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Journal of Physics G Nuclear and Particle Physics

Translationally invariant coupled cluster method in coordinate space for nuclei

I. Moliner, Niels R. Walet, R. F. Bishop

Published Apr 18, 2002• 5 citations
Publisher

Abstract

We study a formulation of the translationally invariant coupled cluster method in coordinate space for finite nuclei. The new formulation remedies convergence problems that plagued previous calculations in configuration space. The method is applied to light nuclei using semi-realistic central interactions.
DOI
10.1088/0954-3899/28/6/305

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The authors have previously formulated a general translationally invariant version of the coupled cluster theory. The authors investigate the linearized approximation referred to as the second-order translationally invariant cluster method. In particular, they algebraically formulate and numerically solve this method at the optimal Euler-Lagrange level for finite bosonic systems. The method proves to be highly superior to the standard configuration-interaction calculations of the shell-model type, and confirms the validity of the Gaussian geminal expansion of the two-body cluster C 2 .
The widely used coupled cluster method (CCM) in quantum many-body theory has recently provided very accurate descriptions of a large number of extended systems. Although its earlier applications to closed-shell and neighboring finite nuclei were also very successful, they have been shrouded in algebraic and technical complexity. Furthermore, they are difficult to compare with more traditional calculations of generalized shell-model theory since, at least at the important level of two-body correlations, they have been largely implemented in relative-coordinate space rather than the more usual oscillator configuration space. The CCM is reviewed here in the precise context of applications to simple finite systems. Special attention is paid to formulate it in such a way that comparison may be made with generalized shell-model or configuration-interaction (CI) theories. Particular regard is paid to an exact incorporation of translational invariance, so that any spuriosity associated with the center-of-mass motion is always avoided. An important side benefit is that the number of many-body configurations in the usual oscillator basis is dramatically reduced. We are thereby able to present both CI and CCM calculations on 4He^{4}\mathrm{He}4He up to the essentially unprecedented level of 60\ensuremath{\Elzxh}\ensuremath{\omega} in oscillator excitation energy, for two popular and quasirealistic choices of the nucleon-nucleon interaction for which exact Monte Carlo results are available for this nucleus. Although even our simplest approximations attain about 95% of the total binding energy, the convergence in the oscillator configuration space is shown to be both very slow and of a complicated nonuniform nature. Strong implications are drawn for standard implementations of generalized shell-model techniques for heavier nuclei.
A general coupled-cluster method valid for arbitrary multideterminantal reference states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and applied by Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and collaborators. The connected nature of the cluster operators and the effective interaction is proven in the case when the reference space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete reference spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approximate schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.