The generator coordinate method as an approximation to the shell model: A comparison on 24Ne

The applicability of the generator coordinate method as an approximation to the complete shell model diagonalization is tested in the sd shell on ²⁴Ne. We use the quadrupole moment and the pairing energy as generator coordinates and choose as generating function a constrained Hartree-Bogoliubov solution projected on the subspace of good angular momentum and good proton and neutron number. The spectra obtained by the generator coordinate method and the complete diagonalization show good overall agreement. Also we draw some conclusions about the nature of some low-lying states in ²⁴Ne, interpreted in terms of vibrations.     

On the analytical unfolding of generator coordinate kernels for unequal fragments

The generator coordinate formalism breaks down for scattering and reactions involving fragments of unequal size. Analytical expressions are given through which, even in that case, the resonating group kernels can be extracted from those of the generator coordinate method. The possibility of defining a generator coordinate density matrix is discussed.     

The generator coordinate representation in an natural state formalism

The generator coordinate method for bound states is reformulated using the classical Fredholm theory of linear integral equations. A natural state formalism is developed to investigate the properties of the variational Hill-Wheeler equations. A generator coordinate representation of the one-dimensional harmonic oscillator is worked out analytically.     

Relationship between the interacting boson model of Arima and Iachello and the collective model of Bohr and Mottelson

The generator coordinate method is used to relate the interacting boson model and the collective model through an isometric transformation. It associated complex parameters to the original boson operators whereas the ultimate collective variables are real. The mapping of operators of the interacting boson model onto those of the collective model turns out to be of Holstein-Primakoff type.     

A generator coordinate approach for the description of pairing vibrations in the seniority-zero space

The application of the generator coordinate method is extended to non-harmonic systems. The many-dimensional Hill-Wheeler integral equation is reduced to a one-dimensional integral equation by expressing all independent parameters in the generator function by a single parameter. It is shown that the subspace spanned by a proper single-parameter family is the same as that spanned by the many-parametric family of generator functions.     

CONTINUOUS VARIABLE REPRESENTATION OF ELLIOTT SU(3) MODEL

The continuous variable representation and the geometrical interpretation of the Elliott SU(3) model are given by virtue of the generator coordinate method. It is pointed out that the carrier subspace of collective states of the dynamical group SU(3) in the Elliott model is spanned by the β and γ vibration states as well as the rotation states.     

Collective pairing Hamiltonian in the GCM approximation

Using the generator coordinate method and the gaussian overlap approximation we derived the collective Schrödinger-type equation starting from a microscopic single-particle plus pairing hamiltonian for one kind of particle. The BCS wave function was used as the generator function. The pairing energy-gap parameter Δ and the gauge transformation angle ø were taken as the generator coordinates. Numerical results have been obtained for the full and the mean-field pairing hamiltonians and compared with the cranking estimates. A significant role played by the zero-point energy correction in the collective pairing potential is found. The ground-state energy dependence on the pairing strength agrees very well with the exact solution of the Richardson model for a set of equidistant doubly-degenerate single-particle levels.     

On the implicit integral character of Roothaan's expansion

The molecular generator coordinate Hartree-Fock method is reviewed. The connection between a quadrature solution of the generator coordinate Hartree-Fock equations and Roothaan's equations is stressed. The relation between linear expansion coefficients and generator coordinate weight functions is discussed and a numerical and analytical example is provided for the 1s orbital of the hydrogen atom represented as the integral transform of a Gaussian function. For the same example, the Gauss-Labatto quadrature is employed to emphasize the implicit integral character of Roothaan's equations. As a major conclusion, the interpretation that every LCAO calculation is actually performing integrations of the Griffin-Wheeler equations is advanced. Basis sets are therefore abscissas of the implicit quadrature used in the integration, whereas the linear coefficients automatically incorporate the corresponding weights. Subsequently, it is shown how to extract the generator coordinate weight function from the LCAO coefficients which has the advantage of being a characteristic of the physical system under study and not of the particular calculation being carried out. As such, basis set design becomes how to efficiently sample the weight function. Received: 13 June 1998 / Received in final form: 12 August 1998 / Accepted: 14 September 1998     

On the relation between the interacting boson model of Arima and Iachello and the collective model of Bohr and Mottelson

The generator coordinate method is used to relate the interacting boson model of Arima and Iachello and the collective model of Bohr and Mottelson through an isometric transformation. It associates complex parameters to the original boson operators whereas the ultimate collective variables are real. The absolute squares of the collective wave functions can be given a direct probability interpretation. The lowest order Bohr-Mottelson hamiltonian is obtained in the harmonic approximation to the interacting boson model; anharmonic coupling terms render the collective potential to be velocity-dependent.     

Generator coordinate calculations on 28Si

The spectrum of ²⁸Si is calculated by the generator coordinate method for various choices of generating functions obtained by solving the HF equations within the s-d shell with a constraint on the quadrupole or the hexadecapole moment, or with a constraint of the Holzwarth-Yukawa type. The influence of the symmetry of the generating functions is studied. The spectrum if quite sensitive to the choice of generator coordinate. A comparison with the shell-model calculations of Soyeur and Zuker, and of Whitehead and Watt, is attempted.     

On the connection between the generator coordinate method and boson expansions for odd-particle systems

The generator coordinate method is used to derive a boson expansion for odd-particle systems which is equivalent to the Yamamura and Marshalek expansions.     

The interacting boson approximation in nuclei with a few valence nucleons

In order to investigate microscopically nuclear collective models, the generator coordinate method as a means for deriving the effective hamiltonian for nuclear collective states is discussed. A simple case is treated as an illustration. Perfect agreement of the result with the exact solution testifies to the adequacy of the method. This method is applied to nuclei with a few valence nucleons and gives us the results of the interacting model in the SU(5) limit. The possibility of application to nuclei with many valence nucleons is also briefly discussed.     

The generator coordinate method as an approximation to the exact diagonalization in a given model space

The low-lying states of the nuclei ¹⁶O and ²⁰Ne are calculated with the generator coordinate method. For ¹⁶O two generator coordinates are used, which are related to the quadrupole and to the octupole deformation of the single-particle potential. For ²⁰Ne the generator coordinates are related to the parameters β and D of the Nilsson potential. In both cases six pairs of coordinate values are chosen. The results are compared with those of the complete diagonalization. The agreement is, for both cases, better than 0.5 MeV for most low-lying states.     

Derivation of a wave equation for collective motion

The generator coordinate method based on states derived from the random phase approximation, together with a strong assumption, leads to a differential equation for collective motion similar to the Bohr Hamiltonian with inertial parameters given by a formula related to the cranking model and with a potential almost equal to the ground state energy in the random phase approximation for each deformation.     

Collective potential and mass parameters derived from the generator coordinate method

The collective hamiltonian for the axial quadrupole vibrations was derived from theQQ+PP model. The generator coordinate method was applied and the results obtained through the symmetric moments expansion and the gaussian overlap approximation were compared. It was found that the collective potential and the average magnitude of the mass parameter obtained in both approximations are close to each other.     

A generator coordinate theory of nuclear reactions

The generator coordinate formalism is generalized to incorporate optical potentials, coupled waves, and a distorted-wave approximation. An estimate of the importance of recoil is proposed. An analytical illustrative example related to the generator coordinate DWBA is given.     

Cutting DNA: Mechanics of the topoisomerase

The framework of relativistic self-consistent mean-field models is extended to include correlations related to restoration of broken symmetries and to fluctuations of collective variables. The generator coordinate method is used to perform configuration mixing of angular-momentum and particle-number projected relativistic wave functions. The model, currently restricted to axially symmetric shapes, employs a relativistic point-coupling (contact) nucleon-nucleon effective interaction in the particle-hole channel, and a δ-interaction in the pairing channel. Both bulk and spectroscopic nuclear properties are explored.     

On the relation between the interacting boson model of Arima and Iachello and the collective model of Bohr and Mottelson. II

The generator coordinate method is used to relate the interacting boson model of Arima and lachello and the collective model of Bohr and Mottelson through an isometric transformation. It associates complex parameters to the original boson operators whereas the ultimate collective variables are real. The absolute squares of the collective wave functions can be given a direct probability interpretation. The lowest order Bohr-Mottelson hamiltonian is obtained in the harmonic approximation to the interacting boson model; unharmonic coupling terms render the collective potential to be velocity dependent. The mapping of operators of the interacting boson model onto those of the Bohr-Mottelson model turns out to be of Holstein-Primakoff type.     

Note on the applicability of the Peierls-Yoccoz method

A simple criterion for the approximate validity of the single projection Peierls-Yoccoz method is discussed. A procedure for chosing the generator coordinate is proposed.