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:1807.04549 · Physical review. B./Physical review. B

Optimized correlations inspired by perturbation theory

Martin Panholzer, Raphael Hobbiger, Helga M. Böhm

Published May 31, 2019• 5 citations
arXivHTMLPDF

Abstract

We study the accuracy of analytical wave-function-based many-body methods derived by energy minimization of a Jastrow-Feenberg ansatz for electrons (the ``Fermi hypernetted chain/Euler Lagrange'' approach). Approximations to avoid the complexity of the fermion problem are chosen to parallel successful boson theories and to be computationally efficient. For the three-dimensional homogeneous electron gas, we calculate the correlation energy, the pair distribution function, and the static structure function in comparison with simulation results. We also present a variant of theory which is interpreted as an approximate, self-consistent sum of ladder and ring diagrams of perturbation theory. The theory performs particularly well in the highly dilute density regime.

DOI
10.1103/physrevb.99.195156

Related papers

Optimal three-body correlations and elementary diagrams in liquid He4

E. Krotscheck

Within the framework of the variational theory for strongly interacting Bose liquids, we study the optimal determination of three-body correlation factors. The Euler-Lagrange equation for the three-body correlations suggests an iterative procedure in which the three-body correlation factor is expressed as a series of diagrams containing only the pair distribution function. The long-wavelength behavior of the three-body structure function agrees formally with the prediction of quantum hydrodynamics. The simplest approximation for the optimized three-body correlation function is identical to the prediction of second-order correlated-basis-function perturbation theory. We show that this simplest approximation sums essentially the same sets of diagrams that are included in non-optimized hypernetted-chain calculations with three-body correlations. Along with the three-body correlations we include those elementary diagrams of fourth and fifth order which have a comparable topological structure. The predictions for the ground-state energy and the pair distribution function are substantially improved compared with hypernetted-chain calculations restricted to two-body correlations.

Physical review. B, Condensed matter•Mar 1, 1986•91 citations

Variational problem in Jastrow theory

E. Krotscheck

The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\ensuremath{\rightarrow}0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O({r}^{\ensuremath{-}2})$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.

Physical Review A•Jan 1, 1977•77 citations

Approximate Euler-Lagrange Equations for the Ground-State Electron Gas

M. A. Pokrant, F. A. Stevens

The ground-state wave function for a quantum electron gas is approximated by $\ensuremath{\psi}=\ensuremath{\Phi}D$, where $D$ is the ground-state ideal-gas wave function and $\ensuremath{\Phi}=\mathrm{exp}[\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{i<j}^{}u({r}_{\mathrm{ij}})]$. Three-body correlation functions in the energy are approximated so that the energy is a functional of only pair-correlation functions. The variational principle is applied by utilizing the functional derivative of the pair-correlation function with respect to $u(r)$. The three- and four-body correlation functions in the functional derivative are approximated in a consistent manner. Use of the Kirkwood superposition approximation for the many-body correlation functions in the functional derivative gives unphysical results. A new approximation to the functional derivative is obtained by using the convolution approximation for the many-body terms. This gives a reasonable result for the asymptotic form of the wave function. The hypernetted-chain functional derivative combined with the convolution approximation for the energy gives the same asymptotic form. When the random-phase approximation to the energy is used with the Gaskell-Broyles-Sahlin-Carley functional derivative, the equation may be solved analytically, giving the same result obtained by Gaskell. The asymptotic form is the same as the two previous methods, namely, $u\ensuremath{\sim}(\frac{e}{\ensuremath{\hbar}})\frac{{(\frac{m}{\ensuremath{\pi}\ensuremath{\rho}})}^{\frac{1}{2}}}{r}$.

Physical Review A•May 1, 1973•17 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

Linked-cluster expansion for Jastrow-type wave functions and its application to the electron-gas problem

James D. Talman

A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like $1+\frac{A}{{r}^{2}}$. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like $1+\frac{A}{{r}^{2}}$ for large $r$. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.

Physical Review A•Oct 1, 1974•24 citations

Number-conserving model for boson pairing

Stefano Fantoni, T. M. Nguyen, Subodh R. Shenoy +1 more

An independent-pair ansatz is developed for the many-body wave function of dilute Bose systems. The pair correlation is optimized by minimizing the expectation value of the full Hamiltonian (rather than the truncated Bogoliubov one) providing a rigorous energy upper bound. In contrast with the Jastrow model, hypernetted chain theory provides closed-form exactly solvable equations for the optimized pair correlation. The model involves both condensate and coherent pairing with number conservation and kinetic-energy sum rules satisfied exactly and the compressibility sum rule obeyed at low density. We compute, for bulk boson matter at a given density and zero temperature, (i) the two-body distribution function, (ii) the energy per particle, (iii) the sound velocity, (iv) the chemical potential, (v) the momentum distribution and its condensate fraction, and (vi) the pairing function, which quantifies the off-diagonal long-range order resulting from the structural properties of the two-particle density matrix. The connections with the low-density expansion and Bogoliubov theory are analyzed at different density values, including the density and scattering length regime of interest of trapped-atom Bose-Einstein condensates. Comparison with the available diffusion Monte Carlo results is also made.

arXiv:cond-mat/0205282•Physical Review A•Sep 11, 2002•6 citations
We study the accuracy of analytical wave-function-based many-body methods derived by energy minimization of a Jastrow-Feenberg ansatz for electrons (the ``Fermi hypernetted chain/Euler Lagrange'' approach). Approximations to avoid the complexity of the fermion problem are chosen to parallel successful boson theories and to be computationally efficient. For the three-dimensional homogeneous electron gas, we calculate the correlation energy, the pair distribution function, and the static structure function in comparison with simulation results. We also present a variant of theory which is interpreted as an approximate, self-consistent sum of ladder and ring diagrams of perturbation theory. The theory performs particularly well in the highly dilute density regime.
Within the framework of the variational theory for strongly interacting Bose liquids, we study the optimal determination of three-body correlation factors. The Euler-Lagrange equation for the three-body correlations suggests an iterative procedure in which the three-body correlation factor is expressed as a series of diagrams containing only the pair distribution function. The long-wavelength behavior of the three-body structure function agrees formally with the prediction of quantum hydrodynamics. The simplest approximation for the optimized three-body correlation function is identical to the prediction of second-order correlated-basis-function perturbation theory. We show that this simplest approximation sums essentially the same sets of diagrams that are included in non-optimized hypernetted-chain calculations with three-body correlations. Along with the three-body correlations we include those elementary diagrams of fourth and fifth order which have a comparable topological structure. The predictions for the ground-state energy and the pair distribution function are substantially improved compared with hypernetted-chain calculations restricted to two-body correlations.
The ground-state wave function of an infinite system of fermions is approximated by the state-independent Jastrow ansatz. In order to optimize the pair correlation function, the Euler-Lagrange equations of the variational problem for the energy expectation value are derived. The short- and long-range behaviors of the optimum pair correlation function are discussed. Application of graphical techniques and use of rigorous results on the connection between the slope of the static structure function as $k\ensuremath{\rightarrow}0$+ and long-range Jastrow correlations allow one to prove that the optimum pair correlation function behaves like $1+O({r}^{\ensuremath{-}2})$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$. A connection is derived between the weight of the long-range correlations and the Landau parameters. As simple examples the limit of Bose statistics and the electron-gas problem are considered. The consequences of these investigations on numerical calculations and their relation to alternative expansion methods are investigated.
The ground-state wave function for a quantum electron gas is approximated by $\ensuremath{\psi}=\ensuremath{\Phi}D$, where DDD is the ground-state ideal-gas wave function and $\ensuremath{\Phi}=\mathrm{exp}[\ensuremath{-}(\frac{1}{2})\ensuremath{\Sigma}{i<j}^{}u({r}_{\mathrm{ij}})]$. Three-body correlation functions in the energy are approximated so that the energy is a functional of only pair-correlation functions. The variational principle is applied by utilizing the functional derivative of the pair-correlation function with respect to u(r)u(r)u(r). The three- and four-body correlation functions in the functional derivative are approximated in a consistent manner. Use of the Kirkwood superposition approximation for the many-body correlation functions in the functional derivative gives unphysical results. A new approximation to the functional derivative is obtained by using the convolution approximation for the many-body terms. This gives a reasonable result for the asymptotic form of the wave function. The hypernetted-chain functional derivative combined with the convolution approximation for the energy gives the same asymptotic form. When the random-phase approximation to the energy is used with the Gaskell-Broyles-Sahlin-Carley functional derivative, the equation may be solved analytically, giving the same result obtained by Gaskell. The asymptotic form is the same as the two previous methods, namely, $u\ensuremath{\sim}(\frac{e}{\ensuremath{\hbar}})\frac{{(\frac{m}{\ensuremath{\pi}\ensuremath{\rho}})}^{\frac{1}{2}}}{r}$.
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.
A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.
An independent-pair ansatz is developed for the many-body wave function of dilute Bose systems. The pair correlation is optimized by minimizing the expectation value of the full Hamiltonian (rather than the truncated Bogoliubov one) providing a rigorous energy upper bound. In contrast with the Jastrow model, hypernetted chain theory provides closed-form exactly solvable equations for the optimized pair correlation. The model involves both condensate and coherent pairing with number conservation and kinetic-energy sum rules satisfied exactly and the compressibility sum rule obeyed at low density. We compute, for bulk boson matter at a given density and zero temperature, (i) the two-body distribution function, (ii) the energy per particle, (iii) the sound velocity, (iv) the chemical potential, (v) the momentum distribution and its condensate fraction, and (vi) the pairing function, which quantifies the off-diagonal long-range order resulting from the structural properties of the two-particle density matrix. The connections with the low-density expansion and Bogoliubov theory are analyzed at different density values, including the density and scattering length regime of interest of trapped-atom Bose-Einstein condensates. Comparison with the available diffusion Monte Carlo results is also made.