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 A

Representation of the Kato electron-electron cusp condition by wavelet-based density-operator expansions

Jānos Pipek, Szilvia Nagy

Published Oct 12, 2001• 10 citations
Publisher

Abstract

Since Kato proved his singularity condition for Coulomb potentials in 1957, there has been interest in the creation of wave functions that meet the prescriptions of the cusp conditions, necessary for high-precision quantum-mechanical calculations. It is well known, that wave-function expansions based on Slater determinants of one-electron functions are poorly convergent with respect to satisfying the electron-electron cusp condition. In this contribution we show that with the wavelet expansion of density operators even the local form of the electron-electron cusp condition is easily representable by Slater determinants of one-electron wavelet functions with a proper asymptotics of the expansion coefficients, which is explicitly calculated for Haar wavelets.

DOI
10.1103/physreva.64.052506

Related papers

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

Diagrammatic multiresolution analysis for electron correlations

Heinz‐Jürgen Flad, Wolfgang Hackbusch, Hongjun Luo +1 more

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.

Physical Review B•Mar 18, 2005•15 citations

Factorization of the wave-function coefficients in configuration-interaction expansions

Stefano Evangelisti

A factorization of the wave-function coefficients obtained from configuration-interaction expansions is presented. For this purpose, the system orbitals are partitioned into a set of overlapping domains and a factor is associated with a given electron distribution in each domain. The configuration-interaction coefficient of each Slater determinant is given by the product of all the domain factors that are associated with this determinant. This expansion is asymptotically exact, which means that the exact wave function is recovered in the limit of large domains. Moreover, the number of factors needed to define a wave function has a slow growth as a function of the system size. The idea is illustrated through applications to two model systems, i.e. a collection of non-interacting electron pairs and a Néel-type wave function. A possible strategy for the calculation of the configuration factors is also briefly discussed.

Molecular Physics•Mar 20, 2005•0 citations

Deciphering the coalescence behavior of Coulomb-Schrödinger atomic wave functions from an operator-product expansion

Y. S. Huang, Yu Jia, Rui Yu

We revisit the coalescence behavior of the atomic Schr\"odinger wave functions from the angle of an operator-product expansion (OPE) within the nonrelativistic Coulomb-Schr\"odinger effective-field theory. We take the electron-nucleus coalescence as an explicit example to demonstrate our formalism, by proving an exact OPE relation to all orders in perturbation theory. As a consequence, the celebrated Kato's cusp condition can be readily reproduced. Our approach can be readily extended to ascertain the multiparticle coalescence behaviors of atomic wave functions, which are otherwise difficult to achieve from the conventional Schr\"odinger equation approach. Moreover, by incorporating relativistic effects, our OPE approach appears to be able to provide key insight for solving a 90-year-old puzzle concerning the peculiar coalescence behavior of the wave function for a Dirac hydrogen atom.

arXiv:1809.09023•Physical Review A•Mar 19, 2025•1 citation

Properties and $N$-representability conditions of the two-body density matrix

Nicholas J. Cox

A summary is presented of the properties of the coefficient matrices formed by expanding the two-body reduced density matrix in a complete set of two-electron wave functions. Calculating the relationship between the many electron wave function and these so-called two-body density matrices enables the derivation of necessary $N$-representability conditions that ensure a given matrix obeys the postulates of many-body quantum mechanics.

arXiv:2112.11405•Dec 21, 2021•0 citations

Density functional theory

Philip L. Taylor, Olle Heinonen

The Hohenberg–Kohn theorem In Chapter 2 we explored some of the consequences of electron–electron interactions, albeit in some simple perturbative approaches and within the random phase approximation. There we found that the problem of treating these interactions is exceedingly difficult, even in the case where there is no external one–particle potential applied to the system. We have also explored some of the properties of noninteracting electrons in an external potential, in this case the periodic lattice potential. This led to the concepts of electron bands and band structure, subjects of fundamental importance in understanding the physics of metals, insulators, and semiconductors. Of course, in the real world, electrons in matter are subjected both to electron–electron interactions and to external potentials. How to include systematically and correctly the electron–electron interactions in calculations of real systems is truly a formidable problem. Why that is so is easily demonstrated. Suppose that we want to solve the problem of N electrons interacting in some external potential. The N -electron wavefunction can be expanded in Slater determinants of some suitable single-particle basis such as plane waves. We can describe the Slater determinants by occupation numbers in our second–quantized notation. Suppose furthermore that we have a basis of a total of N k plane wave states at our disposal. Here N k must be large enough that all reasonable “wiggles” of the many-body wavefunction can be included.

Cambridge University Press eBooks•Feb 28, 2002•4 citations
Since Kato proved his singularity condition for Coulomb potentials in 1957, there has been interest in the creation of wave functions that meet the prescriptions of the cusp conditions, necessary for high-precision quantum-mechanical calculations. It is well known, that wave-function expansions based on Slater determinants of one-electron functions are poorly convergent with respect to satisfying the electron-electron cusp condition. In this contribution we show that with the wavelet expansion of density operators even the local form of the electron-electron cusp condition is easily representable by Slater determinants of one-electron wavelet functions with a proper asymptotics of the expansion coefficients, which is explicitly calculated for Haar wavelets.
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.
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.
A factorization of the wave-function coefficients obtained from configuration-interaction expansions is presented. For this purpose, the system orbitals are partitioned into a set of overlapping domains and a factor is associated with a given electron distribution in each domain. The configuration-interaction coefficient of each Slater determinant is given by the product of all the domain factors that are associated with this determinant. This expansion is asymptotically exact, which means that the exact wave function is recovered in the limit of large domains. Moreover, the number of factors needed to define a wave function has a slow growth as a function of the system size. The idea is illustrated through applications to two model systems, i.e. a collection of non-interacting electron pairs and a Néel-type wave function. A possible strategy for the calculation of the configuration factors is also briefly discussed.
We revisit the coalescence behavior of the atomic Schr\"odinger wave functions from the angle of an operator-product expansion (OPE) within the nonrelativistic Coulomb-Schr\"odinger effective-field theory. We take the electron-nucleus coalescence as an explicit example to demonstrate our formalism, by proving an exact OPE relation to all orders in perturbation theory. As a consequence, the celebrated Kato's cusp condition can be readily reproduced. Our approach can be readily extended to ascertain the multiparticle coalescence behaviors of atomic wave functions, which are otherwise difficult to achieve from the conventional Schr\"odinger equation approach. Moreover, by incorporating relativistic effects, our OPE approach appears to be able to provide key insight for solving a 90-year-old puzzle concerning the peculiar coalescence behavior of the wave function for a Dirac hydrogen atom.
Properties and NNN-representability conditions of the two-body density matrix
A summary is presented of the properties of the coefficient matrices formed by expanding the two-body reduced density matrix in a complete set of two-electron wave functions. Calculating the relationship between the many electron wave function and these so-called two-body density matrices enables the derivation of necessary NNN-representability conditions that ensure a given matrix obeys the postulates of many-body quantum mechanics.
The Hohenberg–Kohn theorem In Chapter 2 we explored some of the consequences of electron–electron interactions, albeit in some simple perturbative approaches and within the random phase approximation. There we found that the problem of treating these interactions is exceedingly difficult, even in the case where there is no external one–particle potential applied to the system. We have also explored some of the properties of noninteracting electrons in an external potential, in this case the periodic lattice potential. This led to the concepts of electron bands and band structure, subjects of fundamental importance in understanding the physics of metals, insulators, and semiconductors. Of course, in the real world, electrons in matter are subjected both to electron–electron interactions and to external potentials. How to include systematically and correctly the electron–electron interactions in calculations of real systems is truly a formidable problem. Why that is so is easily demonstrated. Suppose that we want to solve the problem of N electrons interacting in some external potential. The N -electron wavefunction can be expanded in Slater determinants of some suitable single-particle basis such as plane waves. We can describe the Slater determinants by occupation numbers in our second–quantized notation. Suppose furthermore that we have a basis of a total of N k plane wave states at our disposal. Here N k must be large enough that all reasonable “wiggles” of the many-body wavefunction can be included.