Mulliken population analysis
Mulliken population analysis uses Mulliken’s partitioning scheme to partition the number operator.
In the orthonormal basis, the matrix elements of the number operator are the elements of the identity matrix: \[\begin{equation} \require{physics} \hat{N} = \sum_P \hat{a}^\dagger_P \hat{a}_P \thinspace , \end{equation}\] so Mulliken’s partitioning scheme leads to the following form of the (Hermitianized) Mulliken-partitioned number operator: \[\begin{equation} \hat{w}_A = \sum_{PQ} w_{A, \thinspace PQ} \hat{a}^\dagger_P \hat{a}_Q \thinspace , \end{equation}\] with the matrix elements: \[\begin{equation} \vb{w}_A = \frac{1}{2} \qty( \vb{C}^\dagger \boldsymbol{\mathcal{P}}_A \vb{C}^{-1, \dagger} + \vb{C}^{-1} \boldsymbol{\mathcal{P}}_A \vb{C} ) \thinspace . \end{equation}\] These expressions are in line with the relations found in (Carbó-Dorca and Bultinck 2004) (Soriano and Palacios 2014) (Mayer and Hamza 2005).
We should note that indeed the sum of all Mulliken operators of all fragments is the total number operator: \[\begin{equation} \hat{N} = \sum_A \hat{w}_A \thinspace . \end{equation}\]
If we apply the Mulliken partition scheme to the \(\sigma\)-“number” operator, we find that its matrix elements (still in general spinor basis) are given by: \[\begin{equation} \vb{w}^{\sigma}_A = \frac{1}{2} \qty( \vb{C}^{-1} \boldsymbol{\mathcal{P}}_A \vb{C} \vb{C}^{\sigma, \dagger} \vb{S}^{\text{AO}} \vb{C}^\sigma + \vb{C}^{\sigma, \dagger} \vb{S}^{\text{AO}} \vb{C}^\sigma \vb{C}^\dagger \boldsymbol{\mathcal{P}}_{A} \vb{C}^{-1, \dagger} ) \thinspace . \end{equation}\]
In order to discuss the Mulliken operator in a restricted spin-orbital basis, we must examine the partitioning matrix \(\boldsymbol{\mathcal{P}}_A\) first. Since the AOs initially represent non-orthogonal spin-orbitals, \(\boldsymbol{\mathcal{P}}_A\) expressed for the spinor basis will have entries that are paired, since the scalar basis functions may contribute to both an \(\alpha\)- and a \(\beta\)-spin-orbital.
Therefore, in a restricted spin-orbital basis, the matrix elements \(\hat{w}_A\) (expressed in 2-component spinor basis) can be expressed as a tensorial product with \(\vb{I}_2\), so the Mulliken operator \(\hat{w}_A\) reduces to: \[\begin{equation} \hat{w}_A = \frac{1}{2} \sum_{pq} \qty( \vb{C}^\dagger \boldsymbol{\mathcal{P}}_A \vb{C}^{-1, \dagger} + \vb{C}^{-1} \boldsymbol{\mathcal{P}}_A \vb{C} )_{pq} \hat{E}_{pq} \thinspace , \end{equation}\] where the matrices are now expressed with respect to the spatial orbitals/scalar basis functions.
In a(n) (unrestricted) spin-orbital basis, the Mulliken operators for \(\sigma\)-spin reduce to: \[\begin{equation} \hat{w}^\sigma_A = \frac{1}{2} \sum_{pq} \qty( \vb{C}^{\sigma, -1} \boldsymbol{\mathcal{P}}_A \vb{C}^\sigma + \vb{C}^{\sigma, \dagger} \boldsymbol{\mathcal{P}}_{A} \vb{C}^{\sigma, -1, \dagger} )_{pq} \hat{a}^\dagger_{p \sigma} \hat{a}_{q \sigma} \thinspace . \end{equation}\]