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arXiv:2410.06666 · Molecular Physics

Low-rank decomposition on the antisymmetric product of geminals for strongly correlated electrons

Airi Kawasaki, Naoki Nakatani

Published Jan 21, 2025• 5 citations

Abstract

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.

DOI: 10.1080/00268976.2024.2449186

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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 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.

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 H

_2

O and N

_2

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 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.

We analyze wave functions constructed as a sum of product of two-electron functions, or as a polynomial of geminals, to investigate their ability to represent the ground state of a strongly correlated few-body system. The known difficulty associated with variational determination of the total energy is overcome by applying a tensor decomposition method called Waring decomposition. Convergence speed of the total energy is compared for various polynomial types. The result provides information bridging between geminal product wave functions and the full-CI in the strongly correlated regime, thereby enriching knowledge on the hierarchy of molecular orbital theories of electron pairs.

The antisymmetrized geminal power (AGP) wave function has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP-based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the regions of Hilbert space. This correlation method called J -CI provides excellent accuracy in energies when applied to the pairing Hamiltonian.

To avoid the combinatorial computational cost of configuration interaction (CI), we previously introduced the symmetric tensor decomposition CI (STD-CI) method, which takes advantage of the antisymmetric nature of the electronic wave function and expresses the CI coefficients compactly as a series of Kronecker product states (STD series) [W. Uemura and O. Sugino, Phys. Rev. Lett. 109, 253001 (2012)]. Here we extend the variational degrees of freedom by using different molecular orbitals for different terms in the STD series. This scheme is equivalent to the linear combination of the Hartree-Fock-Bogoliubov state or the antisymmetrized geminal powers (AGPs). The total energy converges very rapidly within 0.72 $\ensuremath{\mu}\mathrm{hartree}$ taking only 10 terms for the water molecule, and the convergence is likewise fast for Hubbard tetramers. The computational cost scales as the fifth power of the number of electrons and the square of the number of terms in the STD series, indicating the promise of this AGP-based scheme for highly accurate and efficient computation of quantum systems.