Seniority
The seniority operator counts the number of orbitals that are singly occupied in a Slater determinant. It is a Hermitian operator and can be defined (Bytautas et al. 2011) as \[\begin{equation} \label{eq:seniority_operator} \require{physics} \hat{\Omega} % = \sum_p^K \hat{\Omega}_p % \thinspace , \end{equation}\] where \(\hat{\Omega}_p\) can be written in many different Van Raemdonck et al. (2015) ways: \[\begin{align} \hat{\Omega}_p % &= \sum_\sigma % \hat{a}^\dagger_{p \sigma} \hat{a}_{p \sigma} % - \sum_{\sigma \tau} % \hat{a}^\dagger_{p \sigma} \hat{a}^\dagger_{p \tau} % \hat{a}_{p \tau} \hat{a}_{p \sigma} \\ &= \hat{E}_{pp} - \hat{e}_{pppp} \\ &= \hat{a}^\dagger_{p \alpha} \hat{a}_{p \alpha} % + \hat{a}^\dagger_{p \beta} \hat{a}_{p \beta} % - 2 \hat{a}^\dagger_{p \alpha} \hat{a}^\dagger_{p \beta} % \hat{a}_{p \beta} \hat{a}_{p \alpha} \\ &= \hat{N}_{p \alpha} + \hat{N}_{p \beta} % - 2 \hat{N}_{p \alpha} \hat{N}_{p \beta} \\ &= (\hat{N}_{p \alpha} - \hat{N}_{p \beta})^2 % \label{eq:seniority_number_operators_squared} \\ &= \frac{4}{3} \hat{S}^2_p % \label{eq:seniority_casimir_S} \\ &= -\frac{4}{3} \hat{C}_p + 1 % \label{eq:seniority_casimir_P} \thinspace . \end{align}\]
The vacuum state has seniority zero: \[\begin{equation} \hat{\Omega} \ket{\text{vac}} % = 0 % \thinspace . \end{equation}\] The seniority of a general wave function \(\ket{\Psi}\) can be calculated as \[\begin{equation} \ev{ \hat{\Omega} }{\Psi} % = N - \sum_p^K d_{pppp} % \thinspace , \end{equation}\] in which \(\vb{d}\) is the 2-DM.
As the \(p\)-th orbital seniority operator can be written in linear terms of the \(\hat{S}\) and \(\hat{P}\) Casimir operators, i.e. \(\hat{S}^2_p\) and \(\hat{C}_p\), the following commutators are also zero: \[\begin{equation} \label{eq:commutator_seniority_P} \comm{ % \hat{\Omega}_p % }{ % \hat{P}^+_q % } % = \comm{ % \hat{\Omega}_p % }{ % \hat{P}^-_q % } % = \comm{ % \hat{\Omega}_p % }{ % \hat{P}^\circ_q % } = 0 % \thinspace , \end{equation}\] and \[\begin{equation} \label{eq:commutator_seniority_S} \comm{ % \hat{\Omega}_p % }{ % \hat{S}^+_q % } % = \comm{ % \hat{\Omega}_p % }{ % \hat{S}^-_q % } % = \comm{ % \hat{\Omega}_p % }{ % \hat{S}^z_q % } = 0 % \thinspace . \end{equation}\] Furthermore, since orbital occupation number operators commute among themselves, we have \[\begin{equation} \label{eq:commutator_seniority_on_spin} \comm{ % \hat{N}_{p \sigma} % }{ % \hat{\Omega}_q % } = 0 \end{equation}\] and therefore \[\begin{equation} \label{eq:commutator_seniority_on} \comm{ % \hat{N}_p % }{ % \hat{\Omega}_q % } = 0 % \thinspace . \end{equation}\] A single Slater determinant (an occupation number vector \(\ket{\vb{k}}\)) is an eigenvector of the seniority operator: \[\begin{equation} \hat{\Omega} \ket{\vb{k}} % = \Omega_{\vb{k}} \ket{\vb{k}} % \thinspace , \end{equation}\] with \[\begin{equation} \Omega_{\vb{k}} % = \sum_p^K \qty( % k_{p \alpha} - k_{p \beta} % )^2 % \thinspace , \end{equation}\] which means that an arbitrary linear combination of occupation number vectors, \[\begin{equation} \ket{\vb{c}} % = \sum_{\vb{k}} % c_{\vb{k}} \ket{\vb{k}} % \thinspace , \end{equation}\] can be divided in so called seniority sectors (Bytautas et al. 2011), i.e. sets of occupation number vectors (Slater determinants) that belong to a certain seniority number. Seniority being a good quantum number, or preserving the seniority number means that if an operator \(\hat{A}\), acts on for example a seniority zero wave function, the corresponding seniority number \(\Omega=0\) will not change: \[\begin{equation} \hat{A} \ket{\Omega = 0} % \propto % \ket{\Omega = 0} % \qquad \iff \qquad % \comm{ % \hat{A} % }{ % \hat{\Omega} % } = 0 % \thinspace . \end{equation}\] In other words, \(\hat{A} \ket{\Omega = 0}\) remains in the seniority zero sector, or equivalently: the amount of broken pairs (for singlet wave functions) remains the same. Another way to say this is that \(\hat{\Omega}\) is a symmetry or constant of motion for \(\hat{A}\). We have already seen examples of seniority-preserving operators: \(\hat{P}^+_p, \hat{P}^-_p\) and \(\hat{P}^\circ_p\) (see above: the commutators with \(\hat{\Omega}\) vanish).
The dimension of the Fock space spanned by the determinants having \(N_\alpha\) \(\alpha\)-electrons and \(N_\beta\) \(\beta\)-electrons and is equal to \[\begin{equation} \dim \mathcal{F}(2K, N_\alpha, N_\beta) % = \binom{K}{N_\alpha} \binom{K}{N_\beta} % \thinspace , \end{equation}\] in which \(K\) is the number of spatial orbitals. We can arrivate at since every electron by realizing that every \(\alpha\)-electron can only occupy one of the \(K\) spin orbitals, and equivalently for the \(\beta\)-electrons. Note that this is not equal to \[\begin{equation*} \binom{2K}{N_\alpha + N_\beta} % \thinspace , \end{equation*}\] an expression that could have been derived from equation \(\eqref{eq:dim_fock_M_N}\). The reason for inequality is because the latter way of looking at it is by looking at \(N = N_\alpha + N_\beta\) electrons as a whole, while the former looks at \(N_\alpha\) \(\alpha\)-electrons and \(N_\beta\) \(\beta\)-electrons separately (i.e. with a defined spin projection \(S_z\)). Given an equal amount of \(\alpha\)- and \(\beta\)-electrons (in total summed up to \(N\)), Bytautas (Bytautas et al. 2011) writes the number of seniority \(\Omega=2k\) determinants as \[\begin{equation} D_{\Omega = 2k} % = \binom{K}{N/2} \binom{N/2}{k} \binom{K - N/2}{k} % \thinspace . \end{equation}\] Alcoba (Alcoba et al. 2014) extended this formula to \[\begin{equation} D_{\Omega = N - 2r} % = \binom{K}{N/2 + S_z} % \binom{N/2 + S_z}{r} % \binom{K - (N/2 + S_z)}{N/2 - S_z - r} % \thinspace , \end{equation}\] which in terms of \(N_\alpha\), \(N_\beta\), \(K\) and the seniority \(\Omega\) is \[\begin{equation} D(\Omega, K, N_\alpha, N_\beta) % = \binom{K}{N_\alpha} % \binom{N_\alpha}{ % \frac{ N_\alpha + N_\beta - \Omega }{2} % } % \binom{K-N_\alpha}{ % \frac{ \Omega - N_\alpha + N_\beta }{2} % } % \thinspace . \end{equation}\]