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

Physical Review B

Diagrammatic multiresolution analysis for electron correlations

Heinz‐Jürgen Flad, Wolfgang Hackbusch, Hongjun Luo, D. Kolb

Published Mar 18, 2005• 15 citations
Publisher

Abstract

We present a multiscale treatment of electron correlations based on hyperbolic wavelet expansions of Jastrow-type correlation functions. Wavelets provide hierarchical basis sets that can be locally adapted to the length- and energy-scales of physical phenomena. Combined with hyperbolic tensor products and local adaptive refinement near the interelectron cusp, these wavelet bases enable sparse representations of Jastrow factors. The computational efficiency of wavelets in electronic structure calculations is demonstrated within the coupled electron-pair approximation (local ansatz). Based on a diagrammatic multiresolution analysis, we discuss various kinds of sparsity features for matrix elements required by the local ansatz. Sparsity originates from the hierarchical structure and vanishing moments property of wavelet bases. This led us to a recurrence scheme for the evaluation of matrix elements with almost linear computational complexity with respect to the size of the underlying isotropic 3d-wavelet basis. Numerical studies for selected diagrams are presented for a homogeneous electron gas.

DOI
10.1103/physrevb.71.125115

Related papers

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

Dmitry I. Lyakh

While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.

International Journal of Quantum Chemistry•Jul 7, 2014•4 citations

Perturbative calculation of Jastrow factors

Hongjun Luo, D. Kolb, Heinz‐Jürgen Flad +1 more

We present a perturbative treatment of Jastrow-type correlation factors which focuses on an accurate description of short-range correlations. Our approach is closely related to coupled cluster perturbation theory with the essential difference that we start from a variational formulation for the energy. Such kind of perturbation theory is especially suited for multiscale bases, like wavelets, which provide sparse representations for Jatrow factors. Envisaged applications in solid state physics are confined many-particle systems like electrons or multi excitons in quantum dots. The resulting Jastrow factors can be further used as trial wavefunctions in quantum Monte Carlo calculations for these systems. First applications to a screened homogeneous Fermi gas model demonstrate that already first order Jastrow factors recover 95 % of the correlation energy in variational Monte Carlo calculations over a fairly large range of densities and screening parameters. The corresponding second and third order perturbation energies turned out to be more sensitive to the specific choice of the model parameters. Furthermore, we have compared our first order Jastrow factors with those obtained from Fermi hypernetted chain calculations, where excellent agreement at short and intermediate inter-particle distances has been observed. 1

•Apr 25, 2007•2 citations

Perturbative calculation of Jastrow factors: Accurate description of short-range correlations

Hongjun Luo, D. Kolb, Heinz‐Jürgen Flad +1 more

We present a perturbative treatment of Jastrow-type correlation factors which focus on an accurate description of short-range correlations. Our approach is closely related to coupled cluster perturbation theory with the essential difference that we start from a variational formulation for the energy. Such kind of perturbation theory is especially suited for multiscale bases, such as wavelets, which provide sparse representations for Jastrow factors. Envisaged applications in solid-state physics are confined many-particle systems such as electrons or multiexcitons in quantum dots. The resulting Jastrow factors can be further used as trial wave functions in quantum Monte Carlo calculations for these systems. First applications to a screened homogeneous Fermi gas model demonstrate that first-order Jastrow factors already recover 95% of the correlation energy in variational Monte Carlo calculations over a fairly large range of densities and screening parameters. The corresponding second- and third-order perturbation energies turned out to be more sensitive to the specific choice of the model parameters. Furthermore, we have compared our first-order Jastrow factors with those obtained from Fermi hypernetted chain calculations, where excellent agreement at short and intermediate interparticle distances has been observed.

Physical Review B•Mar 16, 2007•6 citations

Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties

Ahmed E. Ismail, Gregory C. Rutledge, George Stephanopoulos

The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a 4×4 Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.

arXiv:physics/0212079•The Journal of Chemical Physics•Feb 25, 2003•51 citations

Local expansion of N-representable one-particle density matrices yielding a prescribed electron density

Jānos Pipek, Szilvia Nagy

Multiresolution (or wavelet) analysis offers a strictly local basis set for a systematic introduction of new details into Hilbert space operators. Using this tool we have previously developed an expansion method for density matrices. The set of density operators providing a given electron density plays an essential role in density functional theory, in the minimization of energy expectation values with the constraint that the electron density is fixed. In this contribution, using multiresolution analysis, we present an excellent quality density matrix expansion yielding a prescribed electron density, and compare it to other known methods. Due to the strictly local nature of the applied basis functions, our construction has the specific advantage that the resulting density matrix is correlated and N-representable in the infinite resolution limit. As a further consequence of this scheme we can conclude that the deviation of the exact kinetic energy functional from the Weizsäcker term is not a necessary consequence of the particle statistics.

The Journal of Chemical Physics•Oct 16, 2003•2 citations

Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties

Ahmed E. Ismail, Gregory C. Rutledge, George Stephanopoulos

The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a $4\\times 4$ Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.

•Dec 19, 2002•53 citations
We present a multiscale treatment of electron correlations based on hyperbolic wavelet expansions of Jastrow-type correlation functions. Wavelets provide hierarchical basis sets that can be locally adapted to the length- and energy-scales of physical phenomena. Combined with hyperbolic tensor products and local adaptive refinement near the interelectron cusp, these wavelet bases enable sparse representations of Jastrow factors. The computational efficiency of wavelets in electronic structure calculations is demonstrated within the coupled electron-pair approximation (local ansatz). Based on a diagrammatic multiresolution analysis, we discuss various kinds of sparsity features for matrix elements required by the local ansatz. Sparsity originates from the hierarchical structure and vanishing moments property of wavelet bases. This led us to a recurrence scheme for the evaluation of matrix elements with almost linear computational complexity with respect to the size of the underlying isotropic 3d-wavelet basis. Numerical studies for selected diagrams are presented for a homogeneous electron gas.
While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.
We present a perturbative treatment of Jastrow-type correlation factors which focuses on an accurate description of short-range correlations. Our approach is closely related to coupled cluster perturbation theory with the essential difference that we start from a variational formulation for the energy. Such kind of perturbation theory is especially suited for multiscale bases, like wavelets, which provide sparse representations for Jatrow factors. Envisaged applications in solid state physics are confined many-particle systems like electrons or multi excitons in quantum dots. The resulting Jastrow factors can be further used as trial wavefunctions in quantum Monte Carlo calculations for these systems. First applications to a screened homogeneous Fermi gas model demonstrate that already first order Jastrow factors recover 95 % of the correlation energy in variational Monte Carlo calculations over a fairly large range of densities and screening parameters. The corresponding second and third order perturbation energies turned out to be more sensitive to the specific choice of the model parameters. Furthermore, we have compared our first order Jastrow factors with those obtained from Fermi hypernetted chain calculations, where excellent agreement at short and intermediate inter-particle distances has been observed. 1
We present a perturbative treatment of Jastrow-type correlation factors which focus on an accurate description of short-range correlations. Our approach is closely related to coupled cluster perturbation theory with the essential difference that we start from a variational formulation for the energy. Such kind of perturbation theory is especially suited for multiscale bases, such as wavelets, which provide sparse representations for Jastrow factors. Envisaged applications in solid-state physics are confined many-particle systems such as electrons or multiexcitons in quantum dots. The resulting Jastrow factors can be further used as trial wave functions in quantum Monte Carlo calculations for these systems. First applications to a screened homogeneous Fermi gas model demonstrate that first-order Jastrow factors already recover 95% of the correlation energy in variational Monte Carlo calculations over a fairly large range of densities and screening parameters. The corresponding second- and third-order perturbation energies turned out to be more sensitive to the specific choice of the model parameters. Furthermore, we have compared our first-order Jastrow factors with those obtained from Fermi hypernetted chain calculations, where excellent agreement at short and intermediate interparticle distances has been observed.
The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a 4×4 Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.
Multiresolution (or wavelet) analysis offers a strictly local basis set for a systematic introduction of new details into Hilbert space operators. Using this tool we have previously developed an expansion method for density matrices. The set of density operators providing a given electron density plays an essential role in density functional theory, in the minimization of energy expectation values with the constraint that the electron density is fixed. In this contribution, using multiresolution analysis, we present an excellent quality density matrix expansion yielding a prescribed electron density, and compare it to other known methods. Due to the strictly local nature of the applied basis functions, our construction has the specific advantage that the resulting density matrix is correlated and N-representable in the infinite resolution limit. As a further consequence of this scheme we can conclude that the deviation of the exact kinetic energy functional from the Weizsäcker term is not a necessary consequence of the particle statistics.
The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a 4times44\\times 44times4 Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.