I was reading about the self-consistent extended Hückel method (SC-EHT), and stumbled upon various formulae for the diagonal elements of the Hamiltonian matrix. I noticed that many formulae depend on either the gross atomic population or Mulliken atomic charges. I know the formula for the Mulliken population analysis, but what is the Mulliken atomic charge and how can I evaluate it?



Mulliken atomic charges can be defined as[1]:

$$q_A = Z_A-\sum_{\mu\in A}\left( \mathbf{P\cdot S} \right)_{\mu\mu} \tag{Szabo 3.196}$$

Here I have used the same notation as in Szabo[1], with $\mathbf{P}$ being the density matrix and $\mathbf{S}$ beign the overlap matrix. $Z$ is the nuclear charge, and using the greek letter $\mu$ as indicies indicates that we are working in atomic orbital basis (not in in molecular orbital basis).

The sum runs over $\mu\in A$ meaning, that we only consider atomic orbitals that are centered on the $A$th atom. we can therefore note that Mulliken charges are only defined when we used atomic-centered basis functions (which we most often do).

[1] : Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Attila Szabo and Neil S. Ostlund

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