Gaussian Overlap Approximation Research Articles

The generator coordinate method, path integrals, and quantized time-dependent mean field motion

The two parameter generator coordinate (GCM) method is used to derive the equation of motion for time-dependent mean fields associated to large amplitude vibrations. To achieve this a semclassical assumption and the gaussian overlap approximation is introduced into the GCM before the variational principle is applied. The result consists in time-dependent Hartree-Fock equations with periodic boundary conditions and a semiclassical quantization rule to select those TDHF orbits which correspond to the stationary states. The prescription turns out to be identical to the one obtained recently by path integral techniques and gauge invariant quantization.

Collective Hamiltonians in the generator coordinate method: A numerical procedure

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 $^{4}\mathrm{He}$ nucleus.NUCLEAR STRUCTURE Numerical derivation of collective Hamiltonians. Generator coordinate method, Weyl procedure. Comparison with analytical results.

Coherent-State Representation of Many-Fermion Quantum Mechanics

An exact coherent-state representation is given for fermion systems which avoids the Gaussian overlap approximation in the reduction of the Hill-Wheeler-Griffin integral equation to a differential equation. An application shows that the random-phase-approximation ground-state correlation energy, derived with use of the Gaussian overlap approximation, is a factor of 2 too large.

A self-consistent microscopic approach to the 12C+ 12C reaction at low energy

A completely microscopic approach based on the full Hartree-Fock-Bogolyubov procedure and 'the gaussian overlap approximation of the generator coordinate method is applied to the 12C+ 12C reaction at low energy. No accurate reproduction of the experimental results is looked for. The aim of this work is rather to obtain the general behaviour of the quantities of interest by using an approach as free as possible from phenomenological hypotheses.

The kinematics of generator co-ordinates

The generator co-ordinate approach is used to construct a collective subspace of the full many-body Hilbert space. The construction has a purely kinematical character, given the apriori (e.g. phenomenological) specification of a set of generator many-body states. It is based on the analysis of the properties of the overlaps of the generator states and on the use of the standard spaces of square-integrable functions of quantum mechanics. Some well-known misbehaviours of the generator co-ordinate weight functions are clearly identified as of kinematical origin. A standard representation in the collective phase space is introduced which eliminates them. It is also indicated how appropriate collective dynamical variables can be defined aposteriori. Hilbert-Schmidt overlap kernels and the Gaussian overlap approximation are treated as special examples.

Systematic relations between collective energies from the GCM and RPA methods

It has previously been found numerically that GCM and RPA methods using the same forces can give almost the same collective energies. We point out that there are systematic reasons for this. When the GCM approach is used with the usual Gaussian Overlap Approximation, the energy has a lower limit in terms of RPA energies. If oscillator properties are assigned to the orbitals, the inequalities are converted into equalities. It is shown that some formulations of the GCM approach imply violation of sum-rules, so should be used with caution.

Collective Hamiltonian in the Generator Coordinate Method with Local Gaussian Approximation

A collective Hamiltonian is derived by using the local Gaussian overlap approximation in the generator coordinate method. Three Euler angles ω(θφψ), the gauge angle χ, the gap energy Δ and the deformation parameters β and γ are employed as generator coordinates. The metric tensor gµν(ξ) and the second-moment tensor of energy aµν(ξ) are calculated by introducing infinitesimal operators for the BCS-type trial wave function. The moments of inertia are obtained for an axially asymmetric rotor.

On the Rotation of Non-Axial nuclei

The rotational properties of non-axial nuclei are studied by means of the generator coordinate method, and in particular it is shown that using the angular momentum projection technique and assuming the Gaussian overlap approximation for the Hamiltonian and overlap matrix elements, the Hill-Wheeler integral equation may be transformed into an equivalent Schrodinger equation for a non-axial rotator.

An extension of Wong's theory for complex generator coordinates

An extension of Wong's generator coordinate theory is presented in terms of a reciprocal basis. The new formalism permits the derivation of explicit conditions under which the solutions of the variational equation of Griffin, Hill and Wheeler are entire functions, or even elements of Bargmann's Hubert space. The well known extension to complex parameters in the Gaussian overlap approximation then appears as a special case.

Mass parameter in nuclear quadrupole motion

The self-cranked generator coordinate (SCGC) formalism described earlier for computing the mass parameter for collective nuclear motion is cast into a form containing no explicit reference to the Hamiltonian. The expression is then specialized to the nuclear model of deformed harmonic oscillators. The usual Gaussian overlap approximation is eschewed in favor of direct evaluation of the appropriate matrix elements. Exchange terms are handled by a diagrammatic technique. The validity of certain assumptions made in the derivation of the mass formula is tested numerically. The effects of pairing and of short-range Jastrow correlations are investigated. The SCGC mass parameter is computed for several $N=Z$ nuclei and found to be smaller than either the cranking or the irrotational values, if no correlations are included. The inclusion of short-range correlations is shown to lead to important changes in the value of the mass parameter.NUCLEAR STRUCTURE Quadrupole mass parameter calculated. Self-cranked generator coordinate formalism. Correlations included. Compared to irrotational flow values.

On the time evolution of generator coordinates

The parameters of the generator coordinate theory are allowed to be complex and time-dependent. Their time-dependences are found by a variational principle which automatically yields a normalisation and energy which are independent of time. The parameters satisfy equations of motion analogous to Hamilton's equations of classical mechanics. This analogy is exact if the overlap function is gaussian. Combinations of the parameters may be found which form conjugate pairs; a procedure which may be useful in defining appropriate variables for the construction of a collective hamiltonian. Conditions are derived under which the parameters may be chosen to be real. It is shown that for a cranking model wavefunction the collective parameters may be made real if the collective hamiltonian is harmonic. Reality of the parameters with a general hamiltonian, however, is shown to require a restricted set of generator coordinate spaces which embraces the gaussian overlap approximation.

A new particle-hole approach to collective states

The relevance of the particle-hole space is demonstrated by showing that in some commonly used formalisms the first excited state lies entirely within the particle-hole space generated from the correlated ground state. This property is proved for several cases of angularmomentum projection — projected Hartree-Fock method (PHF) — and for the generator coordinate method in the Gaussian overlap approximation, while in other cases it has been verified only numerically. A new method is presented for the approximate calculation of energies and transition amplitudes of particle-hole excited states. Only hermitian one-body operators are used to generate the excited states. The two-body density matrix of a correlated state approximating the ground state is required as input data. The formula is tested on the 2 + and 3 − states of 8Be and 12C by using the PHF ground state. Where comparison is possible the method gives better agreement with PHF and experiment than the extended random-phase approximation.

Theory of anharmonic collective oscillations

An expression for the anharmonic collective Hamiltonian containing a variable inertia coefficient is derived, on the basis of the generator coordinate method and the Gaussian overlap approximation. The anharmonic model obtained is tested with Lipkin's model. A classical discussion of anharmonic oscillations based on the time-dependent variational principle is also presented and compared with the quantal description.

Corrections to the Gaussian overlap approximation

It is observed that the validity of the Gaussian overlap approximation does not imply the validity of the harmonic approximation. A simple procedure for converting the integral eigenvalue equation of the generator coordinate method into a differential eigenvalue equation is presented. The method proposed assumes the validity of the Gaussian overlap approximation, but may be applied to highly anharmonic situations. A test of the method has been made with the Lipkin model.

Corrections to the gaussian overlap approximation: A new boson expansion

The Gaussian overlap approximation is applied to Lipkin's solvable model, both for a weak and for a strong interaction, in order to obtain insight on the conditions for the validity of that approximation. By carrying the expansions in the generator coordinate to higher orders, corrections to the harmonic approximation are obtained and applied, also with good results, to Lipkin's model.

Gaussian overlap approximation in the projected Hartree–Fock method

AbstractA method for the approximate calculation of matrix elements with respect to projected Hartree–Fock wave functions is proposed. The method is tested on some calculations in the many‐parameter AMO method. It is found that the approximation reduces the amount of work, involved in the evaluation of the energy, by a factor of five and that it reproduces the exact values to within a few per cent.

The generator-coordinate theory as a flexible formulation of the many-body Schrödinger equation

By using generator coordinates as state labels we formulate the generator-coordinate theory of collective motion as non-orthogonal representations of the many-body Schrödinger equation in a subspace of the Hilbert space of many-body state vectors. The weight function of the usual generator-coordinate theory is generalized to become the components of the state vector in one of the two bi-orthogonal representations labelled by generator coordinates. The well-known case of the Gaussian-overlap approximation is studied in order to show how the new formalism also permits a solution of the problem using the original real generator coordinates. The concept of generator coordinates is clarified by studying (i) the connection between the redundancy of generator coordinates and the linear dependence of base vectors labelled by these generator coordinates, and (ii) the construction of states having the required properties under translation and rotation. Finally, the consideration of rotational properties leads to a double-projection method for constructing internal states of good angular momentum and for removing spurious states of c.m. motion.

The generator coordinate theory of collective motion

Collective motion in many-particle systems is studied by using the generator-coordinate method of Griffin, Hill and Wheeler. The equations of the theory are solved in the Gaussian overlap approximation and are shown to be equivalent to the quantal equations of a set of coupled harmonic oscillators. General properties of the solutions are studied, including those resulting from invariance principles and intrinsic symmetries. Similarities with the Tamm-Dancoff approximation and the random phase approximation are pointed out. Normally the generator-coordinate method is formulated in terms of generating functions depending on real parameters. Various difficulties of the “real” approach can be overcome by making an extension to complex parameters. Relations between the real and complex representation are established. To illustrate the general formalism the method is applied to the alpha-particle model of 20Ne.

The generator-coordinate method and the random phase approximation

The equations of generator coordinate method in the Gaussian overlap approximation are solved in a way which suggests a relation with the random phase approximation.