G. Do Dang

Reaction paths and generalized valley approximation

The generalized valley approximation has been developed as a method of approximately decoupling one or a few low‐frequency nonlinear modes from the remaining higher frequency modes of a multiparticle system. This decoupling will be best when the difference in frequencies is large; this is the case of adiabatic motion. We describe the application of this method to chemical reactions, relying in some measure on our earlier work, and contrast it with reaction‐path theories. We give an algorithm for the incorporation of our method in a chemical calculation of the Born–Oppenheimer type. Detailed calculations are reported for several standard models that couple a double well to a harmonic oscillator. The decoupling procedure leads to an effective or renormalized one‐dimensional double‐well problem. The energy splitting of the lowest doublet in this well is contrasted with the exact splitting obtained by numerical integration of the two‐dimensional Schrodinger equation. Results are good when the adiabatic condit...

Classical theory of collective motion in the large amplitude, small velocity regime

Abstract A classical theory of collective motion is developed for the large amplitude, small velocity limit, i.e., for a hamiltonian that is at most quadratic in the momenta, allowance being made for a mass tensor that is a general function of the coordinates. It is based on the identification of decoupled motions that are confined to submanifolds of the full configuration space. Conditions for decoupling are derived and then transformed into several different sets of equivalent conditions, more useful for practical applications. Algorithms are given for constructing manifolds that are exactly decoupled if a given dynamical system admits such motions and that can be utilized as well when there is approximate decoupling, as evidenced by criteria that are established. Some examples are worked out. The connection to previous research on this problem is described.

GROUND-STATE CORRELATIONS AND RESTORATION OF BROKEN SYMMETRY TO NUCLEAR MEAN FIELD-THEORY

Abstract We reconsider the long-standing problem that the ground-state correlations predicted by the standard quasiboson (random phase) approximation are too large by a factor of two in the limit of weak residual interaction, and are thus inconsistent with the Pauli exclusion principle. We solve this problem by noting that for the derivation of the equations of the random phase approximation (RPA), which determine a set of matrix elements and excitation energies, it is unnecessary to assume that the ground state is the vacuum of a set of boson excitations, whereas in the faculty calculation of the ground-state correlation energy, this assumption is made. For the study of excitation energies and the restoration of symmetries broken by an initial mean-field solution, we apply a symmetry-preserving form of the equations of motion for fermion particle-hole excitation operators rather than for bosons. We illustrate this for the case of translations. We then describe a separate calculation of the ground-state energy that is consistent with the Pauli principle, but can nevertheless be evaluated in terms of the solutions of the RPA equations of motion. In contrast to some earlier work, we treat direct and exchange terms on an equal footing throughout.

Relation between the local harmonic formulation and the generalized valley formulation of the exact decoupling conditions in the adiabatic theory of large amplitude collective motion

Abstract In recent work, partially recapitulated, we have shown how the conditions for the existence of exactly decoupled classical motion in the adiabatic limit can be transformed into a constructive procedure, designated the generalized valley method, for the determination of the submanifold on which the motion occurs. We prove, for any number of decoupled coordinates, that where the conditions for exact decoupling in the adiabatic limit are satisfied, this formulation is equivalent to the local harmonic one which is widely discussed in the literature. The two formulations are however equivalent, in the general case where the decoupling is not exact, only for the one-dimensional submanifold. An alternative form of the local harmonic formulation, which does not utilize the metric properties associated with the mass tensor, and can therefore be extended to a general hamiltonian, is also discussed briefly.

Uniqueness of the collective submanifold in the adiabatic theory of large amplitude collective motion

Abstract It is emphasized that the conditions for the existence of a collective submanifold which follow from adiabatic time-dependent Hartree-Fock theory are precisely the conditions for the existence of a manifold of solutions of Hamiltons equations confined to a surface of reduced dimensionality. A constructive procedure, valid in any number of dimensions and involving the concept of the multidimensional valley, is developed to determine whether a given system admits such a manifold. It is extended to include the idea of the approximate manifold, and an application to a generalized landscape model is described.

Heating and fluctuations in elastoplastic systems

Abstract Starting from the equations of motion of a simple system possessing the properties of elastic and plastic bodies, we construct its Lagrangian and Hamiltonian functions and also the Rayleigh dissipation function. This allows us to find the rate of heating of the system and to analyze the fluctuations of basic observables. Introducing into the Hamilton-Rayleigh equation of motion a random force producing on average the same effects as a dissipation function, we arrive first at the Langevin equations describing the fluctuations and then at a kinetic equation for the distribution function defined in the space of the collective variables. In this way a rather general scheme is established for solving dynamical problems in different and more complex elastoplastic systems, in nuclear physics and maybe even in physics of molecules and atomic clusters. In a preliminary study, the model is applied to estimate the probability of the quasi-fission process coming from the thermal fluctuations of the nuclear shape.

Relativistic effects for the spin—orbit magnetic moment and in the dipole sum rule

Abstract Using the non-relativistic expansion of the meson exchange potentials, we examine the role of relativistic effects for the isoscalar magnetic moment and in the dipole sum rule. Cancellation coming from ω -exchange is established in the first case while genuine effects remain in the second.

Stochastic aspects of large amplitude collective motion

Abstract We consider externally driven many-body systems with complex spectra of intrinsic states. The effects of the coupling to the external world are analyzed by assuming time-dependent random matrix properties for the intrinsic system. We derive and solve evolution equations for intrinsic state population probabilities, average excitation energy and its fluctuations. The diffusive process is likely to be dominated by memory effects and an unexpected type of fluctuations.

QUANTUM CORRECTIONS TO THE CRANKING MODEL

A method is described for the restoration of translation or rotation symmetry to a system of fermions, starting from a self-consistent cranking solution and valid when either momentum or angular momentum is large enough so that semi-classical approximations are valid. The quantum fluctuations that restore the broken symmetry are described in terms of the particle-hole degrees of freedom of the original system rather than by mapping these variables onto a boson space, as in most previous work. Only the leading quantum correction to the mean field solution is worked out in detail. New results include the treatment of direct and exchange effects on an equal footing and a method for computing transition rates.

Coupling Between Slow and Fast Degrees of Freedom in Systems with Complex Spectra

Due to space limitation we shall skip the presentation of the motivation for our study, the historical perspective, a rather large number of relevant references to earlier work along with a significant amount of details of our approach. We refer the reader to Ref. 1 and references therein for additional information. We assume that the Hamiltonian describing the intrinsic system has the following structure