Rethinking the X (cid:2) + CH 3 Y [X = OH, SH, CN, NH 2 , PH 2 ; Y = F, Cl, Br, I] S N 2 reactions †

Moving beyond the textbook mechanisms of bimolecular nucleophilic substitution (S N 2) reactions, we characterize several novel stationary points and pathways for the reactions of X (cid:2) [X = OH, SH, CN, NH 2 , PH 2 ] nucleophiles with CH 3 Y [Y = F, Cl, Br, I] molecules using the high-level explicitly-correlated CCSD(T)-F12b method with the aug-cc-pV n Z(-PP) [ n = D, T, Q] basis sets. Besides the not-always-existing traditional pre- and post-reaction ion-dipole complexes, X (cid:2) (cid:3)(cid:3)(cid:3) H 3 CY and XCH 3 (cid:3)(cid:3)(cid:3) Y (cid:2) , and the Walden-inversion transition state, [X–CH 3 –Y] (cid:2) , we find hydrogen-bonded X (cid:2) (cid:3)(cid:3)(cid:3) HCH 2 Y (X = OH, CN, NH 2 ; Y a F) and front-side H 3 CY (cid:3)(cid:3)(cid:3) X (cid:2) (Y a F) complexes in the entrance and hydrogen-bonded XH 2 CH (cid:3)(cid:3)(cid:3) Y (cid:2) (X = SH, CN, PH 2 ) and H 3 CX (cid:3)(cid:3)(cid:3) Y (cid:2) (X = OH, SH, NH 2 ) complexes in the exit channels depending on the nucleophile and leaving group as indicated in parentheses. Retention pathways via either a high-energy front-side attack barrier, XYCH 3 (cid:2) , or a novel double-inversion transition state, XH (cid:3)(cid:3)(cid:3) CH 2 Y (cid:2) , having lower energy for X = OH, CN, and NH 2 and becoming submerged (barrier-less) for X = OH and Y = I as well as X = NH 2 and Y = Cl, Br, and I, are also investigated.


I. Introduction
Since the discovery of an optical inversion in a chemical reaction by Paul Walden in 1896, 1 bimolecular nucleophilic substitution (S N 2) has become a widely known reaction class in organic chemistry. The atomic-level mechanisms of S N 2 reactions were already described in the 1930s by Ingold and co-workers. 2 The traditional S N 2 Walden-inversion pathway goes through a double-well potential featuring ion-dipole complexes in the entrance-and exit-channel wells separated by a central transition state (first-order saddle point). In a typical S N 2 reaction X À + CH 3 Y, the nucleophile (X À ) attacks the back side (methyl side) of the CH 3 Y molecule and while a new X-C bond forms and the C-Y bond breaks an umbrella motion of the H atoms inverts the configuration around the carbon center. The above-described Walden-inversion mechanism of S N 2 reactions has become textbook material and is probably the best-known stereo-specific reaction pathway in chemistry. However, in 2016 Xie and Hase 3 published a perspective article ''rethinking the S N 2 reaction'', because ''the gas-phase dynamics of a paradigm organic reaction are more complex than expected''. This statement on the complexity of S N 2 reactions was based on recent experimental and theoretical findings revealing several novel and unexpected reaction mechanisms for these paradigm systems. [4][5][6][7][8][9] For example, S N 2 reactions can proceed with direct rebound, stripping, and front-side attack mechanisms as well as via indirect ion-dipole, hydrogen-bond, and front-side complex forming, roundabout, 4 and double-inversion 7 pathways. 3 Among the above mechanisms front-side attack is a retention pathway, which proceeds via a high-energy XYCH 3 À -like transition state with X-C-Y angle of around 801. 10 This front-side attack retention pathway has been known from the early 1930s; 2 however, a quantitative description of the front-side attack dynamics was just recently achieved in our group utilizing newly developed analytical ab initio potential energy surfaces. 7,11,12 Furthermore, our reaction dynamics simulations on these analytical potentials revealed a new retention mechanism called double inversion, 7 where a protonabstraction induced inversion (first inversion) is followed by a Walden inversion (second inversion) resulting in retention of the initial configuration. The non-traditional front-side complex (H 3 CYÁ Á ÁX À ) formation was also quantitatively characterized by our trajectory simulations 13 motivated by the joint work with the experimental group of Wester and co-workers. 8 Another nontraditional pre-reaction complex stabilized by a hydrogen-bond between X À and the methyl group of CH 3 Y was also found first by Hase and co-workers 14 and later in our theoretical studies. 7,11,12,15 In the present work we focus on ''rethinking'' the gas-phase X À + CH 3 Y [X = OH, SH, CN, NH 2 , PH 2 ; Y = F, Cl, Br, I] S N 2 reactions using high-level ab initio methods. Among the 20 possible fundamental S N 2 reactions with the above-defined 5 different nucleophiles (X À ) and 4 leaving groups (Y), there are only a few which were studied previously in the gas [16][17][18][19][20][21][22][23][24] and/or condensed [25][26][27][28][29] phases, and almost none of them in view of the recent non-traditional mechanisms. Focusing on the gas-phase studies, in the case of Y = F Gonzales et al. 16,17 characterized 3 stationary points, i.e., pre-and post-reaction complexes and a Walden-inversion transition state, for each reaction; however, front-side complex formation, a front-side attack transition state, and double inversion were not investigated, partially due to the fact that some of these were not known at that time. For OH À + CH 3 F Hase and co-workers 18,19 described the same 3 stationary points as Gonzales et al. 16,17 and reported interesting direct dynamics results showing that trajectories usually avoid the deep post-reaction minimum. 18 Considering ligands other than Y = F, Tachikawa and co-workers 20 investigated again 3 stationary points for the OH À + CH 3 Cl reaction and recently Wester and co-workers 21 reported crossed-beam experiments for the CN À + CH 3 I system and characterized 3 stationary points for each of the C-C and C-N bond forming S N 2 reactions. In the case of X = OH and SH, the X À + CH 3 Y [Y = F, Cl, Br] reactions were investigated by Longo and co-workers 22 reporting again 3 stationary points for each system. Perhaps the most thoroughly studied reaction is OH À + CH 3 I, for which H-bonded and frontside complex formation were studied in the Hase group 24 and experiments were performed in the groups of Viggiano 23 and Wester. 24 In 2018 we reported a high-level ab initio study on the OH À + CH 3 Y [Y = F, Cl, Br, I] S N 2 reactions applying coreand post-CCSD(T) correlation corrections, which are usually neglected in the literature, and revealing many stationary points involving H-bonded and front-side complexes as well as front-side attack and double-inversion transition states. 30 In the present work we extend our previous studies 9, 30 considering SH À , CN À , NH 2 À , and PH 2 À nucleophiles besides OH À , which is described again for completeness and comparison. For the first time for X = SH, CN, NH 2 , and PH 2 we use the high-level modern explicitly-correlated CCSD(T)-F12b method 31 to obtain benchmark structures and relative energies for the stationary points. However, instead of presenting technical details of a high-level electronic structure study, the present work aims to provide new qualitative insights into the reaction pathways of 20 different S N 2 reactions, showing that these systems ''are more complex than expected'', 3 supporting the views of the above-mentioned ''rethinking'' perspective article. 3

II. Computational details
The energies, geometries, and harmonic vibrational frequencies of the stationary points are computed using second-order Møller-Plesset perturbation theory 32 (MP2) with the correlationconsistent aug-cc-pVDZ basis set 33 and the explicitly correlated coupled cluster singles, doubles, and perturbative triples CCSD(T)-F12b method 31 with the aug-cc-pVDZ and the aug-cc-pVTZ basis sets. 33 To obtain the benchmark classical relative energies at the CCSD(T)-F12b/aug-cc-pVTZ geometries singlepoint computations are performed using the CCSD(T)-F12b method with the quadruple-z aug-cc-pVQZ basis set. 33 For bromine and iodine relativistic small-core effective core potentials 34 and the corresponding pseudo-potential basis sets 34 are employed. The benchmark adiabatic relative energies are determined as:

DE[CCSD(T)-F12b/aug-cc-pVQZ]
where DE is the benchmark classical relative energy and DZPE is the harmonic zero-point energy correction. Note that if the CCSD(T)-F12b/aug-cc-pVTZ or the CCSD(T)-F12b/aug-cc-pVDZ geometry optimizations do not converge the CCSD(T)-F12b/ aug-cc-pVDZ or the MP2/aug-cc-pVDZ structures are utilized, respectively. All the structural parameters, harmonic frequencies, absolute energies, and relative energies can be found in the ESI. † For the best stationary-point geometries Cartesian coordinates are also given in the ESI † in a software-friendly format. We note that in the case of some minima small artificial imaginary frequencies (o100i cm À1 ) are found most likely due to the uncertainty of the numerical gradient and/or Hessian computations. For the present ab initio study the Molpro program package 35 is used.

III. Results and discussion
The OH À + CH 3 Y [Y = F, Cl, Br, I] S N 2 reactions are exothermic and may proceed via back-side attack inversion, front-side attack retention, or double-inversion retention pathways as shown in Fig. 1. Back-side attack is a collective name for several direct and indirect inversion mechanisms such as rebound, stripping, ion-dipole complex formation, hydrogen-bond complex formation, front-side complex formation, and roundabout. The traditional picture says that back-side attack Walden inversion proceeds via a pre-reaction ion-dipole complex (PreMIN), a Walden-inversion transition state (WaldenTS), and a postreaction ion-dipole complex HOCH 3 Á Á ÁY À . However, instead of the traditional HOCH 3 Á Á ÁY À complex, Y À connects to the OH group with a single hydrogen bond forming a PostHMIN complex for all the four leaving groups, as shown in Fig. 1. Thus, Walden inversion of OH À + CH 3 F proceeds as PreMIN -WaldenTS -PostHMIN. (Note that besides PostHMIN a hydrogen-bonded F À Á Á ÁHCH 2 OH complex also exists, which is not shown in Fig. 1, but presented in the ESI. †) This is not the end of the nontraditional mechanisms, because for Y = Cl, Br, and I a hydrogen-bonded pre-reaction complex (HMIN), where OH À connects to one of the H atoms of the methyl group, and a transition state (HTS) connecting HMIN and PREMIN come into play, where HMIN is below PREMIN by about 2 kcal mol À1 . In the case of Y = Cl and Br, OH À + CH 3 Y Walden inversion occurs via HMIN -HTS -PreMIN -WaldenTS -PostHMIN. However, for Y = I the traditional PreMIN and WaldenTS do not exist and the OH À + CH 3 I inversion proceeds as HMIN -HTS -PostHMIN. Furthermore, for Y = Cl, Br, and I a front-side complex (FSMIN), where OH À connects to Y, can also be found, which in the case of Y = I is actually a significantly deeper minimum than HMIN, suggesting a steering effect into a nonreactive orientation, thereby making the OH À + CH 3 I reaction indirect as previous experiments 5, 8 and our dynamics simulations 13 suggested for F À + CH 3 I. Unlike back-side attack inversion which has submerged stationary points, front-side attack retention proceeds via a highenergy transition state (FSTS) with adiabatic barrier heights of 43.1 (F), 29.2 (Cl), 22.9 (Br), and 17.7 (I) kcal mol À1 , whereas the corresponding double-inversion transition state (DITS) barrier heights are only 16.9 (F), 3.5 (Cl), 1.0 (Br), and À3.9 (I) kcal mol À1 . Thus, double inversion opens a barrier-less retention pathway for the OH À + CH 3 I S N 2 reaction.
If we replace the OH À nucleophile with SH À , the S N 2 reactions become less exothermic and, moreover, endothermic for Y = F, as shown in Fig. 2. Unlike OH À , SH À does not form a hydrogenbonded pre-reaction complex, except for Y = I as shown in the ESI, † and the back-side attack inversion pathway proceeds as PreMIN -WaldenTS -PostHMIN. The WaldenTSs are below the reactants by 1.4, 6.0, and 8.5 kcal mol À1 , for Y = Cl, Br, and I, respectively, whereas the SH À + CH 3 F S N 2 reaction has a positive adiabatic barrier height of 13.7 kcal mol À1 . Similar to the reactions of OH À , the global minimum of the potential energy surfaces of the SH À + CH 3 Y systems is PostHMIN, but in the present case Y À Á Á ÁHC bonded (PostHMIN2) and traditional ion-dipole post-reaction (WaldenPostMIN) complexes are also found with similar energies within 1 kcal mol À1 and above POSTHMIN by a few kcal mol À1 , except for Y = F, where PostHMIN2 is just slightly bonded and WaldenPostMIN does not exist, as shown in Fig. 2. The dissociation energies of the PostHMIN complexes are around 10 kcal mol À1 for Y = Cl, Br, and I, whereas much larger, about 38 kcal mol À1 , for Y = F. In the case of X = OH, the Y = Cl, Br, and I PostHMIN complexes are more stable by a few kcal mol À1 , however, the Y = F complex has a dissociation energy of only about 30 kcal mol À1 . These energy trends are in accord with the Y À Á Á ÁHX distances, as these are slightly longer for X = SH than OH if Y = Cl, Br, and I, whereas the F À Á Á ÁHS distance (0.985 Å) is significantly shorter than the F À Á Á ÁHO distance (1.345 Å) as shown in Fig. 1 and 2. Similar to the OH À + CH 3 Y [Y = Cl, Br, I] reactions, FSMIN complexes are also found for X = SH systems, but with longer YÁ Á ÁX À distances and consequently with less stability. Here the H 3 CIÁ Á ÁSH À complex is just similarly stable to PreMIN, whereas FSMIN is a significantly deeper minimum than PREMIN in the case of the OH À complex. We find qualitatively similar DITS and FSTS structures for the reactions of OH À and SH À , with significantly higher barrier heights for SH À . Furthermore, unlike in the case of the OH À systems, the DITSs of the SH À nucleophile are always above the corresponding FSTSs.
The CN À nucleophile can form either a C-C or a C-N bond in the S N 2 reactions with the CH 3 Y molecules. In the present work we study the energetically favored, more exothermic C-C bond formation channel leading to Y À + CH 3 CN products. The CN À + CH 3 Y reactions are more and less exothermic than the corresponding SH À and OH À reactions, respectively. Since the CN ligand does not have a hydrogen atom, PostHMIN-type postreaction complexes do not exist. Here the global minimum is either a traditional ion-dipole WaldenPostMIN (C 3v symmetry) or a hydrogen-bonded PostHMIN2 (C s symmetry) complex, View Article Online whose energies agree within 1 kcal mol À1 as shown in Fig. 3. Note that for Y = F, only PostHMIN2 is found. Traditional PreMINs and WaldenTSs are found for all the four leaving groups and unlike in the case of OH À and SH À , these stationary points have C 3v symmetry with collinear Y-C-C-N arrangements. WaldenTSs are clearly submerged for Y = Br and I with classical (adiabatic) energies of À4.2 (À3.7) and À6.1 (À5.6) kcal mol À1 , respectively, relative to the reactants. For Y = Cl the barrier height is close to zero, i.e. À0.2 (0.2) kcal mol À1 , whereas for Y = F Walden inversion has a positive barrier of 12.2 (12.3) kcal mol À1 , similar to the SH À + CH 3 F reaction. Besides the ion-dipole PreMIN complexes, we find hydrogen-bonded complexes (HMIN1) for Y = Br and I and a transition state connecting HMIN1 and PreMIN for Y = I as seen in Fig. 3. Unlike in the OH À case, here HMIN1 is  slightly above PreMIN. Furthermore, for all the four CN À + CH 3 Y systems we find pre-reaction complexes (HMIN2), where the C and N atoms of CN À are connected to the H and C atoms of CH 3 Y, respectively, as seen in Fig. 3. These HMIN2 complexes are slightly more stable than the PREMINs; thus, HMIN2 configurations are the deepest regions of the entrance channels. Similar to the OH À and SH À reactions, front-side FSMIN complexes are found for Y = Cl, Br, and I, but unlike in the former cases, the H 3 CYÁ Á ÁCN À complexes have C 3v symmetry. In the Y = I case the depth of FSMIN is similar, 10 AE 1 kcal mol À1 , to the other minima in the entrance channel, whereas for Y = Br the classical (adiabatic) FSMIN depth is only 3.4 (3.1) kcal mol À1 . For Y = Cl FSMIN is unbonded, with an energy slightly above the reactant asymptote. The retention pathways have large barriers between 30 and 60 kcal mol À1 depending on the leaving group. Similar to the OH À + CH 3 Y systems, DITSs are always below the corresponding FSTSs, but whereas the OH À + CH 3 I reaction has a negative double-inversion barrier, the classical (adiabatic) barrier height of the CN À + CH 3 I reaction is 31.9 (29.9) kcal mol À1 .
The S N 2 reactions of NH 2 À with CH 3 Y are the most exothermic among the systems studied in this work; the enthalpies of the NH 2 À + CH 3 Y reactions are more negative by 15-20 kcal mol À1 than those of the corresponding, also highly exothermic OH À + CH 3 Y processes. Similar to the reactions of OH À , the global minima of the NH 2 À + CH 3 Y potential energy surfaces correspond to PostHMIN complexes, where Y À is connected to the NH 2 group of CH 3 NH 2 with a single hydrogen bond as seen in  PreMIN -WaldenTS -PostHMIN inversion pathway featuring a slightly submerged transition state with a classical (adiabatic) energy of À3.0 (À1.9) kcal mol À1 relative to the reactants. For Y = Cl a hydrogen-bonded pre-reaction complex (HMIN) is also found with similar energy, around À14 kcal mol À1 , to that of PreMIN and WaldenTS is at À12.8 (À12.1) kcal mol À1 . In the case of Y = Br we could not find a WaldenTS, and HMIN is deeper than PreMIN by about 1 kcal mol À1 . For Y = I neither a traditional PreMIN nor WaldenTS is found, but HMIN and a pre-reaction transition state (PreTS) exist as seen in Fig. 4. The submerged PreTS with a classical (adiabatic) energy of À15.6 (À15.2) kcal mol À1 relative to the reactants differs from WaldenTS in the orientation of the NH 2 unit and the relative stretching of the C-Y distance, because C-F and C-Cl WaldenTS distances are stretched by 0.324 and 0.220 Å relative to the corresponding bond lengths of the CH 3 Y molecule, whereas the C-I PreTS distance is just longer by 0.107 Å than the same bond in CH 3 I. Therefore, PreTS may be viewed as an early WaldenTS. As always front-side complexes are found for Y = Cl, Br, and I with D e (D 0 ) dissociation energies of 1.7 (1.1), 12.5 (11.5), and 25.8 (24.4) kcal mol À1 , respectively. Thus, the stability of the front-side complex is negligible for Y = F, comparable to that of HMIN in the case of Y = Br, and Results indexed by * correspond to MP2/aug-cc-pVDZ structures. ''Experimental'' 0 K reaction enthalpies, with AE0.1 kcal mol À1 uncertainties, are À34.7, À66.1, À73.7, and À79.9 kcal mol À1 for Y = F, Cl, Br, and I, respectively. 36 significantly stronger than that of HMIN for Y = I. Note that H 3 CIÁ Á ÁNH 2 À is the most strongly bonded front-side complex investigated in the present study. The retention pathways of the title reactions are the most favored in the case of the NH 2 À nucleophile. All the DITS and FSTS barrier heights are below the corresponding, also favorable, OH À barriers by about 10 kcal mol À1 . Furthermore, all the DITS barriers are below the corresponding FSTS ones by about 20 kcal mol À1 ; thus, double inversion opens a significantly lower energy retention pathway compared to the more traditional front-side attack. Moreover, for Y = Cl, Br, and I double inversion has negative classical (adiabatic) barrier heights of À3.6 (À2.6), À6.3 (À5.2), and À11.7 (À10.3) kcal mol À1 , respectively, opening retention pathways which may become competitive with inversion especially for the iodine leaving group.
The PH 2 À + CH 3 Y reactions are less exothermic than the analogous NH 2 À + CH 3 Y systems due to the weaker C-P bonds compared to the binding energy of a C-N bond. For the reactions of PH 2 À we find a PostHMIN complex only for Y = F; however, its structure differs from the geometry of the PostHMIN complexes of the other nucleophiles. Here, the PÁ Á ÁH distance is significantly stretched, whereas the F-H distance is close to the bond length in the HF molecule; thus, this PostHMIN is better viewed as a post-reaction proton-abstraction (proton transfer from the PH 2 group to F À ) complex. For all the four leaving groups we find hydrogen-bonded complexes in the exit channel (PostHMIN2), where Y À connects to one of the H atoms of the methyl group as seen in Fig. 5. Note that the protonabstraction-like PostHMIN, found for Y = F, is a deeper minimum than PostHMIN2, but the enthalpy of the HF + CH 3 PH À channel (À8.4 kcal mol À1 ) is less exothermic than that corresponding to the F À + CH 3 PH 2 S N 2 products (À10.3 kcal mol À1 ). A traditional ion-dipole complex (WaldenPostMIN) is only found for Y = Br, but with slightly less stability than the corresponding PostHMIN2.
In the entrance channel hydrogen-bonded complexes are not found unlike for the analogous NH 2 À reactions. For Y = F, Cl, and Br traditional ion-dipole complexes (PreMIN) and WaldenTSs are found as shown in Fig. 5

IV. Summary and conclusions
Motivated by a recent perspective article entitled ''Rethinking the S N 2 reaction'' 3 and our previous findings about a novel double-inversion 7 pathway and front-side complex formation, 13 we have investigated the stationary points and possible reaction mechanisms of 20 different gas-phase S N 2 reactions involving OH À , SH À , CN À , NH 2 À , and PH 2 À nucleophiles and methylhalide molecules (CH 3 Y, Y = F, Cl, Br, I). Previous work usually reported three stationary points, i.e., pre-and post-reaction complexes and a central transition state, for some of these S N 2 systems. In the present comprehensive high-level explicitlycorrelated ab initio study we reveal that these reactions are much more complex and many stationary points play a role in the dynamics. In the entrance channel, besides the traditional ion-dipole complex and Walden-inversion transition state, hydrogen-bonded and front-side complexes are found. Pre-reaction hydrogen-bond formation usually occurs between OH À , CN À , and NH 2 À nucleophiles and CH 3 Y, where Y = Cl, Br, and I, and the hydrogen-bonded complex is more stable than the traditional ion-dipole complex for OH À and NH 2 À . Front-side complexes, where the nucleophile connects to the leaving group, are found in all cases except Y = F. For X = OH, NH 2 , and PH 2 , the H 3 CIÁ Á ÁX À complex is the deepest region in the entrance channel suggesting indirect dynamics, because the deep front-side minimum may steer the reactants into a non-reactive orientation. In the exit channel a hydrogenbonded H 3 CXÁ Á ÁY À complex may be formed for X = OH, SH, and NH 2 , which is the global minimum of the potential energy surface. In the case of X = CN and PH 2 , the Y À rather forms a hydrogen bond with one of the H atoms of the methyl group or in some cases a traditional ion-dipole complex is also found. Besides the Walden-inversion mechanisms, we report, in most cases for the first time, front-side attack and double-inversion retention pathways for the title reactions. Front-side attack always has a positive, usually large, barrier, whereas the double-inversion barrier heights are below the front-side attack ones in the case of the OH À , CN À , and NH 2 À nucleophiles.
Moreover, for the OH À + CH 3 I and the NH 2 À + CH 3 Y [Y = Cl, Br, I] reactions the double-inversion transition state is submerged, thereby opening a barrier-less retention pathway for these S N 2 reactions. We hope that the present study motivates future experimental and theoretical investigations on the dynamics of the title reactions and shapes the way how we think about S N 2 reactions.

Conflicts of interest
There are no conflicts of interest to declare.