Mulliken partitioning
We can generalize the theory of Mulliken population analysis (Mulliken 1955) (Carbó-Dorca and Bultinck 2004) and come up with a general way to partition (Soriano and Palacios 2014) (Mayer and Hamza 2005) (Vyboishchikov and Salvador 2006) second-quantized operators, based on the assignment of different AOs to a certain fragment of a molecule. Since the partitioning happens on the level of the AO basis, we will make extensive use of the theory of non-orthogonal spinor bases.
The end goal of this scheme is to partition the second-quantized one-electron operator \[\begin{equation} \require{physics} \hat{f} = \sum_{PQ} f_{PQ} \hat{a}^\dagger_P \hat{a}_Q \end{equation}\] into operators \(\hat{f}_A\) (each belonging to a certain partitioning/fragment \(A\)), such that: \[\begin{equation} \hat{f} = \sum_A \hat{f}_A \thinspace . \end{equation}\]
Using the non-orthogonal elementary operators, we can write a Hermitian one-electron operator as \[\begin{align} \hat{f} &= \sum_{PQ} f_{PQ} \hat{a}^\dagger_P \hat{a}_Q \\ &= \sum_{\mu \nu} ( \vb{C} \vb{f} \vb{C}^\dagger )_{\mu \nu} \hat{b}^\dagger_\mu \hat{b}_\nu \thinspace . \end{align}\]
It is in this, non-orthogonal, basis that Mulliken’s partitioning scheme is applied, assigning the basis functions of the AO basis to disjoint sets/fragments. We can subsequently write: \[\begin{equation} \hat{f} = \sum_A \sum_{\mu} \sum_{\nu \in A} ( \vb{C} \vb{f} \vb{C}^\dagger )_{\mu \nu} \hat{b}^\dagger_\mu \hat{b}_\nu \thinspace , \end{equation}\] where the summation is over all fragments/partitions \(A\) and basis function indices \(\nu \in A\) (i.e. that belong to the partition \(A\)). We may then rewrite the summation as: \[\begin{equation} \hat{f} = \sum_A \sum_{\mu \nu} ( \vb{C} \vb{f} \vb{C}^\dagger )_{\mu \nu} \delta_{\nu \in A} \hat{b}^\dagger_\mu \hat{b}_\nu \thinspace , \end{equation}\] using the Kronecker-delta-like symbol \(\delta_{\nu \in A}\), which only has a value equal to \(1\) if \(\nu\) belongs to the partition \(A\). We will now introduce a small mathematical trick, introducing a summation and a Kronecker-delta: \[\begin{equation} \hat{f} = \sum_A \sum_{\mu \nu \lambda} ( \vb{C} \vb{f} \vb{C}^\dagger )_{\mu \nu} \delta_{\nu \in A} \delta_{\nu \lambda} \hat{b}^\dagger_\mu \hat{b}_\lambda \thinspace , \end{equation}\] and we subsequently define partitioning matrices \(\set{ \boldsymbol{\mathcal{P}}_A }\) that partition the set of basis functions (AOs) into disjoint sets. For a fragment \(A\), the partition matrix \(\boldsymbol{\mathcal{P}}_A\) a diagonal matrix, whose entries consists of ones if the related basis function belongs to the set of basis functions associated with fragment \(A\) and zeros otherwise: \[\begin{equation} \mathcal{P}_{A, \thinspace \mu \nu} = \delta_{\mu \nu} \delta_{\mu \in \text{A}} \thinspace . \end{equation}\] Recognizing the elements of the partitioning matrices in the previous expression for the second-quantized operator, we may write: \[\begin{equation} \hat{f} = \sum_A \sum_{\mu \lambda} ( \vb{C} \vb{f} \vb{C}^\dagger \boldsymbol{\mathcal{P}}_{A} )_{\mu \lambda} \hat{b}^\dagger_\mu \hat{b}_\lambda \thinspace . \end{equation}\]
Rewriting this operator back using the orthonormal creation and annihilation operators, we finally find: \[\begin{equation} \hat{f} = \sum_A \sum_{PQ} ( \vb{f} \vb{C}^\dagger \boldsymbol{\mathcal{P}}_{A} \vb{C}^{-1, \dagger} )_{PQ} \hat{a}^\dagger_P \hat{a}_Q \thinspace , \end{equation}\] such that we can naturally define the Mulliken-partitioned operator \(\hat{f}_A\) as: \[\begin{equation} \hat{f}_A = \sum_{PQ} ( \vb{f} \vb{C}^\dagger \boldsymbol{\mathcal{P}}_{A} \vb{C}^{-1, \dagger} )_{PQ} \hat{a}^\dagger_P \hat{a}_Q \thinspace . \end{equation}\]
Unfortunately, this operator isn’t necessarily Hermitian, i.e. \[\begin{equation} \vb{f} \vb{C}^\dagger \boldsymbol{\mathcal{P}}_{A} \vb{C}^{-1, \dagger} \overset{?}{=} \vb{C}^{-1} \boldsymbol{\mathcal{P}}_A \vb{C} \vb{f} \thinspace , \end{equation}\] but, we can introduce the (Hermitianized) operator \(\hat{u}\): \[\begin{equation} \hat{u}_A = \frac{1}{2} \sum_{PQ} \qty( \vb{C}^{-1} \boldsymbol{\mathcal{P}}_A \vb{C} \vb{f} + \vb{f} \vb{C}^\dagger \boldsymbol{\mathcal{P}}_{A} \vb{C}^{-1, \dagger} )_{PQ} \hat{a}^\dagger_P \hat{a}_Q \thinspace , \end{equation}\] which still satisfies \[\begin{equation} \hat{f} = \sum_A \hat{u}_A \thinspace , \end{equation}\] since the following relation holds for the partition matrices \[\begin{equation} \sum_A \boldsymbol{\mathcal{P}}_A = \vb{I} \thinspace , \end{equation}\] i.e. every basis function only belongs to one fragment. Furthermore, the Hermitianized operator \(\hat{u}_A\) has the same expectation value for real wave functions and atomic orbitals: \[\begin{equation} \ev{\hat{f}_A}{\Psi} = \ev{\hat{u}_A}{\Psi} \thinspace . \end{equation}\]
Partitioning the number operator according to Mulliken’s partitioning scheme leads to Mulliken population analysis.