The “|Δμ| big is good” rule, the maximum hardness, and minimum electrophilicity principles

  • Ramón Alain Miranda-QuintanaEmail author
  • Paul W. Ayers
Regular Article


We show the relation between the “|Δμ| big is good” rule and the maximum hardness and the minimum electrophilicity principles. We focus on a double-exchange acid–base (e.g., charge transfer) reaction. We then prove that the species favored by the “|Δμ| big is good” rule are those such that the multiplication of their hardnesses is the biggest, while the validity of the minimum electrophilicity principle requires other conditions.


Conceptual density functional theory Reactivity principles Chemical potential Chemical hardness Electrophilicity 



We thank NSERC, the Canada Research Chairs, Canarie, and Compute Canada for their support. RAMQ acknowledges funding from York University in the form of a York Science Fellowship.


  1. 1.
    Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford UP, New YorkGoogle Scholar
  2. 2.
    Parr RG (1994) Companions in the search. Int J Quantum Chem 49:739–770CrossRefGoogle Scholar
  3. 3.
    Geerlings P, De Proft F, Langenaeker W (2003) Conceptual density functional theory. Chem Rev 103:1793–1873. CrossRefPubMedGoogle Scholar
  4. 4.
    Johnson PA, Bartolotti LJ, Ayers PW, Fievez T, Geerlings P (2012) Charge density and chemical reactivity: a unified view from conceptual DFT”. In: Gatti C, Macchi P (eds) Modern charge density analysis. Springer, New York, pp 715–764Google Scholar
  5. 5.
    Miranda-Quintana RA (2018) Density functional theory for chemical reactivity. In: Islam N, Kaya S (eds) Conceptual density functional theory and its applications in the chemical domain. Apple Academic Press, New Jersey, pp 15–44CrossRefGoogle Scholar
  6. 6.
    Miranda-Quintana RA, Zadeh FH, Ayers PW (2018) Elementary derivation of the “|Δμ| big is good” rule. J Phys Chem Lett 9(15):4344–4348. CrossRefPubMedGoogle Scholar
  7. 7.
    Pearson RG (1963) Hard and soft acids and bases. J Am Chem Soc 85:3533–3539CrossRefGoogle Scholar
  8. 8.
    Pearson RG (1966) Acids and bases. Science 151:172–177CrossRefGoogle Scholar
  9. 9.
    Pearson RG (1967) Hard and soft acids and bases. ChemBr 3(3):103–107Google Scholar
  10. 10.
    Ayers PW (2005) An elementary derivation of the hard/soft-acid/base principle. J Chem Phys 122:141102CrossRefGoogle Scholar
  11. 11.
    Ayers PW, Parr RG, Pearson RG (2006) Elucidating the hard/soft acid/base principle: a perspective based on half-reactions. J Chem Phys 124:194107CrossRefGoogle Scholar
  12. 12.
    Ayers PW (2007) The physical basis of the hard/soft acid/base principle. Faraday Discuss 135:161–190CrossRefGoogle Scholar
  13. 13.
    Ayers PW, Cardenas C (2013) Communication: a case where the hard/soft acid/base principle holds regardless of acid/base strength. J Chem Phys 138:181106CrossRefGoogle Scholar
  14. 14.
    Cardenas C, Ayers PW (2013) How reliable is the hard-soft acid-base principle? An assessment from numerical simulations of electron transfer energies. PCCP 15(33):13959–13968. CrossRefPubMedGoogle Scholar
  15. 15.
    Chattaraj PK, Lee H, Parr RG (1991) HSAB principle. J Am Chem Soc 113:1855–1856CrossRefGoogle Scholar
  16. 16.
    Gazquez JL, Mendez F (1994) The hard and soft acids and bases principle: an atoms in molecules viewpoint. J Phys Chem 98:4591–4593CrossRefGoogle Scholar
  17. 17.
    Mendez F, Gazquez JL (1994) Chemical-reactivity of enolate ions—the local hard and soft acids and bases principle viewpoint. J Am Chem Soc 116:9298–9301CrossRefGoogle Scholar
  18. 18.
    Miranda-Quintana RA, Kim TD, Cardenas C, Ayers PW (2017) The HSAB principle from a finite temperature grand-canonical perspective. Theor Chem Acc 136:135CrossRefGoogle Scholar
  19. 19.
    von Szentpály L (2017) Hardness maximization or equalization? New insights and quantitative relations between hardness increase and bond dissociation energy. J Mol Model 23(7):217. CrossRefGoogle Scholar
  20. 20.
    Parr RG, Chattaraj PK (1991) Principle of maximum hardness. J Am Chem Soc 113:1854–1855CrossRefGoogle Scholar
  21. 21.
    Pearson RG (1987) Recent advances in the concept of hard and soft acids and bases. JChemEduc 64:561–567Google Scholar
  22. 22.
    Ayers PW, Parr RG (2000) Variational principles for describing chemical reactions: the Fukui function and chemical hardness revisited. J Am Chem Soc 122:2010–2018. CrossRefGoogle Scholar
  23. 23.
    Pearson RG, Palke WE (1992) Support for a principle of maximum hardness. JPhysChem 96:3283–3285Google Scholar
  24. 24.
    Chattaraj PK (1996) The maximum hardness principle: an overview. Proc Indian Nat Sci Acad Part A 62:513–531Google Scholar
  25. 25.
    Torrent-Sucarrat M, Luis JM, Duran M, Sola M (2001) On the validity of the maximum hardness and minimum polarizability principles for nontotally symmetric vibrations. J Am Chem Soc 123:7951–7952CrossRefGoogle Scholar
  26. 26.
    Torrent-Sucarrat M, Luis JM, Duran M, Sola M (2002) Are the maximum hardness and minimum polarizability principles always obeyed in nontotally symmetric vibrations? J Chem Phys 117:10561–10570CrossRefGoogle Scholar
  27. 27.
    Chattaraj PK, Sarkar U, Roy DR (2006) Electrophilicity index. Chem Rev 106:2065–2091CrossRefGoogle Scholar
  28. 28.
    Morell C, Labet V, Grand A, Chermette H (2009) Minimum electrophilicity principle: an analysis based upon the variation of both chemical potential and absolute hardness. PCCP 11:3414. CrossRefGoogle Scholar
  29. 29.
    Parthasarathi R, Elango M, Subramanian V, Chattaraj PK (2005) Variation of electrophilicity during molecular vibrations and internal rotations. Theor Chem Acc 113:257–266. CrossRefGoogle Scholar
  30. 30.
    Chamorro E, Chattaraj PK, Fuentealba P (2003) Variation of the electrophilicity index along the reaction path. J Phys Chem A 107:7068–7072. CrossRefPubMedGoogle Scholar
  31. 31.
    Noorizadeh S (2007) Is there a minimum electrophilicity principle in chemical reactions? Chin J Chem 25:1439–1444CrossRefGoogle Scholar
  32. 32.
    Noorizadeh S (2007) Minimum electrophilicity principle in photocycloaddition formation of oxetanes. J Phys Org Chem 20:514–524CrossRefGoogle Scholar
  33. 33.
    Miranda-Quintana RA (2017) Thermodynamic electrophilicity. J Chem Phys 146:214113. CrossRefPubMedGoogle Scholar
  34. 34.
    Miranda-Quintana RA, Chattaraj PK, Ayers PW (2017) Finite temperature grand canonical ensemble study of the minimum electrophilicity principle. J Chem Phys 147:124103. CrossRefPubMedGoogle Scholar
  35. 35.
    Miranda-Quintana RA (2017) The minimum electrophilicity and the hard/soft acid/base principles. J Chem Phys 146:046101. CrossRefPubMedGoogle Scholar
  36. 36.
    Miranda-Quintana RA, Ayers PW (2018) Maximum hardness and minimum electrophilicity principles. J Chem Phys 148:196101CrossRefGoogle Scholar
  37. 37.
    Parr RG, Donnelly RA, Levy M, Palke WE (1978) Electronegativity: the density functional viewpoint. J Chem Phys 68:3801–3807. CrossRefGoogle Scholar
  38. 38.
    Parr RG, Pearson RG (1983) Absolute hardness: companion parameter to absolute electronegativity. J Am Chem Soc 105:7512–7516. CrossRefGoogle Scholar
  39. 39.
    Parr RG, von Szentpály L, Liu SB (1999) Electrophilicity index. J Am Chem Soc 121:1922–1924. CrossRefGoogle Scholar
  40. 40.
    Miranda-Quintana RA, Ayers PW (2016) Interpolation of property-values between electron numbers is inconsistent with ensemble averaging. J Chem Phys 144:244112. CrossRefPubMedGoogle Scholar
  41. 41.
    Heidar Zadeh F, Miranda-Quintana RA, Verstraelen T, Bultinck P, Ayers PW (2016) When is the Fukui function not normalized? The danger of inconsistent energy interpolation models in density functional theory. J Chem Theory Comp 12:5777–5787. CrossRefGoogle Scholar
  42. 42.
    Miranda-Quintana RA, Ayers PW (2018) Grand-canonical interpolation models. In: Islam N, Kaya S (eds) Conceptual density functional theory and its applications in the chemical domain. Apple Academic Press, New Jersey, pp 61–88CrossRefGoogle Scholar
  43. 43.
    Chattaraj PK, Ayers PW (2005) The maximum hardness principle implies the hard/soft acid/base rule. J Chem Phys 123:086101CrossRefGoogle Scholar
  44. 44.
    Pearson RG (1968) J Chem Educ 45:981Google Scholar
  45. 45.
    Miranda-Quintana RA, Kim TD, Zadeh FH, Ayers PW (2019) On the Impossibility of Unambiguously Selecting the Best Model for Fitting Data. J Math Chem:(under review)Google Scholar
  46. 46.
    Miranda-Quintana RA (2017) Perturbed reactivity descriptors: the chemical hardness. Theor Chem Acc 136:76. CrossRefGoogle Scholar
  47. 47.
    Miranda-Quintana RA, Ayers PW (2016) Fractional electron number, temperature, and perturbations in chemical reactions. PCCP 18:15070–15080. CrossRefPubMedGoogle Scholar
  48. 48.
    Miranda-Quintana RA, Ayers PW (2016) Charge transfer and chemical potential in 1,3-dipolar cycloadditions. Theor Chem Acc 135:172. CrossRefGoogle Scholar
  49. 49.
    Miranda-Quintana RA, González MM, Ayers PW (2016) Electronegativity and redox reactions. PCCP 18:22235–22243. CrossRefPubMedGoogle Scholar
  50. 50.
    Miranda-Quintana RA, Ayers PW (2018) Dipolar cycloadditions and the “|Δμ| big is good” rule: a computational study. Theor Chem Acc 137:177. CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Ramón Alain Miranda-Quintana
    • 1
    • 2
    Email author
  • Paul W. Ayers
    • 2
  1. 1.Department of ChemistryYork UniversityTorontoCanada
  2. 2.Department of Chemistry and Chemical BiologyMcMaster UniversityHamiltonCanada

Personalised recommendations