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Physical Review Letters

Variational Monte Carlo Calculations for Spin-Aligned Deuterium

Robert M. Panoff, J. W. Clark, Michael A. Lee, Kevin E. Schmidt, Malvin H. Kalos, Geoffrey V. Chester

Published Jun 14, 1982• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.48.1675

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Journal of the Physical Society of Japan•Jun 10, 2013•3 citations
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 b3\ensuremathΣu+b{^{3}\ensuremath{\Sigma}_{\mathrm{u}}}^{+}b3\ensuremathΣu​+ 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.
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.
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.
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.
The bulk, ground state properties of atomic hydrogen, deuterium, and tritium systems are calculated assuming that all pair interactions occur via the 3Σ+u atomic triplet potential. The conditions required to obtain this system, including inhibition of recombination through the energetically favorable singlet interaction, are discussed. The internal energy, pressure, and compressibility are calculated over the volume range 40 cm3/mole ? V ⩽200 cm3/mole applying the Monte Carlo technique with a quantum mechanical variational wavefunction. The system studied consisted of 32 atoms in a box with periodic boundary conditions. Results show that atomic triplet hydrogen and deuterium remain gasous at 0 K; i.e., the internal energy is positive at all molar volumes considered. Under the same conditions, tritium forms a liquid with a binding energy of E0 ? −0.75 K per atom, at a volume of V ? 130 cm3/mole. The pair distribution function for these systems is also calculated. A brief discussion of the predicted superfluid behavior of 3Σ+u hydrogen and tritium is presented.
The correlated-basis methods applied earlier to nuclear-spin-polarized liquid 3He^{3}\mathrm{He}3He are extended to electron-spin-aligned deuterium with one available nuclear spin state (D${\ensuremath{\downarrow}}_{1}$). Results are obtained for the linear response function and zero-sound mode using the local correlated random-phase approximation. The self-energy, single-particle spectrum and effective mass are calculated within the same framework.
Variational quantum Monte Carlo (VMC) method applied to interacting electron systems has played a pivotal role for the studies of electronic states of atoms, molecules and condensed matters. First, the many-body wave functions optimized by the VMC method provide good guiding functions in more accurate, diffusion quantum Monte Carlo (DMC) calculations for identical systems. Second, the VMC calculations themselves can give a good upper bound for the exact ground-state energy of many-electron systems when an appropriate form for the variational wave function is employed. In addition to a conventional form for the variational wave function that is expressed by the product of the Jastrow factor and the single Slater determinant, recent investigations have found that the expression in terms of many Slater determinants can provide a very accurate description of electronic states, especially for intrinsically multi-reference systems, which corresponds to the configuration interaction (CI) expansion method in conventional quantum chemistry. Although the CI method needs an extremely large number of expansion coefficients as variational parameters, the number of parameters can be reduced efficiently by using an idea of (molecular) orbital pair correlations. The purpose of the present paper is to examine the feasibility and usefulness of this idea within the framework of VMC method with the Jastrow–Slater wave function, where the ground state of a water molecule is investigated as a test case. In this study we rely on a variational wave function composed of a product of the Jastrow and Slater parts as 1⁄4 J S; ð1Þ