Electronic Stress as a Guiding Force for Chemical Bonding

  • Alfredo Guevara-García
  • Paul W. AyersEmail author
  • Samantha Jenkins
  • Steven R. Kirk
  • Eleonora Echegaray
  • Alejandro Toro-Labbe
Part of the Topics in Current Chemistry book series (TOPCURRCHEM, volume 351)


In the electron-preceding picture of chemical change, the paramount problem is identifying favorable changes in electronic structure. The electronic stress tensor provides this information; its eigenvectors represent electronic normal modes, pointing the way towards energetically favorable (or unfavorable) chemical rearrangements. The resulting method is well founded in both density functional theory and the quantum theory of atoms in molecules (QTAIM). Stress tensor analysis is a natural way to extend the QTAIM to address chemical reactivity. The definition and basic properties of the electronic stress tensor are reviewed and the inherent ambiguity of the stress tensor is discussed. Extending previous work in which the stress tensor was used to analyze hydrogen-bonding patterns, this work focuses on chemical bonding patterns in organic reactions. Other related material (charge-shift bonding, links to the second-density-derivative tensor) is summarized and reviewed. The stress tensor provides a multifaceted characterization of bonding and can be used to predict and describe bond formation and migration.


Chemical reaction prediction Conceptual density functional theory Ehrenfest force Electronic stress tensor Reaction force partitioning Quantum theory of atoms in molecules 



PWA thanks the Canada Research Chairs and NSERC for funding. Computational facilities were provided by Sharcnet. AG-G was supported by a fellowship from CONACYT (Mexico). The Knowledge Foundation (grant number 2004/0284) and the Hundred Talents Foundation of Hunan Province are gratefully acknowledged for the support of SJ and SRK. EE and ATL acknowledge financial support from Fondecyt through project No. 1090460


  1. 1.
    Nakatsuji H (1974) Common nature of the electron cloud of a system undergoing change in nuclear configuration. J Am Chem Soc 96:24–30CrossRefGoogle Scholar
  2. 2.
    Nakatsuji H (1974) Electron-cloud following and preceding and the shapes of molecules. J Am Chem Soc 96:30–37CrossRefGoogle Scholar
  3. 3.
    Nichols J, Taylor H, Schmidt P, Simons J (1990) Walking on potential-energy surfaces. J Chem Phys 92:340–346CrossRefGoogle Scholar
  4. 4.
    Decius JC (1963) Compliance matrix and molecular vibrations. J Chem Phys 38:241–248CrossRefGoogle Scholar
  5. 5.
    Swanson BI, Satija SK (1977) Molecular vibrations and reaction pathways: minimum energy coordinates and compliance constants for some tetrahedral and octahedral complexes. J Am Chem Soc 99:987–991CrossRefGoogle Scholar
  6. 6.
    Nalewajski RF (2000) Coupling relations between molecular electronic and geometrical degrees of freedom in density functional theory and charge sensitivity analysis. Comput Chem 24:243–257CrossRefGoogle Scholar
  7. 7.
    Nalewajski RF (2006) Probing the interplay between electronic and geometric degrees-of-freedom in molecules and reactive systems. Adv Quantum Chem 51:235–305CrossRefGoogle Scholar
  8. 8.
    Tachibana A (2005) A new visualization scheme of chemical energy density and bonds in molecules. J Mol Model 11:301–311CrossRefGoogle Scholar
  9. 9.
    Szarek P, Tachibana A (2007) The field theoretical study of chemical interaction in terms of the rigged QED: new reactivity indices. J Mol Model 13:651–663CrossRefGoogle Scholar
  10. 10.
    Szarek P, Sueda Y, Tachibana A (2008) Electronic stress tensor description of chemical bonds using nonclassical bond order concept. J Chem Phys 129:094102CrossRefGoogle Scholar
  11. 11.
    Bader RFW (1990) Atoms in molecules: a quantum theory. Clarendon, OxfordGoogle Scholar
  12. 12.
    Popelier PLA (2000) Atoms in molecules: an introduction. Pearson, HarlowGoogle Scholar
  13. 13.
    Bader RFW, Nguyendang TT, Tal Y (1979) Quantum topology of molecular charge-distributions. 2. Molecular-structure and its change. J Chem Phys 70(9):4316–4329CrossRefGoogle Scholar
  14. 14.
    Bader RFW, Tal Y, Anderson SG, Nguyen-Dang TT (1980) Quantum topology: theory of molecular structure and its change. Isr J Chem 19:8–29CrossRefGoogle Scholar
  15. 15.
    Bader RFW, Nguyendang TT (1981) Quantum-theory of atoms in molecules: Dalton revisited. Adv Quantum Chem 14:63–124CrossRefGoogle Scholar
  16. 16.
    Tal Y, Bader RFW, Nguyendang TT, Ojha M, Anderson SG (1981) Quantum topology. 4. Relation between the topological and energetic stabilities of molecular structures. J Chem Phys 74:5162–5167CrossRefGoogle Scholar
  17. 17.
    Bone RGA, Bader RFW (1996) Identifying and analyzing intermolecular bonding interactions in van der Waals molecules. J Phys Chem 100:10892–10911CrossRefGoogle Scholar
  18. 18.
    Bader RFW, Laidig KE (1991) The prediction and calculation of properties of atoms in molecules. Theochem 80:75–94CrossRefGoogle Scholar
  19. 19.
    Collard K, Hall GG (1977) Orthogonal trajectories of the electron density. Int J Quantum Chem 12:623–637CrossRefGoogle Scholar
  20. 20.
    Jenkins S, Morrison I (1999) Characterization of various phases of ice on the basis of the charge density. J Phys Chem B 103:11041–11049CrossRefGoogle Scholar
  21. 21.
    Jenkins S, Heggie MI (2000) Quantitative analysis of bonding in 90 degrees partial dislocation in diamond. J Phys Condens Matter 12:10325–10333CrossRefGoogle Scholar
  22. 22.
    Jenkins S, Morrison I (2000) The chemical character of the intermolecular bonds of seven phases of ice as revealed by ab initio calculation of electron densities. Chem Phys Lett 317:97–102CrossRefGoogle Scholar
  23. 23.
    Jenkins S, Morrison I (2001) The dependence on structure of the projected vibrational density of states of various phases of ice as calculated by ab initio methods. J Phys Condens Matter 13:9207–9229CrossRefGoogle Scholar
  24. 24.
    Jenkins S, Kirk SR, Ayers PW (2006) Topological transitions between ice phases. In: Kuhs WF (ed) Physics and chemistry of ice (PCI-2006). Royal Society of Chemistry, pp 249–256Google Scholar
  25. 25.
    Jenkins S, Kirk SR, Ayers PW (2006) The importance of O-O bonding interactions in various phases of ice. In: Kuhs WF (ed) Physics and chemistry of ice (PCI-2006). Royal Society of Chemistry, pp 257–263Google Scholar
  26. 26.
    Jenkins S, Kirk SR, Ayers PW (2006) The chemical character of very high pressure ice phases. In: Kuhs WF (ed) Physics and chemistry of ice (PCI-2006). Royal Society of Chemistry, pp 265–272Google Scholar
  27. 27.
    Jenkins S, Kirk SR, Ayers PW (2006) Real-space study of mechanical instability of ice XI on a ‘bond-by-bond’ basis. In: Kuhs WF (ed) Physics and chemistry of ice (PCI-2006). Royal Society of Chemistry, pp 273–280Google Scholar
  28. 28.
    Ayers PW, Jenkins S (2009) An electron-preceding perspective on the deformation of materials. J Chem Phys 130:154104CrossRefGoogle Scholar
  29. 29.
    Holas A, March NH (1995) Exact theorems concerning noninteracting kinetic-energy density functional in D dimensions and their implications for gradient expansions. Int J Quantum Chem 56:371–383CrossRefGoogle Scholar
  30. 30.
    Bader RFW (1980) Quantum topology of molecular charge-distributions. 3. The mechanics of an atom in a molecule. J Chem Phys 73:2871–2883CrossRefGoogle Scholar
  31. 31.
    Jenkins S et al. (2011) The mechanics of charge-shift bonds: A perspective from the electronic stress tensor. Chem Phys Lett 510:18–20Google Scholar
  32. 32.
    Guevara-García A et al. (2011) Pointing the way to the products? Comparison of the stress tensor and the second-derivative tensor of the electron density. J Chem Phys 134:234106Google Scholar
  33. 33.
    Schrödinger E (1927) Der energieimpulssatz der materiewellen. Ann Phys 387:265–272CrossRefGoogle Scholar
  34. 34.
    Pauli W (1958) Handbuch der physik. Springer, BerlinGoogle Scholar
  35. 35.
    Epstein ST (1975) Coordinate invariance, the differential force law, and the divergence of the stress-energy tensor. J Chem Phys 63:3573–3574CrossRefGoogle Scholar
  36. 36.
    Bartolotti LJ, Parr RG (1980) The concept of pressure in density functional theory. J Chem Phys 72:1593–1596CrossRefGoogle Scholar
  37. 37.
    Deb BM, Bamzai AS (1978) Internal stresses in molecules. 1. One-electron systems. Mol Phys 35:1349–1367CrossRefGoogle Scholar
  38. 38.
    Deb BM, Bamzai AS (1979) Internal stresses in molecules. 2. Local view of chemical-binding in the H2 molecule. Mol Phys 38:2069–2097CrossRefGoogle Scholar
  39. 39.
    Deb BM, Ghosh SK (1979) Some local force densities and stress tensors in molecular quantum mechanics. J Phys B At Mol Opt Phys 12:3857–3871CrossRefGoogle Scholar
  40. 40.
    Ghosh SK, Berkowitz M (1985) A classical fluid-like approach to the density-functional formalism of many-electron systems. J Chem Phys 83:2976–2983CrossRefGoogle Scholar
  41. 41.
    Nielsen OH, Martin RM (1983) 1st-Principles calculation of stress. Phys Rev Lett 50:697–700CrossRefGoogle Scholar
  42. 42.
    Nielsen OH, Martin RM (1985) Quantum-mechanical theory of stress and force. Phys Rev B 32:3780–3791CrossRefGoogle Scholar
  43. 43.
    Tachibana A (2001) Electronic energy density in chemical reaction systems. J Chem Phys 115:3497–3518CrossRefGoogle Scholar
  44. 44.
    Tachibana A (2004) Spindle structure of the stress tensor of chemical bond. Int J Quantum Chem 100:981–993CrossRefGoogle Scholar
  45. 45.
    Ichikawa K, Tachibana A (2009) Stress tensor of the hydrogen molecular ion. Phys Rev A 80:062507CrossRefGoogle Scholar
  46. 46.
    Ichikawa K, Wagatsuma A, Kusumoto M, Tachibana A (2010) Electronic stress tensor of the hydrogen molecular ion: comparison between the exact wave function and approximate wave functions using Gaussian basis sets. J Mol Struct Theochem 951:49–59CrossRefGoogle Scholar
  47. 47.
    Tachibana A (2010) Energy density concept: a stress tensor approach. J Mol Struct Theochem 943:138–151CrossRefGoogle Scholar
  48. 48.
    Henry DJ et al (2011) Reactivity and regioselectivity of aluminum nanoclusters: insights from regional density functional theory. J Phys Chem C 115:1714–1723CrossRefGoogle Scholar
  49. 49.
    Fukushima A, Senami M, Tsuchida Y, Tachibana A (2010) Local dielectric property of cubic hafnia. Jpn J Appl Phys 49:111504CrossRefGoogle Scholar
  50. 50.
    Ichikawa K et al (2009) A theoretical study on a reaction of iron(III) hydroxide with boron trichloride by ab initio calculation. J Mol Struct Theochem 915:1–10CrossRefGoogle Scholar
  51. 51.
    Szarek P et al. (2009) Regional DFT: electronic stress tensor study of aluminum nanostructures for hydrogen storage. In: Wei DQ, Wang XJ (eds) Theory and applications of computational chemistry – 2008, AIP conference proceedings, vol 1102, pp 299–305Google Scholar
  52. 52.
    Tao JM, Vignale G, Tokatly IV (2008) Quantum stress focusing in descriptive chemistry. Phys Rev Lett 100:206405CrossRefGoogle Scholar
  53. 53.
    Tokatly IV (2005) Quantum many-body dynamics in a Lagrangian frame: I. Equations of motion and conservation laws. Phys Rev B 71:165104CrossRefGoogle Scholar
  54. 54.
    Maranganti R, Sharma P (2010) Revisiting quantum notions of stress. Proc R Soc A Math Phys Eng Sci 466:2097–2116CrossRefGoogle Scholar
  55. 55.
    Maranganti R, Sharma P, Wheeler L (2007) Quantum notions of stress. J Aerospace Eng 20:22–37CrossRefGoogle Scholar
  56. 56.
    Holas A, March NH (1995) Exact exchange-correlation potential and approximate exchange potential in terms of density matrices. Phys Rev A 51:2040–2048CrossRefGoogle Scholar
  57. 57.
    Nagy A, March NH (1997) Differential and local virial theorem. Mol Phys 91:597–602CrossRefGoogle Scholar
  58. 58.
    Ehrenfest P (1927) Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Z Phys A 45:455–457CrossRefGoogle Scholar
  59. 59.
    Slater JC (1951) A simplification of the Hartree-Fock method. Phys Rev 81(3):385–390CrossRefGoogle Scholar
  60. 60.
    Yang ZZ, Davidson ER (1997) Evaluation of a characteristic atomic radius by an ab initio method. Int J Quantum Chem 62(1):47–53CrossRefGoogle Scholar
  61. 61.
    Rogers CL, Rappe AM (2002) Geometric theory of stress fields for quantum systems at finite temperature. In: Landau DP, Lewis SP, Schuttler HB (eds) Computer simulation studies in condensed-matter physics xiv, Springer Proceedings in Physics, vol 89, pp 209–213Google Scholar
  62. 62.
    Rogers CL, Rappe AM (2002) Geometric formulation of quantum stress fields. Phys Rev B 65:224117CrossRefGoogle Scholar
  63. 63.
    Godfrey MJ (1988) Stress-field in quantum systems. Phys Rev B 37:10176–10183CrossRefGoogle Scholar
  64. 64.
    Anderson JSM, Ayers PW, Hernandez JIR (2010) How ambiguous is the local kinetic energy? J Phys Chem A 114:8884–8895CrossRefGoogle Scholar
  65. 65.
    Morante S, Rossi GC, Testa M (2006) The stress tensor of a molecular system: an exercise in statistical mechanics. J Chem Phys 125:034101CrossRefGoogle Scholar
  66. 66.
    Nelson DF, Lax M (1976) Asymmetric total stress tensor. Phys Rev B 13:1770–1776CrossRefGoogle Scholar
  67. 67.
    Das A (1978) Stress tensor in a class of gauge theories. Phys Rev D 18:2065–2067CrossRefGoogle Scholar
  68. 68.
    Cohen L (1979) Local kinetic energy in quantum mechanics. J Chem Phys 70:788–789CrossRefGoogle Scholar
  69. 69.
    Cohen L (1984) Representable local kinetic energy. J Chem Phys 80:4277–4279CrossRefGoogle Scholar
  70. 70.
    Cohen L (1996) Local values in quantum mechanics. Phys Lett A 212:315–319CrossRefGoogle Scholar
  71. 71.
    Ayers PW, Parr RG, Nagy A (2002) Local kinetic energy and local temperature in the density-functional theory of electronic structure. Int J Quantum Chem 90:309–326CrossRefGoogle Scholar
  72. 72.
    Cohen L (1966) Generalized phase-space distribution functions. J Math Phys 7:781–786CrossRefGoogle Scholar
  73. 73.
    Cohen L (1966) Can quantum mechanics be formulated as classical probability theory. Philos Sci 33:317–322CrossRefGoogle Scholar
  74. 74.
    Becke AD (1988) Density-functional exchange-energy approximation with correct asymptotic-behavior. Phys Rev A 38:3098–3100CrossRefGoogle Scholar
  75. 75.
    Becke AD (1993) Density-functional thermochemistry. 3. The role of exact exchange. J Chem Phys 98:5648–5652CrossRefGoogle Scholar
  76. 76.
    Lee C, Yang W, Parr RG (1988) Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys Rev B 37:785–789CrossRefGoogle Scholar
  77. 77.
    Miehlich B, Savin A, Stoll H, Preuss H (1989) Results obtained with the correlation-energy density functionals of Becke and Lee, Yang and Parr. Chem Phys Lett 157(3):200–206CrossRefGoogle Scholar
  78. 78.
    Frisch MJ et al (2004) Gaussian03, Revision D.01. Gaussian Inc., Wallingford, CTGoogle Scholar
  79. 79.
    Frisch MJ et al (2009) Gaussian 09, Revision A.1. Gaussian Inc., Wallingford CTGoogle Scholar
  80. 80.
    Keith TA (2010) AIMAll., 10.09.12.Google Scholar
  81. 81.
    Shaik S, Danovich D, Wu W, Hiberty PC (2009) Charge-shift bonding and its manifestations in chemistry. Nat Chem 1:443–449CrossRefGoogle Scholar
  82. 82.
    Shaik S, Maitre P, Sini G, Hiberty PC (1992) The charge-shift bonding concept: electron-pair bonds with very large ionic-covalent resonance energies. J Am Chem Soc 114:7861–7866CrossRefGoogle Scholar
  83. 83.
    Zhang LX, Ying FM, Wu W, Hiberty PC, Shaik S (2009) Topology of electron charge density for chemical bonds from valence bond theory: a probe of bonding types. Chem Eur J 15:2979–2989CrossRefGoogle Scholar
  84. 84.
    Hirshfeld FL (1977) Theor Chim Act 44:129–138CrossRefGoogle Scholar
  85. 85.
    Hiberty PC, Ramozzi R, Song LC, Wu W, Shaik S (2007) The physical origin of large covalent-ionic resonance energies in some two-electron bonds. Faraday Discuss 135:261–272CrossRefGoogle Scholar
  86. 86.
    Toro-Labbe A (1999) Characterization of chemical reactions from the profiles of energy, chemical potential and hardness. J Phys Chem A 103:4398–4403CrossRefGoogle Scholar
  87. 87.
    Bulat FA, Toro-Labbe A (2003) An extension of the Hammond postulate. Structural effects on the classification of chemical reactions. J Phys Chem A 107:3987–3994CrossRefGoogle Scholar
  88. 88.
    Toro-Labbé A, Gutierrerez-Oliva S, Murray JS, Politzer P (2007) A new perspective on chemical and physical processes: the reaction force. Mol Phys 105:2619–2625CrossRefGoogle Scholar
  89. 89.
    Toro-Labbe A, Gutierrerez-Oliva S, Politzer P, Murray JS (2009) The reaction force: a rigorously defined approach to analyzing chemical and physical processes. In: Chattaraj PK (ed) Chemical reactivity theory: a density functional view. CRC, Boca Raton, pp 293–302Google Scholar
  90. 90.
    Politzer P et al (2005) The reaction force: three key points along an intrinsic reaction coordinate. J Chem Sci 117:467–472CrossRefGoogle Scholar
  91. 91.
    Wang SG, Qiu YX, Schwarz WHE (2010) Antibond breaking – the formation and decomposition of He@Adamantane: descriptions, explanations, and meaning of concepts. Chem Eur J 16:9107–9116CrossRefGoogle Scholar
  92. 92.
    Bader RFW (2009) Bond paths are not chemical bonds. J Phys Chem A 113:10391–10396CrossRefGoogle Scholar
  93. 93.
    Wang SG, Qiu YX, Schwarz WHE (2009) Bonding or nonbonding? Description or explanation? “Confinement bonding” of He@adamantane. Chem Eur J 15:6032–6040CrossRefGoogle Scholar
  94. 94.
    Grimme S et al (2009) When do interacting atoms form a chemical bond? Spectroscopic measurements and theoretical analyses of dideuteriophenanthrene. Angew Chem Int Ed 48:2592–2595CrossRefGoogle Scholar
  95. 95.
    Cerpa E, Krapp A, Flores-Moreno R, Donald KJ, Merino G (2009) Influence of endohedral confinement on the electronic interaction between He atoms: a He-2@C20H20 case study. Chem Eur J 15:1985–1990CrossRefGoogle Scholar
  96. 96.
    Cerpa E, Krapp A, Vela A, Merino G (2008) The implications of symmetry of the external potential on bond paths. Chem Eur J 14:10232–10234CrossRefGoogle Scholar
  97. 97.
    Becke AD, Edgecombe KE (1990) A simple measure of electron localization in atomic and molecular systems. J Chem Phys 92:5397–5403CrossRefGoogle Scholar
  98. 98.
    Savin A, Nesper R, Wengert S, Fassler TF (1997) ELF: the electron localization function. Angew Chem 36:1809–1832CrossRefGoogle Scholar
  99. 99.
    Savin A et al (1991) A new look at electron localization. Angew Chem 30:409–412CrossRefGoogle Scholar
  100. 100.
    Gillespie RJ, Bytheway I, Dewitte RS, Bader RFW (1994) Trigonal bipyramidal and related molecules of the main-group elements: investigation of apparent exceptions to the VSEPR model through the analysis of the Laplacian of the electron density. Inorg Chem 33:2115–2121CrossRefGoogle Scholar
  101. 101.
    Bader RFW, Gillespie RJ, Macdougall PJ (1988) A physical basis for the VSEPR model of molecular geometry. J Am Chem Soc 110:7329–7336CrossRefGoogle Scholar
  102. 102.
    Schmider HL, Becke AD (2002) Two functions of the density matrix and their relation to the chemical bond. J Chem Phys 116(8):3184–3193CrossRefGoogle Scholar
  103. 103.
    Schmider HL, Becke AD (2000) Chemical content of the kinetic energy density. Theochem J Mol Struct 527:51–61CrossRefGoogle Scholar
  104. 104.
    Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford University Press, New YorkGoogle Scholar
  105. 105.
    Geerlings P, De Proft F, Langenaeker W (2003) Conceptual density functional theory. Chem Rev 103:1793–1873CrossRefGoogle Scholar
  106. 106.
    Ayers PW, Anderson JSM, Bartolotti LJ (2005) Perturbative perspectives on the chemical reaction prediction problem. Int J Quantum Chem 101:520–534CrossRefGoogle Scholar
  107. 107.
    Gazquez JL (2008) Perspectives on the density functional theory of chemical reactivity. J Mex Chem Soc 52:3–10Google Scholar
  108. 108.
    Cohen MH, Ganduglia-Pirovano MV, Kudrnovsky J (1995) Reactivity kernels, the normal modes of chemical reactivity, and the hardness and softness spectra. J Chem Phys 103:3543–3551CrossRefGoogle Scholar
  109. 109.
    Nalewajski RF (1995) Chemical reactivity concepts in charge sensitivity analysis. Int J Quantum Chem 56(5):453–476CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alfredo Guevara-García
    • 1
  • Paul W. Ayers
    • 1
    Email author
  • Samantha Jenkins
    • 2
    • 3
  • Steven R. Kirk
    • 2
    • 3
  • Eleonora Echegaray
    • 4
  • Alejandro Toro-Labbe
    • 4
  1. 1.Department of Chemistry and Chemical BiologyMcMaster UniversityHamiltonCanada
  2. 2.College of Chemistry and Chemical EngineeringHunan Normal UniversityChangshaChina
  3. 3.Department of EngineeringUniversity WestTrollhättanSweden
  4. 4.Laboratorio de Química Teórica Computacional (QTC), Facultad de QuímicaPontificia Universidad Católica de ChileSantiagoChile

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