Advertisement

Journal of Mathematical Chemistry

, Volume 43, Issue 1, pp 285–303 | Cite as

The dependence on and continuity of the energy and other molecular properties with respect to the number of electrons

  • Paul W. AyersEmail author
Article

It was recently shown that the size consistency of the energy implies that, for any system with a rational number of electrons, the energy is given by the weighted average of the two systems with the nearest integer numbers of electrons. Specifically, E[N+P/Q] =  (1−P/Q)E[N] + (P/Q)E[N+1]. This paper extends that analysis, showing that the same result holds for irrational numbers of electrons. This proves that the energy is a continuous function of the number of electrons, and justifies differentiation with respect to electron number, providing a rigorous justification or the density-functional theoretic approaches to chemical concepts like the electronegativity and the Fukui function. Similar results hold for properties other than the energy. Specific emphasis is placed on molecular response properties associated with the density-functional theory of chemical reactivity.

Keywords

zero-temperature grand canonical ensemble derivative discontinuity conceptual density functional theory chemical reactivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Parr R.G., Yang W. (1989) Density-Functional Theory of Atoms and Molecules. Oxford University Press, New YorkGoogle Scholar
  2. 2.
    Dreizler R.M., Gross E.K.U. (1990) Density Functional Theory: An Approach to the Quantum Many-Body Problem. Springer-Verlag, BerlinGoogle Scholar
  3. 3.
    Kohn W., Becke A.D., Parr R.G. (1996). J. Phys. Chem. 100:12974CrossRefGoogle Scholar
  4. 4.
    Kohn W. (1999). Rev. Mod. Phys. 71:1253CrossRefGoogle Scholar
  5. 5.
    Perdew J.P., Parr R.G., Levy M., Balduz J.L. Jr (1982). Phys. Rev. Lett. 49:1691CrossRefGoogle Scholar
  6. 6.
    Chan G.K.L. (1999). J. Chem. Phys. 110:4710CrossRefGoogle Scholar
  7. 7.
    Yang W., Zhang Y., Ayers P.W. (2000). Phys. Rev. Lett. 84:5172CrossRefGoogle Scholar
  8. 8.
    Parr R.G., Yang W.T. (1995). Ann. Rev. Phys. Chem. 46:701CrossRefGoogle Scholar
  9. 9.
    Geerlings P., De Proft F., Langenaeker W. (2003). Chem. Rev. 103:1793CrossRefGoogle Scholar
  10. 10.
    Chermette H. (1999). J. Comp. Chem. 20:129CrossRefGoogle Scholar
  11. 11.
    Ayers P.W., Anderson J.S.M., Bartolotti L.J. (2005). Int. J. Quantum Chem. 101:520CrossRefGoogle Scholar
  12. 12.
    Parr R.G., Donnelly R.A., Levy M., Palke W.E. (1978). J. Chem. Phys. 68:3801CrossRefGoogle Scholar
  13. 13.
    Mulliken R.S. (1934). J. Chem. Phys. 2:782CrossRefGoogle Scholar
  14. 14.
    Parr R.G., Pearson R.G. (1983). J. Am. Chem. Soc. 105:7512CrossRefGoogle Scholar
  15. 15.
    Pearson R.G. (1999). J. Chem. Educ. 76:267CrossRefGoogle Scholar
  16. 16.
    Pearson R.G. (1987). J. Chem. Educ. 64:561Google Scholar
  17. 17.
    Parr R.G., Yang W. (1984). J. Am. Chem. Soc. 106:4049CrossRefGoogle Scholar
  18. 18.
    Yang W., Parr R.G., Pucci R. (1984). J. Chem. Phys. 81:2862CrossRefGoogle Scholar
  19. 19.
    Ayers P.W., Levy M. (2000). Theor. Chem. Acc. 103:353Google Scholar
  20. 20.
    Ayers P.W., Parr R.G. (2000). J. Am. Chem. Soc. 122:2010CrossRefGoogle Scholar
  21. 21.
    Ayers P.W., Levy M. (2001). J. Chem. Phys. 115:4438CrossRefGoogle Scholar
  22. 22.
    Zhang Y., Yang W. (2000). Theor. Chem. Acc. 103:346Google Scholar
  23. 23.
    Della Sala F., Gorling A. (2002). Phys. Rev. Lett. 89:033003CrossRefGoogle Scholar
  24. 24.
    Della Sala F., Gorling A. (2002). J. Chem. Phys. 116:5374CrossRefGoogle Scholar
  25. 25.
    Wu Q., Ayers P.W., Yang W.T. (2003). J. Chem. Phys. 119:2978CrossRefGoogle Scholar
  26. 26.
    Parks J.M., Parr R.G. (1958). J. Chem. Phys. 28:335CrossRefGoogle Scholar
  27. 27.
    Ayers P.W., Parr R.G. (2001). J. Am. Chem. Soc. 123:2007CrossRefGoogle Scholar
  28. 28.
    J.S.M. Anderson, J. Melin and P.W. Ayers, J. Chem. Theor. and Comput. 2006 (accepted).Google Scholar
  29. 29.
    Senet P. (1996). J. Chem. Phys. 105:6471CrossRefGoogle Scholar
  30. 30.
    Ayers P.W. (2001). Theor. Chem. Acc. 106:271CrossRefGoogle Scholar
  31. 31.
    Berkowitz M. (1987). J. Am. Chem. Soc. 109:4823CrossRefGoogle Scholar
  32. 32.
    Parr R.G., Bartolotti L.J. (1982). J. Am. Chem. Soc. 104:3801CrossRefGoogle Scholar
  33. 33.
    Chattaraj P.K., Lee H., Parr R.G. (1991). J. Am. Chem. Soc. 113:1855CrossRefGoogle Scholar
  34. 34.
    Ayers P.W. (2005). J. Chem. Phys. 122:141102CrossRefGoogle Scholar
  35. 35.
    Gazquez J.L. (1997). J. Phys. Chem. A 101:4657CrossRefGoogle Scholar
  36. 36.
    Gazquez J.L., Mendez F. (1994). J. Phys. Chem. 98:4591CrossRefGoogle Scholar
  37. 37.
    Mendez F., Gazquez J.L. (1994). J. Am. Chem. Soc. 116:9298CrossRefGoogle Scholar
  38. 38.
    Ayers P.W., Parr R.G., Pearson R.G. (2006). J. Chem. Phys. 124:194107CrossRefGoogle Scholar
  39. 39.
    P.W. Ayers, Faraday Discussions 2006 (accepted). DOI: 10.1007/s00214-006-0165-6.Google Scholar
  40. 40.
    W. Langenaeker, K. Demel and P. Geerlings, THEOCHEM 80 (1991) 329.Google Scholar
  41. 41.
    P.W. Ayers and J. Melin, Theor. Chem. Acc. 2006 (accepted).Google Scholar
  42. 42.
    Melin J., Ayers P.W., Ortiz J.V. (2005). J. Chem. Sci. 117:387CrossRefGoogle Scholar
  43. 43.
    Bartolotti L.J., Ayers P.W. (2005). J. Phys. Chem. A 109:1146CrossRefGoogle Scholar
  44. 44.
    Parr R.G., Chattaraj P.K. (1991). J. Am. Chem. Soc. 113:1854CrossRefGoogle Scholar
  45. 45.
    Morell C., Grand A., Toro-Labbe A. (2006). Chem. Phys. Lett. 425:342CrossRefGoogle Scholar
  46. 46.
    Morell C., Grand A., Toro-Labbe A. (2005). J. Phys. Chem. A 109:205CrossRefGoogle Scholar
  47. 47.
    F. De Proft, P.W. Ayers, S. Fias and P. Geerlings, J. Chem. Phys. 2006 (accepted).Google Scholar
  48. 48.
    P.W. Ayers, C. Morell, F. De Proft and P. Geerlings (in preparation) 2006.Google Scholar
  49. 49.
    C. Morell, A. Grand, A. Toro-Labbe and P.W. Ayers (in preparation) 2006.Google Scholar
  50. 50.
    R.G. Parr personal communication.Google Scholar
  51. 51.
    Cohen M.H., Wasserman A. (2003). Israel J. Chem. 43:219CrossRefGoogle Scholar
  52. 52.
    S.M. Valone and S.R. Atlas Philos. Mag. 86 (2006) 2683.Google Scholar
  53. 53.
    Morales J., Martinez T.J. (2004). J. Phys. Chem. A 108:3076CrossRefGoogle Scholar
  54. 54.
    Valone S.M., Atlas S.R. (2004). J. Chem. Phys. 120:7262CrossRefGoogle Scholar
  55. 55.
    Morales J., Martinez T.J. (2001). J. Phys. Chem. A 105:2842CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of ChemistryMcMaster UniversityHamiltonCanada

Personalised recommendations