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arXiv:2305.17649

The asymptotic behaviour of the many-body coupled cluster amplitudes

Saar Beck, R. Weiss, Nir Barnea

Published May 28, 2023• 0 citations
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Abstract

We analyze the asymptotic behaviour of the coupled cluster many-body wave-function in the limit of highly excited two- and three-particles states. We find that in this limit the different coupled cluster amplitudes exhibit a recurring behaviour, factorizing into a common asymptotic two- or three-body term. These asymptotic terms depend on the potential and in general are system specific. We also suggest that the knowledge of the asymptotic behaviour can potentially help solving the coupled cluster equations in a more efficient way.

DOI
10.48550/arxiv.2305.17649

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We analyze the asymptotic behaviour of the coupled cluster many-body wave-function in the limit of highly excited two- and three-particles states. We find that in this limit the different coupled cluster amplitudes exhibit a recurring behaviour, factorizing into a common asymptotic two- or three-body term. These asymptotic terms depend on the potential and in general are system specific. We also suggest that the knowledge of the asymptotic behaviour can potentially help solving the coupled cluster equations in a more efficient way.
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We present a systematic analysis of the nuclear two- and three-body short-range correlations and their relations to the zero-energy eigenstates of the Schr\"odinger equation. To this end we analyze the doublet and triplet coupled-cluster amplitudes in the high momentum limit, and show that they obey universal equations independent of the number of nucleons and their state. Furthermore, we find that these coupled-cluster amplitudes coincide with the zero-energy Bloch-Horowitz operator. These results illuminate the relations between the nuclear many-body theory and the generalized contact formalism, introduced to describe the nuclear two-body short range correlations, and they might also be helpful for general coupled-cluster computations as the asymptotic part of the amplitudes is given and shown to be universal.
The fundamentality of the exponential representation of a second-quantised correlated wave function is emphasised with an accent on the physical sense of cluster amplitudes as cumulants of the correlated ansatz. Three main wave function formalisms, namely, the configuration-interaction theory, the coupled-cluster approach, and the many-body perturbation theory (as well as their extensions, e.g. the equation-of-motion coupled-cluster method, multireference schemes, etc.), are represented in an exponential form, leading to a formulation of the working equations in terms of cluster amplitudes. By expressing the corresponding many-body tensor equations in terms of cluster amplitudes, we could unambiguously check connectivity types and the asymptotic behaviour of all tensors/scalars involved (in the formal limit of an infinite number of correlated particles). In particular, the appearance of disconnected cluster amplitudes corresponds to unphysical correlations. Besides, we demonstrate that the equation-of-motion coupled-cluster approach, as well as certain excited-state configuration-interaction methods, can be recast in a fully connected (exponential) form, thus breaking the common belief that all truncated configuration-interaction methods violate connectivity. Our work is based on the recently developed algebraic framework which can be viewed as a complement to the classical diagrammatic analysis.
Abstract To describe long-range behaviour of one particle removed from a few- or a many-body system, a hyperspherical cluster model has been developed. It has been applied to the ground and first excited states of helium drops with five, six, eight and ten atoms interacting via a two-body soft gaussian potential. Convergence of the hyperspherical cluster harmonics expansion is studied for binding energies, root-mean-squared radii and overlaps of the wave functions of two helium drops differing by one atom. It was shown that with increasing model space the functional form of such overlaps at large distances converges to the correct asymptotic behaviour. The asymptotic normalization coefficients that quantify the overlaps’ amplitudes in this region are calculated. It was also shown that in the first excited state one helium atom stays far apart from the rest forming a two-body molecule, or a halo. The probability of finding the halo atom in the classically-forbidden region of space depends on the definition of the latter and on the valence atom binding energy. The total norm of the overlap integrals, the spectroscopic factor, represents the number of partitions of a many-body state into a chosen state of the system with one particle removed. The spectroscopic factors have been calculated and their sum rules are discussed giving a further insight into the structure of helium drops.
The factorization of amplitudes of multiple excitations from those of elementary (single and double) excitations is only reliable when excitation energies are nearly additive. Such deviations from additivity may be important, especially in solid-state physics problems. Instead of introducing many-body operators in the coupled-cluster expansion, one may take into account the effect of these deviations from the additivity of excitation energies by proper energy-dependent corrections to the usual product of amplitudes. Their efficiency is illustrated on several model problems where the traditional approach gives poor results. © 1995 John Wiley & Sons, Inc.