Algebraic geminal creators
A geminal is an electron pair state \(\require{physics} \ket{\gamma}\): \[\begin{equation} \ket{\gamma} = \hat{\Gamma}^+_{\gamma} \ket{\text{vac}} \thinspace , \end{equation}\] where \(\hat{\Gamma}^+_{\gamma}\) is a yet unspecified geminal creation operator for the geminal with label \(\gamma\).1 Since the geminal creation operator \(\hat{\Gamma}^+_{\gamma}\) creates an electron pair, its commutator with the number operator \(\hat{N}\) must be: \[\begin{equation} \comm{ \hat{N} }{ \hat{\Gamma}^+_{\gamma} } = 2 \hat{\Gamma}^+_{\gamma} \thinspace . \end{equation}\] Any operator \(\hat{\Gamma}^+_{\gamma}\) that satisfies this commutation relation will automatically describe an electron pair, so this equation might serve as a very abstract definition of a geminal creation operator.
For non-relativistic applications, we introduce a set of \(M\) orthonormal spinors \(\set{\phi_P(\vb{r})}\), which are the electron single particle states to which the elementary operators \(\hat{a}^\dagger_P\) and \(\hat{a}_P\) are linked. If we exclude annihilators from \(\hat{\Gamma}^+_\gamma\), we may write the most general algebraic geminal creator as: \[\begin{equation} \hat{\Gamma}_\gamma^+ = \sum_{PQ} G^{PQ}_\gamma \hat{a}_P^\dagger \hat{a}_Q^\dagger \thinspace . \end{equation}\] Here, we have introduced a composite particle creation operator for the “particle” \(\gamma\), in which all possible combinations of pair creations \(\hat{a}_P^\dagger \hat{a}_Q^\dagger\) are allowed, each weighed by a certain geminal coefficient \(G_\gamma^{PQ}\). The inclusion or exclusion of certain geminal coefficients \(G_\gamma^{PQ}\) leads to the so-called notions of geminal pairing schemes. The composite particle \(\ket{\gamma}\) is called a generalized electron pair, or geminal.
Due to the fermion anticommutation rules, we immediately have that the geminal coefficients must be antisymmetric: \[\begin{equation} G^{QP}_\gamma = - G^{PQ}_\gamma \thinspace , \end{equation}\] from which automatically follows that the contributions of the form \[\begin{equation} G^{PP}_\gamma \hat{a}^\dagger_P \hat{a}^\dagger_P \end{equation}\] vanish automatically, which was to be expected from the Pauli principle that is encoded into the fermion anticommutation relations. Taking into account these previously discussed redundant parameters, we can equally write this geminal creation operator as \[\begin{equation} \hat{\Gamma}_\gamma^+ = 2 \sum_{P>Q}^M G_\gamma^{PQ} \hat{a}_P^\dagger \hat{a}_Q^\dagger \thinspace , \end{equation}\] as is done in (Péter. R. Surján 1999), but he incorporates the factor of 2 inside the geminal coefficient.
As an example, we may describe a two-electron system using three spinors using the following geminal creator: \[\begin{align} \hat{\Gamma}^+ &= G^{12} \hat{a}^\dagger_1 \hat{a}^\dagger_2 + G^{13} \hat{a}^\dagger_3 \hat{a}^\dagger_3 + G^{21} \hat{a}^\dagger_2 \hat{a}^\dagger_1 + G^{23} \hat{a}^\dagger_2 \hat{a}^\dagger_3 + G^{31} \hat{a}^\dagger_3 \hat{a}^\dagger_1 + G^{32} \hat{a}^\dagger_3 \hat{a}^\dagger_2 \\ &= 2 G^{21} \hat{a}^\dagger_2 \hat{a}^\dagger_1 + 2 G^{31} \hat{a}^\dagger_3 \hat{a}^\dagger_1 + 2 G^{32} \hat{a}^\dagger_3 \hat{a}^\dagger_2 \thinspace , \end{align}\] where we have dropped the geminal label since we only require one of them for the description of a two-electron system.
As opposed to working with spinors of a general type, we can also provide a description when we are working in a spin-orbital basis. We may then write the geminal creator as: \[\begin{equation} \hat{\Gamma}^+_\gamma = \sum_{pq} \sum_{\sigma \tau} G^{p \sigma, q \tau}_\gamma \hat{a}^\dagger_{p \sigma} \hat{a}^\dagger_{q \tau} \thinspace , \end{equation}\] where the geminal coefficients are still antisymmetric upon exchange of spin-orbital labels: \[\begin{equation} G^{q \tau, p \sigma}_\gamma = - G^{p \sigma, q \tau}_\gamma \thinspace . \end{equation}\] Using the two-electron spin tensor creation operators, together with the change of variables: \[\begin{align} & G^{0,0; \thinspace pq}_\gamma = \frac{1}{\sqrt{2}} \qty( G^{p \alpha, q \beta}_\gamma - G^{p \beta, q \alpha}_\gamma ) \\ % & G^{1,1; \thinspace pq}_\gamma = G^{p \alpha, q \alpha}_\gamma \\ % & G^{1,0; \thinspace pq}_\gamma = \frac{1}{\sqrt{2}} \qty( G^{p \alpha, q \beta}_\gamma + G^{p \beta, q \alpha}_\gamma ) \\ % & G^{1,-1; \thinspace pq}_\gamma = G^{p \beta, q \beta}_\gamma \thinspace , \end{align}\] we can rewrite the geminal creator as: \[\begin{equation} \hat{\Gamma}^+_\gamma = \sum_{pq} G^{0,0; \thinspace pq}_\gamma \hat{Q}^{0,0}_{pq} + \sum_{M=-1}^1 \sum_{pq} G^{1,M; \thinspace pq}_\gamma \hat{Q}^{1,M}_{pq} \thinspace . \end{equation}\] From the change of variables, we see that the triplet geminal coefficients are antisymmetric: \[\begin{equation} G^{1,M; \thinspace qp}_\gamma = - G^{1,M; \thinspace pq}_\gamma \thinspace , \end{equation}\] while the singlet geminal coefficients are symmetric: \[\begin{equation} G^{0,0; \thinspace qp}_\gamma = G^{0,0; \thinspace pq}_\gamma \thinspace . \end{equation}\] Alternative treatments of spin in geminal theories are described by Surján (P. R. Surján 2016).
The simplest spin-orbital geminal creator is then given by the singlet geminal creator: \[\begin{equation} \hat{\Gamma}^{0, 0; \thinspace +}_\gamma = \sum_{pq} G^{0,0; \thinspace pq}_\gamma \hat{Q}^{0,0}_{pq} \thinspace , \end{equation}\] with the symmetric singlet geminal coefficients \(G^{0,0; \thinspace pq}_\gamma\).
The pair functions \(\omega_R(\vb{r}_1, \vb{r}_2)\) and \(\omega_S(\vb{r}_1, \vb{r}_2)\) are called weakly orthogonal (P. R. Surján 2016) if their overlap integral is zero: \[\begin{equation} \int \int \dd{\vb{r}_1} \dd{\vb{r}_2} % \omega^*_R(\vb{r}_1, \vb{r}_2) % \omega_S(\vb{r}_1, \vb{r}_2) = 0 % \thinspace . \end{equation}\] This can be because (a) they do not overlap because they are localized in different regions of space, or (b) because they are not localized in different parts of space, but their nodal structure cancels out the overlap integral, like the \(2p\)-functions do. (P. R. Surján 2016) The pair functions are then called strongly orthogonal (P. R. Surján 2016) if the following relation holds: \[\begin{equation} \label{eq:strong_orthogonality} \int \dd{\vb{r}} % \omega^*_R(\vb{r}, \vb{r}_1) % \omega_S(\vb{r}, \vb{r}_2) % = 0 % \thinspace . \end{equation}\]
References
For a \(N = 2 N_P\)-electron system, \(\gamma = 1, 2, \ldots, N_P\) with \(N\) the number of electrons and \(N_P = N/2\) the number of electron pairs.↩︎