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Physical Review C

Time-dependent coupled-cluster approximation to nuclear dynamics. II. General formulation

Pervez Amirali Hoodbhoy, John W. Negele

Published Apr 1, 1979• 38 citations
Publisher

Abstract

A general formulation is presented of the time-dependent coupled-cluster hierarchy of equations introduced previously. Truncations appropriate to potentials with strongly repulsive cores are specified and the properties of the resulting equations of motion are discussed.NUCLEAR STRUCTURE. Time-dependent approximations to nuclear dynamics. Generalized time-dependent Hartree-Fock and Bethe-Goldstone equations.

DOI
10.1103/physrevc.19.1971

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A general formulation is presented of the time-dependent coupled-cluster hierarchy of equations introduced previously. Truncations appropriate to potentials with strongly repulsive cores are specified and the properties of the resulting equations of motion are discussed.NUCLEAR STRUCTURE. Time-dependent approximations to nuclear dynamics. Generalized time-dependent Hartree-Fock and Bethe-Goldstone equations.
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A generalization of many-fermion Feynman-Goldstone diagrams is derived based on a time-dependent unperturbed Hamiltonian along with a time-dependent reference state. In lowest order, the time-dependent Hartree-Fock equations follow immediately from the same arguments which lead to static Hartree-Fock when the basis is time independent. Systematic corrections to the time-dependent Hartree-Fock mean-field approximation are obtained through higher order diagrams.[NUCLEAR STRUCTURE Generalization of Feynman-Goldstone diagrams to a time-dependent reference state. Diagrammatic derivation of the time-dependent Hartree-Fock equations and their systematic corrections.]
From the quantum kinetic equation of Bogoliubov, Gurov, Born, and Green, the equations of motion for macroscopic variables are examined in many different ways. First, in the configuration space a hierarchy of generalized fluid-dynamical equations can be obtained by taking the appropriate limits of the quantum kinetic equation. The resultant equations of continuity, of momentum, and of energy are similar in form to those one encounters in classical fluid dynamics with the exception of additional terms proportional to ${\ensuremath{\hbar}}^{2}$ Secondly, the quantum kinetic equation is examined in phase space. The same set of equations of continuity, of momentum, and of energy can be derived by taking the first three moments of the quantum kinetic equation. The exact results we obtained are utilized to form the starting point for many simplifying approximations for the investigation of the dynamics of quantum many-body systems such as the elastic response limit, the hydrodynamical limit, Landau's Fermi-liquid theory, and finally the time-dependent Hartree-Fock and the multideterminant time-dependent Hartree-Fock approximations. The fact that all these different dynamical descriptions can be traced to a common origin provides a unifying viewpoint to the present approach with the quantum kinetic equation.NUCLEAR STRUCTURE Dynamics of nuclear fluid, quantum kinetic equation, and generalized fluid-dynamical equation. Elastic response, hydrodynamical limit, Landau's Fermi-liquid theory, time-dependent Hartree-Fock approximation, and multideterminant time-dependent Hartree-Fock approximation.
A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator U(tf,ti)U({t}_{f},{t}_{i})U(tf​,ti​). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of UUU is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.
A dynamical formulation of coupled cluster theory is derived using a variational principle. By allowing time-dependent single-particle functions, a high degree of adaptivity is introduced, allowing complex systems to be simulated with high accuracy. Equations of motion are derived which are shown to be suitable for computer implementation. The method, called adaptive time-dependent coupled cluster, is a strict generalization of the formulation used in standard coupled cluster response theory, and also represents a systematic hierarchy of size-consistent approximations, including standard time-dependent Hartree--Fock as a trivial case.