1 Introduction

The formulation of compact wave functions, be it exact or approximate formulations, is a central part of both quantum physics and chemistry. The guiding principles behind these formulations are often vague though attempts at stricter measures of compactness from entropy measures are currently being developed [1,2,3,4,5]. While correlation between entropy and compactness is seen for the methods investigated with the entropy measures these methods are, unlike full configuration interaction (FCI), not invariant to orbital rotations, and the result may therefore be dependent on the choice of orbitals. The usefulness of the entropy as a black box measure of compactness of a wave function is therefore not straightforward. The more traditional way of analysing the quality and compactness of a wave function is the statistical analysis of direct numerical comparisons [6, 7]. While the statistical analysis is straightforward, it is, however, difficult to extend to large systems since there is no exact result to compare against and the comparisons are therefore limited in the number of electrons and basis functions.

Creating compact representations of the wave function, expanded in a one-particle basis set, which capture both static and dynamic correlation, has proven to be challenging. The classic methods such as multi-configurational self-consistent field (MCSSCF) and the specific type of MCSCF, namely complete active space self-consistent field (CASSCF) and the more modern methods such as density matrix renormalization group (DMRG) and diffusion quantum Monte Carlo (DQMC) all have proven very good in capturing the static correlation with system sizes up to around 100 orbitals for the modern methods [8,9,10,11]. For the dynamic correlation of MCSCF, CASSCF, DMRG and DQMC, these methods often rely on second-order perturbation theory [12,13,14]. The straightforward application of perturbation theory on these large active spaces, however, require higher-order reduced density matrices which unfortunately makes the perturbation theory the bottleneck. For methods like multi-reference configuration interaction (MRCI) and the many flavours of multi-reference coupled cluster (MRCC) where both static and dynamic correlation is treated simultaneous, very accurate calculations on larger systems is hampered by the rapid scaling increase with every improvement in the correlation level [15,16,17].

Wave-function ansätze based on geminals, which are two-electron functions, present a promising alternative to conventional state-of-the-art electronic structure methods to model both static and dynamic correlation. Tecmer and Boguslawski recently reviewed the current progress in creating gemninal ansätze to deal with strong correlation, missing dynamical correlation, correlation extensions, excites states and open shells [18]. The review clearly demonstrates that geminal wave function methods are a growing niche. We will here not focus on any of these many interesting extensions but more on the Lie algebra of geminals, the transformation from the one-particle basis to the geminal basis and working consistently in a geminal basis.

Recently, it was shown that the necessary and sufficient conditions, or more aptly the stationary conditions, for the exact wave function always can be written as a generalized Brillouin theorem irrespective of the order of interaction in the Hamiltonian [19]. For a Hamiltonian containing only one-electron terms, the exact wave function can be found from the Brillouin theorem by minimizing the energy with respect to rotations between the occupied and virtual orbitals in a one-particle basis set [20]. We will here formulate the stationary conditions, for the exact wave function in a geminal basis. From the stationary conditions, we examine the optimization effect of geminal rotations and geminal replacement operator for different geminal wave function ansätze.

The variational optimization using geminal rotations and geminal replacement operators of the two simplest geminal wave function ansätze, which are the antisymmetrized geminal power (AGP) [21,22,23,24,25] and the antisymmetrized product of geminals (APG) [26,27,28,29,30,31,32,33,34,35,36], will be examined. The full geminal product (FGP), which is a linear combinations of all possible geminal products, is the equivalent of the full configuration interaction (FCI) in a geminal basis. We will here show that the FGP is exact by virtue of completeness of the expansion, exactly like the FCI in a one-particle basis set. The aim is not to derive working equations for any of the geminal methods but to connect the geminal methods to the exact stationary conditions in a common way.

We will throughout use \(p,q,\ldots\) and \(\mu , \nu ,\ldots\) as general indices and \(a, b, \ldots\) and \(\alpha , \beta , \ldots\) as occupied indices when referring to orbital and geminal indices, respectively, unless stated otherwise. We will here use real orbitals and a real coefficient matrix, but this is straightforward to extend to include complex algebra.

2 Geminal basis and algebra

In order to obtain a proper pair rotation, a geminal basis is needed since the general pair rotation in a one-particle basis does not fulfil a Lie algebra [37, 38]. The geminal basis has been explored rather extensively in quantum chemistry though always in a very approximate form [24,25,26,27,28,29, 39, 40]. In nuclear physics, a restricted geminal basis or pair fermion basis has been applied extensively in Dyson or other boson-fermion type mappings [30,31,32,33,34,35,36, 41].

In this section, we will sketch the familiar general geminal algebra in some detail where the aim is to show the Lie algebra and how the structure constants from any geminal resulting from nested commutators easily can be found by projecting onto the geminal basis. The notation from Surján, where \(\Psi ^+_{\mu }\) and \(\Psi ^-_{\nu }\) are the general geminal creation and annihilation of geminal \(\mu\) and \(\nu\), respectively, will be used [42, 43].

2.1 The geminal basis

In a spin-orbital basis, the most general way of writing a geminal \(\Psi ^{+}_{\mu }\) is

$$\begin{aligned} \Psi _{\mu }^{+} = \frac{1}{2} \sum _{pq, \sigma \upsilon }^{m,s} C_{p\sigma q\upsilon } a^{\dagger }_{p \sigma } a^{\dagger }_{q \upsilon } \end{aligned}$$
(1)

where \(C_{p\sigma q\upsilon }\) is an element in a skew-symmetric coefficient matrix, and the sum is over all orbitals m and their respective spin-functions s. The annihilation geminal is here defined as the Hermitian adjoint of \(\Psi ^{+}_{\mu }\)

$$\begin{aligned} \Psi _{\mu }^{-} = \frac{1}{2} \sum _{pq, \sigma \upsilon }^{m,s} C_{p\sigma q\upsilon } a_{q \upsilon } a_{p \sigma }. \end{aligned}$$
(2)

The natural form of a geminal is often used due to the simplicity in the diagonal of the natural geminal. Since we here are interested in expressing any wave function and Hamiltonian in a complete geminal basis, the natural geminal is here not of interest here since in it is not possible to transform all geminals in the basis to the natural form at the same time as discussed in Appendix A.

We will here keep the general geminal and not restrict ourselves to geminals with definite S\(_z\), even if the general algebra is more cumbersome, since we are not interested in creating working equations but rather focus on transformations and stationary conditions. Both singlet and triplet geminals with definite S\(_z\) can be obtained by making appropriate linear combinations and restrictions similar to those seen for the AGP-Jastrow [44, 45]

2.2 Normalization

Exactly like orbitals, which are able to form an orthonormal set, we will likewise demand that the geminals form an orthonormal set

$$\begin{aligned} \langle vac | \Psi _{\nu }^- \Psi _{\mu }^+ | vac \rangle = \delta _{\nu \mu }. \end{aligned}$$
(3)

The orthonormality condition in Eq. 3 is usually, in quantum chemistry, referred to as the weak orthogonality condition and is just the Frobenius inner product of the coefficient matrices

$$\begin{aligned} \frac{1}{2} tr(\textbf{C}^{T}_{\nu } \textbf{C}_{\mu }) = \delta _{\nu \mu }. \end{aligned}$$
(4)

The Frobenius inner product is a component-wise inner product of two matrices which treat the matrices as vectors with an inner product.

In quantum chemistry, it has been customary to use the strong orthogonality instead of the weak orthogonality. While the strong orthogonality condition simplifies the geminal algebra [39, 40], due to Arai’s theorem where an orbital will only belong to a single geminal [46,47,48,49], we will here focus on the general case where the weak orthogonality is the only constraint on the geminal basis.

2.3 Lie algebra and commutation relations

We will here present the most important parts of the familiar geminal algebra. Since the geminals are composite bosons they do not form a nice algebra like real bosons [42, 43, 50] and show neither real Bose-Einstein nor Fermi-Dirac statistics [22, 23].

The commutation of the creation and annihilation geminal operators among themselves follows that of regular bosons

$$\begin{aligned}{}[ \Psi _{\nu }^+ , \Psi _{\mu }^+ ] = [ \Psi _{\nu }^- , \Psi _{\mu }^- ] = 0. \end{aligned}$$
(5)

The commutation between the creation and annihilation geminals produces an additional term R

$$\begin{aligned}{}[ \Psi _{\nu }^- , \Psi _{\mu }^+] = Q_{\nu \mu } = \delta _{\nu \mu } + R_{\nu \mu } \end{aligned}$$
(6)

where R, in general, is nonzero and need not commute with any other operator

$$\begin{aligned} R_{\nu \mu } = \sum _{pqs} C_{pq}^{\nu } C_{qs}^{\mu } a^{\dagger }_s a_p. \end{aligned}$$
(7)

On way of constructing a simpler geminal algebra is therefore finding a suitable R that will give simple commutation relations with the geminals. Much work has focused on eliminating R since this restores the regular boson algebra. Eliminating R unfortunately only seems possible when using Arai’s theorem for the strong orthogonality condition [46, 47]. We will not introduce any attempts at making the algebra simpler but simply show the consequences of the regular geminal algebra.

Since R, in general, is nonzero, it is therefore of interest to investigate the commutator between \(\Psi ^{+}\) and R

$$\begin{aligned}{}[[ \Psi _{\nu }^{-} , \Psi _{\mu }^{+}], \Psi _{\gamma }^{+}] = [ R_{\nu \mu }, \Psi _{\gamma }^{+}] = \sum _{\tau } c^{\mu \gamma }_{\nu \tau } \Psi _{\tau }^{+} = \Psi _{\nu \mu \gamma }^{+}. \end{aligned}$$
(8)

The commutator gives a new geminal \(\Psi _{\nu \mu \gamma }^{+}\) where \(c^{\mu \gamma }_{\nu \tau }\) are the structure constants for the expansion in the geminal basis. \(\Psi _{\nu \mu \gamma }^{+}\) can be expanded exactly in the geminal when all possible geminals created from m orbitals are included in the geminal basis. The structure constants can easily be found by applying the Frobenius inner product and will show the following symmetry [51]

$$\begin{aligned} c^{\mu \gamma }_{\nu \tau } = c^{\gamma \mu }_{\nu \tau } = c^{\mu \gamma }_{\tau \nu } = (c_{\mu \gamma }^{\nu \tau })^* . \end{aligned}$$
(9)

A similar relation can be derived from the pair annihilator from Eq. 2

$$\begin{aligned}{}[[ \Psi _{\nu }^{+} , \Psi _{\mu }^{-} ], \Psi _{\gamma }^{-}] = \sum _{\tau } c_{\mu \gamma }^{\nu \tau } \Psi _{\tau }^{-} = \Psi _{\nu \mu \gamma }^{-} . \end{aligned}$$
(10)

Since the geminals anticommute and fulfil the Jacobi identity, the geminals form a Lie algebra [32, 35].

The nested commutator for a single geminal, which could be used for internal occupied rotations,

$$\begin{aligned}{}[[ \Psi _{\mu }^{-} , \Psi _{\mu }^{+}], \Psi _{\mu }^{+}] = \sum _{\tau } c^{\mu \mu }_{\mu \tau } \Psi _{\tau }^{+} \end{aligned}$$
(11)

does not give great simplifications to Eq. 8 and shows that even internal rotations of geminals produce additional terms for the general geminal algebra.

3 Rotations, Hamiltonian and parameters in a geminal basis

We will here define the simplest possible geminal basis and show the effect of orbital and geminal rotations for this simple geminal basis and other bases. From the geminal rotations, we will write down the Hamiltonian in an arbitrary geminal basis and show that the stationary conditions for the exact wave function in a geminal basis are a generalization of the known Brillouin theorem in a geminal basis.

3.1 A simple geminal basis

Starting from a one-particle basis set, we will introduce the simplest possible geminal basis, for which will use the labels i,j,\(\ldots\),

$$\begin{aligned} \Psi _{i}^+ = \frac{1}{2} ( C_{pq} a^{\dagger }_p a^{\dagger }_q + C_{qp} a^{\dagger }_q a^{\dagger }_p) \qquad C_{pq} = \left\{ \begin{array}{rl} 1 &{}\text {if } p > q, \\ -1 &{}\text {if } p < q, \\ 0 &{}\text {if } p = q, \end{array} \right. \end{aligned}$$
(12)

where we join the two spin-orbital indices to a single label in the geminals basis. We here define an annihilation geminal from the Hermitian adjoint of Eq. 12

$$\begin{aligned} \Psi _{i}^- = \frac{1}{2} ( C_{pq} a_q a_p + C_{qp} a_p a_q) \qquad C_{pq} = \left\{ \begin{array}{rl} 1 &{}\text {if } p > q, \\ -1 &{}\text {if } p < q, \\ 0 &{}\text {if } p = q. \end{array} \right. \end{aligned}$$
(13)

With this definition, a complete orthonormal geminal basis is defined, according to the Frobenius inner product, where the number of geminals in the basis will be \(m(m-1)/2\) for m spin-orbitals since we enforce skew symmetry of the coefficient matrix. This simple basis can be considered a unit basis for geminals.

Any geminal from the basis defined in Sec. 2.1 can now be expanded in this simple basis

$$\begin{aligned} \Psi _{\mu }^{+} = \sum _{i} c_{\mu i} \Psi _{i}^+ = \sum _{pq} C_{pq}^{\mu } a^{\dagger }_p a^{\dagger }_q \end{aligned}$$
(14)

and likewise for the annihilation geminal

$$\begin{aligned} \Psi _{\mu }^{-} = \sum _{i} c_{\mu i} \Psi _{i}^- = \sum _{pq} C_{pq}^{\mu } a_q a_p, \end{aligned}$$
(15)

remembering that p, q is here in a spin-orbital basis.

3.2 Orbital and geminal rotations

The simple geminal basis as shown in Eq. 14 will of course change with the rotation of one-electron functions. We will here therefore relate two simple geminal bases where the one-particle function have been rotated. In the rotated basis, denoted with a bar, the simple geminal can be written as

$$\begin{aligned} \bar{\Psi }_{i}^+ = \frac{1}{2} ( C_{pq} \bar{a}^{\dagger }_p \bar{a}^{\dagger }_q + C_{qp} \bar{a}^{\dagger }_q \bar{a}^{\dagger }_p). \end{aligned}$$
(16)

Since the orbitals are related by some unitary transformation, the pair of orbitals are related as

$$\begin{aligned} \bar{a}^{\dagger }_p \bar{a}^{\dagger }_q = \sum _{rs} U_{pr} a^{\dagger }_r U_{qs} a^{\dagger }_s \end{aligned}$$
(17)

and therefore the also simple geminals

$$\begin{aligned} \bar{\Psi }_{i}^+ = \sum _j U_{ij} \Psi _{j}^+ \end{aligned}$$
(18)

where every column of \(U_{ij}\) is a vectorized coefficient matrix consistent with the definition of the Frobenius inner product.

Since the Frobenius inner product is used on the coefficient matrix to create an orthonormal geminal basis, the expansion coefficients in Eq. 14 will also be a simple unitary transformation to a new basis

$$\begin{aligned} \Psi _{\mu }^{+} = \sum _{i} U_{\mu i} \Psi _{i}^+ = \sum _{i} \bar{U}_{\mu i} \bar{\Psi }_{i}^+ = \sum _{\nu } \bar{U}_{\mu \nu } \bar{\Psi }_{\nu }^{+} \end{aligned}$$
(19)

irrespective of the choice of one-particle basis.

3.3 Hamiltonian in a geminal basis

Knowing the transformation from the one-particle basis to the simple and the general geminal basis writing, the Hamiltonian in any geminal basis is straightforward

$$\begin{aligned} \hat{H}= & {} \sum _{pqrs} f_{pq} a_p^{\dagger } a_q + \sum _{pqrs} g_{pqrs} a_p^{\dagger } a_r^{\dagger } a_s a_q \nonumber \\= & {} \sum _{pqrs} (\frac{f_{pq} \delta _{rs}}{N-1} + g_{pqrs} ) a_p^{\dagger } a_r^{\dagger } a_s a_q = \sum _{pqrs} \omega _{pqrs} a_p^{\dagger } a_r^{\dagger } a_s a_q \nonumber \\= & {} \sum _{ij} \omega _{ij} \Psi _{i}^+ \Psi _{j}^- = \sum _{ij} \omega _{ij} \sum _{\mu } U_{i \mu } \Psi _{\mu }^+ \sum _{\nu } U_{i \nu } \Psi _{\nu }^- \nonumber \\= & {} \sum _{\mu \nu } \omega _{\mu \nu } \Psi _{\mu }^+ \Psi _{\nu }^-. \end{aligned}$$
(20)

The stationary conditions for the exact wave function in a geminal basis are simply

$$\begin{aligned} 0 = \langle \Psi | (\hat{H} - E) \omega _{\mu \nu } \Psi _{\mu }^+ \Psi _{\nu }^- | \Psi , \rangle . \end{aligned}$$
(21)

where Eq. 21 is seen to be identical to the stationary conditions for the exact wave function in a single-particle basis [19, 52]. The stationary conditions for the wave function, in a geminal basis, are therefore a Brillouin’s theorem of geminals.

4 Geminal wave function ansätze

The two simplest geminals ansätze, namely the AGP and APG, and a full geminal product (FGP) ansatz, will be investigated separately after brief overview of the stationary conditions and variational optimization.

4.1 Wave function requirements

In the AGP and APG, it is the geminals in the reference that are being optimized, and the requirements are therefore similar to the requirements seen for a Hamiltonian with only one-body interactions where the orbitals in a configuration-state function (CSF) is being optimized. For the FGP, the requirements are on the coefficients in front of every geminal product similar to FCI.

Variations of the reference \(\Psi _{0}\) should ideally give

$$\begin{aligned} \Psi _{0} \rightarrow \Psi _{0} + \delta \Psi _0 = \Psi _0 + \Psi _0 (\Psi _{\alpha }^+ \rightarrow \sum _{\mu } \eta _{\mu \alpha } \Psi _{\mu }^+ ) + \mathcal {O}_2 + \ldots \end{aligned}$$
(22)

a first-order linearly independent variation \(\delta \Psi\) to \(\Psi _{0}\) where the geminal \(\Psi _{\alpha }^+\) in \(\Psi _{0}\) is replaced by a sum over of geminals with some expansion coefficient \(\eta _{\mu \alpha }\), as indicated by \(\Psi _0 (\Psi _{\alpha }^+ \rightarrow \sum _{\mu } \eta _{\mu \alpha } \Psi _{\mu }^+ )\), along with some higher-order corrections in \(\eta _{\mu \alpha }\) which disappears when the function is optimized. The energy is in this way variational

$$\begin{aligned} \langle \Psi _{0} | (\hat{H} - E) | \delta \Psi _0 \rangle = 0 \end{aligned}$$
(23)

and the first derivative with respect to the variational coefficients \(\eta _{\mu \alpha }\) should, if the wave function is exact, reproduce the stationary conditions of the exact wave function.

4.1.1 The stationary conditions of the exact wave function

The initial conditions for the derivative from Nakatsuji [52]

$$\begin{aligned} \left. \frac{\partial \Psi _{0}}{\partial C_{qs}^{pr}} \right| _{C =0} = a_p^{\dagger } a_r^{\dagger } a_s a_q | \Psi _{0} \rangle \end{aligned}$$
(24)

can be replaced by a weaker conditions as shown by Mukherjee and Kutzelnigg [37]

$$\begin{aligned} \left. \frac{\partial \Psi _{0}}{\partial C_{qs}^{pr}} \right| _{C =0} = \sum _{t>u,v>x} a_t^{\dagger } a_u^{\dagger } a_v a_x | \Psi _{0} \rangle . \end{aligned}$$
(25)

The stricter conditions for a wave function in a geminal basis, as seen from Eq. 21, is

$$\begin{aligned} \left. \frac{\partial \Psi _{0}}{\partial \eta _{\mu \nu }} \right| _{\eta =0} = \Psi _{\nu }^+ \Psi _{\mu }^- | \Psi _{0} \rangle \end{aligned}$$
(26)

and following Eq. 25, the weaker conditions in a geminal basis is

$$\begin{aligned} \left. \frac{\partial \Psi _{0}}{\partial \eta _{\mu \nu }} \right| _{\eta =0} = \sum _{\varphi \upsilon } \Psi _{\upsilon }^+ \Psi _{\varphi }^- | \Psi _{0} \rangle . \end{aligned}$$
(27)

We will here use the stricter conditions in Eq. 26 when relating the replacement operators from the derivative to geminal rotations. The stricter conditions is the geminal equivalent of Brillouin’s theorem where the replacement operator of geminals works on the reference.

4.2 The antisymmetrized geminal power (AGP)

The AGP [21, 24, 53], or Pfaffian, wave function has been used successfully for the properties of the solid state and in cooperative phenomena such as for superconductivity and superfluidity in BCS theory [21,22,23] but is much less explored for molecules though recently there has been a renewed interest in combining the AGP with CI [54,55,56,57].

The AGP is a simple tensor product of of N/2 identical geminals

$$\begin{aligned} \Psi _\text {AGP} = \prod ^{N/2} \Psi _{\alpha }^+ | vac \rangle = (\Psi _{\alpha }^+)^{N/2} | vac \rangle \end{aligned}$$
(28)

where N is the number of electrons, N is here assumed even.

4.2.1 Variations of the geminals in the AGP

Varying the geminal in the AGP

$$\begin{aligned} \Psi _{\text {AGP}}= & {} \prod ^{N/2} \tilde{\Psi }_{\alpha }^+ | vac \rangle \nonumber \\= & {} \prod ^{N/2} ( \Psi _{\alpha }^+ + \sum _{\nu \ne \alpha } \eta _{\nu \alpha } \Psi _{\nu }^+) | vac \rangle \end{aligned}$$
(29)

gives the AGP wave function in the transformed geminal basis \(\tilde{\Psi }_{\alpha }^+\). Taking the first derivative of the transformed wave function with respect to the variational coefficients \(\eta _{\nu \alpha }\) should, if exact, reproduce the stationary conditions

$$\begin{aligned} \left. \frac{\partial \Psi _{\text {AGP}}}{\partial \eta _{\nu \alpha }} \right| _{\eta =0} = N/2 \Psi _{\nu }^+ \prod ^{N/2-1} \Psi _{\alpha }^+ | vac \rangle . \end{aligned}$$
(30)

In order to see if the stationary conditions of the exact wave function are reproduced, we will first examine the effect of \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) on the AGP

$$\begin{aligned}{} & {} \Psi _{\nu }^{+} \Psi _{\mu }^{-} \prod ^{N/2} \Psi _{\alpha }^{+} | vac \rangle \nonumber \\{} & {} \quad = \Psi _{\nu }^{+} (\delta _{\mu \alpha } + R_{\mu \alpha } + \Psi _{\alpha }^{+} \Psi _{\mu }^{-}) \prod ^{N/2 - 1} \Psi _{\alpha }^{+} | vac \rangle \nonumber \\{} & {} \quad = \delta _{\mu \alpha } \Psi _{\nu }^{+} \prod ^{N/2 - 1} \Psi _{\alpha }^{+} | vac \rangle + \Psi _{\nu }^{+} (\Psi _{\alpha }^{+} R_{\mu \alpha } + \Psi _{\mu \alpha \alpha }^{+}) \prod ^{N/2 - 2} \Psi _{\alpha }^{+} | vac \rangle \nonumber \\{} & {} \qquad + \Psi _{\nu }^{+} \Psi _{\alpha }^+ (\Psi _{\alpha }^{+} \Psi _{\mu }^{-} + \delta _{\mu \alpha } + R_{\mu \alpha } ) \prod ^{N/2 - 2} \Psi _{\alpha }^{+} | vac \rangle \nonumber \\{} & {} \quad = \delta _{\mu \alpha } \frac{N}{2} \Psi _{\nu }^{+} \prod ^{N/2 - 1} \Psi _{\alpha }^{+} | vac \rangle \nonumber \\{} & {} \qquad + \frac{1}{8}(N^2 -2N ) \Psi _{\nu }^{+} \Psi _{\mu \alpha \alpha }^{+} \prod ^{N/2 - 2} \Psi _{\alpha }^{+} | vac \rangle . \end{aligned}$$
(31)

In Eq. 31, two terms appear. The first term is a simple replacement of \(\Psi _{\alpha }^{+}\) with \(\Psi _{\nu }^{+}\) which also comes from the derivative of the variations of the geminals in Eq. 30. In the second term, two geminals are replaced meaning that the variation in the form of \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) on the AGP cannot be considered a simple rotation of the geminals in the AGP as in Eq. 30. We notice that the second term will disappear for \(N=2\).

From Eq. 30, we see that the energy of the AGP wave function can be minimized by varying the geminal \(\Psi _{\alpha }^{+}\), but from Eq. 31, it follows that the application of \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) on the AGP wave function, the minimization does not reproduce the stricter conditions for the exact wave function in Eq. 26. The AGP wave function is therefore not invariant to variations in the form of \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) since these geminal variations rotate the AGP wave function to a non-AGP wave function. While the AGP does not fulfil the stationary conditions for the exact wave function, it is still a relatively inexpensive correlation method which describe basic correlation.

4.3 The antisymmetrized product of geminals (APG)

The APG is a tensor product of N/2 different geminals and has been used extensively in both quantum chemistry and nuclear physics [30,31,32,33,34,35,36, 41] though often in connection with the strong orthogonality. The antisymmetrized product of strongly orthogonal geminals (APSG) [39, 40], using Arai’s theorem [46, 47], gives a compact wave function with a reasonable accuracy [58].

The APG wave function is built from the tensor product of N/2 different geminals [27,28,29, 59]

$$\begin{aligned} \Psi _{APG} = \prod _{\alpha }^{N/2} \Psi _{\alpha }^+ | vac \rangle . \end{aligned}$$
(32)

4.3.1 Variations of the APG

Varying the geminal in the APG

$$\begin{aligned} \Psi _{APG}= & {} \prod ^{N/2}_{\alpha } \tilde{\Psi }_{\alpha }^+ | vac \rangle \nonumber \\= & {} \prod ^{N/2}_{\alpha } ( \Psi _{\alpha }^+ + \sum _{\nu \ne \alpha } \eta _{\nu \alpha } \Psi _{\nu }^+) | vac \rangle \end{aligned}$$
(33)

gives the APG wave function in the transformed geminal basis \(\tilde{\Psi }_{\alpha }^+\). Taking the first derivative of the transformed wave function with respect to the variational coefficients \(\eta _{\nu \alpha }\) should, if exact, reproduce the stationary conditions

$$\begin{aligned} \left. \frac{\partial \Psi _{APG}}{\partial \eta _{\nu \alpha }} \right| _{\eta =0} = \Psi _{\nu }^+ \prod ^{N/2-1}_{\beta \ne \alpha } \Psi _{\beta }^+ | vac \rangle . \end{aligned}$$
(34)

In order to see if the stationary conditions of the exact wave function is reproduced, we will first examine the effect of \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) on the APG

$$\begin{aligned}{} & {} \Psi _{\nu }^{+} \Psi _{\mu }^{-} \prod ^{N/2}_{\alpha } \Psi _{\alpha }^+ | vac \rangle \nonumber \\{} & {} \quad = \Psi _{\nu }^{+} (\delta _{\mu \alpha } + R_{\mu \alpha } +\Psi _{\alpha }^{+} \Psi _{\mu }^{-}) \prod ^{N/2 - 1}_{\beta \ne \alpha } \Psi _{\beta }^{+} | vac \rangle \nonumber \\{} & {} \quad = \delta _{\mu \alpha } \Psi _{\nu }^{+} \prod ^{N/2 - 1}_{\beta \ne \alpha } \Psi _{\beta }^{+} | vac \rangle + \Psi _{\nu }^{+} (\Psi _{\beta }^{+} R_{\mu \alpha } + \Psi _{\mu \alpha \beta }^{+}) \prod ^{N/2 - 2}_{\gamma \ne \alpha ,\beta } \Psi _{\gamma }^{+} | vac \rangle \nonumber \\{} & {} \qquad + \Psi _{\nu }^{+} \Psi _{\alpha }^+ (\Psi _{\beta }^{+} \Psi _{\mu }^{-} + \delta _{\mu \beta } + R_{\mu \beta } ) \prod ^{N/2 - 2}_{\gamma \ne \alpha ,\beta } \Psi _{\gamma }^{+} | vac \rangle \nonumber \\{} & {} \quad = \Psi _{\nu }^{+} \sum _{\alpha } \delta _{\mu \alpha } \prod ^{N/2 - 1}_{\beta \ne \alpha } \Psi _{\beta }^{+} | vac \rangle + \Psi _{\nu }^{+} \sum _{\alpha < \beta } \Psi _{\mu \alpha \beta }^{+} \prod ^{N/2 - 2}_{\gamma \ne \alpha ,\beta } \Psi _{\gamma }^{+} | vac \rangle \end{aligned}$$
(35)

where we have assumed a canonical ordering of the geminals. Again for the APG, the first term is a simple replacement of any geminal with index \(\mu\) with \(\Psi _{\nu }^{+}\) corresponding to a variational rotation of the geminal in the APG. In the second term, two geminals are replaced when applying \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) on the APG.

Just like the AGP can the energy of the APG wave function can be minimized by varying the occupied geminals but from Eq. 35 it follows that the application of \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) on the APG wave function, the minimization does not appear to reproduce the stricter conditions for the exact wave function in Eq. 26 unless the linear combination in the second term is zero or can be added to the first term as part of a rotation. The APG wave function, in an orthogonal geminal basis, therefore does not appear to be invariant to geminal variations from \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) since these geminal variations appear to rotate the APG wave function to a non-APG wave function.

4.4 The full geminal product (FGP)

The FGP ansatz is simply

$$\begin{aligned}{} & {} \Psi _\text {FGP} = \sum _{\nu _1 \le \nu _2 \le \ldots \nu _{N/2}} C_{\nu _1 \nu _2 \ldots \nu _{N/2}} \Psi _{\nu _1}^{+} \Psi _{\nu _2}^{+} \ldots \Psi _{\nu _{N/2}}^{+} | vac \rangle \nonumber \\{} & {} \quad = \sum _p C_p \Psi _p \end{aligned}$$
(36)

the sum of all possible tensor products of the geminals with the expansion coefficient C. Here, \(\Psi _p\) is the p tensor product with the corresponding expansion coefficient \(C_p\). Since the geminals commute, as seen in Eq. 5, the tensor product of geminals in Eq. 36 is written in canonical ordering in order to avoid duplicate products.

The FGP as written in Eq. 36 contain some redundancy which can easily be seen by inserting the simple geminal basis from Sec. 3.1 since all tensor products with repeating geminals will be trivially zero in the simple basis. Secondly, all products where one or more spin-orbitals are repeated in the tensor product of the simple geminals will also be zero. Finding these redundancies in the tensor product of the general geminal are significantly more difficult and all terms in Eq. 36 will therefore be kept, even if there are redundancies.

4.4.1 Variations of the FGP

For the FGP, we will follow a proof for the FCI where the configurations are swapped with tensor products of geminals [52]. From the stationary conditions in Eq. 23, we find

$$\begin{aligned} \langle \Psi _{\text {FGP}} | (\hat{H} - E) | \Psi _p \rangle = 0 \end{aligned}$$
(37)

from the derivative of Eq. 36. Since \(\Psi _{\text {FGP}}\) is spanned by all possible geminal tensor products \(\Psi _{\text {FGP}}\) is complete and any function \(\Psi _{\nu }^{+} \Psi _{\mu }^{-} \Psi _p\) can therefore be written as \(\Psi _k\), which is a linear combination of the functions spanned by \(\Psi _{\text {FGP}}\). Because \(\Psi _k\) can be expanded in the set of \(\Psi _p\) we can therefore write

$$\begin{aligned} \langle \Psi _{\text {FGP}} | (\hat{H} - E) \Psi _{\nu }^{+} \Psi _{\mu }^{-} |\Psi _k \rangle = 0 \end{aligned}$$
(38)

from which follows that the FGP is exact

$$\begin{aligned} \langle \Psi _{\text {FGP}} | (\hat{H} - E) \Psi _{\nu }^{+} \Psi _{\mu }^{-} |\Psi _{\text {FGP}} \rangle = 0. \end{aligned}$$
(39)

That a complete product of geminals is exact is not surprising since Røeggen have shown that the extended geminal model is exact [48, 60,61,62].

The FGP not only fulfil the stationary conditions for the exact wave function in Eq. 21, as expressed in Eq. 39, but also higher-order geminal products

$$\begin{aligned} \langle \Psi _{\text {FGP}} | (\hat{H} - E) \Psi _{\nu }^{+} \Psi _{\xi }^{+} \Psi _{\gamma }^{-} \Psi _{\mu }^{-} |\Psi _{\text {FGP}} \rangle=\, & {} 0 \nonumber \\ \langle \Psi _{\text {FGP}} | (\hat{H} - E) \Psi _{\nu }^{+} \Psi _{\xi }^{+} \Psi _{\pi }^{+} \Psi _{\rho }^{-} \Psi _{\gamma }^{-} \Psi _{\mu }^{-} |\Psi _{\text {FGP}} \rangle=\, & {} 0 \nonumber \\ \ldots \end{aligned}$$
(40)

since these higher products also can be expanded in the set of \(\Psi _p\). That the FGP also fulfil higher order products is simply due to having a complete set of products, exactly as seen for the FCI with a complete set of configurations [52].

The FGP does not appear to have any numerical advantage over the FCI, even if redundant terms in the FGP are not included in the expansion, due to the more complicated algebra. A truncated version of the FGP could prove to be significantly more compact than CI since the first term in the FGP is of either AGP or APG quality. The multiple Pfaffians method from Bajdichet al. [63] is an example of a truncated FGP wave function which shows numerical promise [64].

5 Summary and prospects

We have here shown the transformation from a one-particle basis to a geminal basis by defining a unit geminal basis. In this way, both the wave function and Hamiltonian are written in the geminal basis. It is shown that the necessary and sufficient conditions, herein referred to as the stationary conditions, of the exact wave function [52, 65] can be written as a Brillouin theorem of geminals which is consistent with the generalized Brillouin theorem derived previously [19].

A significant amount of space have been dedicated to the well-known Lie algebra of geminals in order to compare the geminal rotations with the effect of, what in a one-particle basis would be called, the replacement operator \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\). We here show that \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) introduce primary and secondary rotations. The primary rotations reproduce the regular rotations of the geminals but the secondary rotations rotate two geminals in the reference at the same time where one of the geminals is a linear combination of multiple geminals. The origin of the secondary rotations are the composite boson behaviour of the geminals where the commutation between the creator and annihilator geminals produce an additional term R, which does not commute with any other operators.

We have gone through the simplest ansätze for a wave function in a geminal basis and compared the variation of the geminals to the exact stationary conditions. For the antisymmetrized geminal power (AGP), we show that the secondary rotations from \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\) rotate the AGP away from a pure AGP wave function. The AGP is therefore not invariant to variations from \(\Psi _{\nu }^{+} \Psi _{\mu }^{-}\). The antisymmetrized product of geminals (APG) appears to be rotated away from a pure APG though this is less obvious than for the AGP. The full geminal product (FGP), which is a tensor product of all possible geminals, is shown to be exact. Using all terms in the FGP does not give any advantage over the full configuration interaction (FCI), however, a truncated version of the FGP could give a significantly more compact representation of the wave function since the lowest order of the FGP can be chosen to be the APG or AGP. Furthermore, developing a hierarchy of geminal-based wave functions would be desirable in order to have a systematic way of improving any calculation. Some work along this direction with the multiple Pfaffians method where the FGP is approximated by a sum of AGP’s have shown promising numerical results [63, 64].

The clear advantage of the these general geminal wave function ansätze is the ability of including all determinants in the FCI in the wave function at the same time without the exponential scaling. The coefficients in front of all determinants in the geminal wave functions of course cannot vary freely, like in the FCI, but are constrained by the wave function ansatz. Despite these constraints, this should make geminal wave functions more agnostic with respect to static and dynamic correlation and simply include the most important correlation, a feat that has proven very difficult for wave functions expanded in a one-particle basis set.