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\section{L\"owdin and Mulliken population analysis}

We have $| \phi_\mu\rangle$ atomic (or atomic-like) 
orbitals, not necessarily orthonormal, and we want
to write KS orbitals for our system:
$|\psi_\alpha\rangle$, as sums over said
atomic orbitals:
$$
|\psi_\alpha\rangle = \sum_\mu c_\mu^{(\alpha)} | \phi_\mu \rangle.
$$
The generalized orthonormality relations for KS orbitals
is written as:
$$
\langle\psi_\alpha| \hat S |\psi_\beta\rangle = \delta_{\alpha\beta}.
$$
where $\hat S$ is an operator defined in the US PP framework.
The charge density $\rho({\bf r})$ is given by
$$
\rho({\bf r}) = \sum_{\mu,\nu} P_{\mu\nu} 
\left (\phi^*_\nu \hat S \phi_\mu\right) ({\bf r})
$$
where 
$$
P_{\mu\nu} = \sum_\alpha  c_\mu^{(\alpha)} c_\nu^{(\alpha)*}
$$
define an operator $\hat P$, and
$$
\left (\phi^*_\nu \hat S \phi_\mu\right) ({\bf r}) =
\phi^*_\nu({\bf r}) \phi_\mu({\bf r}) +
\sum_{lm} \langle\phi_\nu|\beta_l\rangle q_{lm}({\bf r})
          \langle\beta_m|\phi_\mu\rangle
$$
where the $\beta$'s and $q$'s are components of the US PP.

\subsection{Mulliken population analysis}

We write the total number of electrons $N$ as:
$$
N = \int \rho({\bf r}) d{\bf r}
  = \mbox{Tr} \hat P\hat S = \sum_\mu (\sum_\nu P_{\mu\nu} S_{\nu\mu})
         \equiv \sum_\mu q_\mu
$$
where $q_\mu$ is the {\em Mulliken charge} associated to state $\mu$ :
$$
q_\mu = \sum_\nu \sum_\alpha c_\mu^{(\alpha)} c_\nu^{(\alpha)*} S_{\nu\mu}
$$
and
$$
S_{\nu\mu} = \langle\phi_\nu|\hat S|\phi_\mu\rangle
           = \int \phi^*_\nu({\bf r}) \phi_\mu({\bf r}) d{\bf r} +
             \sum_{lm} \langle\phi_\nu|\beta_l\rangle Q_{lm}
             \langle\beta_m|\phi_\mu\rangle
$$
with
$$
Q_{lm}=\int q_{lm}({\bf r}) d{\bf r}.
$$
In general, this matrix is not diagonal, even with NC PP.
The coefficients $c_\mu^{(\alpha)}$ are obtained by inverting the linear 
system:
$$
\langle \phi_\nu | \hat S | \psi_\alpha\rangle = 
  \sum_\mu c_\mu^{(\alpha)} S_{\nu\mu}
$$
that is
$$
  c_\mu^{(\alpha)} = (\hat S^{-1})_{\mu\nu}
  \langle \phi_\nu | \hat S | \psi_\alpha\rangle 
$$
and finally
$$
q_\mu  = \sum_{\alpha\nu} 
         \langle \psi_\alpha | \hat S | \phi_\mu\rangle
         (\hat S^{-1})_{\mu\nu} 
         \langle \phi_\nu | \hat S | \psi_\alpha\rangle.
$$

\subsection{L\"owdin population analysis}

The total number of electrons $N$ can be alternatively written as
$$
N = \mbox{Tr} \left[\hat S^{1/2}\hat P \hat S^{1/2}\right] 
  = \sum_\mu \tilde q_\mu.
$$
where $\tilde q_\mu$ is called {\em L\"owdin charge} associated to state 
$\mu$. Let us introduce an auxiliary set of atomic orbitals 
$\tilde\phi$ via : 
$$
 |\tilde\phi_\mu\rangle = \sum_\nu (\hat S^{-1/2})_{\nu\mu}| \phi_\nu\rangle,
\qquad
 | \phi_\mu\rangle = \sum_\nu (\hat S^{1/2})_{\nu\mu}|\tilde\phi_\nu\rangle,
$$
for which the generalized orthonormality relation
$$
 \langle\tilde\phi_\mu|\hat S| \tilde\phi_\nu\rangle = \delta_{\mu\nu}
$$
holds. The KS orbitals for our systems can be rewritten as
$$
|\psi_\alpha\rangle = \sum_\mu c_\mu^{(\alpha)} | \phi_\mu \rangle
                    = \sum_\mu \tilde c_\mu^{(\alpha)} | \tilde\phi_\mu \rangle
$$
where
$$
\tilde c_\mu^{(\alpha)} = \sum_\nu (\hat S^{1/2})_{\nu\mu} c_\nu^{(\alpha)}.
$$
By comparison with the above expression of $\tilde q_\mu$ we get
$$
\tilde q_\mu = \sum_\alpha |\tilde c_\mu^{(\alpha)}|^2
       = |\langle \tilde\phi_\mu | \hat S | \psi_\alpha\rangle|^2
$$

Reference: {\em Modern Quantum Chemistry}, A. Szabo and N. Ostlund (Dover, NY 1996),
p. 153
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