A valence bond perspective of the reaction force formalism

  • Luis RinconEmail author
  • F. Javier Torres
  • Jose R. Mora
  • Cesar H. Zambrano
  • Vladimir Rodriguez
Regular Article
Part of the following topical collections:
  1. Chemical Concepts from Theory and Computation


The reaction force formalism represents a convenient approach to analyze the course of a reaction step. From this analysis, the reaction path can be separated in a number of regions that are associated to either structural changes or electronic reorganization. This empirical observation is rationalized in this work on the basis of a simple two-state valence bond correlation diagram. We demonstrate that the ratio between the integrated reaction force and the region of interest (\(w_{\text{ii}}/w_{\text{i}}\) for the forward reaction and \(w_{\text{iii}}/w_{\text{iv}}\) for the backward reaction) increases with the ratio between the quantum mechanical resonance energy and the energy required to reach the crossing point at the transition state, we call to this ratio the strength of the resonance. This observation means that the size of the transition region (region ii and iii), that includes the transition state, depends on the strength of the resonance, and the structural zones (region i and iv), that are regions associated with the pure valence bond state curves (no resonance). We propose a simple analytical relationship for \(w_{\text{ii}}/w_{\text{i}}\) and \(w_{\text{iii}}/w_{\text{iv}}\) based on three parameters: (i) the quantum mechanical resonance energy, (ii) the energy of the reaction and (iii) the overlap between the VB structures at the transition state. The previous conclusions were supported by a reaction force analysis of a \({\text{S}}_{N}2\) reactions, \({\text{X}}^{-} + {\text{CH}}_{3}{-}{\text{Y}} \rightarrow {{\text{X}}{-}{\text{CH}}}_{3} + {\text{Y}}^{-} ({\text{X}} = {\text{F}}, {\text{Cl}}, {\text{Br}})\). The valence bond parameters for these reactions are estimated from empirical considerations. A very good agreement is found between the computed reaction force ratios and the predicted one.


Reaction force Valence bond correlation diagrams Valence bond resonance energy \({\text{S}}_{N}2\) mechanism 



This work has been performed by employing the resources of the USFQ’s High Performance Computing system (HPC-USFQ). The authors would like to thank to the 2019 USFQ’s collaboration grants and Poli-grants program for financial support.


  1. 1.
    Fukui K (1970) Formulation of the reaction coordinate. J Phys Chem 74(23):4161–4163. CrossRefGoogle Scholar
  2. 2.
    Fukui K (1981) The path of chemical reactions-the irc approach. Acc Chem Res 14(12):363–368. CrossRefGoogle Scholar
  3. 3.
    Toro-Labbe A (1999) Characterization of chemical reactions from the profiles of energy, chemical potential, and hardness. J Phys Chem A 103(22):4398–4403. CrossRefGoogle Scholar
  4. 4.
    Politzer P, Toro-Labbe A, Gutierrez-Oliva S, Herrera B, Jaque P, Concha M, Murray J (2005) The reaction force: three key points along the intrinsic reaction coordinate. J Chem Sci 117(5):467–472. CrossRefGoogle Scholar
  5. 5.
    Toro-Labbe A, Gutierrez-Oliva S, Murray J, Politzer P (2007) A new perspective on chemical and physical processes: the reaction force. Mol Phys 105(19–22):2619–2625. CrossRefGoogle Scholar
  6. 6.
    Toro-Labbe A, Gutierrez-Oliva S, Murray J, Politzer P (2009) The reaction force and the transition region of a reaction. J Mol Model 15(6):707–710. CrossRefPubMedGoogle Scholar
  7. 7.
    Murray J, Toro-Labbe A, Clark T, Politzer P (2009) Analysis of diatomic bond dissociation and formation in terms of the reaction force and the position-dependent reaction force constant. J Mol Model 15(6):701–706. CrossRefPubMedGoogle Scholar
  8. 8.
    Politzer P, Reimers J, Murray J, Toro-Labbe A (2010) Reaction force and its link to diabatic analysis: a unifying approach to analyzing chemical reactions. J Chem Phys Lett 1(19):2858–2862. CrossRefGoogle Scholar
  9. 9.
    Politzer P, Toro-Labbe A, Gutierrez-Oliva S, Murray J (2012) Perspectives on the reaction force. Adv Quantum Chem 64:189–209. CrossRefGoogle Scholar
  10. 10.
    Jaque P, Toro-Labbe A, Politzer P, Geerling P (2008) Reaction force constant and projected force constants of vibrational modes along the path of an intermolecular proton transfer reaction. Chem Phys Lett 456:135–140. CrossRefGoogle Scholar
  11. 11.
    Yepez D, Murray J, Politzer P, Jaque P (2012) The reaction force constant: an indicator of the synchronicity in double proton transfer reactions. Phys Chem Chem Phys 14(31):11125–11134. CrossRefGoogle Scholar
  12. 12.
    Politzer P, Murray J, Jaque P (2013) Perspectives on the reaction force constant. J Mol Model 19(10):4111–4118. CrossRefPubMedGoogle Scholar
  13. 13.
    Yepez D, Donoso-Tauda O, Perez P, Murray J, Politzer P, Jaque P (2013) The reaction force constant as an indicator of synchronicity/nonsynchronicity in \([4+2]\) cycloaddition processes. Phys Chem Chem Phys 15(19):7311–7320. CrossRefGoogle Scholar
  14. 14.
    Cortes-Arriaga D, Toro-Labbe A, Mora J, Rincon L, Mereau R, Torres F (2017) Theoretical analysis of c-f bond cleavage mediated by cob[i]alamin-based structures. J Mol Model 23:264. CrossRefGoogle Scholar
  15. 15.
    Politzer P, Burda J, Concha M, Lane P, Murray J (2006) Analysis of the reaction force for a gas phase \({\text{S}}_{N}2\) process. J Phys Chem A 110:756–761. CrossRefPubMedGoogle Scholar
  16. 16.
    Burda J, Toro-Labbe A, Gutierrez-Oliva S, Murray J, Politzer P (2007) Reaction force decomposition of activation barriers to elucidate solvent effects. J Phys Chem A 111(13):2455–2457. CrossRefPubMedGoogle Scholar
  17. 17.
    Giri S, Echegaray E, Ayers P, Nuñez A, Lund F, Toro-Labbe A (2012) Insights into the mechanism of an S\(_{N}\)2 reaction from the reaction force and the reaction electronic flux. J Phys Chem A 116(40):10015–10026. CrossRefPubMedGoogle Scholar
  18. 18.
    Burda J, Morray J, Toro-Labbe A, Gutierrez-Oliva S, Politzer P (2009) Analysis of solvent effects in the addition of hcl to propene. J Phys Chem A 115:6500–6505. CrossRefGoogle Scholar
  19. 19.
    Jaque P, Toro-Labbe A, Geerling P, De Proft F (2009) Theoretical study of the regioselectivity of [2+2] photocycloaddition reaction of acrolein with olefins. J Phys Chem A 113:332–344. CrossRefPubMedGoogle Scholar
  20. 20.
    Toro-Labbe A, Gutierrez-Oliva S, Concha M, Murray J, Politzer P (2004) Analysis of two intramolecular proton transfer processes in terms of the reaction force. J Chem Phys 121(10):4570–4576. CrossRefPubMedGoogle Scholar
  21. 21.
    Herrera B, Toro-Labbe A (2004) The role of the reaction force to characterize local specific interactions that activate the intramolecular proton transfer in dna bases. J Chem Phys 121:7096–7102. CrossRefPubMedGoogle Scholar
  22. 22.
    Rincon E, Jaque P, Toro-Labbe A (2006) A reaction force analysis of the effect of Mg(II) on the 1,3 intramolecular hydrogen transfer in thymmine. J Phys Chem A 120:9478–9485. CrossRefGoogle Scholar
  23. 23.
    Yepez D, Murray J, Santos J, Toro-Labbe A, Politzer P, Jaque P (2013) Fine structure in the transition region: reaction force analyses of water-assisted proton transfer. J Mol Model 19(7):2689–2697. CrossRefGoogle Scholar
  24. 24.
    Inostrosa-Rivera R, Herrera B, Toro-Labbe A (2014) Using the reaction force and the reaction electronic flux on the proton transfer of formamide derived systems. Phys Chem Chem Phys 16:14489–14495. CrossRefGoogle Scholar
  25. 25.
    Murray J, Lane P, Nieder A, Klapotke T, Politzer P (2009) Enhanced detonation sensitivities of silicon analogs of petn: Reaction force analysis and the role of \(\sigma\)-hole interactions. Theor Chem Acc 127(4):345–354. CrossRefGoogle Scholar
  26. 26.
    Murray J, Lane P, Gobel M, Klapotke T, Politzer P (2009) Reaction force analysis of nitro-aci tautomerizations of trinitromethane, the elusive trinitromethanol, picric acid and 2,4-dinitro-1h-imidazole. Theor Chem Acc 124:355–363. CrossRefGoogle Scholar
  27. 27.
    Labet V, Morrel A, Grand A, Toro-Labbe A (2008) Theoretical study of cytosine deamination from the perspective of the reaction force analysis. J Phys Chem A 112:11487–11494. CrossRefPubMedGoogle Scholar
  28. 28.
    Cortes-Arriagada D, Gutierrez-Oliva S, Herrera B, Soto K, Toro-Labbe A (2014) The mechanism of chemisorption of hydrogen atom on graphene: insights from the reaction force and the reaction electronic flux. J Chem Phys 141:134701. CrossRefPubMedGoogle Scholar
  29. 29.
    Villegas-Escobar N, Toro-Labbe A, Becera M, Real-Enriquez M, Mora J, Rincon L (2017) A dft study of hydrogen and methane activation by \(b(c_{6}f_{5})_{3}/p(t-bu)_{3}\) and \(al(c_{6}f_{5})_{3}/p(t-bu)_{3}\) frustrated lewis pairs. J Mol Model 23:234. CrossRefPubMedGoogle Scholar
  30. 30.
    Polanyi J, Zewail A (1995) Direct observation of the transition state. Acc Chem Res 28:199–132. CrossRefGoogle Scholar
  31. 31.
    Shaik S (1981) What happens to molecules as they react? A valence bond approach to reactivity. J Am Chem Soc 103(13):3692–3701. CrossRefGoogle Scholar
  32. 32.
    Pross A, Shaik S (1983) A qualitative valence-bond approach to organic reactivity. Acc Chem Res 16(10):363–370. CrossRefGoogle Scholar
  33. 33.
    Shaik S (1985) The collage of \({\text{S}}_{N}2\) reactivity patterns: a state correlation diagram model. Prog Phys Org Chem 15:197. CrossRefGoogle Scholar
  34. 34.
    Shaik S, Schlegel H, Wolfe S (1992) Theoretical aspects of physical organic chemistry. The S\(_{N}\)2 mechanism. Wiley, New YorkGoogle Scholar
  35. 35.
    Shaik S, Hiberty P (1995) Valence bond mixing and curve crossing diagrams in chemical reactivity and bonding. Adv Quantum Chem 26:99–163. CrossRefGoogle Scholar
  36. 36.
    Shaik S, Shurki A (1999) Vb diagrams and chemical reactivity. Angew Chem Int Ed 38(5):586–625;2-T CrossRefGoogle Scholar
  37. 37.
    Shaik S, Hiberty P (2008) A chemist’s guide to valence bond theory. Wiley, New YorkGoogle Scholar
  38. 38.
    Martinez J, Toro-Labbe A (2004) Energy and chemical force profiles from the marcus equation. Chem Phys Lett 392(1–3):132–139. CrossRefGoogle Scholar
  39. 39.
    Gutierrez-Oliva S, Herrera B, Toro-Labbe A (2017) An estension of the Marcus equation: the Marcus potential energy function. J Mol Model 24:104. CrossRefGoogle Scholar
  40. 40.
    Marcus RA (1964) Chemical and electrochemical electron-transfer theory. Annu Rev Phys Chem 15:155–196. CrossRefGoogle Scholar
  41. 41.
    Marcus RA (1992) Electron transfer reactions in chemistry. Theory and experiment. Rev Mod Phys 65:599–610. CrossRefGoogle Scholar
  42. 42.
    Zhao Y, Truhlar D (2008) Exploring the limit of accuracy of the global hybrid meta density functional for main-group thermochemistry, kinetics, and noncovalent interactions. J Chem Theory Comput 4:1849. CrossRefPubMedGoogle Scholar
  43. 43.
    Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Petersson GA, Nakatsuji H, Li X, Caricato M, Marenich AV, Bloino J, Janesko BG, Gomperts R, Mennucci B, Hratchian HP, Ortiz JV, Izmaylov AF, Sonnenberg JL, Williams-Young D, Ding F, Lipparini F, Egidi F, Goings J, Peng B, Petrone A, Henderson T, Ranasinghe D, Zakrzewski VG, Gao J, Rega N, Zheng G, Liang W, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Throssell K, Montgomery JA Jr, Peralta JE, Ogliaro F, Bearpark MJ, Heyd JJ, Brothers EN, Kudin KN, Staroverov VN, Keith TA, Kobayashi R, Normand J, Raghavachari K, Rendell AP, Burant JC, Iyengar SS, Tomasi J, Cossi M, Millam JM, Klene M, Adamo C, Cammi R, Ochterski JW, Martin RL, Morokuma K, Farkas O, Foresman JB, Fox DJ (2016) Gaussian\(^{\sim }\)16 Revision C.01. Gaussian Inc., WallingfordGoogle Scholar
  44. 44.
    Schlegel HB (1987) Optimization of equilibrium geometries and transition structures. Adv Chem Phys 67:249. CrossRefGoogle Scholar
  45. 45.
    Hratchian HP, Schlegel HB (2004) Accurate reaction paths using a Hessian based predictor–corrector integrator. J Chem Phys 120:9918–9924. CrossRefPubMedGoogle Scholar
  46. 46.
    Jones E, Oliphant T, Peterson P et al (2001) Scipy: open source scientific tools for python.
  47. 47.
    Wu W, Su P, Shaik S, Hiberty P (2011) Classical valence bond approach by modern methods. Chem Rev 111(11):7557–7593. CrossRefPubMedGoogle Scholar
  48. 48.
    Song L, Wu W, Hiberty PC, Shaik S (2006) Identity \({\text{S}}_{N}2\) reactions \(\text{x}^{-} + \text{ch}_{3}\text{x} \rightarrow \text{x-ch}_{3} + \text{x}^{-}\) (x = f, cl, br and i) in vacuum and in aqueos solution: a valence bond study. Chem Eur J 12:7458–7466. CrossRefPubMedGoogle Scholar
  49. 49.
    Shaik SS, Duzy E, Bartuv A (1990) The quantum mechanical resonance energy of transition states. An indicator of transition state grometry and electronic structure. J Phys Chem 94:6574–6582. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Grupo de Química Computacional y Teórica, Departamento de Ingeniería QuímicaUniversidad San Francisco de QuitoQuitoEcuador
  2. 2.Instituto de Simulación ComputacionalUniversidad San Francisco de QuitoQuitoEcuador
  3. 3.Departamento de MatemáticasUniversidad San Francisco de QuitoQuitoEcuador

Personalised recommendations