Generator Coordinate Method Research Articles

Role of momentum in the generator-coordinate method applied to barrier penetration

Role of momentum in the generator-coordinate method applied to barrier penetration

Relativistic Prolapse-Free Gaussian Basis Sets of Double- and Triple-ζ Quality for s- and p-Block Elements: (aug-)RPF-2Z and (aug-)RPF-3Z.

This study presents two new relativistic Gaussian basis sets without variational prolapse of double- and triple-ζ quality, RPF-2Z and RPF-3Z, along with augmented versions including additional diffuse functions, aug-RPF-2Z and aug-RPF-3Z, which are available for all s and p block elements from Hydrogen to Oganesson. The exponents of the Correlation/Polarization (C/P) functions are obtained from a polynomial version of the generator coordinate Dirac-Fock method (p-GCDF). The choice of C/P functions was guided by multireference configuration interaction calculations with single and double excitations (MR-CISD) based on a valence active space. Finally, calculations of fundamental properties done for atomic and molecular systems (bond lengths, vibrational frequencies, dipole moments, and electron affinities) ensure the expected quality of these new basis sets, which may also exhibit some computational efficiency advantages. Additionally, the prolapse-free feature of these sets must provide a reliable description of properties more dependent on core electron distributions, as well.

Ab initio description of monopole resonances in light- and medium-mass nuclei

The paper is the third of a series dedicated to the ab initio description of monopole giant resonances in mid-mass closed- and open-shell nuclei via the so-called projected generator coordinate method. The present focus is on the computation of the moments mk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_k$$\\end{document} of the monopole strength distribution, which are used to quantify its centroid energy and dispersion. First, the capacity to compute low-order moments via two different methods is developed and benchmarked for the m1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_1$$\\end{document} moment. Second, the impact of the angular momentum projection on the centroid energy and dispersion of the monopole strength is analysed before comparing the results to those obtained from consistent quasi-particle random phase approximation calculations. Next, the so-called energy weighted sum rule (EWSR) is investigated. First, the appropriate ESWR in the center-of-mass frame is derived analytically. Second, the intrinsic EWSR is tested in order to quantify the (unwanted) local-gauge symmetry breaking of the presently employed chiral effective field theory (χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}EFT) interactions. Finally, the infinite nuclear matter incompressibility associated with the employed χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document}EFT interactions is extracted by extrapolating the finite-nucleus incompressibility computed from the monopole centroid energy.

Emulating the generator coordinate method with extended eigenvector continuation for the Lipkin-Meshkov-Glick model

Emulating the generator coordinate method with extended eigenvector continuation for the Lipkin-Meshkov-Glick model

Ab initio description of monopole resonances in light- and medium-mass nuclei

Giant resonances (GRs) are a striking manifestation of collective motions in atomic nuclei. The present paper is the second in a series of four dedicated to the use of the projected generator coordinate method (PGCM) for the ab initio determination of the isoscalar giant monopole resonance (GMR) in closed- and open-shell mid-mass nuclei. While the first paper was dedicated to quantifying various uncertainty sources, the present paper focuses on the first applications to three doubly-open shell nuclei, namely 46Ti\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{46}\\hbox {Ti}$$\\end{document}, 28Si\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{28}\\hbox {Si}$$\\end{document} and 24Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{24}\\hbox {Mg}$$\\end{document}. In particular, the goal is to investigate, starting from chiral effective field theory nuclear interactions, (i) the coupling of the GMR with the giant quadrupole resonance (GQR) in intrinsically-deformed nuclei, (ii) the possible impact of shape coexistence and shape mixing on the GMR, (iii) the GMR based on shape isomers and (iv) the impact of anharmonic effects on the monopole response. The latter is studied by comparing PGCM results to those obtained via the quasi-particle random phase approximation (QRPA), the traditional many-body approach to giant resonances, performed in a consistent setting. Eventually, PGCM results for sd-shell nuclei are in good agreement with experimental data, which is attributed to the capacity of the PGCM to capture the important fragmentation of the monopole response in light, intrinsically-deformed systems. Still, the comparison to data in 28Si\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{28}\\hbox {Si}$$\\end{document} and 24Mg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{24}\\hbox {Mg}$$\\end{document} illustrates the challenge (and the potential benefit) of extracting unambiguous experimental information.

Lambda motion and cluster states of [formula omitted] predicted via neural networks guided microscopic calculation

We investigate the Λ particle motion and cluster states of BeΛ9−11 using a novel multiple cooling approach guided by the Control Neural Networks method (Ctrl.NN). The numerical results are well reproduced, and the Ctrl.NN method shows rapid convergence concerning the set of the basis states in comparison to traditional generator coordinate method (GCM) calculations. Taking advantage of this merit, we calculate ground state energies and rotational bands of BeΛ9−11 which agree well with experimental observations. By analyzing the density distributions of the main basis, a slight short-range repulsive correlation between Λ particle and α clusters in directions perpendicular to the axis of the Be8 core is revealed. The Λ particle presents a significant broad distribution as a consequence of this correlation, which we call an “analog π-orbit” structure. Furthermore, we prove that this correlation originates from the competition between kinetic energy and Λ-N interaction, as indicated by the energy curves for different configurations of BeΛ9. In the ground states of BeΛ10 and BeΛ11, the Λ particle is confined in a cage-like structure of core nuclei formed by two α clusters and valence neutrons. These structures lead to the development of more compact Λ particle configurations, and the shrinkage of α-α relative distance is also well reproduced.

Ab initio description of monopole resonances in light- and medium-mass nuclei

Giant resonances (GRs) are a striking manifestation of collective motions in mesoscopic systems such as atomic nuclei. Until recently, theoretical investigations have essentially relied on the (quasiparticle) random phase approximation ((Q)RPA), and extensions of it, based on phenomenological energy density functionals (EDFs). As part of a current effort to describe GRs within an ab initio theoretical scheme, the present work promotes the use of the projected generator coordinate method (PGCM). This method, which can handle anharmonic effects while satisfying symmetries of the nuclear Hamiltonian, displays a favorable (i.e. mean-field-like) scaling with system’s size. Presently focusing on the isoscalar giant monopole resonance (GMR) of light- and medium-mass nuclei, PGCM’s potential to deliver wide-range ab initio studies of GRs in closed- and open-shell nuclei encompassing pairing, deformation, and shape coexistence effects is demonstrated. The comparison with consistent QRPA calculations highlights PGCM’s unique attributes and sheds light on the intricate interplay of nuclear collective excitations. The present paper is the first in a series of four and focuses on technical aspects and uncertainty quantification of ab initio PGCM calculations of GMR using the doubly open-shell 46\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{46}$$\\end{document}Ti as an illustrative example. The second paper displays results for a set of nuclei of physical interest and proceeds to the comparison with consistent (deformed) ab initio QRPA calculations. While the third paper analyzes useful moments of the monopolar strength function and different ways to access them within PGCM calculations, the fourth paper focuses on the effect of the symmetry restoration on the monopole strength function.

Quantum-Number Projected Generator Coordinate Method for 21Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction

In this paper, we report a study of the low-lying states of deformed 21Ne within the framework of the quantum-number projected generator coordinate method (PGCM), starting from a chiral two-nucleon-plus-three-nucleon (NN+3N) interaction. The wave functions of states are constructed as a linear combination of a set of axially deformed Hartree–Fock–Bogliubov (HFB) wave functions with different quadrupole deformations. These HFB wave functions are projected onto different angular momenta and the correct neutron and proton numbers for 21Ne. The results of the calculations based on the effective Hamiltonians derived by normal-ordering the 3N interaction with respect to three different reference states, including the quantum-number projected HFB wave functions for 20Ne, 22Ne, and an ensemble of them with equal weights, are compared. This study serves as a key step towards ab initio calculations of odd-mass deformed nuclei with the in-medium GCM.

Solving the Lipkin model using quantum computers with two qubits only with a hybrid quantum-classical technique based on the generator coordinate method

Solving the Lipkin model using quantum computers with two qubits only with a hybrid quantum-classical technique based on the generator coordinate method

Generalized time-dependent generator coordinate method for induced fission dynamics

Generalized time-dependent generator coordinate method for induced fission dynamics

Non-empirical description of nuclear collective motion with optimized basis for multi-reference density functional theory

The generator coordinate method (GCM) has been a well-known method to describe nuclear collective motions. In this method, one a priori specifies collective degrees of freedom as inputs of the method based on empirical and/or phenomenological assumptions. We here present an extension of the GCM, in which both the basis Slater determinants and weight factors are optimized in a non-empirical manner. The result for 16O nucleus with the Skyrme functional suggests that a collective coordinate should be determined in a more complex way than what has been assumed so far.

Electromagnetic properties of Λ hypernuclei with a beyond relativistic mean-field approach

We present a microscopic study of electromagnetic properties of the low-lying states of single-Λ hypernucleus Λ9Be using a quantum-number projected generator coordinate method (HyperGCM) based on covariant density functional theory. All the configurations are restricted to have axial symmetry, with the Λ hyperon in the lowest-energy state. The results are compared to the predictions of a simple particle-rotor model (PRM), for which analytic expressions are available. We show that the electromagnetic properties of Λ9Be from the HyperGCM calculation are close to those by the PRM. In particular, it is found that the magnetic moments and M1 transition strengths are less sensitive to the coupling strengths of the ΛN interaction than the electric quadrupole transition strengths. The latter may provide additional constraints on the ΛN interaction.

Extension of the generator coordinate method with basis optimization

Extension of the generator coordinate method with basis optimization

Generator coordinate method for nuclear octupole excitations: Status and perspectives

Strong octupole correlations have been observed in the low-lying states of atomic nuclei across various mass regions. In this paper, we provide an overview of Beyond Mean-Field (BMF) studies of nuclear octupole collective motions with Generator Coordinate Method (GCM) in combination with quantum-number projections that are implemented to restore the broken symmetries in nuclear mean-field states. We highlight recent developments within this framework and their applications to excitation spectra and electromagnetic transition rates in octupole-shaped nuclei and hypernuclei. We discuss the novel phenomena of nucleon clustering in light nuclei. Additionally, we explore the phase transition from octupole vibrations to rotational motions as spin increases in heavy nuclei. Lastly, we examine the status and future prospects of studies on octupole deformation effects in nuclear Schiff moments. These studies, along with the upper limits of atomic Electric Dipole Moment (EDM), impose stringent constraints on beyond-standard-model time-reversal-violating nucleon–nucleon interactions.

Dynamics in Nuclei

The nucleus is a complex many-body system with some remarkable emergent collective properties of multiple nucleons acting together. Bohr and Mottelson [1] provided a description of collective motion in nuclei based on geometrical shapes with superimposed oscillations around those shapes. Later, Lie algebras and symmetries were used to describe nuclear dynamics [2], followed by advances in the shell model approach [3] with new effective nucleon-nucleon two- and three-body interactions, and more recently with Hartree-Fock-Bogoliubov approximations within the extended generator coordinate method [4]. Yet, the underlying science question has remained the same. In nuclei, where there is explicit deformation in the ground state, “are the low-lying 0+ states collective vibrations built on the ground state or are they minima of a coexisting shape?” Ref. [4] has shown that for a significant percentage of K = 0+ excitations built on the deformed ground state (g.s.) should, in fact, be a collective vibration. The question has remained open due to sufficiently convincing experimental data with lifetimes, transfer reaction cross sections, and E0 transitions [5]. This paper summarizes the experimental situation regarding the lifetimes of 0+ states.

Generator coordinate method for 1D contacting bosons in harmonic trap

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.

Spectroscopic factors with generator coordinate method and application to 25Mg states

Nuclei can be studied through the analysis of their shell structure, that provides important information on the wavefunctions useful for interpreting many phenomena, from excitations to transfer reactions. Spectroscopic factors provide the degree of single particle behaviour of a nuclear state and are used to represent the shell structure of a nucleus.In this manuscript, we expand our recently developed model, which uses the generator coordinate method with an effective Hamiltonian, to calculate spectroscopic factors of states of odd deformed nuclei. Here we provide the results for 25Mg.

Towards microscopic optical potentials in deformed nuclei

A microscopic and consistent description of both nuclear structure and reactions is instrumental to extend the predictivity of models calculating scattering observables. In particular, this is crucial in the case of exotic nuclei not yet discovered.In this manuscript, we will present the plan of the Lund effort for a symmetry breaking description of bound and scattering observables using a generator coordinate method model based on an effective Hamiltonian constrained with a (Skyrme) functional. Following this, we will illustrate the steps to construct an optical potential for deformed nuclei from microscopic wavefunctions obtained with projected generator coordinate method from Hartree-Fock-Bogoliubov basis.

Relativistic adapted Gaussian basis sets free of variational prolapse of small and medium size for cesium through radon.

Relativistic adapted Gaussian basis sets of small and medium sizes are presented in this study for all elements from cesium to radon, including some alternative electron configurations. Both basis sets are made free of variational prolapse, being developed by means of a polynomial version of the generator coordinate Dirac-Fock method. In addition, these sets were designed to be promptly used with two popular finite nuclear models, uniform sphere and Gaussian nuclei. The largest basis set errors found with the uniform sphere nucleus are 27.3 and 10.6 mHartree, respectively, for the small- and medium-size sets. The largest basis set errors obtained with the Gaussian nuclear model are smaller, reaching 23.2 and 7.1 mHartree for the small- and medium-size sets, respectively. Soon, these basis sets will be augmented with polarization functions to be properly used in molecular calculations.

Generalized time-dependent generator coordinate method for small- and large-amplitude collective motion

Generalized time-dependent generator coordinate method for small- and large-amplitude collective motion