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Journal of Physics A Mathematical and General

Many-particle systems. IX. RIP and SHRIMP models

R J M Carr, H R Post

Published Apr 1, 1977• 7 citations
Publisher

Abstract

For pt.VIII see ibid., vol.4, p.665 (1971). The energy lower bound models HIP and SHIP may be substantially improved for small numbers of fermions by a simple modification to the limiting procedure in their derivation. It is shown that for intermediate values of N the modified SHIP model gives in some cases the best available lower bounds.

DOI
10.1088/0305-4470/10/4/002

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Journal of Physics A Mathematical and General•Dec 1, 1979•3 citations
For pt.VIII see ibid., vol.4, p.665 (1971). The energy lower bound models HIP and SHIP may be substantially improved for small numbers of fermions by a simple modification to the limiting procedure in their derivation. It is shown that for intermediate values of N the modified SHIP model gives in some cases the best available lower bounds.
The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.
Lower bound energy formulae for the N-fermion problem of the type used by Hall (1967) and Post (1956) are considered in terms of a reduced density operator for the interparticle separation. In bounds of this type the N-body energy is bounded below by a sum of eigen-energies of states of relative motion weighted by occupations for the relative states. It is shown that the wavefunctions of relative motion are subject to constraints and in particular must be zero when the particle separation is zero. This modifies the energy spectrum of the reduced Hamiltonian of relative motion and improves the lower bound due to Hall. The improvement is demonstrated for some simple model problems. Furthermore it is shown that the bounds of this type currently known do not exhibit saturation regardless of the interaction details. This leads to some speculation on improved lower bounds.