A dual vision on geminals
Introduction
We know that a general APIG state can be written as the creation of \(N_P\) generalized pairs (= geminals) on the vacuum and : \[\begin{align*} \require{physics} \ket{\text{APIG}} &= \prod_\gamma^{N_P} \hat{\Gamma}^+_\gamma \ket{\text{vac}} \\ &= \prod_\gamma^{N_P} \qty( \sum_p^K G^p_\gamma \hat{P}^+_p ) \thinspace . \end{align*}\] Dually (with respect to particle-hole duality), we would then write a dual representation of APIG as \[\begin{equation} \label{eq:dual_APIG} \ket*{\widetilde{\text{APIG}}} = \prod_\lambda^{K-N_P} \hat{L}^+_\lambda \ket{\widetilde{\text{vac}}} \thinspace , \end{equation}\] where \(\hat{L}_\lambda\) is the hole-like of \(\hat{\Gamma}_\lambda\), so that equation \(\eqref{eq:dual_APIG}\) means that \(K-N_P\) generalized hole pairs are being created on the hole vacuum. We would then suggest the generalized hole pair creator \(\hat{L}_\lambda\) to be of the form \[\begin{equation} \label{eq:generalized_hole_pair} \hat{L}^+_\lambda = \sum_p^K \tilde{G}^p_\lambda \hat{\xi}^+_p \thinspace , \end{equation}\] in which \(\hat{\xi}^+_p\) would then take the role of a hole creation operator in the spatial orbital \(i\).
The hole vacuum - hole operators
Let us first address the hole-vacuum \(\ket{\widetilde{\text{vac}}}\). We all know that the Fock vacuum \(\ket{\text{vac}}\) is the state in which no electrons (or pairs) are present. Analogously, we could then define the hole vacuum as being the state in which no holes (or hole pairs) are present, which is actually the state that is full with particles (or pairs). DOCI-wise, we would write: \[\begin{equation} \ket{\widetilde{\text{vac}}} = \prod_p^K \hat{P}^+_p \ket{\text{vac}} \thinspace , \end{equation}\] or equivalently: \[\begin{equation} \ket{\text{vac}} = \prod_p^K \hat{P}^-_p \ket{\widetilde{\text{vac}}} \thinspace . \end{equation}\]
Creating a hole pair in the spatial orbital \(p\) is then as easy as annihilating an electron pair in that spatial orbital \(p\). To this end, let us introduce the operators \[\begin{align} & \hat{\xi}^+_p = \hat{P}^-_p \\ & \hat{\xi}^-_p = \hat{P}^+_p \\ & \hat{\xi}^\circ_p = - \hat{P}^\circ_p = \frac{1}{2} \qty(1 - \hat{N}_p) \thinspace , \end{align}\] that obey the \(\mathfrak{su}(2)\) commutation relations: \[\begin{align} & \comm{\hat{\xi}^+_p}{\hat{\xi}^-_q} = 2 \delta_{pq} \hat{\xi}^\circ_p = \delta_{pq} (1 - \hat{N}_p) \\ & \comm{\hat{\xi}^\circ_p}{\hat{\xi}^\pm_q} = \pm \delta_{pq} \hat{\xi}^\pm_p \thinspace . \end{align}\]
We cannot annihilate a hole pair in the hole vacuum: \[\begin{equation} \hat{\xi}^-_p \ket{\widetilde{\text{vac}}} = 0 \thinspace , \end{equation}\] and we can also define a hole pair excitation operator as \[\begin{equation} \hat{\chi}_a^i = \hat{\xi}_i^+ \hat{\xi}^-_a \thinspace , \end{equation}\] which promotes a hole pair from orbital \(a\) to orbital \(i\).
Particle-hole duality in geminal wave function forms
A fully paired reference determinant can be described in a dual way: \[\begin{align} \ket{\Phi_0} &= \prod_i^{N_P} \hat{P}^+_i \ket{\text{vac}} \\ &= \prod_{a=N_P+1}^K \hat{\xi}^+_a \ket{\widetilde{\text{vac}}} \\ &= \ket*{\tilde{\Phi}_0} \thinspace , \end{align}\] and so can a single pair excited reference determinant: \[\begin{align} \ket{\Phi_i^a} &= \hat{P}^+_a \hat{P}^-_i \prod_j^{N_P} \hat{P}^+_j \ket{\text{vac}} \\ &= \hat{\xi}^-_a \hat{\xi}^+_i \prod_{j=N_P+1}^K \hat{\xi}^+_j \ket{\widetilde{\text{vac}}} \\ &= \hat{\chi}_a^i \ket*{\tilde{\Phi}_0} \thinspace . \end{align}\]
As a first case trying to describe geminal wave functions in their dual form, let us consider an AGP wave function : \[\begin{align} \ket{\text{AGP}} &= \qty( \hat{\Gamma}^+ )^{N_P} \ket{\text{vac}} \\ &= \qty( \sum_p^K G_p \hat{P}^+_p )^{N_P} \ket{\text{vac}} \label{eq:AGP_coeff} \thinspace . \end{align}\] A dual representation of AGP would then be \[\begin{equation} \ket*{\widetilde{\text{AGP}}} = \qty( \hat{L}^+ )^{K - N_P} \ket{\widetilde{\text{vac}}} \thinspace , \end{equation}\] with the generalized hole pair creation operator \[\begin{equation} \label{eq:L_hole_coeff} \hat{L}^+ = \sum_p^K \tilde{G}_p \hat{\xi}^+_p \thinspace . \end{equation}\] We can see that the coefficient matrix now becomes a \((K-N_P) \times K\) matrix with equal rows, leading to \(K\) unique dual coefficients.
Indeed, we can show that \[\begin{equation} \ket{\text{AGP}} = \ket*{\widetilde{\text{AGP}}} \thinspace , \end{equation}\] in which the coefficients of equations \(\eqref{eq:AGP_coeff}\) and \(\eqref{eq:L_hole_coeff}\) are linked by the equation \[\begin{equation} N_P! \prod_i^{N_P} G_i = (K - N_P)! \prod_{a=N_P+1}^K \tilde{G}_a \thinspace , \end{equation}\] where it is understood that the \(N_P\) chosen coefficients \(G_i\) on the left-hand side do not appear in the \(K-N_P\) chosen dual coefficients \(\tilde{G}_a\). The existence of the solution \[\begin{equation} \label{eq:G_Gtilde_AGP} G_i \tilde{G}_i = \qty[ \frac{N_P!}{(K-N_P)!} \prod_j^K G_j ]^{\frac{1}{K-N_P}} \thinspace , \end{equation}\] ensures AGP is self-dual (Paul Andrew Johnson 2014).
As an example, let us work out the case of 1 electron pair in 3 orbitals (i.e. \(N_P = 1\), \(K = 3\)). We can then write \[\begin{equation} \ket{\text{AGP}} = \qty( G_1 \hat{P}^+_1 + G_2 \hat{P}^+_2 + G_3 \hat{P}^+_3 ) \ket{\text{vac}} \end{equation}\] and \[\begin{equation} \ket*{\widetilde{\text{AGP}}} = \qty( 2 \tilde{G}_1 \tilde{G}_2 \hat{\xi}^+_1 \hat{\xi}^+_2 + 2 \tilde{G}_1 \tilde{G}_3 \hat{\xi}^+_1 \hat{\xi}^+_3 + 2 \tilde{G}_2 \tilde{G}_3 \hat{\xi}^+_2 \hat{\xi}^+_3 ) \ket{\text{vac}} \thinspace . \end{equation}\] If they are equal, this leads to the following system of equations: \[\begin{equation} \begin{cases} G_1 = 2 \tilde{G}_2 \tilde{G}_3 \\ G_2 = 2 \tilde{G}_1 \tilde{G}_3 \\ G_3 = 2 \tilde{G}_1 \tilde{G}_2 \end{cases} . \end{equation}\] If the dual representation is known (meaning that \(\tilde{G}_1, \tilde{G}_2\) and \(\tilde{G}_3\) are known), the solution is trivially seen from the system of equations. On the other hand, if the regular representation is known (meaning that \(G_1, G_2,\) and \(G_3\) are known), the system of equations has the solution \[\begin{equation} (\tilde{G}_1, \tilde{G}_2, \tilde{G}_3) = \qty( \frac{\sqrt{2 G_1 G_2 G_3}}{2 G_1} , \frac{\sqrt{2 G_1 G_2 G_3}}{2 G_2} , \frac{\sqrt{2 G_1 G_2 G_3}}{2 G_3} ) \end{equation}\] in agreement with equation \(\eqref{eq:G_Gtilde_AGP}\).
In the general case, there are \(K\) equations in \(K\) unknowns.
The most general wave function form we will face is APIG: \[\begin{align*} \ket{\text{APIG}} &= \prod_p^{N_P} \hat{\Gamma}^+_p \ket{\text{vac}} \\ &= \prod_p^{N_P} \qty( \sum_i^K G^i_p \hat{P}^+_i ) \ket{\text{vac}} \thinspace , \end{align*}\] with the dual representation \[\begin{align} \ket*{\widetilde{\text{APIG}}} &= \prod_\lambda^{K-N_P} \hat{L}^+_\lambda \ket{\widetilde{\text{vac}}} \\ &= \prod_\lambda^{K-N_P} \qty( \sum_i^K \tilde{G}^i_h \hat{\xi}^+_i ) \ket{\widetilde{\text{vac}}} \thinspace . \end{align}\] The dual coefficient matrix now becomes a \((K-N_P) \times K\) matrix, with as many unique coefficients.
By connecting both representations, we find that they are equal if \[\begin{equation} \label{eq:G_Gtilde_APIG} |\vb{G}(\vb{m})|_+ = |\tilde{\vb{G}}(\tilde{\vb{m}})|_+ \thinspace , \end{equation}\] where the meaning of \(|\vb{G}(\vb{m})|_+\) is explained in section . \(|\tilde{\vb{G}}(\tilde{\vb{m}})|_+\) has a similar meaning, but with the dual coefficients \(\tilde{G}^i_h\), and \(\tilde{\vb{m}}\) means the dual pair occupancy vector (i.e. where every 0 is replaced by 1 and vice versa).
Let us also work out the case of 1 electron pair in 3 orbitals (i.e. \(N_P = 1\), \(K = 3\)). According to equation \(\eqref{eq:G_Gtilde_APIG}\), APIG is self-dual if we can solve the following system of equations: \[\begin{equation} \begin{cases} G^1_1 = \tilde{G}^2_1 \tilde{G}^3_2 + \tilde{G}^2_2 \tilde{G}^3_1 \\ G^2_1 = \tilde{G}^1_1 \tilde{G}^3_2 + \tilde{G}^1_2 \tilde{G}^3_1 \\ G^3_1 = \tilde{G}^1_1 \tilde{G}^2_2 + \tilde{G}^1_2 \tilde{G}^2_1 \end{cases} \thinspace . \end{equation}\] When the dual representation is known, the coefficients in the regular representation can be trivially found from the system of equations. If, however, we are looking to find the dual representation when the regular representation is known, we are trying to solve a (non-linear) system of 3 equations in 6 unknowns, which could signify that APIG is not, in general, self-dual.
In general, there are \(\binom{K}{N_P}\) equations (number of unique occupancy vectors) in \(N_P \times K\) (number of APIG coefficients) or \((K-N_P) \times K\) (number of dual APIG coefficients) unknowns, depending on the problem we are facing.