Introduction

Chemical reactions lie at the heart of chemistry. From the electronic structure point of view, a chemical reaction can be viewed as the movement of the nuclei over the potential energy surface (PES) created by the quantum dynamics of the electrons. Therefore, the resistance, or lack thereof, of a molecule to a chemical reaction should be understandable in terms of the forces experienced by the atomic nuclei in transit from reactants to products [1,2,3,4,5,6,7,8,9,10,11,12]. In absence of electronic degeneracies [13], either natural or induced by radiation fields [14, 15], the force on a given nucleus \(\alpha \) is given by the Hellman-Feynman theorem [16]:

$$\begin{aligned} \mathbf {F_{\alpha }}=&-\frac{\partial {E}}{\partial {{\textbf{R}}}}=\int {\frac{Z_\alpha ({\textbf{r}}-{\textbf{R}}_\alpha )}{\left|{{\textbf{r}}-{\textbf{R}}_\alpha }\right|^3}{\rho (\textbf{r})}d{\textbf{r}}}\nonumber \\ &-\sum _{\beta \ne \alpha }\frac{Z_{\alpha }Z_{\beta }({\textbf{R}}_\beta -{\textbf{R}}_\alpha )}{\left|{{\textbf{R}}_\alpha -{\textbf{R}}_\beta }\right|^3}. \end{aligned}$$
(1)

However, Eq. 1 together with Hohenberg and Kohn’s theorem [17] (HK) allows us to establish that the electron density, \({\rho (\textbf{r})}\), itself determines the forces on the atoms and therefore the course of the reaction. Two possibilities arise, characterizing the course of a reaction in terms of the response of either the force or the density as the reaction proceeds. These two possibilities are related to Nakatsuji’s classification of chemical processes based on which whether the electron-following (EF) or electron-preceding (EP) perspective is most elucidative [18]. In the electron-following perspective, the nuclei move, inducing a change in the electron density with a (possibly infinitesimal) “delay,” thereby generating a restoring force on the nuclei. That is, the density responds to changes in the position of the nuclei. The alternative, electron-preceding perspective is that a perturbation of the electron density, \({\rho (\textbf{r})}\), “anticipates” the nuclear motion.

Characterization of reactivity based on response functions of the electron density (EF) is widely used to elucidate chemical reactivity within the framework of conceptual density functional theory (c-DFT) [19,20,21,22,23,24,25,26,27,28,29,30,31]. For example, the Fukui functions [32,33,34,35] measure the change in electron density due to a gain (+) or loss of electrons (\(-\)) while the external potential, \(v({\textbf{r}})\), remains fixed:

$$\begin{aligned} f({\textbf{r}})^{\pm } = \left( \frac{\partial {{\rho (\textbf{r})}}}{\partial {N}} \right) ^{\pm }_{v({\textbf{r}})}. \end{aligned}$$
(2)

Similarly, the local softness [36] measures the change in electron density due to a change in chemical potential [37]:

$$\begin{aligned} s({\textbf{r}}) = \left( \frac{\partial {{\rho (\textbf{r})}}}{\partial {\mu }} \right) _{v({\textbf{r}})}. \end{aligned}$$
(3)

Although less popular, in c-DFT, there are also force-based descriptors that are vectors instead of scalars [38,39,40,41,42,43,44], including nuclear reactivity descriptors that measure the response in the force on the atoms [5, 6, 45,46,47,48,49,50]. In analogy to the Fukui function, the nuclear Fukui function of an atom \(\alpha \) is defined as [45]

$$\begin{aligned} \mathbf {\Phi }_{\alpha } = \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {N}} \right) _{v({\textbf{r}})}=\int {\frac{({\textbf{r}}-{\textbf{R}}_\alpha )}{\left|{{\textbf{r}}-{\textbf{R}}_\alpha }\right|^3}f({\textbf{r}}) d{\textbf{r}}}, \end{aligned}$$
(4)

and nuclear softness as [46],

$$\begin{aligned} \mathbf {\sigma }_{\alpha } = \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\mu }} \right) _{v({\textbf{r}})}=\int {\frac{({\textbf{r}}-{\textbf{R}}_\alpha )}{\left|{{\textbf{r}}- {\textbf{R}}_\alpha }\right|^3}s({\textbf{r}}) d{\textbf{r}}}. \end{aligned}$$
(5)

The conceptualization of reactions through the change of the force as the reaction progresses is an idea widely developed by Professor Alejandro Toro-Labbé and those who have worked in his group [51,52,53,54,55,56,57,58,59,60,61,62,63]. To better understand the way the reaction proceeds, Toro-Labbe [51, 55, 64,65,66,67] introduced the concept of reaction force, \(F(\zeta )\), as the negative of derivative of the potential energy, V, along a significant trajectory that connects reactants with products. This trajectory is usually taken to be the intrinsic reaction coordinate, \(\zeta \), or minimum-energy path [68,69,70,71]

$$\begin{aligned} F(\zeta )=-\frac{\partial {V}}{\partial {\zeta }}. \end{aligned}$$
(6)

For a single-step reaction, the energy profile has three stationary points, namely the transition state (TS), reactants, and products. At these points, the net force is zero. Therefore, the force profile naturally shows a minima (\(\zeta _{\text {min}}\)) at the inflection point of the energy between reactants and the TS and a maxima (\(\zeta _{\text {max}}\)) between the TS and the product (Fig. 1). At these two points, a qualitative shift in the relationship between changes in molecular geometry and electronic structure is observed [64]. Subsequently, Toro-Labbé et al. [51,52,53, 64] have proposed that a single-step reaction path can be broken into four regions characterized by different extents of the geometric and electronic rearrangements [53].

Fig. 1
figure 1

Profiles of the potential energy (black), reaction force (blue), and reaction force constant (ochre) along a reaction coordinate

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In the region between reactants and the minimum of the force (\(\zeta _R \Rightarrow \zeta _{\text {min}}\)) (i.e., the initial stage of the reaction), the changes in nuclear positions are relatively large compared to the electronic changes. The reaction force is negative in this region, indicating that the species are constrained by the stable geometric rearrangement of the reactant configuration towards instability and become most constrained at the minima of the force profile. In Nakatsuji’s classification, this region is likely dominated by electron following. Between the minima and TS (\(\zeta _{\text {min}} \Rightarrow \zeta _{TS}\)), the net force seeks to overcome the constraints. At this stage, electronic structure rearrangements act as a driving force to the reaction. In Nakatsuji’s classification, this region is more likely one of electron preceding. In the stage after the TS (\(\zeta _{TS} \Rightarrow \zeta _{\text {max}}\)), the electronic rearrangements are still large, resulting in a net positive force that drives the chemical process towards the final structure. In the final stage (\(\zeta _{\text {max}} \Rightarrow \zeta _{P}\)), the net force is again dominated by the geometric relaxation process and reduces to zero at the conversion to the products.

Higher-order derivatives of the reaction profile have also been investigated by Jaque et al. [65] introducing the concept of reaction force constant, \(\kappa (\zeta )\), as the curvature of the energy profile along the reaction coordinate:

$$\begin{aligned} \kappa (\zeta )=\frac{\partial ^2 V}{\partial \zeta ^2}=-\frac{\partial {F(\zeta )}}{\partial {\zeta }}. \end{aligned}$$
(7)

For a single-step reaction, it is clear that \(\kappa (\zeta )\) must be negative between the reaction force minimum and maximum, as in that region, the curvature of \(V(\zeta )\) is negative. Similarly, \(\kappa (\zeta )\) must be positive elsewhere. Therefore, the reaction force constant is negative in the region where electronic structure rearrangement is supposed to take place.

In this paper, we will show that the reaction force constant can be written in terms of nuclear descriptors of c-DFT reactivity. Specifically, we rewrite the reaction force constant in terms of the nuclear softness, Eq. 5, and the variations of the chemical potential. In addition, we give arguments in favor of a separation of \(\kappa \) into terms dominant in the electron-rearrangement regions and others dominant in the geometry-rearrangement regions.

Results and discussion

Let us begin by writing the reaction force in terms of the contributions (projections) of each nuclear coordinate on a parametric curve that defines the reaction coordinate:

$$\begin{aligned} F(\zeta )=-\sum _{\alpha }{\frac{\partial {V(\zeta )}}{\partial {{\textbf{R}}_{\alpha }}}\cdot \frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}}= \sum _{\alpha }{\textbf{F}_{\alpha }\cdot \frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}}, \end{aligned}$$
(8)

where the derivative \(\frac{\partial }{\partial {{\textbf{R}}_{\alpha }}}\) should be understood as the gradient with respect to the coordinate \({\textbf{R}}_{\alpha }\). The reaction force constant results from taking the next derivative and multiplying it by \(-1\):

$$\begin{aligned} \kappa (\zeta )=-\left[ \sum _{\alpha }{\textbf{F}_{\alpha }\cdot \frac{\partial ^2{\textbf{R}}_{\alpha }}{\partial \zeta ^2}} + \sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{N}} \right] . \end{aligned}$$
(9)

As we shall see, it is convenient to express derivatives at fixed N (micro-canonical ensemble) in terms of derivatives at fixed \(\mu \) (grand-canonical ensemble). For this purpose, we use the differentiation rules for derivatives of implicit functions commonly used in thermodynamics:

$$\begin{aligned} \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{N}=\left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }+\left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\mu }}\right) _{N}\left( \frac{\partial {\mu }}{\partial {\zeta }}\right) _{N} \end{aligned}$$
(10)

Inserting Eqs. 10 into 9, one gets

$$\begin{aligned} \kappa (\zeta )=-&\left[ \sum _{\alpha }{\textbf{F}_{\alpha }\cdot \frac{\partial ^2{\textbf{R}}_{\alpha }}{\partial \zeta ^2}} + \sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }}\right. \nonumber \\ &\left. + \left( \frac{\partial {\mu }}{\partial {\zeta }}\right) _{N} \sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\mu }}\right) _{N}} \right] . \end{aligned}$$
(11)

Now, note that \(\left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\mu }}\right) _{N}\) is just the nuclear softness, Eq. 5,

$$\begin{aligned} \kappa (\zeta )=-&\left[ \sum _{\alpha }{\textbf{F}_{\alpha }\cdot \frac{\partial ^2{\textbf{R}}_{\alpha }}{\partial \zeta ^2}} + \sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }}\right. \nonumber \\ &\left. + \left( \frac{\partial {\mu }}{\partial {\zeta }}\right) _{N}\sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \mathbf {\sigma }_{\alpha }} \right] . \end{aligned}$$
(12)

This equation is one of the central results of this work because it establishes a relationship between the reaction force constant and descriptors of c-DFT reactivity, such as nuclear softness and changes of the chemical potential (electronegativity). Therefore, it is necessary to analyze all its elements in detail. In each term of Eq. 12, there is a vector \(\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\). This vector measures how much an atom’s coordinate contributes to the reaction coordinate. If an atom is not part of the reaction center, this derivative will be negligible. In fact, in large molecules, most atoms are merely “spectators” to a given reaction, and it is mainly the atoms between which bonds are created/broken which have appreciable values for the derivative \(\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\). For example, at the transition state, \(\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\) will be non-negligible only for those atoms that contribute significantly to the imaginary-frequency vibrational mode. On the other hand, the sign of each term in Eq. 12 will depend on the angle between the dot product of the vectors and the sign of \(\left( \frac{\partial {\mu }}{\partial {\zeta }}\right) _{N}\). Let us take as a reference direction an infinitesimal displacement in the increasing direction of the reaction coordinate. When the reaction proceeds \(\Delta \zeta \) in the direction of the products, there will be atoms in the reaction center that move in the same direction as the reaction coordinate and others that move in the opposite direction. A simple example is the transfer of hydrogen between a donor atom (D) to an acceptor atom (A) [59, 72, 73]. One can take as the reaction coordinate the distance D-H (see Fig. 2). When this distance increases, H changes its position by an amount \(\Delta {\textbf{R}}_{\alpha }\) that is practically parallel to \(\Delta \mathbf {\zeta }\). In addition, in this example, \(\Delta {\textbf{R}}_{\alpha }\) and \(\Delta \mathbf {\zeta }\) have almost the same magnitude,Footnote 1 so that \(\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\) is close to a unit vector in the direction of the products. If this relationship between \(\Delta {\textbf{R}}_{\alpha }\) and \(\Delta \mathbf {\zeta }\) remains similar throughout the reaction coordinate, then it is plausible to assume that \(\frac{\partial ^2{\textbf{R}}_{\alpha }}{\partial \zeta ^2}\) is negligible and that

$$\begin{aligned} \kappa (\zeta )\approx -&\left[ \sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }}\right. \nonumber \\ &\left. + \left( \frac{\partial {\mu }}{\partial {\zeta }}\right) _{N}\sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \mathbf {\sigma }_{\alpha }} \right] . \end{aligned}$$
(13)

This expression is exact in the reactants, the products, and the TS, where the force vanishes by definition.

Fig. 2
figure 2

Schematic of a hydrogen migration reaction, indicating the changes in H-atom position and the reaction coordinate, \(\zeta \)

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Electronic structure rearrangement

The second term in Eq. 13 captures mostly electronic structure rearrangements. It will be large only for atoms with a large nuclear softness. The soft regions of a molecule are those which respond with a large change in density to a perturbation in the chemical potential (electron preceding). Soft regions are characterized by being more polarizable [74,75,76,77]. In a broken bond, the electron density is more labile, so that as the reaction progresses to stages where bonds are broken and created, the magnitude of nuclear softness should increase. The same can be understood by the principle of minimum polarizability (or maximum hardness), which states that in stable (unstable) configurations, molecules tend to have minimum polarizability (maximum hardness) [78,79,80,81,82,83,84,85]. In fact, there is abundant evidence that in a reaction, the hardness is minimal around the TS [51, 86,87,88,89,90].

Since \(\mathbf {\sigma }_{\alpha }\) points in the direction in which the force on an atom changes as \(\mu \) varies due to the advance in the reaction coordinate, it is likely that in the atoms of the reaction center, \(\mathbf {\sigma }_{\alpha }\) points either in the same direction or opposite to \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\) that of the advance towards the products. The direction of sigma with respect to \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\) will then depend on whether \(\mu \) increases or decreases along the reaction path. This is more clearly seen from the first-order change of the chemical potential due to a shift in the atom’s positions along the reaction coordinate, \(\Delta \textbf{Q}_{\alpha }\) [91],

$$\begin{aligned} \Delta \mu = -\sum _{\alpha }{\Phi _\alpha \cdot \Delta \textbf{Q}_{\alpha }}, \end{aligned}$$
(14)

which can be written in terms of the nuclear softness and \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\),

$$\begin{aligned} \Delta \mu = -\frac{\Delta \zeta }{S}\sum _{\alpha }{\mathbf {\sigma }_\alpha \cdot \frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}}. \end{aligned}$$
(15)

In passing from Eqs. 14 to 15 we have made use of \(\Delta \textbf{Q}_{\alpha }=\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\Delta \zeta \) and \(\mathbf {\sigma }_\alpha =S\mathbf {\Phi }_\alpha \). For simplicity, let us assume that only one atomic coordinate changes significantly along the reaction path (the hydrogen in a proton transfer). Then, according to Eq. 15, if \(\Delta \mu \) is negative, \(\sigma _{\alpha }\) must point in the same direction as \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\) and vice versa. Usually, the chemical potential (which is mostly determined by the average of the HOMO and LUMO energies) decreases from the reactants to somewhere near the TS. From there, it grows again. See for example Fig. 3, which shows the chemical potential of SNOH along the internal reaction path of the proton transfer from O to S (S=NOH\(\rightarrow \) HSN=O). So between the reactants and the minimum of the chemical potential (close to the TS), \(\sigma _{\alpha }\) must point in the direction of the products (\(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\)) and thereafter in the direction of the reactants (opposite to \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\)). This can be confirmed in Fig.  4, where we have plotted the nuclear softness and \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\) vectors at key points along the intrinsic reaction pathway for this proton transfer.

Fig. 3
figure 3

Chemical potential (in kcal/mol) along the internal reaction coordinate (IRC) for the reaction S=NOH\(\rightarrow \) HSN=O. Gray dashed lines indicate the points where the reaction force attains its minimum (\(\zeta _{\text {min}}=-0.63\)) and maximum values \(\zeta _{\text {max}}=0.73\)) along the reaction path. The transition state (TS) is located at \(\zeta =0\). Note that the chemical potential reaches its minimum value close to the TS. Calculations of the neutral, anion, and cation were performed with DFT and B3LYP/6-311++g(d,2p) scheme. \(\mu \) is calculated in each point of the IRC as \(-(I+A)/2\) (see reference [92])

Full size image
Fig. 4
figure 4

Nuclear softness, \(\mathbf {\sigma }_{\alpha }\), at critical points of the reaction force for the internal proton transfer S=NOH\(\rightarrow \) HSN=O. Atoms are depicted in yellow, blue, red, and gray for sulfur, nitrogen, oxygen, and hydrogen, respectively. \(\frac{\mathbf {\sigma }_{\alpha }}{\sqrt{M_{\alpha }}}\) is represented with vectors with the same color as the atoms, except for H in which black is used. Nuclear softness is divided by the \(\sqrt{M_{\alpha }}\) to better represent the displacement that a force would produce in each atom. Green vectors correspond to \(\frac{\partial {\textbf{R}_\alpha }}{\partial {\zeta }}\), which is very small for all atoms except for H. In the potential energy profile, gray dashed lines indicate the points where the reaction force attains its minimum (\(\zeta _{\text {min}}=-0.63\)) and maximum values \(\zeta _{\text {max}}=0.73\)). The transition state is located at \(\zeta =0\)

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Finally, the link between the change in the chemical potential along a reaction path and the direction of the nuclear Fukui function (14) has also been studied by Ordon et al. [91] for a set of reactions. Their findings are equivalent to ours.

It is interesting that the derivative of the chemical potential with respect to the reaction coordinate appears naturally as one of the determinants of the magnitude and sign of \(\kappa \), since its negative is proportional to the reaction electronic flux, \(J=-Q\left( \frac{\partial {\mu }}{\partial {\zeta }}\right) _{N}\), another concept introduced by Toro-Labbé and Herrera [54, 62, 93, 94]. According to the authors, positive values of J “entail spontaneous changes in electronic density, indicating that bond strengthening or forming processes drive the reaction. In contrast to this, negatives values of J indicate nonspontaneous electronic reordering driven by weakening or bond breaking processes.” Note that both, the reaction electronic flux and nuclear softness, are expected to be large in the transition state region. Therefore, J will be mainly positive between \(\zeta _{\text {min}}\) and \(\zeta _{TS}\) and negative between \(\zeta _{TS}\) and \(\zeta _{\text {max}}\).

Constrained geometric rearrangement

The reaction path is a parametric curve in the space of nuclear positions. A change in \(\zeta \) necessarily implies a change in the external potential, \(v({\textbf{r}})\). According to the HK theorems and the Euler-Lagrange equation derived from them [95],

$$\begin{aligned} \mu =v({\textbf{r}})+\frac{\delta {F[\rho ]}}{\delta {\rho }}, \end{aligned}$$
(16)

an arbitrary change in the external potential must be accompanied by a change in the chemical potential [96,97,98]. Therefore, if \(\mu \) remains constant, the derivative \(\left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }\) is well defined if the number of electrons can vary. This does not mean that in a gas phase reaction, the number of electrons of reactants plus products does not remain constant. The change from constant N to constant \(\mu \) was introduced for convenience in Eq. 10. However, one is free to consider the reaction center as an open system (of a given \(\mu \)) that exchanges electrons with other parts of the reactants and products or with an idealized solvent. In such situations, \(\left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }\) measures how much the force on the atoms changes as \(\zeta \) changes, where at the same time, there is a flow of electrons such that the chemical potential remains constant. This force should be greatest in the regions of greatest hardness of the reaction path, i.e., in the vicinity of the products and reactants (\(\zeta _R \rightarrow \zeta _{\text {max}}\) and \(\zeta _{\text {min}} \rightarrow \zeta _{P}\)). In these regions of higher resistance to electronic flow, large electronic rearrangements are less likely. This picture is in agreement with Nakatsuji’s classification of these regions as ones of electron following. In summary, the term \(\sum _{\alpha }{\frac{\partial {{\textbf{R}}_{\alpha }}}{\partial {\zeta }}\cdot \left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }}\) in Eq. 13 will be predominantly large in regions of constrained but large geometric rearrangement.

By definition, \(\kappa \) is positive where \(\zeta _R< \zeta < \zeta _{\text {min}}\) and \(\zeta _{\text {max}}< \zeta < \zeta _{P}\). On the other hand, we have seen that between the reactants and close to the TS (the minimum of \(\mu (\zeta )\)), \(\frac{\partial {\mu }}{\partial {\zeta }}\) is negative and \(\frac{\partial {{\textbf{R}}_\alpha }}{\partial {\zeta }}\) and \(\mathbf {\sigma }_{\alpha }\) point in the same direction. Therefore, the second term in Eq. 13 is negative. As the first term predominates in \(\zeta _R< \zeta < \zeta _{\text {min}}\) (see Fig. 4), it should be presumably predominantly negative in that region for \(\kappa \) to be positive. That is, around the reactants, \(\left( \frac{\partial {\textbf{F}_{\alpha }}}{\partial {\zeta }}\right) _{\mu }\) seems to oppose the progress of the reaction as it points towards the structure of the reactants. The balance between both terms in Eq. 13 is something that deserves further investigation in a broad set of reactions. This is, however, beyond the scope of this first work.

Conclusions

Over the years, Professor Toro-Labbé has introduced a framework for understanding chemical reactions in terms of the potential energy profile and its derivatives along a reaction coordinate. The first and second derivatives are the reaction force and the reaction force constant, respectively. Both allow the reaction path to be split into regions where electronic or geometrical rearrangements tend to dominate.

In this work, we have shown that it is possible to establish a link between the reaction force constant and nuclear reactivity descriptors of the conceptual DFT. In particular, we gave arguments to justify why in the regions where the reaction force constant is negative the largest electronic structure rearrangements occur, while where it is positive, the dominant rearrangements are geometrical. We argue that in the electronic rearrangement region, they are dominated by large values of the nuclear softness and the reaction electronic flux, while the regions of higher hardness correspond to those of geometrical rearrangements. All these conclusions are based on the mathematical structure of the reaction force constant. It is certainly worth exploring further the structure of Eq. 13 (or Eq. 12) in a larger set of reactions. To do so, it is necessary to establish a method of calculating each of its terms, a question that we certainly do not address here.