QUANTUM COMPUTATIONAL STUDIES ON PORPHYCENE - Theoretical and Computational Research in the 21st Century - page 167

9.2.3 LOCAL CHARGES

9.2.3.1 MULLIKEN CHARGE

Mulliken charges [31-33] 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
orbital molecular orbital method. The charge thus arises from the Mulliken
population analysis.
Let us consider C μi is the coeffi cients of the basis functions in the mo-
lecular orbital for the μ'th basis function in the i 'th molecular orbital.
Then, the density matrix terms are given by

∑

D μν =2 ci μ C * νi (28)

i

For a closed-shell system, where each molecular orbital is doubly occu-
pied, the population matrix P has terms:

P μν = D μν S μν (29)

where S is the overlap matrix of the basis functions and the sum of all
terms of P μν is N , which is the total number of electrons.
The Mulliken population analysis aims fi rst to divide N among all the
basis functions. This is done by taking the diagonal element of P μν and then
dividing the off-diagonal elements equally between the two appropriate
basis functions. As the off-diagonal terms include P μν and P μν , this sim-
plifi es to just the sum of a row. This defi nes the gross orbital population
(GOP) as

∑

v Pv

μ

(GOP) μ = (30)

The (GOP) μ terms sum to N and thus divide the total number of electrons
between the basis functions. It remains to sum these terms over all basis
functions on a given atom A to give the gross atom population (GAP). The
sum of (GAP) A terms is also N . The charge, Q A , is then defined as the dif-
ference between the number of electrons on the isolated free atom, which
is the atomic number Z A , and the GAP:

Q A = Z A − (GAP) A (31)