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Translationally-invariant coupled-cluster method for finite systems

R. Guardiola, I. Moliner, Jesús Navarro, Portesi, M

Published Dec 15, 1997• 6 citations
OpenAlex

Abstract

The translational invariant formulation of the coupled-cluster method is presented here at the complete SUB(2) level for a system of nucleons treated as bosons. The correlation amplitudes are solution of a non-linear coupled system of equations. These equations have been solved for light and medium systems, considering the central but still semi-realistic nucleon-nucleon S3 interaction.

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•Jan 1, 2020•0 citations
The translational invariant formulation of the coupled-cluster method is presented here at the complete SUB(2) level for a system of nucleons treated as bosons. The correlation amplitudes are solution of a non-linear coupled system of equations. These equations have been solved for light and medium systems, considering the central but still semi-realistic nucleon-nucleon S3 interaction.
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.
Cluster expansion of the many-body Schr\"odinger wave function is manifested through the projected Faddeev-Yakubovsky equations. It is shown that using these equations one can formulate a systematic expansion for the wave function of finite systems in a translational invariant manner. The solution of the N-body problem at a given order becomes equivalent to solving the few-body Faddeev-Yakubovsky equations. The first two orders in this expansion are given explicitly and then solved for the case of N bosons interacting through the nucleon-nucleon Afnan-Tang S3 and Malfliet-Tjon MT-V potentials.
We derive coupled-cluster equations for three-body Hamiltonians. The equations for the one- and two-body cluster amplitudes are presented in a factorized form that leads to an efficient numerical implementation. We employ low-momentum two- and three-nucleon interactions and calculate the binding energy of 4He^{4}\mathrm{He}4He. The results show that the main contribution of the three-nucleon interaction stems from its density-dependent zero-, one-, and two-body terms that result from the normal ordering of the Hamiltonian in coupled-cluster theory. The residual three-body terms that remain after normal ordering can be neglected.