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arXiv:nucl-th/0304059 · Physical Review C

Quadrupole correlation energy by the generator coordinate method

K. Hagino, G. F. Bertsch, P.‐G. Reinhard

Published Aug 14, 2003• 16 citations
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Abstract

We investigate the accuracy of several schemes to calculate ground-state correlation energies using the generator coordinate technique. Our test bed for the study is the sd-interacting boson model, equivalent to a six-level Lipkin-type model. We find that the simplified projection of a triaxial generator coordinate state using the ${S}_{3}$ subgroup of the rotation group is not very accurate in the parameter space of the Hamiltonian of interest. On the other hand, a full rotational projection of an axial generator coordinate state gives remarkable accuracy. We also discuss the validity of the simplified treatment using the extended Gaussian overlap approximation, and show that it works reasonably well when the number of boson is 4 or larger.

DOI
10.1103/physrevc.68.024306

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We investigate the accuracy of several schemes to calculate ground-state correlation energies using the generator coordinate technique. Our test bed for the study is the sd-interacting boson model, equivalent to a six-level Lipkin-type model. We find that the simplified projection of a triaxial generator coordinate state using the S3{S}_{3}S3​ subgroup of the rotation group is not very accurate in the parameter space of the Hamiltonian of interest. On the other hand, a full rotational projection of an axial generator coordinate state gives remarkable accuracy. We also discuss the validity of the simplified treatment using the extended Gaussian overlap approximation, and show that it works reasonably well when the number of boson is 4 or larger.
We develop and test efficient approximations to estimate ground state correlations associated with low- and zero-energy modes. The scheme is an extension of the generator coordinate method (GCM) within Gaussian overlap approximation (GOA). We show that the GOA fails in non-Cartesian topologies and presents a topologically correct generalization of the GOA (topGOA). A random-phase-approximation like (RPA-like) correction is derived as the small amplitude limit of topGOA, called topRPA. Using exactly solvable models, the topGOA and topRPA schemes are compared with conventional approaches (GCM-GOA, RPA, Lipkin-Nogami projection) for rotational-vibrational motion and for particle-number projection. The results shows that the new schemes perform very well in all regimes of coupling.
In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small (emax=4e_{\mathrm{max}}=4emax​=4) model space demonstrate that exact binding energies of closed- (\nucl{O}{16}) and open-shell (\nucl{O}{18}, \nucl{Ne}{20}) nuclei are reproduced within 0.5−1.5%0.5-1.5\%0.5−1.5% at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.
We study the pairing energy in the ground state of a system of N particles occupying a symmetric two-level model space with a level degeneracy \ensuremath{\Omega}. Assuming a pairing Hamiltonian, we calculate the energies of the projected BCS states, using the exact, the Kamlah, and the Lipkin-Nogami particle number projection methods. We find that conclusions regarding the quality of the BCS approximation as well as various approximate projection methods, drawn from studying the N=\ensuremath{\Omega} case alone, are not valid in the more general N\ensuremath{\ne}\ensuremath{\Omega} case when the pairing interaction is weak.
The optimized-inner-projection (OIP) method is implemented and applied to determine the lower bounds to the ground-state correlation energy for the Pariser-Parr-Pople and Hubbard models of cyclic polyenes CN{\mathrm{C}}_{\mathit{N}}CN​HN{\mathrm{H}}_{\mathit{N}}HN​, N=4\ensuremath{\nu}+2, \ensuremath{\nu}=3--5, in the whole range of the coupling constant. In actual calculations, the orthogonally spin-adapted form of the OIP equations is used, and full advantage is taken of the high symmetry of the cyclic polyene model, so that systems with N\ensuremath{\ge}14 can be explored. Comparison is made with the results of both full and limited configuration-interaction calculations and with the correlation energies obtained with certain approximate coupled-cluster schemes. Contrary to the six- and ten-membered rings, the OIP lower bounds to the ground-state correlation energy of cyclic polyenes with larger number of sites are very poor and cannot serve as a meaningful source of information. Since the eigenvalues of the intermediate Hamiltonian, in terms of which the OIP technique is defined, are highly degenerate even in the weakly correlated region, and in view of the random behavior of the OIP bracketing function, the OIP method becomes computationally very demanding for N\ensuremath{\ge}14 rings. Moreover, simple iterative schemes usually diverge and sophisticated root searching procedures have to be applied. The usefulness of the OIP technique for the determination of lower bounds to energy eigenvalues in larger systems is thus questioned.
A numerical procedure is introduced which allows us to extract a collective Hamiltonian, expressed in terms of a collective canonical pair $\stackrel{^}{p}$ and $\stackrel{^}{q}$. The starting point is a microscopic many-body approach, namely the generator coordinate method, and the Weyl formalism permits the description of the collective dynamics in the $p\ensuremath{-}q$ space. As an illustration we compare numerical calculations with exact results, obtained analytically, for the quadratic energy kernel with the Gaussian overlap approximation case and the Goldhaber-Teller dipole model applied to the 4He^{4}\mathrm{He}4He nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.
Abstract We propose a new method, termed generator coordinate method (GCM)-correlated pair wave function (CPWF), for studying one-dimensional bosons confined in harmonic potentials with contact repulsive interactions. Our approach involves using the effective CPWF as a basis, combined with the GCM to handle complex many-particle correlations accurately. We demonstrate the reliability of our GCM-CPWF wave functions by comparing ground energy and one-body density with those obtained by other numerical methods. Moreover, we present the energy spectrum up to six particles and the occupation number on the harmonic oscillator state. Utilizing these GCM-CPWF wave functions, we explore the properties of the ground and excited states of the many-particle system. Our GCM-CPWF framework is highly flexible and can be generalized to investigate more complex many-particle systems.