Configuration state functions

The exact wave function, which is an eigenfunction of the non-relativistic Hamiltonian, should have quantum numbers \(S\) and \(M\) in Fock space, as the Hamiltonian is a singlet operator and thus commutes with both \(\hat{S}^z\) and \(\hat{S}^2\). Since the basis vectors of Fock space are SSDs (determinants, ON vectors), it makes sense to look at their spin properties. It can be shown that SSDs are eigenfunctions of projected spin: \[\begin{equation} \require{physics} \hat{S}^z \ket{\vb{k}} % = \frac{1}{2} (n_\alpha - n_\beta) \ket{\vb{k}} % \thinspace , \end{equation}\] in which \[\begin{equation} n_\sigma = \sum_p^K k_{p \sigma} (1 - k_{p \bar{\sigma}}) \thinspace , \end{equation}\] but they are not, in general, eigenfunctions of total spin. Only for closed-shell systems (in which all orbitals are doubly occupied) and high-spin states (in which all singly occupied orbitals have the same spin), they are. For closed shell-systems: \[\begin{equation} \hat{S}^2 \ket{\vb{k}} = 0 % \thinspace , \end{equation}\] and for high-\(\sigma\)-spin states: \[\begin{equation} \hat{S}^2 \ket{\vb{k}} % = \frac{n_\sigma}{2} \qty( \frac{n_\sigma}{2} + 1 ) \ket{\vb{k}} % \thinspace . \end{equation}\]

Configuration state functions then are a basis of functions that are simultaneously eigenfunctions of projected spin, total spin and the orbital ON operators. An orbital configuration is the set of all determinants with the same orbital ONs, in a sense an orbital configuration is degenerate w.r.t. the orbital ON operators. We can show that \(\hat{S}^2\) does not couple determinants from different orbital configurations, as \[\begin{equation} (k_p - m_p) \matrixel{\vb{k}}{\hat{S}^2}{\vb{m}} = 0 % \thinspace . \end{equation}\] The advantages of using CSFs as basis functions instead of SSDs are that they immediately impose the correct spin symmetry and lead to a shorter wave function expansion.

The genealogical coupling scheme arrives at an \(N\)-electron CSF after \(N\) steps. Since there is no need to consider doubly occupied orbitals explicitly, we will only consider \(N_\text{open}=N\) singly occupied orbitals. We can represent a genealogical coupling by a vector \(\vb{T}\) of length \(N\), where each element \(T_i\) indicates the total spin resulting from the coupling of the first \(i\) electrons. Equivalently, we can use the vector \(\vb{t}\) that represents the genealogical coupling via the intermediate spin couplings (\(t_i = T_i - T_{i-1}\) and \(t_1=T_1\)). For determinants, we can represent each determinant by a vector \(\vb{P}\), where \(P_i\) gives the spin projection of the first \(i\) spin orbitals, or we can equivalently introduce \(\vb{p}\), where \(p_i\) is the spin projection of the \(i\)-th spin orbital.

We will expand the \(N\)-electron CSF with total spin \(S\) and projected spin \(M\) in the determinants with projected spin \(M\) belonging to the same orbital configuration: \[\begin{equation} \label{eq:csf_expansion} \ket{\vb{t}}^\text{csf} % = \sum_i d_i \ket{{}^i\vb{p}}^\text{det} % \thinspace , \end{equation}\] where we can write the determinants as \[\begin{equation} \ket{\vb{t}}^\text{det} % = \hat{a}^\dagger_{1 p_1} \cdots \hat{a}^\dagger_{N p_N} % \ket{\text{cs}} \thinspace , \end{equation}\] where \(\ket{\text{cs}}\) represents the closed-shell core electrons. We can also write \[\begin{equation} \ket{\vb{t}}^\text{csf} = \hat{O}^{S,M}_N \ket{\text{cs}} % \thinspace , \end{equation}\] where \(\hat{O}^{S,M}_N\) generates a normalized \(N\)-electron spin eigenfunction as specified by \(\vb{t}\). The genealogical coupling is then represented by the following equation: \[\begin{align} \hat{O}^{S,M}_N(\vb{t}) \ket{\text{cs}} % &= \sum_{\sigma = \pm \frac{1}{2}} % C^{S,M}_{t_N, \sigma} % \hat{O}^{S-t_N,M-\sigma}_{N-1}(\vb{t}) % \hat{a}^\dagger_{N \sigma} % \ket{\text{cs}} \\ &= C^{S,M}_{t_N, 1/2} % \hat{O}^{S-t_N,M-1/2}_{N-1}(\vb{t}) % \hat{a}^\dagger_{N \alpha} \ket{\text{cs}} % + C^{S,M}_{t_N, -1/2} % \hat{O}^{S-t_N,M+1/2}_{N-1}(\vb{t}) % \hat{a}^\dagger_{N \beta} \ket{\text{cs}} % \label{eq:coupling_creators} \thinspace , \end{align}\] in which \(C^{S,M}_{t_N, \sigma}\) are the Clebsch-Gordan coefficients: \[\begin{align} & C^{S, M}_{1/2, \sigma} = \sqrt{\frac{S + 2 \sigma M}{2S}} \\ & C^{S, M}_{-1/2, \sigma} % = -2 \sigma \sqrt{ \frac{S + 1 - 2\sigma M}{2(S + 1)} } \thinspace . \end{align}\]

The coupling (expansion) coefficients from equation \(\eqref{eq:csf_expansion}\) can be calculated by \[\begin{equation} ^{\text{det}} \braket{\vb{p}}{\vb{t}}^\text{csf} % = \prod_{n=1}^N C^{T_n, P_n}_{t_n, p_n} % \thinspace , \end{equation}\] in which \(\ket{\vb{p}}^\text{det}\) is an unspecified Slater determinant, and where it is understood that \[\begin{equation} |P_N| \leq T_N \thinspace . \end{equation}\] At any coupling step, we may use a coupling with annihilators \[\begin{equation} \hat{O}^{S,M}_N(\vb{t}) \ket{\text{cs}} = C^{S,M}_{t_N, -1/2} \hat{O}^{S-t_N,M+1/2}_{N-1}(\vb{t}) \hat{a}_{N \alpha} \ket{\text{cs}} - C^{S,M}_{t_N,1/2} \hat{O}^{S-t_N,M-1/2}_{N-1}(\vb{t}) \hat{a}_{N \beta} \ket{\text{cs}} \thinspace , \end{equation}\] instead of with creators as in equation \(\eqref{eq:coupling_creators}\).