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.
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References
Parr R.G., Yang W. (1989) Density-Functional Theory of Atoms and Molecules. Oxford University Press, New York
Dreizler R.M., Gross E.K.U. (1990) Density Functional Theory: An Approach to the Quantum Many-Body Problem. Springer-Verlag, Berlin
Kohn W., Becke A.D., Parr R.G. (1996). J. Phys. Chem. 100:12974
Kohn W. (1999). Rev. Mod. Phys. 71:1253
Perdew J.P., Parr R.G., Levy M., Balduz J.L. Jr (1982). Phys. Rev. Lett. 49:1691
Chan G.K.L. (1999). J. Chem. Phys. 110:4710
Yang W., Zhang Y., Ayers P.W. (2000). Phys. Rev. Lett. 84:5172
Parr R.G., Yang W.T. (1995). Ann. Rev. Phys. Chem. 46:701
Geerlings P., De Proft F., Langenaeker W. (2003). Chem. Rev. 103:1793
Chermette H. (1999). J. Comp. Chem. 20:129
Ayers P.W., Anderson J.S.M., Bartolotti L.J. (2005). Int. J. Quantum Chem. 101:520
Parr R.G., Donnelly R.A., Levy M., Palke W.E. (1978). J. Chem. Phys. 68:3801
Mulliken R.S. (1934). J. Chem. Phys. 2:782
Parr R.G., Pearson R.G. (1983). J. Am. Chem. Soc. 105:7512
Pearson R.G. (1999). J. Chem. Educ. 76:267
Pearson R.G. (1987). J. Chem. Educ. 64:561
Parr R.G., Yang W. (1984). J. Am. Chem. Soc. 106:4049
Yang W., Parr R.G., Pucci R. (1984). J. Chem. Phys. 81:2862
Ayers P.W., Levy M. (2000). Theor. Chem. Acc. 103:353
Ayers P.W., Parr R.G. (2000). J. Am. Chem. Soc. 122:2010
Ayers P.W., Levy M. (2001). J. Chem. Phys. 115:4438
Zhang Y., Yang W. (2000). Theor. Chem. Acc. 103:346
Della Sala F., Gorling A. (2002). Phys. Rev. Lett. 89:033003
Della Sala F., Gorling A. (2002). J. Chem. Phys. 116:5374
Wu Q., Ayers P.W., Yang W.T. (2003). J. Chem. Phys. 119:2978
Parks J.M., Parr R.G. (1958). J. Chem. Phys. 28:335
Ayers P.W., Parr R.G. (2001). J. Am. Chem. Soc. 123:2007
J.S.M. Anderson, J. Melin and P.W. Ayers, J. Chem. Theor. and Comput. 2006 (accepted).
Senet P. (1996). J. Chem. Phys. 105:6471
Ayers P.W. (2001). Theor. Chem. Acc. 106:271
Berkowitz M. (1987). J. Am. Chem. Soc. 109:4823
Parr R.G., Bartolotti L.J. (1982). J. Am. Chem. Soc. 104:3801
Chattaraj P.K., Lee H., Parr R.G. (1991). J. Am. Chem. Soc. 113:1855
Ayers P.W. (2005). J. Chem. Phys. 122:141102
Gazquez J.L. (1997). J. Phys. Chem. A 101:4657
Gazquez J.L., Mendez F. (1994). J. Phys. Chem. 98:4591
Mendez F., Gazquez J.L. (1994). J. Am. Chem. Soc. 116:9298
Ayers P.W., Parr R.G., Pearson R.G. (2006). J. Chem. Phys. 124:194107
P.W. Ayers, Faraday Discussions 2006 (accepted). DOI: 10.1007/s00214-006-0165-6.
W. Langenaeker, K. Demel and P. Geerlings, THEOCHEM 80 (1991) 329.
P.W. Ayers and J. Melin, Theor. Chem. Acc. 2006 (accepted).
Melin J., Ayers P.W., Ortiz J.V. (2005). J. Chem. Sci. 117:387
Bartolotti L.J., Ayers P.W. (2005). J. Phys. Chem. A 109:1146
Parr R.G., Chattaraj P.K. (1991). J. Am. Chem. Soc. 113:1854
Morell C., Grand A., Toro-Labbe A. (2006). Chem. Phys. Lett. 425:342
Morell C., Grand A., Toro-Labbe A. (2005). J. Phys. Chem. A 109:205
F. De Proft, P.W. Ayers, S. Fias and P. Geerlings, J. Chem. Phys. 2006 (accepted).
P.W. Ayers, C. Morell, F. De Proft and P. Geerlings (in preparation) 2006.
C. Morell, A. Grand, A. Toro-Labbe and P.W. Ayers (in preparation) 2006.
R.G. Parr personal communication.
Cohen M.H., Wasserman A. (2003). Israel J. Chem. 43:219
S.M. Valone and S.R. Atlas Philos. Mag. 86 (2006) 2683.
Morales J., Martinez T.J. (2004). J. Phys. Chem. A 108:3076
Valone S.M., Atlas S.R. (2004). J. Chem. Phys. 120:7262
Morales J., Martinez T.J. (2001). J. Phys. Chem. A 105:2842
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Ayers, P.W. The dependence on and continuity of the energy and other molecular properties with respect to the number of electrons. J Math Chem 43, 285–303 (2008). https://doi.org/10.1007/s10910-006-9195-5
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DOI: https://doi.org/10.1007/s10910-006-9195-5