Mulliken population analysis

Mulliken charges arise from the Mulliken population analysis^[1][2] and provide a means of estimating partial atomic charges from calculations carried out by the methods of computational chemistry, particularly those based on the linear combination of atomic orbitals molecular orbital method, and are routinely used as variables in linear regression (QSAR^[3]) procedures^[4]. If the coefficients of the basis functions in the molecular orbital are C_μi for the μ'th basis function in the i'th molecular orbital, the density matrix terms are:

$\mathbf {D_{\mu\nu}} = \mathbf{2}\sum_{i} \mathbf {C_{\mu i}} \mathbf {C_{\nu i}^*}$

for a closed shell system where each molecular orbital is doubly occupied. The population matrix $\mathbf{P}$ then has terms

$\mathbf {P_{\mu\nu}} = \mathbf {D_{\mu\nu}} \mathbf {S_{\mu\nu}}$

$\mathbf{S}$ is the overlap matrix of the basis functions. The sum of all terms of $\mathbf {P_{\nu\mu}}$ summed over $\mathbf {\mu}$ is the gross orbital product for orbital $\mathbf {\nu}$ - $\mathbf {GOP_{\nu}}$. The sum of the gross orbital products is N - the total number of electrons. The Mulliken population assigns an electronic charge to a given atom A, known as the gross atom population: $\mathbf {GAP_{A}}$ as the sum of $\mathbf {GOP_{\nu}}$ over all orbitals $\mathbf {\nu}$ belonging to atom A. The charge, $\mathbf {Q_{A}}$, is then defined as the difference between the number of electrons on the isolated free atom, which is the atomic number $\mathbf {Z_{A}}$, and the gross atom population:

$\mathbf {Q_{A}} = \mathbf {Z_{A}} - \mathbf {GAP_{A}}$

The problem with this approach is the equal division of the off-diagonal terms between the two basis functions. This leads to charge separations in molecules that are exaggerated. In a modified Mulliken population analysis,^[5] this problem can be reduced by dividing the overlap populations $\mathbf {P_{\mu\nu}}$ between the corresponding orbital populations $\mathbf {P_{\mu\mu}}$ and $\mathbf {P_{\nu\nu}}$ in the ratio between the latter. This choice, although still arbitrary, relates the partitioning in some way to the electronegativity difference between the corresponding atoms.

See also

• Partial charge, for other methods used to estimate atomic charges in molecules.

References

1. ^ Mulliken, R. S. (1955). "Electronic Population Analysis on LCAO-MO Molecular Wave Functions. I". The Journal of Chemical Physics 23 (10): 1833–1831. Bibcode 1955JChPh..23.1833M. doi:10.1063/1.1740588.  edit
2. ^ I. G. Csizmadia, Theory and Practice of MO Calculations on Organic Molecules, Elsevier, Amsterdam, 1976.
3. ^ Leach, Andrew R. (2001). Molecular modelling: principles and applications. Englewood Cliffs, N.J: Prentice Hall. ISBN 0-582-38210-6. 
4. ^ Ohlinger, William S.; Philip E. Klunzinger, Bernard J. Deppmeier, and Warren J. Hehre (January 2009). "Efficient Calculation of Heats of Formation". The Journal of Physical Chemistry A (ACS Publications) 113 (10): 2165–2175. doi:10.1021/jp810144q. PMID 19222177. 
5. ^ Bickelhaupt, F. M.; van Eikema Hommes, N. J. R.; Fonseca Guerra, C.; Baerends, E. J. (1996). "The Carbon−Lithium Electron Pair Bond in (CH₃Li)_n (n = 1, 2, 4)". Organometallics 15 (13): 2923–2931. doi:10.1021/om950966x.  edit