ABSTRACT We consider wavefunctions built from antisymmetrized products of two‐electron wavefunctions (geminals), which is arguably the simplest extension of the antisymmetrized product of one‐electron wavefunctions (orbitals) (i.e., a Slater determinant). Extensive use of geminals in wavefunctions has been limited by their high cost stemming from the many combinations of the two‐electron basis functions (orbital pairs) used to build the geminals. When evaluating the overlap of the APG wavefunction with an orthogonal Slater determinant, this cost can be interpreted as the cost of evaluating the permanent, resulting from the symmetry with respect to the interchange of orbital pairs, and the cost of assigning the occupied orbitals to the orbital pairs of the wavefunction. Focusing on the latter, we present a graphical interpretation of the Slater determinant and utilize the maximum weighted matching algorithm to estimate the combination of orbital pairs with the largest contribution to the overlap. Then, the cost due to partitioning the occupied orbitals in the overlap is reduced from to . Computational results show that many of these combinations are not necessary to obtain an accurate solution to the wavefunction. Because the APG wavefunction is the most general of the geminal wavefunctions, this approach can be applied to any of the simpler geminal wavefunction ansätze. In fact, this approach may even be extended to generalized quasiparticle wavefunctions, opening the door to tractable wavefunctions built using components of arbitrary numbers of electrons, not just two electrons.
ABSTRACT We consider wavefunctions built from antisymmetrized products of two‐electron wavefunctions (geminals), which is arguably the simplest extension of the antisymmetrized product of one‐electron wavefunctions (orbitals) (i.e., a Slater determinant). Extensive use of geminals in wavefunctions has been limited by their high cost stemming from the many combinations of the two‐electron basis functions (orbital pairs) used to build the geminals. When evaluating the overlap of the APG wavefunction with an orthogonal Slater determinant, this cost can be interpreted as the cost of evaluating the permanent, resulting from the symmetry with respect to the interchange of orbital pairs, and the cost of assigning the occupied orbitals to the orbital pairs of the wavefunction. Focusing on the latter, we present a graphical interpretation of the Slater determinant and utilize the maximum weighted matching algorithm to estimate the combination of orbital pairs with the largest contribution to the overlap. Then, the cost due to partitioning the occupied orbitals in the overlap is reduced from to . Computational results show that many of these combinations are not necessary to obtain an accurate solution to the wavefunction. Because the APG wavefunction is the most general of the geminal wavefunctions, this approach can be applied to any of the simpler geminal wavefunction ansätze. In fact, this approach may even be extended to generalized quasiparticle wavefunctions, opening the door to tractable wavefunctions built using components of arbitrary numbers of electrons, not just two electrons.
We propose an approach to the electronic structure problem based on noninteracting electron pairs that has similar computational cost to conventional methods based on noninteracting electrons. In stark contrast to other approaches, the wave function is an antisymmetric product of nonorthogonal geminals, but the geminals are structured so the projected Schrödinger equation can be solved very efficiently. We focus on an approach where, in each geminal, only one of the orbitals in a reference Slater determinant is occupied. The resulting method gives good results for atoms and small molecules. It also performs well for a prototypical example of strongly correlated electronic systems, the hydrogen atom chain.
The Lewis electron-pair picture pervades all of chemistry and a substantialdomain of materials science. “Geminals” are electron-pair quantum states whichare used to express approximate solutions of the Schrödinger equation of polyelec-tronic systems. In quantum chemistry, geminal-based methods are potentially moreeffective than the traditional ones based on the orbital picture: electrons occupyorbitals and experience only the average effect of each other. This is especially truefor strongly correlated-systems. However, the computational cost of using arbitrarygeminal wave functions is non-polynomial in the system size.It is therefore an important challenge to find additional constraints on geminalmodels which preserve the accuracy of the wavefunction description while reducingthe calculations to polynomial cost. The Ayers’ group at Mc Master has developedone such successful geminal model called AP1roG because the wavefunction is anantisymmetric product of geminals with one distinct reference orbital occupied ineach geminal [1]. This ansatz relaxes the so-called “strong-orthogonality” constraintbetween geminals. The latter makes wavefunction matrix element computationsvery easy but appears too drastic from the physical point of view, since the geminalelectron pairs are then distinguishable.We have followed a different path to relax the strong-orthogonality constraint. Wehave defined the geometrical concept of n-orthogonality [2] which can be viewed asa graded indistinguishability measure for electronic states: 1-orthogonality coincideswith strong orthogonality, the larger n the less distinguishable the n-orthogonal elec-tronic states will be. We have studied geminal models constrained by n-orthogonalityrelations for different values of n, imposed to different combinations of geminal prod-uct states [3]. However, in this work the scaling of matrix element computations wasstill unsatisfactory. We will present new ideas developed since then to remedy thisproblem.References[1] P. A Limacher, P. W. Ayers, P. A. Johnson, S. De Baerdemacker, D. Van Neck and P. Bultinck, J. of Chem. Theory and Computation 9, 1394-1401 (2013).[2] P. Cassam-Chenaı̈, Phys. Rev. A77, 032103 (2008).[3] P. Cassam-Chenaı̈, V. Rassolov, Chem. Phys. Lett. 487, 147-152 (2010).
The energy of a molecule is not the sum of the energies of its atomic components. Our aim is to compute an accurate approximation of this quantity based on an electron pair model, that is to say by using an antisymmetric product of two-electron wave functions, called “geminals”. In this model, the total wave function can also be described in a more practical way, by a set of matrices, one for each geminal. However, without further restrictions, such a model has a factorial computational complexity with the number of electrons and its applicability is therefore limited to small systems.We have introduced generalized orthogonality constraints between geminals to reduce the computational effort to a polynomial cost, without sacrificing the indistinguishability of the electrons. In other words, the electron pairs corresponding to the geminals are not fully distinguishable, so that our model remains complex, in the sense that geminal products have still to be antisymmetrized according to the Pauli principle to form bona fide electronic wave functions.Our geometrical constraints translate, in terms of geminal matrices, into simple equations involving the traces of products of these matrices. In the simplest non-trivial model, a set of solutions is given by block-diagonal matrices where each block is of size 2x2 and consists of a Pauli matrix or the Identity, multiplied by a complex parameter to be optimized. With this simplified ansatz for geminals, the number of terms in the calculation of the matrix elements of quantum observables is considerably reduced.
In our previous study [Kawasaki and Nakatani, Mol. Phys. 123(16), e2449186 (2025)], we proposed the low-rank antisymmetric product of geminals (APG) method, which reconstructs the wavefunction by extracting only the important eigenvalues from the APG wavefunction. However, its practical application was limited by the high computational cost from an orbital optimization process, making higher-rank calculations difficult. In this work, we reformulate the orbital part of the wavefunction using Givens rotation matrices, enabling an analytical treatment of the variational optimization. By combining the low-rank APG with the direct Givens rotation method, we achieved a significant improvement in optimization efficiency. We applied the developed method to small molecular systems and confirmed that it provides high accuracy, while also significantly reducing the computational time compared to conventional methods.
We investigated some variational methods to compute a wavefunction based on antisymmetric product of geminals (APG). The Waring decomposition on the APG wavefunction leads a finite sum of antisymmetrized geminal power (AGP) wavefunctions, each for which the variational principle can be applied. We call this as AGP-CI method which provides a variational solution of the APG wavefunction efficiently. However, number of AGP wavefunctions in the exact AGP-CI formalism become exponentially large in case of many-electron systems. Therefore, we also investigate the low-rank APG wavefunction, in which the coefficient matrices of geminals are factorized by the Schur decomposition. Interestingly, only a few non-zero eigenvalues (up to half number of electrons) were found from the Schur decomposition on the APG wavefunction. We developed some methods to approximate the APG wavefunction by lowering the ranks of coefficient matrices of geminals, and demonstrate their performance. Our new geminal method based on the low-rank decomposition can drastically reduce the number of variational parameters, although no efficient formalism has been elaborated so far, due to the difficulty of optimization caused by the inability to compute analytical derivatives.
In this review, we discuss the recent progress in developing geminal-based theories for challenging problems in quantum chemistry. Specifically, we focus on the antisymmetrized geminal power, generalized valence bond, antisymmetrized product of strongly orthogonal geminals, singlet-type orthogonal geminals, the antisymmetric product of 1-reference orbital geminal, also known as the pair coupled cluster doubles ansatz, and geminals constructed from Richardson-Gaudin states. Furthermore, we review various corrections to account for the missing dynamical correlation effects in geminal models and possible extensions to target electronically excited states and open-shell species. Finally, we discuss some numerical examples and present-day challenges for geminal-based models, including a quantitative and qualitative analysis of wave functions, and software availability.