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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Generalized variational principle for the time-dependent Hartree-Fock equations for a Slater determinant

Hans-Jörg Kull, Dieter Pfirsch

Published May 1, 2000• 19 citations
Publisher

Abstract

The time-dependent Hartree-Fock equations are derived from a variational principle for the general N-body action of a Slater determinant of single-electron orbitals. The variational principle generalizes commonly used variational treatments based on reduced two-body actions. The self-consistent field equations are found to contain time-dependent corrections to the standard mean-field interactions. Their physical significance is discussed and a time-dependent phase shift to the Slater determinant is obtained that properly accounts for the total interaction energy in the mean-field approach.

DOI
10.1103/physreve.61.5940

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The time-dependent Hartree-Fock equations are derived from a variational principle for the general N-body action of a Slater determinant of single-electron orbitals. The variational principle generalizes commonly used variational treatments based on reduced two-body actions. The self-consistent field equations are found to contain time-dependent corrections to the standard mean-field interactions. Their physical significance is discussed and a time-dependent phase shift to the Slater determinant is obtained that properly accounts for the total interaction energy in the mean-field approach.
A variational principle is formulated to determine the single-particle states, their pairing, and the occupation number distribution for a trial state vector of the Bardeen, Cooper, Schrieffer type. The equations which are derived generalize those of the Hartree-Fock method obtained with a Slater determinant trial wave function. It is pointed out that in a suitable representation the vacuum state of a general quasi-particle transformation has such a trial form which exhibits directly the pairing of single-particle states. Another variational principle determines the excitation energies.Two coupling cases are distinguished: the commutative case in which the self-consistent densities and energies are related to quantities which all commute, and the more general noncommutative case. The latter is of importance in critical-field phenomena. The equations for the commutative case can be written in a matrix form which retains its validity in the more general noncommutative case. The simple matrix commutator equations appear as direct generalizations of the density matrix form of the Hartree-Fock equations.The equations for small oscillations have an equally simple form. Their connection with a diagonal representation of the quasi-particle energies is exhibited in a way which remains valid in the general coupling case. The "unphysical" solutions are excluded by the supplementary condition. The contact with the Green's function approach is established. The generalized matrix form of the Green's function equations shows especially clearly the symmetry properties of the method.
We propose a Time-Dependent Multi-Determinant (TDMD) approach to the description of the time evolution of the nuclear wave functions. We use the Dirac variational principle to derive the equations of motion using as ansatz for the nuclear wave function a linear combination of Slater determinants. We prove explicitly that the norm and the energy of the wave function are conserved during the time evolution. This approach is a generalization of the time-dependent Hartree–Fock method to many Slater determinants. We apply this approach to a case study of 6 Li using the N3LO interaction renormalized to four major harmonic oscillator shells. We solve the TDMD equations of motion using Krylov subspace methods of Lanczos type. As an application, we discuss the isoscalar monopole strength function.
A time-dependent variational procedure is proposed that possesses the same constants of the motion as the exact many-body Schr\"odinger dynamics. The class of trial wave functions is larger than the manifold of Slater determinants that supports the time-dependent Hartree-Fock dynamics. These wave functions can be regarded as superpositions of the eigenfunctions of the conserved observable of interest and the variational equations display the usual parametric structure, with properly admixed energy gradients and the symplectic metric tensor. In an application to the Lipkin-Meshkov-Glick model, significant improvements over the usual mean-field or determinantal dynamics can be achieved.NUCLEAR STRUCTURE Mean field symmetry breaking; symmetry restoration; nondeterminantal wave function; time-dependent variational principle; parametric equations of motion; mean energy metric tensor; canonical coordinates; quasispin models; comparison with time-dependent Hartree-Fock dynamics.
The variational derivation of the time-dependent Hartree-Fock equation is reviewed. When norm-violating variations are included, a unique time-dependent Hartree-Fock Hamiltonian, which differs from that customarily used in time-dependent Hartree-Fock analyses, is implied. This variationally "true" Hartree-Fock Hamiltonian has the same expectation value as the exact Hamiltonian, equal to the average energy of the system. Since this quantity remains constant under time-dependent Hartree-Fock time evolution, we suggest the label "constant 〈H〉〈H〉〈H〉" for this form of time-dependent Hartree-Fock theory.NUCLEAR REACTIONS Unique time-dependent Hartree-Fock Hamiltonian obtained from variational principle by norm-violating variations.
This article examines the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find the TDHF approximation to be accurate when there are sufficiently many particles and the initial many-particle state is a Slater determinant, or any Gibbs equilibrium state for noninteracting fermions. Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the the TDHF approximation vanishes at all times in the mean field limit.