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.     

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.     

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     

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.     

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.     

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.     

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.     

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.     

On calculating scattering phase shifts by the generator coordinate method

The generator coordinate equations for scattering phase shifts are solved for α-α scattering. Calculations are made in the coordinate representation and the results agree well with those obtained by a more laborious procedure which is formulated in the momentum representation. The results disagree with another calculation in the coordinate representation, where the scattering boundary condition is introduced less accurately.     

Microscopic R-Matrix theory in a generator coordinate basis: (I). Theory and application to dineutron-dineutron and α-α scattering

We present a new method to treat the scattering of composite particles by the generator coordinate method. Our treatment is an improvement of the one proposed by Horiuchi. We use more elaborate connection techniques based on a completely antisymmetrized R-matrix theory. Our results are in very good agreement with those obtained by the other generator coordinate methods, but we need a much smaller number of discretization points to solve the Hill-Wheeler equation. Results obtained by Horiuchi's approximation are also presented. They strongly depend on the radius of the internal region and unphysical resonances appear at high energies. These problems have been eliminated in our treatment.     

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.     

On the Q n ? Q p deformation energy surfaces of neutron-rich oxygen isotopes

We present an analysis based on the generator coordinate method that allows one to disentangle the neutron and proton contributions to the properties of low-lying collective states in oxygen isotopes. The text was submitted by the authors in English.     

The connection between the generator coordinate method and Bose expansions

It is shown that the generator coordinate method leads to a Bose expansion which in lowest order agrees with the RPA, and which for infinite order is equivalent to the Marumori expansion.     

On the connection between the hyperspherical harmonic and generator coordinate methods for monopole vibrations

It is shown that, when the generating functions are Slater determinants of harmonic oscillator wave functions, the generator coordinate method for monopole vibrations is identical to the first approximation of the hyperspherical harmonic method.     

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.     

Maximum overlap,atomic coherent states and the generator coordinate method

A version of the maximum overlap approach is presented, based on the generator coordinate method. Recourse is made to techniques developed with reference to atomic coherent states. Shape instabilities in quasi-spin two-level models are studied.     

COMPARISON BETWEEN THE BOHR-MOTTELSON MODEL AND THE INTERACTING BOSON MODEL

The comparison between BMM and IBM is comprehensively discussed in the spirit of the transformation theory by using the generator coordinate method (GCM). Some differences existed between these two essentially equivalent models are pointed out on the phenomenological level, and the possibility of distinguishing them is briefly discussed.