Balloon [70] software (for molecules 9 and 11, using RMSD filtering) or by relaxed energy scans along relevant coordinates (for instance, dihedral angles for rotations around bonds), which were subsequently optimized with the Gaussian 09 [71] software at the M06-2X/aug-cc-pVTZ level of theory using the recommended ‘ultrafine’ integration grid. Note that, for some species with N + 1 electrons, the quadratic QC algorithm has to be used to properly converge the self-consistent field (SCF) energy (leading to correct positive electron affinities). For analogous reasons, the FDL calculation of the hyperhardness was not considered (since it would involve the N + 2 electron species).

Solvent effects (the selected experimental values were actually measured in dimethylsulfoxide (DMSO), a non-coordinating solvent) were taken into account using a polarizable continuum model [72], in the latest implementation of the integral equation formalism (IEFPCM). QTAIM calculations were performed using the AIMALL software [73] (IQA energies are obtained using the M06-2X tag model), while CDFT descriptors were obtained from homemade scripts. Condensation according to the Chamorro-Pérez scheme was performed with a procedure described by these authors and with a program available online (

Atomic units are used for all descriptors, except for μ, η, ω, ΔEnucleo/electro for which electron-volts are more commonly used (see also Table S2 in the supplementary file for the units of all descriptors).

Results and discussion

Our dataset consists of the 35 molecules represented in Scheme 2. They can be classified into three groups: mono-acceptor-substituted ethylenes (11 molecules, in dark blue, numbered 111), phenyl-substituted Michael acceptors (11 molecules, in bright blue, numbered from 1222), and the so-called “other Michael acceptors” (13 molecules, in violet, numbered from 2325).
Scheme 2

The studied dataset

The first step of our study consisted of a conformational analysis. Table S1 in the supplementary information file shows selected conformers and their relative energy (for instance, four minima were identified for compounds 1, 2, and 3). In all cases, the most stable conformer was found to be the same in terms of SCF and standard Gibbs energies, and also corresponded to that reported by the research group of Mayr, who used a slightly different computational protocol (with only one exception, compound 8, but the energy difference between the two minima is much too low to be significant).

Then, for each molecule, 163 descriptors were evaluated as described in the preceding sections, and whose values are gathered in Table S2 in the supplementary information file. One may wonder whether some are redundant. In particular, it is of methodological interest to investigate if FDL and FMO descriptors are correlated. Figure 1 illustrates the discrepancies between both approaches for four selected descriptors (two atomic descriptors relative to the Cβ carbon atom, condensed within QTAIM, and two global descriptors): ω, ∆Enucleo, f+, and f(2).
Fig. 1

Comparison between the finite difference linearization and the frontier molecular orbital (FMO) schemes for the whole set on selected descriptors at the M06-2X/aug-cc-pVTZ PCM level of theory. ωFMOand ΔEnucleoFMO in eV, other descriptors in atomic units

Interestingly, a high correlation between FDL and FMO was observed for the electrophilicity index [with the coefficient of determination (R2) equal to 0.94], which fully vanished for ∆Enucleo, while both descriptors were based on the same ingredients (μ and η). This well illustrates to what extent error compensations may be or not be at work for CDs, with no clear rule to account for them.

For the atomic electrophilic Fukui function f+ (which measures the amount of electrons an atom can gain when one excess electron is spread over the whole molecule), the R2 value remains quite low (0.66) even if both approaches afford a similar qualitative trend within the whole molecular dataset. The case of the dual descriptor is also instructive. Indeed, let us recall that it allows for quantifying the competition between electrophilicity and nucleophilicity: when f(2)(A) is positive, electrophilicity dominates at the atomic scale, while nucleophilicity is predominant for negative values.

The quadrants (b) and (c) in the bottom right panel in Fig. 1 collect points for which the condensed dual descriptor features the same sign for the FDL and FMO treatments. On the other hand, points in quadrants (a) and (d) correspond to different signs depending on the computational protocol. The fact that there is a non-negligible number of compounds in these areas indicates that the two approaches predict opposite reactivity type.

Such an analysis could also be led to unravel the influence of the population scheme chosen in the Yang-Mortier procedure. The results are pictured in Fig. 2 for the condensed dual descriptor. It immediately appears that the five population schemes give results that are not correlated. The highest R2 value between the QTAIM quantities and the others was obtained with the Hirshfeld partition, but remained low (0.50).
Fig. 2

Four top panels Influence of the population analysis in the Yang-Mortier scheme for the condensation of the dual descriptor on carbon β. Two bottom panels Comparison between QTAIM Yang-Mortier and Chamorro-Pérez (CP) schemes. M06-2X/aug-cc-pVTZ PCM level of theory. All values in atomic units

However, one should underline that (with the exception of the Mulliken case, which displays many negative values) most of the points are located in quadrant (b), meaning that the Cβ carbon is predicted to be electrophilic. Henceforth, even if several Yang-Mortier schemes consistently agree from a qualitative point of view, they lead to significantly different quantitative estimates. Conversely, the good agreement (see bottom of Fig. 2) between the QTAIM values (in the FMO approach) and those obtained from the Chamorro-Pérez scheme for f+ and f(2) should be mentioned.

Such a comparison can also be led for atomic charges (see Fig. S1 in the supplementary information file): it appears that, apparently counterintuitively, ChelpG and Mulliken predict negative atomic charge for the electrophilic carbon atom Cβ in 94% of the cases, while the situation is much more balanced for Hirshfeld and QTAIM. The quadrant analysis shows that these both last approaches are consistent with one another about the charge sign, while considerable discrepancies are observed with the other schemes, so that extrapolating reactivity behavior from these atomic charges may remain elusive.

One final comment about methodology is related to the influence of the level of theory. Figure S2 in the supplementary information compares our FMO electrophilicity indices with those reported by Mayr and coworkers at the B3LYP/6–31 + G (d) level of theory [25] with SMD solvation model [74]. High correlation was found (R2 = 0.95), suggesting good transferability of these descriptors with respect to the computational protocol that is used to generate the wavefunction or electron density.

Now, it follows from our previous findings that all selected descriptors will be actually considered as generally independent the one from the other, so that none of them can be excluded a priori. We then investigated monolinear regressions between each of them and Mayr’s experimental values for electrophilicity. Unfortunately, no high correlations were obtained, since the higher R2 value we obtained was with μ+f+, amounting to 0.64, as depicted in Fig. 3, according to:
$$ {E}_{Mayr}=-768.6{\upmu}_{FMO}^{+}{f}_{FMO, QTAIM}^{+}\left({C}_{\beta}\right)-26.00 $$
Fig. 3

Best monolinear regressions for Mayr’s electrophilicity EMayr (corresponding to highest R2 values) and regression lines for the whole set (top left), mono acceptor substituted ethylenes (M.A.C.S.E.; top right), phenyl substituted acceptors (P.S.M.A.; bottom left), and others (O.M.A.; bottom right) subgroups

Disappointingly, the electrophilicity index, which was previously recommended in the literature [23], only gave R2 = 0.40 (see Fig. S3 in the supplementary information file).

These results imply that a single descriptor may not be sufficient to describe the electrophilicity of the whole set of these Michael acceptors. One may ascribe this failure to the fact that one needs to take into account not only the reactivity of the carbon β, but the reactivity of the whole carbon–carbon double bond since carbon α has also a role to play. This can be achieved by considering a combination of the \( \left({f}_{\alpha}^{+}+{f}_{\beta}^{+}\right) \) type. This leads to a descriptor that will be denoted f+(CαCβ), with its natural grand-canonical (and composite) extensions Sf+(CαCβ), μ+f+(CαCβ), ωf+(CαCβ)….

From a QTAIM point of view, the C=C bond can be basically characterized by the delocalization index, DI(CαCβ), between carbons α and β (which can be considered as the QTAIM Cα–Cβ bond order), the electron density ρC and the electron density laplacian values ∇2ρC at the corresponding bond critical point (BCP) that roughly represent bond strength. Unfortunately, none of these Cα–Cβ QTAIM bond descriptors was more successful [higher R2 = 0.21, with DI(CαCβ)].

A more general strategy is then to consider bilinear regressions. They could in fact provide a more thorough description of reactivity. For instance, one descriptor will be linked to orbital control, while another will be related to charge control. In such cases, one will obtain a combination reminiscent of Ayers’ general purpose reactivity descriptor [75, 76]. As we considered 163 descriptors, 13,203 pairs should be considered. Nevertheless, we decided to reduce this number by grouping only descriptors calculated using the same approach (FDL or FMO) or the same population analysis for the sake of consistency. Indeed, we believe that mixing (for instance) a NPA descriptor with a QTAIM one would generate models that could not be interpreted easily in a chemical sense. For such reasons, we considered only 7732 models over the 13,203 possible ones.

The best suitable bilinear model (see Fig. 4) that emerged consisted in the combination of \( {\omega f}_{\beta}^{+} \) and f+(CαCβ) with R2 = 0.82, according to the following equation:
$$ {E}_{Mayr}=52.12\ {\omega}_{FMO}{f}_{FMO, QTAIM}^{+}\left({C}_{\beta}\right)-44.35\ {f}_{FMO, QTAIM}^{+}\left({C}_{\alpha }{C}_{\beta}\right)-21.96 $$
Fig. 4

Best bilinear models for Mayr’s electrophilicity EMayr for the whole set (top left), the M.A.C.S.E. (top right), the P.S.M.A. (bottom left), and others O.M.A. (bottom right) subgroups

Even if substantial improvement was obtained from this bilinear model [the corresponding mean absolute error (MAE) equals 1.6], it fails in being very accurate. Keeping the two descriptors entering Eq. 9, we then performed a trilinear analysis (see Fig. 5). The double bond delocalization index appears to be the more efficient partner, according to Eq. 10 (with R2 = 0.87, see Fig. 3):
$$ {E}_{Mayr}=65.37\ {\omega}_{FMO}{f}_{FMO, QTAIM}^{+}\left({C}_{\beta}\right)-59.94\ {f}_{FMO, QTAIM}^{+}\left({C}_{\alpha }{C}_{\beta}\right)+16.54\ DI\left({C}_{\alpha }{C}_{\beta}\right)-49.74 $$
Fig. 5

Best trilinear model for Mayr’s electrophilicity EMayr for the whole set

The MAE value was found to be equal to 1.3. This might seem still too much high. An alternative approach is to consider regressions from subsets, as already proposed by Domingo [23]. With this aim, we performed independent monolinear regressions for the mono acceptor substituted ethylenes (11 molecules, M.A.C.S.E. family), the phenyl substituted Michael acceptors (11 molecules, P.S.M.A.), and the “other Michael acceptors” (13 molecules, O.M.A.). The best monolinear models (see Fig. 3) were the following:
$$ {E}_{Mayr}=26.80\ {\omega}_{FMO}-57.53\ \left( compounds\ 1-11\right) $$
$$ {E}_{Mayr}\kern0.5em =-24.52\ {\upmu}_{FMO}-139.23\ \left( compounds\ 12-22\right) $$
$$ {E}_{Mayr}=-662.7{\mu}_{FMO}^{+}{f}_{FMO, QTAIM}^{+}\left({C}_{\beta}\right)\hbox{--} 24.42\kern0.5em \left( compounds\ 23-35\right) $$
The high R2 values for the two first groups should be noted: 0.93 for the first group and 0.91 for the second, while it is significantly lower for the third group (0.81). This is in fact quite expected since the “other” family is a heteroclite set, made of very different molecules. Bilinear regressions were also considered (see Fig. 4), with R2 = 0.96 for the first family, 0.94 for the second, and R2 = 0.91 for the third:
$$ {E}_{Mayr}=-\left(1447{\mu}_{FMO}^{+}+212.1\ {S}_{FMO}\right){f}_{FMO, QTAIM}^{+}\left({C}_{\beta}\right)-21.91\kern0.5em \left( compounds\ 1-11\right) $$
$$ {E}_{Mayr}=-\left(2358{\mu}_{FMO}^{+}-904.2\ {S}_{FMO}\right){f}_{FMO, QTAIM}^{+}\left({C}_{\beta}\right)-19.55\kern0.5em \left( compounds\ 12-22\right) $$
$$ {E}_{Mayr}=-5.91{\mu}_{FMO}+1264\ {S}_{FMO}^2{f}_{FMO, CP}^{(2)}\left({C}_{\beta}\right)-49.98\kern0.5em \left( compounds\ 13-35\right) $$

On the other hand, one can wonder whether for practical applications one needs to predict an accurate value or, instead, to be able to classify the reactants. We thus decided to create a more qualitative typology for Michael acceptors. More specifically, as experimental electrophilicity values range from −7.50 to −25, we divided it into three subclasses of reactivity: C1 = good electrophiles (−7 to −13), C2 = medium (from −13 to −19), and C3 = low (from −19 to −25).

In Figs. 4 and 5, rectangles were drawn that correspond to compounds for which the statistical models and the experimental values predict the same reactivity category. Conversely, points outside these rectangles are for molecules for which the subclass predicted by the model differs from that from the experimental value. One can see that almost all points with the trilinear model belong to these three rectangles, enlightening its potential interest for applied chemists.


In this paper, we investigated the ability of descriptors issued from CDFT and the QTAIM to predict Mayr’s experimental electrophilicity values for a series of 35 Michael acceptors. Various calculation schemes (finite difference linearization, FMOs) and condensation techniques (Yang-Mortier applied on different population analyses, CP approach) were first compared and shown to be in general uncorrelated.

Monolinear regressions were then performed, which proved satisfying for the mono acceptor substituted ethylenes and the phenyl substituted subsets, but inadequate for the whole dataset. Bilinear and trilinear regressions afforded consequent improvement, involving mainly the global electrophilicity index ω and the atomic μ+f+ generalized electrophilic Fukui function (which are obviously chemically meaningful), but they fail in being truly predictive for the whole set.

In summary, this study thus proves that these CDFT-QTAIM descriptors can be efficient to quantify the reactivity difference for similar compounds, and to retrieve qualitative ranking for different families. They also allow for establishing a new typology of “Michael acceptance power” that could be finally useful for experimentalists.



The authors would like to gratefully acknowledge the LABEX SynOrg and the Normandy Region for funding and support, and the Centre Régional Informatique et d’Applications Numériques de Normandie (CRIANN) high-performance computing facility. It is our pleasure to contribute to this special issue honoring the fundamental contributions made in chemistry by Prof. Chattaraj, a pioneer in conceptual density functional theory.

Supplementary material

894_2018_3802_MOESM1_ESM.pdf (1.1 mb)
ESM 1 (PDF 1086 kb)


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Copyright information

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

Authors and Affiliations

  1. 1.COBRA UMR 6014 & FR 3038, INSA Rouen, CNRSUniversité de Rouen NormandieMont St AignanFrance

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