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Journal of Physics Condensed Matter

Constrained variational calculations for many-body spin-polarized atomic deuterium

B. Skjetne, Erlend Østgaard

Published Oct 1, 1999• 9 citations
Publisher

Abstract

A lowest-order constrained variational (LOCV) method, with modified conditions of healing on the two-body Jastrow wave function, is investigated in calculations for the ground-state energy levels of many-body spin-polarized atomic deuterium. Results are obtained for the , and phases, corresponding to equal occupations of one, two or three nuclear spin states. Estimates for the optimum healing conditions are obtained by comparison of LOCV results with current Monte Carlo benchmarks. The nature of the phases, i.e., quantum gas or quantum liquid, is discussed, the energy of the phase in our calculations always occurring above the gas-liquid interphase for healing conditions within a physically acceptable range.

DOI
10.1088/0953-8984/11/41/306

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Physical review. B, Condensed matter•Sep 1, 1985•18 citations
A lowest-order constrained variational (LOCV) method, with modified conditions of healing on the two-body Jastrow wave function, is investigated in calculations for the ground-state energy levels of many-body spin-polarized atomic deuterium. Results are obtained for the , and phases, corresponding to equal occupations of one, two or three nuclear spin states. Estimates for the optimum healing conditions are obtained by comparison of LOCV results with current Monte Carlo benchmarks. The nature of the phases, i.e., quantum gas or quantum liquid, is discussed, the energy of the phase in our calculations always occurring above the gas-liquid interphase for healing conditions within a physically acceptable range.
Microscopic prediction of the ground-state phase of electron-spin-aligned bulk atomic deuterium (D\ensuremath{\downarrow}) is attempted, based on the variational Monte Carlo method. The accurate pair potential of Kolos and Wolniewicz is assumed, and three versions of D\ensuremath{\downarrow} are considered, which, respectively, involve one, two, and three equally occupied nuclear spin states. The most definitive results on the zero-temperature equations of state of these systems are obtained with optimized ground-state trial wave functions incorporating Jastrow pair correlations, triplet correlations, and momentum-dependent backflow effects. The species D${\ensuremath{\downarrow}}_{3}$ is bound already at the pure Jastrow level, while the energy expectation value of D${\ensuremath{\downarrow}}_{2}$ dips below zero upon supplementing the Jastrow description by triplets and momentum-dependent backflow. The variational energy of D${\ensuremath{\downarrow}}_{1}$ remains positive under all current refinements of the ground-state trial function. We conclude that the systems D${\ensuremath{\downarrow}}_{3}$ and D${\ensuremath{\downarrow}}_{2}$, if they could be manufactured and stabilized at relevant densities, would be Fermi liquids at sufficiently low temperature; on the other hand, it is likely that D${\ensuremath{\downarrow}}_{1}$ would remain gaseous down to absolute zero.
It is shown that the method of lowest order constrained variational (LOCV) which is based on the cluster expansion theory is a reliable many-body technique to calculate the nuclear matter equation of state. In this respect, the state dependent correlation functions and the effective interactions which have been produced by the LOCV calculation with the Reid and \ensuremath{\Delta}-Reid soft core interactions are used to estimate the size of higher order cluster terms such as the effect of three-body cluster energy on the nuclear matter ground state energy. Finally it is shown that the LOCV normalization constraint plays a major role in the convergence of the cluster expansion and the result of LOCV calculation can be as good as more sophisticated approaches which go beyond lowest order.
The first variational Monte Carlo calculations of the ground-state energy of spin-aligned deuterium D\ensuremath{\downarrow} are presented. Using Slater-Jastrow trial wave functions, the calculations are based on the accurate theoretical pair potential of Kolos and Wolniewicz for the $b{^{3}\ensuremath{\Sigma}_{\mathrm{u}}}^{+}$ state. For D\ensuremath{\downarrow} with equal population of the three nuclear-spin states, the minimum theoretical ground-state energy is found to be negative, implying that this system is a liquid at zero temperature under zero external pressure.
The ground-state energy of spin-polarized hydrogen, deuterium, and tritium is calculated by means of a modified variational lowest order constrained-variation method, and the calculations are done for five different two-body potentials. Spin-polarized H↓ is not self-bound according to our theoretical results for the ground-state binding energy. For spin-polarized D↓, however, we obtain theoretical results for the ground-state binding energy per particle from −0.42 K at an equilibrium particle density of 0.25 σ−3 or a molar volume of 121 cm3/mol to + 0.32 K at an equilibrium particle density of 0.21 σ−3 or a molar volume of 142 cm3/mol, where σ = 3.69 Å (1 Å = 10−10 m). It is, therefore, not clear whether spin-polarized deuterium should be self-bound or not. For spin-polarized T↓, we obtain theoretical results for the ground-state binding energy per particle from −4.73 K at an equilibrium particle density of 0.41 σ−3 or a molar volume of 74 cm3/mol to −1.21 K at an equilibrium particle density of 0.28 σ−3 or a molar volume of 109 cm3/mol.
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.
Ground-state properties of nuclear-spin-aligned deuterium are studied within the method of correlated basis functions (CBF), using Fermi-hypernetted-chain (FHNC) techniques to evaluate the requisite matrix elements of the Hamiltonian and identity attendant to a Jastrow choice of correlation operator. Results are presented for the ground-state energies of three hypothetical species of this system, calculated to low order in CBF perturbation theory, taking account of two- and three-body CBF effective interactions. Also included are data on the Jastrow momentum distribution and radial distribution function, computed using the FHNC scheme. These findings are supplemented by benchmark variational Monte Carlo results on the energy and radial distribution function. The three deuterium species provide examples of strongly interacting Fermi fluids, intermediate in coupling strength between nuclear matter and liquid 3He^{3}\mathrm{He}3He.