For atoms in molecules, difficult to quantify rigorously.
The charges of atoms in molecules may be fractional, which reflects how electrons are redistributed when a bond forms.
The pair of electrons which constitutes the bond may lie between two atomic centers in such a position that there is no electric polarization, or it may be shifted toward one or the other atom in order to give to that atom a negative, and consequently to the other atom a positive charge. But we can no longer speak of any atom as having an integral number of units of charge, except in the case where one atom takes exclusive possession of the bonding pair, and forms an ion.
- Lewis, 1923
Lewis, G. N. J. Am. Chem. Soc. 54 (1932), p. 83; quoted in Jensen, W. B. J. Chem. Educ. 86 (2009), 545.
metal workfunctions (Gordy and Orville Thomas, 1955)
electrode potentials (Kaputinskii, 1960)
“ Gradation of electroaffinity in the Periodic Table” Abegg, R. Z. Anorg. Chem. 39 (1904), 330. Historical review: Jensen, W. B. J. Chem. Ed. 37 (1996), 10; 80 (2003), 279.
1932: Pauling’s electronegativity Pauling, L. J. Am. Chem. Soc. 54 (1932), 3570. Pauling developed his concept of electronegativity as an empirical additive correction to reaction enthalpies.
1934: Mulliken’s electronegativity Mulliken. R. S. J. Chem. Phys. 2 (1934), 782. The first serious attempt to justify an electronegativity scale using quantum mechanical arguments. *in energies + * +
1935: Mulliken’s charges for diatomics Mulliken. R. S. J. Chem. Phys. 3 (1935), 573.
1935: Mulliken’s charges for diatomics i.e. in modern terms , (McWeeny, 1951) density matrix overlap matrix Mulliken, R. S. J. Chim. Phys. 46 (1949) 675. canonical reference: Mulliken, R. S. J. Chem. Phys. 23 (1955) 1833; 1841; 2338; 2343. The general (polyatomic) case was worked out by Mulliken in 1949. McWeeny, R. J. Chem. Phys. 19 (1951) 1614.
localized more bonding more antibonding Attributed to K. Ruedenberg in Mulliken, R.S; Ermler, W. C. Diatomic molecules: results of ab initio calculations . Academic Press, NY, 1977, pp. 33-38. physically unreasonable region
Some useful reviews: Bachrach, S. M. Rev. Comp. Chem. 5 (1994), 171 Meister, J.; Schwarz, W. H. E. J. Phys. Chem. 98 (1994), 8245 Francl, M. M.; Chirlian, L. E. Rev. Comp. Chem. 14 (2000), 1 - ESP charges Rick. S.W.; Stuart, S. J. Rev. Comp. Chem., 18 (2002), 89 - empirical models
Based purely on the density, so that the charges are true density functionals
A A B B Rigid response to perturbation A A B B A’ A’
A density matrix in general, has the derivative Introducing the projection matrix for the basis functions on atom (fragment) A, can rewrite the derivative in block matrix form We want to find the nearest density matrix that has the form
We can find such a density that minimizes where
The density matrix A that minimizes satisfies where are projection matrices Recover the population from the fragment density and finally calculate the charge as which is clearly minimized by This yields Mulliken populations If we neglect S x and S xx , then this reduces to Mulliken charges!