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Orbital Energy-Based Reaction Analysis of S_{N}2 Reactions

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## Abstract

**:**

^{−}+ CH

_{3}I → ClCH

_{3}+ I

^{−}reaction, for which another reaction path called “roundabout path” is proposed, is found to have a precursor process similar to the roundabout path just before this S

_{N}2 reaction process. The orbital energy-based theory indicates that this precursor process is obviously driven by structural change, while the successor S

_{N}2 reaction proceeds through electron transfer between the contributing orbitals. Comparing the calculated results of the S

_{N}2 reactions in gas phase and in aqueous solution shows that the contributing orbitals significantly depend on solvent effects and these orbitals can be correctly determined by this theory.

## 1. Introduction

_{N}2 reactions are noteworthy in this orbital energy-based reaction analysis because the initial reaction processes of several S

_{N}2 reactions are figured out to be structural change-driven on the minimum energy paths despite these reactions being exothermic [4]. For S

_{N}2 reactions, Hase and coworkers have experimentally questioned the optimum reaction path [14,15,16,17]. Based on a time-resolved ion-molecule cross-beam imaging spectroscopy analysis, they suggested the roundabout reaction path for the Cl

^{−}+ CH

_{3}I → CH

_{3}Cl + I

^{−}reaction [15], in which the CH

_{3}group of CH

_{3}I spins around I

^{−}to move to Cl

^{−}. They also studied the CH

_{3}I + F

^{−}→ CH

_{3}F + I

^{−}reaction mechanism using ion imaging experiments and direct chemical dynamics simulations. Consequently, they proposed three pathways: direct rebound, direct stripping, and indirect hydrogen bonding paths. They also indicated that the results of the experimental rapid intramoleclar vibrational energy distributions and reaction energies are inconsistent with the optimum reaction pathway [16]. Szabó, Czakó, and coworkers performed the dynamics simulation of the F

^{−}+ CH

_{3}Cl → CH

_{3}F + Cl

^{−}reaction by determining its full-dimensional potential energy surfaces [18,19,20]. As a result, they found that the direct rebound mechanism dominates at high collision energies, while the indirect mechanism mainly proceeds at low collision energies [18]. Moreover, they suggested the significance of a double-inversion mechanism, in which the hydrogen atom of the methyl group first moves to F

^{−}and the methyl group then transfers to F

^{−}[19]. Comparing the results of a cross-beam imaging experiment with the dynamics simulation calculation, they, however, found that the mechanism significantly depends on the leaving group, which is X of the F

^{−}+ CH

_{3}X → CH${}_{3}$F + X

^{−}reaction (X = Cl or I) [20]. Based on the above-mentioned orbital energy-based reaction analysis, it is recently found that two S

_{N}2 reactions, F

^{−}+ CH

_{3}Cl → Cl

^{−}+ CH

_{3}F and OHCH

_{3}+ F

^{−}→ OH

^{−}+ CH

_{3}F reactions, are also expected to avoid the minimum energy paths because these reactions give large reactivity indices indicating structural change-driven processes for the initial processes. This suggests that these S

_{N}2 reactions also take other reaction paths than the minimum energy paths to avoid initial structural change-driven processes.

_{N}2 reaction calculations. First, the orbital energy-based reaction analysis theory is formed on the basis of the conceptual DFT [3] in Section 2. After detailing the computational methods in Section 3, the orbital energy-based theory is applied to the calculations of the Cl

^{−}+ CH${}_{3}$I → CH${}_{3}$Cl + I${}^{-}$ reaction and two S${}_{\mathrm{N}}$2 reactions, the Menschutkin reaction (NH${}_{3}$ + CH${}_{3}$Cl → NH${}_{3}$CH${}_{3}^{+}$ + Cl${}^{-}$) and Cl${}^{-}$...CH${}_{3}$Cl → ClCH${}_{3}$...Cl${}^{-}$ reaction, to evaluate the applicability of this orbital energy-based theory in Section 4.

## 2. Orbital Energy-Based Reaction Analysis Theory

- Then, the target orbitals are selected to be the occupied and unoccupied orbitals giving the most varied valence orbital energies. This selection is based on the concept of Equation (7) that reactions proceed to maximize the variation of chemical potentials, i.e., outermost orbital energies.
- The normalized reaction diagram is illustrated by plotting the normalized orbital energy gap:$$\begin{array}{ccc}\hfill {\overline{\Delta}}_{\mathrm{gap}}& =& \frac{{\Delta}_{\mathrm{gap}}-{\Delta}_{\mathrm{gap}}^{\mathrm{initial}}}{{\Delta}_{\mathrm{gap}}^{\mathrm{terminal}}-{\Delta}_{\mathrm{gap}}^{\mathrm{initial}}},\hfill \end{array}$$
- Finally, the orbital energy gap gradient in terms of the normalized IRCs, ${\nabla}_{\overline{V}}{\overline{\Delta}}_{\mathrm{gap}}$, is calculated at the initial reaction stage as a “reactivity index”. Reactions are interpreted to be electron transfer-driven when the orbital energy gap gradient is less than a threshold value, which is temporarily 0.250 in this study.

## 3. Computational Details

## 4. Results and Discussion

#### 4.1. Orbital Energy-Based Reaction Analysis of Cl${}^{-}$ + CH${}_{3}$I → CH${}_{3}$Cl + I${}^{-}$ Reaction

#### 4.2. Reaction Path Search for Cl${}^{-}$ + CH${}_{3}$I → ClCH${}_{3}$ + I${}^{-}$ Reaction

#### 4.3. S${}_{\mathrm{N}}$2 Reactions in Aqueous Solution

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Itatani, J.; Levesque, J.; Zeidler, D.; Niikura, H.; Pepin, H.; Kieffer, J.C.; Corkum, P.B.; Vileneuve, D.M. Tomographic imaging of molecular orbitals. Nature
**2004**, 432, 867–871. [Google Scholar] [CrossRef] [PubMed] - Fukui, K.; Yonezawa, T.; Shingu, H. A molecular orbital theory of reactivity in aromatic hydrocarbons. J. Chem. Phys.
**1952**, 20, 722–725. [Google Scholar] [CrossRef] - Nalewajski, R.F.; Parr, R.G. Legendre transforms and Maxwell relations in density functional theory. J. Chem. Phys.
**1982**, 77, 399–407. [Google Scholar] [CrossRef] - Tsuneda, T.; Singh, R.K. Reactivity index based on orbital energies. J. Comput. Chem.
**2014**, 35, 1093–1100. [Google Scholar] [CrossRef] [PubMed] - Tsuneda, T. Density Functional Theory in Quantum Chemistry; Springer: Tokyo, Japan, 2014. [Google Scholar]
- Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. A long-range correction scheme for generalized-gradient-approximation exchange functionals. J. Chem. Phys.
**2001**, 115, 3540–3544. [Google Scholar] [CrossRef] - Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A long-range-corrected time-dependent density functional theory. J. Chem. Phys.
**2004**, 120, 8425–8433. [Google Scholar] [CrossRef] [PubMed] - Tsuneda, T.; Song, J.W.; Suzuki, S.; Hirao, K. On Koopmans’ theorem in density functional theory. J. Chem. Phys.
**2010**, 133, 174101. [Google Scholar] [CrossRef] [PubMed] - Tsuneda, T.; Kamiya, M.; Hirao, K. Regional self-interaction correction of density functional theory. J. Comput. Chem.
**2003**, 24, 1592–1598. [Google Scholar] [CrossRef] [PubMed] - Nakata, A.; Tsuneda, T.; Hirao, K. Modified Regional Self-Interaction Correction Method Based on the Pseudospectral Method. J. Phys. Chem. A
**2010**, 114, 8521–8528. [Google Scholar] [CrossRef] [PubMed] - Nakata, A.; Tsuneda, T. Density functional theory for comprehensive orbital energy calculations. J. Chem. Phys.
**2013**, 139, 064102. [Google Scholar] [CrossRef] [PubMed] - Singh, R.K.; Tsuneda, T. Reaction energetics on long-range corrected density functional theory: Diels–Alder reactions. J. Comput. Chem.
**2013**, 34, 379–386. [Google Scholar] [CrossRef] [PubMed] - Sham, L.J.; Schlüter, M. Density-functional theory of the band gap. Phys. Rev. B
**1985**, 32, 3883–3889. [Google Scholar] [CrossRef] - Hase, W.L. Simulations of Gas-Phase Chemical Reactions: Applications to S
_{N}2 Nucleophilic Substitution. Science**1994**, 266, 998–1002. [Google Scholar] [CrossRef] [PubMed] - Mikosch, J.; Trippel, S.; Eichhorn, C.; Otto, R.; Lourderaj, U.; Zhang, J.X.; Hase, W.L.; Weidemuller, M.; Wester, R. Imaging nucleophilic substitution dynamics. Science
**2008**, 319, 183–186. [Google Scholar] [CrossRef] [PubMed] - Zhang, J.; Mikosch, J.; Trippel, S.; Otto, R.; Weidemüller, M.; Wester, R.; Hase, W.L. F
^{−}+ CH_{3}I → FCH_{3}+ I^{−}reaction dynamics. Nontraditional atomistic mechanisms and formation of a hydrogen-bonded complex. J. Phys. Chem. Lett.**2010**, 1, 2747–2752. [Google Scholar] [CrossRef] - Zhang, J.; Lourderaj, U.; Sun, R.; Mikosch, J.; Wester, R.; Hase, W.L. Simulation studies of the Cl
^{−}+ CH_{3}I S_{N}2 nucleophilic substitution reaction: Comparison with ion imaging experiments. J. Chem. Phys.**2013**, 138, 114309. [Google Scholar] [CrossRef] [PubMed] - Szabó, I.; Császár, A.G.; Czakó, G. Dynamics of the F
^{−}+ CH_{3}Cl → Cl^{−}+ CH_{3}F S_{N}2 reaction on a chemically accurate potential energy surface. Chem. Sci.**2013**, 4, 4362–4370. [Google Scholar] [CrossRef] - Szabó, I.; Czakó, G. Revealing a double-inversion mechanism for the F
^{−}+ CH_{3}Cl S_{N}2 reaction. Nat. Commun.**2015**, 6, 5972–5977. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stei, M.; Carrascosa, E.; Kainz, M.A.; Kelkar, A.H.; Meyer, J.; Szabó, I.; Czakó, G.; Wester, R. Influence of the leaving group on the dynamics of a gas-phase S
_{N}2 reaction. Nat. Chem.**2016**, 8, 151–156. [Google Scholar] [CrossRef] [PubMed] - Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. B
**1964**, 136, 864–871. [Google Scholar] [CrossRef] - Levy, M. Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proc. Natl. Acad. Sci. USA
**1979**, 76, 6062–6065. [Google Scholar] [CrossRef] [PubMed] - Tsuneda, T. Chemical Reaction Analyses Based on Orbitals and Orbital Energies. Int. J. Quantum Chem.
**2015**, 115, 270–282. [Google Scholar] [CrossRef] - Janak, J.F. Proof that ∂E/∂n
_{i}= ϵ in density-functional theory. Phys. Rev. B**1978**, 18, 7165–7168. [Google Scholar] [CrossRef] - Hratchian, H.P.; Schlegel, H.B. Accurate reaction paths using a Hessian based predictor-corrector integrator. J. Chem. Phys.
**2004**, 120, 9918–9924. [Google Scholar] [CrossRef] [PubMed] - Hratchian, H.P.; Schlegel, H.B. Using Hessian updating to increase the efficiency of a Hessian based predictor-corrector reaction path following method. J. Chem. Theory Comput.
**2005**, 1, 61–69. [Google Scholar] [CrossRef] [PubMed] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev. A
**1965**, 140, 1133–1138. [Google Scholar] [CrossRef] - Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys.
**2005**, 7, 3297–3305. [Google Scholar] [CrossRef] [PubMed] - Weigend, F. Accurate Coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys.
**2006**, 8, 1057–1065. [Google Scholar] [CrossRef] [PubMed] - Maeda, S.; Ohno, K.; Morokuma, K. Systematic exploration of the mechanism of chemical reactions: The global reaction route mapping (GRRM) strategy using the ADDF and AFIR methods. Phys. Chem. Chem. Phys.
**2013**, 15, 3683–3701. [Google Scholar] [CrossRef] [PubMed] - Maeda, S.; Taketsugu, T.; Morokuma, K. Exploring transition state structures for intramolecular pathways by the artificial force induced reaction method. J. Comput. Chem.
**2014**, 35, 166–173. [Google Scholar] [CrossRef] [PubMed] - Kamiya, M.; Tsuneda, T.; Hirao, K. A density functional study of van der Waals interactions. J. Chem. Phys.
**2002**, 117, 6010–6015. [Google Scholar] [CrossRef] - Sato, T.; Tsuneda, T.; Hirao, K. Van der Waals interactions studied by density functional theory. Mol. Phys.
**2005**, 103, 1151–1164. [Google Scholar] [CrossRef] - Sato, T.; Tsuneda, T.; Hirao, K. Long-range corrected density functional study on weakly bound systems: Balanced descriptions of various types of molecular interactions. J. Chem. Phys.
**2007**, 126, 1402–1406. [Google Scholar] [CrossRef] [PubMed] - Sato, T.; Nakai, H. Density functional method including weak interactions: Dispersion coefficients based on the local response approximation. J. Chem. Phys.
**2009**, 131, 224104. [Google Scholar] [CrossRef] [PubMed] - Becke, A.D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A
**1988**, 38, 3098–3100. [Google Scholar] [CrossRef] - Tsuneda, T.; Suzumura, T.; Hirao, K. A new one-parameter progressive Colle-Salvetti-type correlation functional. J. Chem. Phys.
**1999**, 110, 10664–10678. [Google Scholar] [CrossRef] - Barone, V.; Cossi, M. Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model. J. Phys. Chem. A
**1998**, 102, 1995–2001. [Google Scholar] [CrossRef] - Lee, C.; Yang, W.; Parr, R.G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B
**1988**, 37, 785–789. [Google Scholar] [CrossRef] - Schmidt, M.W.; Baldridge, K.K.; Boatz, J.A.; Elbert, S.T.; Gordon, M.S.; Jensen, J.H.; Koseki, S.; Matsunaga, N.; Nguyen, K.A.; Su, S.; et al. General atomic and molecular electronic structure system. J. Comput. Chem.
**1993**, 14, 1347–1363. [Google Scholar] [CrossRef] - Parr, R.G.; Chattaraj, P.K. Principle of maximum hardness. J. Am. Chem. Soc.
**1991**, 113, 1854–1855. [Google Scholar] [CrossRef]

**Figure 1.**A normalized reaction diagram, in which both the gaps and intrinsic reaction coordinates are normalized by setting the values of the reactant and product to be zero and one, respectively. For both forward and backward reactions, the orbital energy gap gradients are estimated by fit to polynomial functions.

**Figure 2.**The normalized reaction diagram of the Cl${}^{-}$ + CH${}_{3}$I → CH${}_{3}$Cl + I${}^{-}$ reaction by LC-BOP/Def2-TZVPD calculation. The images of the molecular orbitals mainly contributing to each reaction process are also shown.

**Figure 3.**The potential energy curve of the overall Cl${}^{-}$ + CH${}_{3}$I → CH${}_{3}$Cl + I${}^{-}$ reaction including its precursor process, which is calculated by the reaction path search method in the GRRM program with LC-BOP+LRD/Def2-TZVPD. The molecular structures are also shown for the local minima (Min) and transition states (TS) on the potential energy curve.

**Figure 4.**The normalized reaction diagrams of the three steps of the Cl${}^{-}$ + CH${}_{3}$I → CH${}_{3}$Cl + I${}^{-}$ reaction given in the potential energy curve of Figure 3, which is calculated by LC-BOP+LRD/Def2-TZVPD. The images of the molecular orbitals mainly contributing to each reaction process are also shown.

**Figure 5.**The potential energy curves and the global hardnesses of the Menschutkin reaction (NH${}_{3}$ + CH${}_{3}$Cl → NH${}_{3}$CH${}_{3}^{+}$ + Cl${}^{-}$) in gas phase and in aqueous solution on their intrinsic reaction coordinates, which are calculated by LC-BLYP/aug-cc-pVTZ. The images of the molecular orbitals mainly contributing to each reaction process are also shown.

**Figure 6.**The normalized reaction diagrams of the three steps of the Menschutkin reaction (NH${}_{3}$ + CH${}_{3}$Cl → NH${}_{3}$CH${}_{3}^{+}$ + Cl${}^{-}$) in gas phase and in aqueous solution, which are calculated by LC-BLYP/aug-cc-pVTZ. The images of the molecular orbitals mainly contributing to each reaction process and the calculated orbital energy gap gradient values are also shown.

**Figure 7.**The potential energy curves and the global hardnesses of the Cl${}^{-}$...CH${}_{3}$Cl → ClCH${}_{3}$...Cl${}^{-}$ reaction in gas phase and in aqueous solution on their intrinsic reaction coordinates, which are calculated by LC-BLYP/aug-cc-pVTZ. The images of the molecular orbitals mainly contributing to each reaction process are also shown.

**Figure 8.**The normalized reaction diagrams of the Cl${}^{-}$...CH${}_{3}$Cl → ClCH${}_{3}$...Cl${}^{-}$ reaction in gas phase and in aqueous solution, which are calculated by LC-BLYP/aug-cc-pVTZ. The images of the molecular orbitals mainly contributing to each reaction process and the calculated orbital energy gap gradient values are also shown.

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**MDPI and ACS Style**

Tsuneda, T.; Maeda, S.; Harabuchi, Y.; Singh, R.K.
Orbital Energy-Based Reaction Analysis of S_{N}2 Reactions. *Computation* **2016**, *4*, 23.
https://doi.org/10.3390/computation4030023

**AMA Style**

Tsuneda T, Maeda S, Harabuchi Y, Singh RK.
Orbital Energy-Based Reaction Analysis of S_{N}2 Reactions. *Computation*. 2016; 4(3):23.
https://doi.org/10.3390/computation4030023

**Chicago/Turabian Style**

Tsuneda, Takao, Satoshi Maeda, Yu Harabuchi, and Raman K. Singh.
2016. "Orbital Energy-Based Reaction Analysis of S_{N}2 Reactions" *Computation* 4, no. 3: 23.
https://doi.org/10.3390/computation4030023