We introduce a new paradigm for one-dimensional uniform electron gases (UEGs). In this model,n electrons are confined to a ring and interact via a bare Coulomb operator. We use Rayleigh-Schrödinger perturbation theory to show that, in the high-density regime, the ground-state reduced (i.e., per electron) energy can be expanded as , where rs is the Seitz radius. We use strong-coupling perturbation theory and show that, in the low-density regime, the reduced energy can be expanded as . We report explicit expressions for ε0(n), ε1(n), ε2(n), ε3(n), η0(n), and η1(n) and derive the thermodynamic (large-n) limits of each of these. Finally, we perform numerical studies of UEGs with n = 2, 3, …, 10, using Hylleraas-type and quantum Monte Carlo methods, and combine these with the perturbative results to obtain a picture of the behavior of the new model over the full range of n and rs values.
Received 14 February 2013Accepted 08 April 2013Published online 30 April 2013
The authors thank Neil Drummond and Shiwei Zhang for helpful discussions, the NCI National Facility for a generous grant of supercomputer time. P.M.W.G. thanks the Australian Research Council (Grant Nos. DP0984806, DP1094170, and DP120104740) for funding. P.F.L. thanks the Australian Research Council for a Discovery Early Career Researcher Award (Grant No. DE130101441).
Article outline: I. INTRODUCTION II. PERTURBATIVE METHODS A. High-density expansion 1. Double-bar integrals 2. Zeroth order 3. First order 4. Second order 5. Third order B. Low-density expansion III. EXPLICITLY CORRELATED METHODS A. 2-ringium B. 3-ringium C. 4- and 5-ringium IV. QUANTUM MONTE CARLO METHODS A. Variational Monte Carlo B. Diffusion Monte Carlo C. Trial wave functions D. Fixed-node approximation E. Results and discussion V. CONCLUSIONS
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