Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1810.13143 · Physical Review A

Antisymmetrized geminal powers with larger chemical basis sets

Wataru Uemura, Takahito Nakajima

Published Jan 28, 2019• 7 citations
arXivHTMLPDF

Abstract

In previous research, we tested the wave-function format of a linear combination of several antisymmetrized geminal power states. A numerical problem in the geminal matrices was noted, which made the total energies of electronic systems with large numbers of electrons unstable. The underlying cause was found to be the large cancellation term in the geminal power series. We have obtained a format to resolve this problem for the case of total energies and partly for the first-order derivatives within the antisymmetrized geminal power states. By using this formalism, we have calculated the ground-state energies for several electronic systems, including the usage of a larger chemical basis set. The results are, in some cases, very close to the exact result, especially for one-dimensional Hubbard systems. Our result for a water molecule with the Dunning double zeta basis set is better than the configuration-interaction singles and doubles energy and approaches the coupled cluster singles and doubles energy.

DOI
10.1103/physreva.99.012519

Related papers

Required number of states increases only moderately with the problem size for antisymmetrized geminal powers

Wataru Uemura, Takahito Nakajima

We propose an algorithm to obtain the ground-state energy of a many-electron system using the variational wave function of a linear combination of antisymmetrized geminal powers. We optimized this algorithm to obtain the energy and the other parameters of a many-electron system. Also we clarified the bottleneck of the total calculation in the tensor contraction and successfully reduced the computational time. As a result, we can use an extended number of geminal states to obtain the ground state of the water molecule and Hubbard models. The result for the water molecule with the Dunning double-zeta basis is of the sub-milihartree order above the energy of exact diagonalization. Further, we observe that the result for the one-dimensional Hubbard model with 14 sites shows good tendency to capture the right ground state and that for the two-dimensional Hubbard model still lacks some part of the energy reflecting the large size of the Hilbert space. We conclude that the required number of terms for geminal states for sufficiently accurate energy is only moderately affected by the problem size. We further show other technical details for the numerical algorithms of geminal states in the variation process. It is expected that with the use of more extended computing resources and larger sizes of electronic systems, our algorithm can provide improved results.

arXiv:1906.02478•Jun 6, 2019•0 citations

Configuration interaction with antisymmetrized geminal powers

Wataru Uemura, Shusuke Kasamatsu, Osamu Sugino

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.

arXiv:1504.06250•Physical Review A•Jun 16, 2015•13 citations

Geometrical constraints to reduce complexity in quantum molecular systems

Pérez, Thomas, Patrick Cassam‐Chenaï

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.

•Mar 25, 2019•1 citation

Geminal-based electronic structure methods in quantum chemistry. Toward a geminal model chemistry

Paweł Tecmer, Katharina Bogusławski

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.

Physical Chemistry Chemical Physics•Jan 1, 2022•64 citations

Perspectives of geminal methods for large molecular systems

Andrey M. Tokmachev

Geminals are counterparts of two‐electron chemical bonds and lone pairs in the realm of wave functions. Antisymmetrized products of geminals provide a solid framework for studies of electron pairing in molecular systems. Natural advantages of geminal wave functions such as the correct description of bond breaking and formation make them a powerful model for electronic structure calculations. The concerted efforts to develop geminal‐based methods concentrate on increasing their accuracy and universality. The downside is the high computational cost limiting applications to small and toy systems. In contrast, the inherent local structure of optimal geminals opens pathways to development of methods for large systems. In particular, perspectives for linear scaling and hybrid quantum/classical approaches using geminals are discussed. The same principles are proposed for development of computational schemes for covalently bound and molecular crystals. © 2015 Wiley Periodicals, Inc.

International Journal of Quantum Chemistry•Jun 25, 2015•12 citations

Configuration interaction extension of AGP for incorporating inter-geminal correlations

Airi Kawasaki, Fei Gao, Gustavo E. Scuseria

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.

The Journal of Chemical Physics•Jun 17, 2026•0 citations
In previous research, we tested the wave-function format of a linear combination of several antisymmetrized geminal power states. A numerical problem in the geminal matrices was noted, which made the total energies of electronic systems with large numbers of electrons unstable. The underlying cause was found to be the large cancellation term in the geminal power series. We have obtained a format to resolve this problem for the case of total energies and partly for the first-order derivatives within the antisymmetrized geminal power states. By using this formalism, we have calculated the ground-state energies for several electronic systems, including the usage of a larger chemical basis set. The results are, in some cases, very close to the exact result, especially for one-dimensional Hubbard systems. Our result for a water molecule with the Dunning double zeta basis set is better than the configuration-interaction singles and doubles energy and approaches the coupled cluster singles and doubles energy.
We propose an algorithm to obtain the ground-state energy of a many-electron system using the variational wave function of a linear combination of antisymmetrized geminal powers. We optimized this algorithm to obtain the energy and the other parameters of a many-electron system. Also we clarified the bottleneck of the total calculation in the tensor contraction and successfully reduced the computational time. As a result, we can use an extended number of geminal states to obtain the ground state of the water molecule and Hubbard models. The result for the water molecule with the Dunning double-zeta basis is of the sub-milihartree order above the energy of exact diagonalization. Further, we observe that the result for the one-dimensional Hubbard model with 14 sites shows good tendency to capture the right ground state and that for the two-dimensional Hubbard model still lacks some part of the energy reflecting the large size of the Hilbert space. We conclude that the required number of terms for geminal states for sufficiently accurate energy is only moderately affected by the problem size. We further show other technical details for the numerical algorithms of geminal states in the variation process. It is expected that with the use of more extended computing resources and larger sizes of electronic systems, our algorithm can provide improved results.
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.
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 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.
Geminals are counterparts of two‐electron chemical bonds and lone pairs in the realm of wave functions. Antisymmetrized products of geminals provide a solid framework for studies of electron pairing in molecular systems. Natural advantages of geminal wave functions such as the correct description of bond breaking and formation make them a powerful model for electronic structure calculations. The concerted efforts to develop geminal‐based methods concentrate on increasing their accuracy and universality. The downside is the high computational cost limiting applications to small and toy systems. In contrast, the inherent local structure of optimal geminals opens pathways to development of methods for large systems. In particular, perspectives for linear scaling and hybrid quantum/classical approaches using geminals are discussed. The same principles are proposed for development of computational schemes for covalently bound and molecular crystals. © 2015 Wiley Periodicals, Inc.
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.