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] The method was developed by Robert S. Mulliken, after whom the method is named. 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:

${\displaystyle \mathbf {D_{\mu \nu }} =\mathbf {2} \sum _{i}\mathbf {C_{\mu i}} \mathbf {C_{\nu i}^{*}} }$

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

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

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

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