Collective Hamiltonian in the Generator Coordinate Method with Local Gaussian Approximation

Abstract

A collective Hamiltonian is derived by using the local Gaussian overlap approximation in the generator coordinate method. Three Euler angles {)) (B¢<P), the gauge angle x, the gap energy .:1 and the deformation parameters /3 and r are employed as generator coordinates. The metric tensor g., (~) and the second-moment tensor of energy a.,(~) are calculated by introducing infinitesimal operators for the BCS-type trial wave function. The moment:; of inertia are obtained for an axially asymmetric rotor. § I. lntroduetion The generator coordinate method 11 is based on the variational principle. The trial wave function is characterized by a set of parameters called generator coot-elinates and the basic wave functions with different values of the parameter are not orthogonal. This non-orthogonality sometimes causes serious difficulties 2 ' in solving the Hill-Wheeler integral equation. The narrowing kernel, 31 a transformation kerne 1 from the non-orthogonal space to an orthogonal space, can remove the difficulties. An idea has been proposed to use an approximate narrowing kernel which is obtained by assuming that the overlap kernel behaves like a Gaussian function locally. 41 In order to extend the assumption to a many-dimensional parameter space, the metric tensor is introduced," by which a curvilinear space is defined. The assumption of the local Gaussian overlap leads to an approximate narrowing kernel through which a collective Hamiltonian is derived from the Hill-Wheeler integral equation. The main purpose of tlw present paper is to cle\'elop the local Gaussian approximation to treat nuclear collective motions realistically. In order to describe the rotational motion Euler angles li.J (Oif;<jJ) are employed. 01 The deformation parameters /3 and r are used for the shape \'ibrational motion. By treating these five generator coordinates simultaneously, we can fully take into account the rotation-vibration coupling which is strong in soft nuclei. The BCS-type wayc function is used to construct the trial \\"aYe function and the pairing \'ihration is taken

Gaussian overlap approximation for the quadrupole collective states

“…They resemble the Yoccoz moment of inertia [38,39] and are in form similar to those of a rigid body [31]. Being a quadrupole scalar, the collective potential depends obviously on deformation d only.…”

“…It allows us to reduce the integral equation to a differential one. The Hamiltonian of the rigid rotor was derived earlier using the same method [31]. We compare the method of the Fourier analysis with the method used usually in deriving the collective Hamiltonian from the GCM [12,20,19].…”

Symmetry restoration in mean-field approaches

“…It contains a potential energy V (q) = q| Ĥ|q , a kinetic term with microscopically derived inertia parameters and zero-point corrections (for details see (Libert et al, 1999)). In the case of three dimensional angular momentum projection with the Euler angles one ends up with the Bohr-Hamiltonian for a rigid rotor in these variables (Une et al, 1976), where angular momentum is automatically preserved. From the generator coordinate method ansatz for the parameters β and γ for quadrupole deformations one finds in this approximation the Bohr-Hamiltonian for collective β-and γ-vibrations.…”

Density functional theory with spatial-symmetry breaking and configuration mixing

“…An alternate formalism uses a collective "Bohr" Hamiltonian [35][36][37][38][39][40][41][42], initially developed as a model for a quantum vibrating liquid droplet, as an alternative or approximation to the Hill-Wheeler equations of the GCM. The mean-field energy landscape in the space of deformation parameters is then used as a potential (with or without zero-point energy corrections) and mass parameters determined from the Slater determinants enter a corresponding kinetic operator.…”

“…Let me stress the latter: the cranking formula used above is by no means assumed correct by itself. It could be replaced by a term derived using adiabatic time-dependent Hartree-Fock (ATDHF) [37,41,65,66] or the Gaussian overlap approximation to the GCM (GCM-GOA) [26,[38][39][40]. Even then, such terms rely entirely on the single-particle orbitals at each position in the collective space and cannot be expected to correctly reproduce the physics of the underlying correlated many-body state.…”