A real space picture of the role of steric effects in SN2 reactions

Abstract Within substitution reactions, the Bimolecular Nucleophilic Substitution (SN2) reaction mechanism is one of the most frequently found and studied ones. Among other factors, the easiness of the SN2 pathway is classically considered to be determined by steric hindrance. However, the diffuse nature of the latter inevitably darkens these and other arguments holding the pillars of chemical intuition. In this work, we employ the steric energy (E ST ) descriptor, formulated within the Interacting Quantum Atoms approach, to offer insights regarding this problem. The steric demands of the substrate, nucleophile and leaving group were studied using the gas‐phase SN2 reaction with different organic skeletons (CH3—, CH3CH2—, (CH3)2CH—, (CH3)3C—, (CH3)3CCH2—) and halogens (F, Cl, and Br) as test‐bed systems. Our results show that, according to E ST , the SH experienced along these simple reactions fits, in the general case, the trends predicted by a meticulous and rigorous application of chemical intuition. However, steric clash alone should not be considered as the only argument used to explain the easiness of the SN2 reaction over different electrophiles.


| INTRODUCTION
Substitution reactions, involving the exchange of two functional groups within a given chemical skeleton or substrate, are fundamental chemical processes with huge implications in chemical reactivity and synthetic strategies. [1][2][3][4] One of the apparently simplest, and yet fundamental, plausible mechanism for these chemical transformations is the so-called bimolecular nucleophilic substitution (S N 2), represented in Under common circumstances, the S N 2 mechanism is considered to take place through the backside attack of the electrophile, leading to a characteristic Walden inversion of the substrate, though alternative proposals, such as the frontside S N 2 reaction or other more recently proposed mechanisms, such as the double inversion, have also been formulated. 6,7 Additionally, several side reactions are known 6,8,9 to compete with the substitution process, including elimination or even rearrangements transformations.
Due to its moderate size, simplicity and relevance, this and other similar reactions have been widely employed as model systems to unravel the nature of chemical bonding and reactivity. [10][11][12] The archetypal example involves a substitution reaction of halogen-like groups taking place over a carbon based electrophile, 6 though substitution reactions of electrophiles of different chemical nature have also been explored. [13][14][15][16] Indeed, the S N 2 reaction constitutes one of the basic kernels of classical organic chemistry and it has been widely studied both computationally [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and experimentally. 9,[35][36][37][38] It is generally believed 4 that the feasibility of the bimolecular substitution reactions is determined by a combination of multiple factors, including the nature of the nucleophile and leaving groups, the skeleton of the electrophile and the solvent. Unfortunately, the actual role of some of these effects remains still unclear and under debate. For instance, the steric hindrance (SH) built upon the formation of the penta-coordinated transition state was originally considered to be one of the dominant driving forces leading to the observed reaction barrier. However, this explanation has been seriously questioned in recent years by some authors, including Fernández and co-workers 39 who pointed out the weakening of the stabilizing orbitalic interactions as the main origin of the observed reaction barrier, though such an approach has also been criticized. 40 Furthermore, steric terms have also been classically considered, 4,41 under the eyes of chemical intuition, as the main impediment preventing the bimolecular nucleophilic substitution over crowded electrophiles from taking place. For instance, as the electrophilic center gets increasingly crowded, so does, generally, the reaction barrier associated to this concerted process, something used to explain the lower tendency of secondary and tertiary substituted carbons to undergo S N 2 like transformations. Such a trend has been classically attributed to the steric shielding induced by bulky substituents in the electrophilic center, difficulting the backside attack of the nucleophile and favoring a stepwise mechanism, usually referred to as a S N 1 reaction, 5 in which the C L bond is broken leading to the formation of a quasi-stable planar carbocation intermediate species.
Whether or not steric congestion is actually behind these and similar processes is still under debate, but its relevance in chemistry is beyond question. Unfortunately and despite its fundamental role in classical chemical models, SH is, along with other chemical terms, one of the so-called chemical unicorns, 42 concepts that lack from a direct observable and a rigorous definition but which are apparently well consolidated in the chemist's minds. The loose definition and vague nature of these terms has motivated the scientific community to unravel its true nature, if any. Indeed, a wide variety of computational tools and models have been employed in recent years to elucidate the origin of SH. [43][44][45][46][47] Following this trend, and drawing on the previous work developed by Popelier and co-workers, 43 we have recently shown 12 that a real space descriptor of steric effects, the so-called steric energy E ST , can be easily obtained with the aid of the IQA energy partitioning scheme. 48 Such a descriptor is obtained by freeing the deformation energy (E def ) of an atom or fragment from its charge transfer component (E CT ). It is worth mentioning that our approach prevents any of the well described 49  Considering the success of real space techniques in the analysis of some simple substitution reactions, 11,12 in this work we try to shine light on the actual role that SH plays in the latter, studying in detail the effect of the steric demands of the nucleophile, electrophile and leaving groups. Furthermore, in the latter case, special attention is paid to the distinctive effect of the geminal and vicinal groups belonging to the substrate. For such a purpose, a collection of halogen exchange reactions was studied in the gas phase with different nucleophile and leaving groups (F, Cl, and Br) and substrates (CH 3 , CH 3 CH 2 , (CH 3 ) 2 CH , (CH 3 ) 3 C , and (CH 3 ) 3 CCH 2 ). It is worth mentioning that though substitution reactions are usually carried out in solution and the effect of the solvent in its dynamics is beyond question, 4,50,51 solvent effects will not be addressed in this work.
Modeling the reaction in the vacuum comes as a reasonable approximation in order to study exclusively the effects of SH. Moreover, and although solvent effects have been claimed to modify substantially SH throughout the reaction, 21,52 it has also been stressed that geometries are only modified to a minor extent 52 and thus our model systems should yield chemically representative and meaningful results.
The work is organized as follows: first a brief overview of the concept of steric energies, E ST , within the context of the IQA partitioning scheme is presented. Then the SH and steric effects attributed to the substrate, nucleophile and leaving group along the studied gas-phase transformations are discussed. The final section gathers the conclusions drawn from this work.

| THE STERIC ENERGY (E S T ) DESCRIPTOR OF SH
The diffuse origin and nature of SH has crystallized in the development of a wide variety of tools and models for the measurement of steric congestion. Indeed, many different formalisms have been used for the previous purpose, including Symmetry Adapted Perturbation Theory (SAPT), 53 the Natural Bond Orbital (NBO) method 54 and energetic partitioning schemes, such as the Energy Decomposition Analysis (EDA) 55 and the IQA 48 approaches, to name just a few. It is precisely within the latter approach, where the steric energy, E ST , descriptor was derived. 12 The IQA approach is a real-space based technique, formulated within the Quantum Theory of Atoms in F I G U R E 1 Schematic representation of the general S N 2 reaction mechanism Molecules (QTAIM) theory, 56 which decomposes the total energy of any chemical system as a sum of one-and two-body energy terms, named self and interaction energies, respectively, as: Moreover, the two contributions appearing in the previous expression can be further partitioned in physically sound terms whose discussion is out of the scope of this work. Since the self or net energy of an atom or fragment is inherent to such an entity, it is generally expected to be only slightly modified by a perturbation of its environment. Thus, the evolution of the self-energy upon the interaction between two chemical moieties is more adequately measured with respect to a suitable reference, resulting in the so-called deformation energies, E A def ,: where E A ref is the self-energy of the QTAIM region A in its reference state, as for instance the isolated moiety in the vacuum. Under common scenarios, deformation energies exhibit a positive and exponentially increasing trend with the confinement or compression of a system, something which has led some authors 43 to conclude that such an energy term is actually able to measure steric congestion.
Unfortunately, the strong dependence of the net or self-energy of an atom with a change in its electron count, makes the previous approach not general, something which is clearly evidenced, for instance, in the inability of E def to represent the SH of H atoms. Thus, a more suitable descriptor of steric congestion can be derived by removing the charge transfer contribution, E CT , from the total deformation energy, as: Furthermore, the charge-transfer term can be rigorously computed within the Grand-Canonical DFT formalism 57 from the successive ionization potentials of the species under consideration (see Table S11 for further details). Altogether, this geometrical deformation, or simply steric energy, has been proven 12 to provide a more faithful and general picture of the steric congestion than plain deformation energies, even in the peculiar case of highly charged H atoms.

| COMPUTATIONAL DETAILS
All calculations were performed in the gas phase, without relativistic corrections, at different levels of theory, including HF and DFT in combination with the aug-cc-pVDZ basis set. Geometry optimizations, frequency calculations, single point calculations and intrinsic reaction coordinate (IRC) paths were computed using the Gaussian09 quantum chemistry package. 58 Similarly, the wavefunctions corresponding to each point along the computed reaction path were generated with the aid of Gaussian09. 58 On the other hand, IQA calculations were performed using the in-house made PROMOLDEN code. 59 Further details about the energetic partitioning calculations can be found in the Supporting Information (SI). As far as DFT is regarded, the Minnesota functional M06-2X 60 was employed as implemented in Gaussian09. 58 Since the steric energy is known to be barely dependent on the methodology, 12

| Effect of the substrate
In order to explore the previously mentioned steric shielding induced by the groups borne by the electrophilic center, the halogen selfexchange reaction was analyzed for a series of progressively bulkier substrates. For the sake of convenience and simplicity, all reactions were studied using F as both leaving group and nucleophile. As shown in Figure 2, and in agreement with already reported data, 24,61-63 the starting reactive species corresponds to an ion-molecular complex, leading to the double-well potential energy surface characteristic of gas phase nucleophilic substitutions over carbon based substrates.
This behavior is, however, greatly diminished in polar solvents. 63 Generally speaking, as the electrophilic center gets increasingly crowded, the reaction becomes less favorable, as reflected in Figure 3, that shows the evolution of the total electronic energy during the reaction for different substrates. Such a trend, corresponding to acti- It is worth mentioning that, despite being thermoneutral, the reaction profiles are not always perfectly symmetric with respect to the transition state. Such a slight asymmetry arises from secondary geometrical processes such as structural reorganization phenomena of the vicinal groups and it is hence more evident in electrophilic scaffolds bearing easily rotatable groups, such as methyl moieties. Additionally, the differences between both counterparts of the reaction profiles become even more prominent through the analysis of the reaction force experienced along the chemical transformation (see SI Section 7, Figure S27).  Table S2 and Figures S13-S15 for more details) on L, which can be rationalized attending to the C L bond breaking process and the subsequent decompression of the leaving group. It is interesting to notice, however, that the overall steric release experienced by L is similar for any of the studied substrates, suggesting that it is nearly unaffected by the chemical environment of the electrophile. Such a result could be attributed to the small size and compactness of the fluoride anion (F À ), indicating that the SH experienced by this atom would mainly arise from the direct compression against the electrophilic center (C atom) while not being affected up to a significant extent by the remaining chemical scaffolds, which are moreover oriented towards the opposite face of the electrophile. Additionally, the evolution in the C L bond distance varies very slightly with the nature of the substrate, exhibiting an increase of $ 0.10 Å throughout the series (see SI Section 3, Figures S3-S12).
On the other hand, the evolution of the SH experienced by the nucleophile exhibits a slightly more complex behavior, following a similar trend to that observed for the total deformation energy (see SI Section 5, Figures S16-S24). First of all, the E Nu ST increases along the nucleophilic attack as a result of the nucleophile to electrophile compression but it does not follow, however, a uniform and steady increase rate. Whereas non-, primary-and secondary-substituted centers lead to maximum steric penalties of roughly 12.0 kcal/mol, the tert-butyl skeleton yields a considerably reduced E Nu ST . This result is in partial disagreement with chemical intuition, 4,5 according to which the nucleophile would be expected to face larger steric penalties with increasingly substituted electrophilic centers. Although counter-intuitive at first glance, such a finding can be perfectly rationalized after a meticulous inspection of the two main factors contributing to the total clashing of the nucleophile:  once the perturbative regime is overcome, the rate of increase of the E Nu ST along the reaction coordinate decreases slightly. However, a somewhat anomalous behavior is found for the methyl substituted (CH 3 CH 2 ) substrate as shown in green in Figure 4, which exhibits a plateau in the evolution of E Nu ST , something that is also observed in a much minor extent in the reaction with the tertiary substituted C atom. In order to understand this discrepancy, a closer look to the evolution of the secondary geometrical distortions undergone by this system must be taken. Indeed, the distance of the nucleophile to the closest geminal atom shows two distinctive plateaus (see SI section 3), something which is moreover coupled to an internal rotation of the methyl group, that goes from a quasi-eclipsed to a staggered local disposition with respect to the nucleophile. These observations suggest that the carbon to nucleophile compression is partially counteracted by the reduction in the relative bulkiness of the geminal groups together with the partial planarization of the electrophile. It is precisely the interplay between these two opposite phenomena that in agreement with already reported data. 21,52 Finally, as shown in Figure 4, a very interesting trend is observed in the evolution of the SH suffered by the electrophilic center. During the previously mentioned perturbative regime, there is a significant and steady increase in the steric congestion suffered by the substrate.
Such an effect arises, once again, as a direct consequence of the compression against the nucleophilic species. After this perturbative regime, the planarization of the local geometry of the substrate becomes the dominant factor, which together with the C L bond breaking leads to a partial decongestion of the substrate. Such a result is in well agreement with previous works, 12,39,66,70 in which the effect of the planarization of carbon based species over its steric congestion was addressed. It should be noticed, however, that a slightly more complex behavior can be observed in the steric penalty of the methyl

| Effect of the nucleophile
In order to explore the effect of the steric penalty attributed to the entering nucleophile, the halogen exchange reaction of fluoromethane (CH 3 F) with a variety of nucleophiles was employed as a model system, see Figure 6.
Generally speaking, the reaction barrier increases significantly along the halogen series, exhibiting free activation energies (see SI Section 2) of 14.5, 32.5, and 36.8 kcal/mol, respectively, and indicating that, in agreement with previously reported results, 6,24 F À is the best gas phase nucleophile among those studied. This already reported trend 6 in the activation barriers has been classically explained 4,5 attending to the evolution of the nucleophilicity along the halogen series. The aforementioned term, sometimes defined as a measure of the ability of a nucleophile to displace the leaving group, is generally considered to be dictated by the interplay between the electron density and the charge borne by the nucleophile, its electronegativity and its bulkiness. As far as SH is regarded, it is believed that as the "size" of the nucleophile increases, so will the steric penalty to be overcome in order to allow the attack to the electrophilic center. This would lead to a reduction of the effective nucleophilicity throughout the halogen series (F > Cl > Br).
The general behavior observed in the evolution of E L ST , as collected in Figure 7, is completely analogous to the one mentioned in the previous section. It exhibits the characteristic trends of both the perturbative and the planarization-distortion regimes, and hence it can be rationalized accordingly. Interestingly, a slight increase in the compression of L is observed for the heavier halogens throughout the perturbative regime, yielding maximum congestions penalties of $ 0.3, 2.0, and 2.7 kcal/mol for F, Cl, and Br, respectively. This could be rationalized in terms of the increasing size of the nucleophile, which leads to a larger compression, and hence distortion, of the electrophile skeleton in which the leaving group is embedded. Indeed, this effect is clearly reflected in the internal geometrical features of the CH 3 moiety, that exhibit a subtle but noticeable shortening of the H L F I G U R E 5 Energetically optimized geometries, at the M06-2X/aug-cc-pVDZ level of theory in the gas phase, of the transition state involved in the self substitution reaction with different substrates: CH 3 , CH 3 CH 2 , (CH 3 ) 2 CH , (CH 3 ) 3 C , and (CH 3 ) 3 CCH 2 . Further details can be found in SI (Tables S3-S10) distance in the initial reaction stages (see SI Section 3.2). It is precisely such a decrease in the inter-atomic distance that ultimately leads to a further compression of the leaving group, something which is moreover coupled to a clear change in the rate of growth of the internal H C H angle, that decreases slightly along the halogen series. Furthermore, the net steric decongestion suffered by the leaving group throughout the distortion regime, as a result of the C F bond breaking process, increases together with the bulkiness of the entering halogen. It exhibits total changes in E L ST of À39, À50, and À52 kcal/mol for F, Cl, and Br, respectively. Such a large difference in the net steric decongestion of the leaving group can be easily rationalized as a result of the increase in the C F distance experienced along the transition from the original ion-molecular complex to the TS structure, as reflected in Figure 8. This distance increases by roughly 0.4, 0.6, and 0.7 Å, respectively (see SI Section 3.2).
On the other hand, as far as the nucleophile is regarded, E Nu

| CONCLUSIONS
Chemical intuition often considers SH as one of the crucial driving forces determining the outcome of multiple chemical phenomena, including the archetypal case of S N 2 reactions. Unfortunately, the lack of rigor inherent to chemical thinking, limits the validity of these arguments. In this work, we have thoroughly explored the SH undergone throughout different gas-phase S N 2 reactions taking place between simple electrophiles and halides. Some relevant conclusions can be distilled from our results: • As far as the substrate is regarded, and with the particular exception of the neopentyl group, the steric compression undergone by F I G U R E 1 0 Evolution of the steric energy E ST of the leaving group (top), nucleophile (center) and electrophile (bottom) throughout the F À + CH 3 X ! CH 3 F + X À reaction for X = F, Cl, and Br. Calculations performed at the M06-2X/aug-cc-pVDZ level of theory in the gas phase the electrophile during the initial stages of the reaction shows an increasing trend with the bulkiness of the substrate, in agreement with the trends followed by the activation barriers.
• Generally speaking, steric clashes become more sensitive to the surrounding chemical environment as the bulkiness of the chemical species under consideration increases. Thus, for instance, along the F > Cl > Br series the contribution of secondary effects to the SH of the halogens becomes larger.
• On the transition from reactants to the activated complex, the evolution of E Nu ST is more susceptible to a change in the decoration of the organic skeleton than E L ST , this being a direct consequence of the backside attack accompanying the studied S N 2 mechanism.
Similarly, for small enough Nu or L, the SH seems to arise almost entirely from the direct clashing against the central C atom, being thus quite unaffected by the substitution of the electrophile.
• Counterintuitively, the electrophilic skeleton generally experiences a subtle compression followed by a moderate steric relief. This results from the interplay between a penalizing nucleophile compression and a relieving electrophile planarization.
• Although, as expected, "larger" species would generally result in significant steric compression along a chemical transformation, changes in interatomic distances can yield relevant deviations of E ST from the classically expected trends. As an example, increasing the size of the extruding leaving group (L) results in smaller decompressions of the latter. This is a manifestation of the non-negligible effect of the variable elongation of the C L bond with the nature of L.
• In agreement with the well-known peculiar nature of the F atom within the halogens group, its E F ST shows a somehow distinctive behavior when compared to that of Cl or Br.
• As far as the neopentyl group ((CH 3 ) 3 CCH 2 ) is regarded, the behavior of its E ST fits the general trends found for the remaining substrates. However E E ST experiences a specially pronounced steric decompression during the evolution from the starting reactive complex to the TS. This is attributed to the larger "in-molecule" steric strain inherent to the neopentyl fluoride skeleton, which is attenuated by the planarization phenomenon.
Our results point out that the IQA description of SH matches almost perfectly the trends expected from chemical intuition in S N 2 reactions, reinforcing its validity as a measure of SH. We emphasize that SH should never be the only argument used to rationalize the easiness of S N 2 reactions. Additionally, the steric clashing between two moieties A and B is not necessarily symmetric and special care must be taken when using chemically intuitive SH arguments to understand chemical phenomena. Altogether, the results of this work show that, formally speaking, the concept of SH as it is usually formulated in chemistry can only be isolated from other effects for perturbative like regimes where, indeed, our E ST descriptor seems to follow the trends in reactivity dictated by chemical intuition. On the other hand, the classical picture of SH blurs in strong distortion regimes, such as those found in the vicinity of transition states, where major electronic structure changes make it difficult to tell it from other contributions. Instead, it is likely that in these strongly interacting situations the effects of charge transfer phenomena have been incorrectly attributed to SH. We would also like to highlight that, within the IQA approach, E ST is only one of the three contributions, along with the E int and E CT terms that build up the total energy, suggesting that predicting changes in the latter under the narrow perspective of only one of its components (as it is often the case when SH arguments are used) offers a strongly biased, to say the least, picture of chemistry.

DATA AVAILABILITY STATEMENT
Data available on request from the authors. The data that support the findings of this study are available from the corresponding author upon reasonable request.