Mulliken populations

By default, the density matrix printed is the Coulson matrix, which assumes that the atomic orbitals are orthogonalized.

If the assumption of orthogonality is not made, then the Mulliken density matrix can be constructed. To construct the Mulliken density matrix (also known as the Mulliken population analysis), the M.O.s must first be re-normalized, using the overlap matrix, S:

\begin{displaymath}\psi_i^{'} = \psi_i\times S^{-\frac{1}{2}}.

From these M.O.s, a Coulson population is carried out. The off diagonal terms are simply the Coulson terms multiplied by the overlap:

\begin{displaymath}P_{\lambda\sigma\neq\lambda}'=S_{\lambda\sigma}2\sum_{i=1}^{occ}c_{\lambda i}
c_{\sigma i},

while the on-diagonal terms are given by the Coulson terms, plus half the sum of the off-diagonal elements:

\begin{displaymath}P_{\lambda \lambda}' =S_{\lambda\sigma}2\sum_{i=1}^{occ}c_{\l...
... i}
+ \frac{1}{2}\sum_{\sigma\neq\lambda}P_{\lambda \sigma}'.

A check of the correctness of the Mulliken populations is to add the diagonal terms: these should equal the number of electrons in the system.   As with the Coulson population, the unit of the Mulliken population analysis is the electron.  For the hydrogen atom, the Pss population would be 1.00.