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

Variational calculations of simple Bose systems

Vijay R. Pandharipande, Kevin E. Schmidt

Published Jun 1, 1977• 47 citations
Publisher

Abstract

Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part $v(r>d)$ and a constant part $\ensuremath{\lambda}$ at $r<d$ of the bare interaction are assumed to contribute to the average field. The correlation function $f(r)$ is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance $d$ is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of $d$ in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present $f(r)$. This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.

DOI
10.1103/physreva.15.2486

Related papers

Variational Method for Dense Systems

Vijay R. Pandharipande, Hans Albrecht Bethe

The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid $^{4}\mathrm{He}$, Fermi neutron gas, and liquid $^{3}\mathrm{He}$ are discussed. The correlation functions are parametrized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of the lowest-order contributions. The healing distance has a prescribed value in lowest-order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The many-body cluster contributions in Bose fluids are summed with successive approximations of an integral equation due to van Leeuwen et al. A simple diagrammatic cluster expansion is presented for Fermi liquids, and its direct diagrams are summed with the integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.

Physical Review C•Apr 1, 1973•178 citations

Variational Theory of Quantum Fluids.

Kevin E. Schmidt

"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."

•Aug 1, 1979•0 citations

Variational Method for Dense Systems

Vijay R. Pandharipande

The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He, Fermi neutron gas, and liquid 3He are discussed. The correlation functions are parameterized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of lowest order contributions. The healing distance has a prescribed value in lowest order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The direct many-body cluster diagrams are summed with successive approximations of an integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.

Symposium - International Astronomical Union•Jan 1, 1974•1 citation

Statistical mechanics of strongly correlated Bose quantum fluids

B. E. Clements, E. Krotscheck, James A. Smith +1 more

The finite-temperature properties of Bose quantum fluids are studied by applying a variational density-matrix approach. Our ansatz for the density matrix is similar to that introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. 30, 3728 (1984)] but has been generalized to include three-body correlations. An expression for the finite-temperature elementary-excitation energy, \ensuremath{\epsilon}(k), is derived from an ansatz for the variational entropy. At zero temperature \ensuremath{\epsilon}(k) reduces to the same expression obtained by Brillouin-Wigner perturbation theory. The hypernetted-chain (HNC) equations provide the relations between the n-particle distribution functions and the correlation functions. A substantial set of elementary diagrams have been included to minimize the loss of precision associated with the HNC approximation. We report values for the static structure function, radial distribution function, excitation energies, particle-hole interaction, Helmholtz free energy, entropy, and isothermal sound velocity. We present a detailed study of the spectra, three-body correlations, and elementary diagram dependence of the thermodynamic spinodal curve. For relevant densities, we find that three-body correlations tend to increase the liquid-gas critical temperature ${\mathit{T}}_{\mathit{c}}$ while adding elementary diagrams tends to reduce ${\mathit{T}}_{\mathit{c}}$. Including multiphonon terms in the spectrum produces several features in the variational entropy that were not observed at the Feynman level of excitations. Entropy isotherms show a rather abrupt transition from a low-density regime, where multiphonon contributions are insignificant, to a high-density regime where the isotherms are strongly influenced by multiphonon scattering processes.

Physical review. B, Condensed matter•Mar 1, 1993•26 citations

Energy and structure of dilute hard- and soft-sphere gases

Ferran Mazzanti, A. Polls, Adelchi Fabrocini

The energy and structure of dilute hard- and soft-sphere Bose gases are systematically studied in the framework of several many-body approaches, such as the variational correlated theory, the Bogoliubov model, and the uniform limit approximation, valid in the weak-interaction regime. When possible, the results are compared with the exact diffusion Monte Carlo ones. Jastrow-type correlation provides a good description of the systems, both hard- and soft-spheres, if the hypernetted chain energy functional is freely minimized and the resulting Euler equation is solved. The study of the soft-sphere potentials confirms the appearance of a dependence of the energy on the shape of the potential at gas paremeter values of $x\ensuremath{\sim}0.001.$ For quantities other than the energy, such as the radial distribution functions and the momentum distributions, the dependence appears at any value of x. The occurrence of a maximum in the radial distribution function, in the momentum distribution, and in the excitation spectrum is a natural effect of the correlations when x increases. The asymptotic behaviors of the functions characterizing the structure of the systems are also investigated. The uniform limit approach is very easy to implement and provides a good description of the soft-sphere gas. Its reliability improves when the interaction weakens.

arXiv:cond-mat/0305502•Physical Review A•Jun 30, 2003•22 citations

Fluid-Solid Phase Transition of a Hard-Sphere Bose System

Jean-Pierre Hansen, D. Levesque, Daniel Schiff

The ground-state energy of a system of hard-sphere bosons of diameter $\ensuremath{\sigma}$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$ and ${\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$.

Physical Review A•Feb 1, 1971•59 citations
Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part v(r>d)v(r>d)v(r>d) and a constant part \ensuremathλ\ensuremath{\lambda}\ensuremathλ at r<dr<dr<d of the bare interaction are assumed to contribute to the average field. The correlation function f(r)f(r)f(r) is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance ddd is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of ddd in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present f(r)f(r)f(r). This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.
The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He^{4}\mathrm{He}4He, Fermi neutron gas, and liquid 3He^{3}\mathrm{He}3He are discussed. The correlation functions are parametrized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of the lowest-order contributions. The healing distance has a prescribed value in lowest-order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The many-body cluster contributions in Bose fluids are summed with successive approximations of an integral equation due to van Leeuwen et al. A simple diagrammatic cluster expansion is presented for Fermi liquids, and its direct diagrams are summed with the integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."
The variational method for calculating energy of quantum fluids, and its applications to the Bose liquid 4He, Fermi neutron gas, and liquid 3He are discussed. The correlation functions are parameterized by their healing distance, and can depend on the states occupied by the correlated particles in the model wave function. They are calculated by constrained variation of lowest order contributions. The healing distance has a prescribed value in lowest order calculations, whereas it is sufficiently large in hopefully exact energy calculations. The direct many-body cluster diagrams are summed with successive approximations of an integral equation. The contribution of exchange diagrams is shown to decrease rapidly with the number of exchanges, and their sums are truncated after the energy has converged to within a few percent.
The finite-temperature properties of Bose quantum fluids are studied by applying a variational density-matrix approach. Our ansatz for the density matrix is similar to that introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. 30, 3728 (1984)] but has been generalized to include three-body correlations. An expression for the finite-temperature elementary-excitation energy, \ensuremath{\epsilon}(k), is derived from an ansatz for the variational entropy. At zero temperature \ensuremath{\epsilon}(k) reduces to the same expression obtained by Brillouin-Wigner perturbation theory. The hypernetted-chain (HNC) equations provide the relations between the n-particle distribution functions and the correlation functions. A substantial set of elementary diagrams have been included to minimize the loss of precision associated with the HNC approximation. We report values for the static structure function, radial distribution function, excitation energies, particle-hole interaction, Helmholtz free energy, entropy, and isothermal sound velocity. We present a detailed study of the spectra, three-body correlations, and elementary diagram dependence of the thermodynamic spinodal curve. For relevant densities, we find that three-body correlations tend to increase the liquid-gas critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ while adding elementary diagrams tends to reduce Tc{\mathit{T}}_{\mathit{c}}Tc​. Including multiphonon terms in the spectrum produces several features in the variational entropy that were not observed at the Feynman level of excitations. Entropy isotherms show a rather abrupt transition from a low-density regime, where multiphonon contributions are insignificant, to a high-density regime where the isotherms are strongly influenced by multiphonon scattering processes.
The energy and structure of dilute hard- and soft-sphere Bose gases are systematically studied in the framework of several many-body approaches, such as the variational correlated theory, the Bogoliubov model, and the uniform limit approximation, valid in the weak-interaction regime. When possible, the results are compared with the exact diffusion Monte Carlo ones. Jastrow-type correlation provides a good description of the systems, both hard- and soft-spheres, if the hypernetted chain energy functional is freely minimized and the resulting Euler equation is solved. The study of the soft-sphere potentials confirms the appearance of a dependence of the energy on the shape of the potential at gas paremeter values of $x\ensuremath{\sim}0.001.$ For quantities other than the energy, such as the radial distribution functions and the momentum distributions, the dependence appears at any value of x. The occurrence of a maximum in the radial distribution function, in the momentum distribution, and in the excitation spectrum is a natural effect of the correlations when x increases. The asymptotic behaviors of the functions characterizing the structure of the systems are also investigated. The uniform limit approach is very easy to implement and provides a good description of the soft-sphere gas. Its reliability improves when the interaction weakens.
The ground-state energy of a system of hard-sphere bosons of diameter $\ensuremath{\sigma}$ is calculated by a variational method using a Jastrow wave-function for the fluid phase and a "localized" Jastrow times a Gaussian wave function for the solid phase. The transition densities are ${\ensuremath{\rho}}_{\mathrm{fluid}}=0.23\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$ and ${\ensuremath{\rho}}_{\mathrm{solid}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ particles/${\mathrm{\ensuremath{\sigma}}}^{3}$.