A new truncation scheme for non-orthogonal antisymmetrized products of interacting geminals (APIG) is introduced based on antisymmetrized products of strongly orthogonal geminals (APSG). This wavefunction hierarchy of interacting geminals (IG) allows us to gradually increase the accuracy at which the ground state of a seniority-zero Hamiltonian can be estimated, ranging from APSG up to approximation-free APIG. Mathematical expressions for the lowest four orders (IG0, IG1, IG2, and IG3) are given explicitly and the computational cost to evaluate their transition density matrix is verified to scale only cubically with system size. Exemplary numerical calculations indicate that already a very early truncation level leads to results virtually identical to APIG.
A new truncation scheme for non-orthogonal antisymmetrized products of interacting geminals (APIG) is introduced based on antisymmetrized products of strongly orthogonal geminals (APSG). This wavefunction hierarchy of interacting geminals (IG) allows us to gradually increase the accuracy at which the ground state of a seniority-zero Hamiltonian can be estimated, ranging from APSG up to approximation-free APIG. Mathematical expressions for the lowest four orders (IG0, IG1, IG2, and IG3) are given explicitly and the computational cost to evaluate their transition density matrix is verified to scale only cubically with system size. Exemplary numerical calculations indicate that already a very early truncation level leads to results virtually identical to APIG.
We use symbolic computation, numerical experiments, and mathematical arguments to establish that every seniority-zero state can be expressed as a (number-symmetry-broken) antisymmetric product of interacting geminals (APIG). Moreover, for bosons, the configuration interaction coefficients are exactly parameterized by matrix permanents. This establishes that bosonic wavefunctions can be exactly described by a mean field, specifically the (number-symmetry-broken) symmetric product of boson orbitals. While exact parameterization is extremely costly, we present numerical evidence showing that the APIG wavefunction approaches the true wavefunction exponentially quickly as the number of interacting geminals increases.
A new family of geminal Ansätze called antisymmetrized product of 2D-block geminals (AP2D-BG) has been introduced recently to solve the electronic Schrödinger equation of molecules. They build on an antisymmetrized product of strongly orthogonal geminals (APSG) while lifting the "strong orthogonality" and "seniority zero" restrictions. The combinatorics to select the excitations which can be added to each geminal of the product grows exponentially with the number of geminals. However, in this study on diatomic molecules, we present simple guidelines to drastically reduce the number of excitations, which are useful to lower the electronic energy in variational calculations. Although different regimes to partition the orbital space into 1D- and 2D-blocks in an optimal way are observed along the potential energy curves (PEC), a simple strategy for choosing the block types associated to the geminals in each 2D-block is found successful in all cases. This makes the method quite practical to use. Moreover, the optimal geminals can be easily related to classical Lewis structures of chemistry. The quality of the PECs is assessed by calculating equilibrium geometries and fundamental frequencies; the improvement with respect to APSG is discussed.
Strong orthogonality is an important constraint placed on geminal wavefunctions in order to make variational minimization tractable. However, strong orthogonality prevents certain, possibly important, excited configurations from contributing to the ground state description of chemical systems. The presented method lifts strong orthogonality constraint from geminal wavefunction by computing a perturbative-like correction to each geminal independently from the corrections to all other geminals. The method is applied to the Singlet-type Strongly orthogonal Geminals variant of the geminal wavefunction. Comparisons of this new SS(p)G method are made to the non-orthogonal AP1roG and the unconstrained Geminal Mean-Field Configuration Interaction method using small atomic and molecular systems. The correction is also compared to Density Matrix Renormalization Group calculations performed on long polyene chains in order to assess its scalability and applicability to large strongly correlated systems. The results of these comparisons demonstrate that although the perturbative correction is small, it may be a necessary first step in the systematic improvement of any strongly orthogonal geminal method.
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).
In this paper, we develop a class of antisymmetrized geminal power configuration interaction (AGP-CI) wave functions that extend the AGP framework by incorporating inter-geminal correlations through a CI expansion. To make these wavefunctions computationally tractable, we evaluate them by rewriting the AGP-CI ansatz as a linear combination of AGPs (LC-AGP), for which overlaps and Hamiltonian matrix elements can be computed with standard AGP machinery. Motivated by border-rank decompositions, we further reorganize this ansatz into a compact linear combination of AGPs depending on a small deformation parameter τ, which controls how closely the truncated expansion approximates the full AGP-CI state. Benchmark applications to the Hubbard model and to the small molecules H2O and N2 demonstrate that the proposed wavefunctions achieve consistently high accuracy and outperform the LC-AGP, particularly for systems with more electrons and in strongly correlated regimes.
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.