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The Journal of Chemical Physics

Dynamic mean-field models with correlated modes

Benjamin G. Vekhter, Mark A. Ratner, R. Benny Gerber

Published Nov 15, 1993• 17 citations
Publisher

Abstract

We discuss a generalized dynamic mean-field method combining the advantages of explicit pair correlations and of configuration interaction. The approximate dynamical method, which we call time-dependent self-consistent-field configuration interaction (TDSCF2-CI), is constructed by including N(N−1)/2 TDSCF2 configurations. In each configuration a given pair of N coupled modes is directly (not in the mean-field approach) correlated; the N(N−1)/2 configurations include all such choices of pairs. As such, it has both the usual advantages of TDSCF and improvements due to some inclusion of correlations (exact results for any two-mode problem, improved descriptions of dynamical corrections, and greater accuracy). A three-mode model Hamiltonian is analyzed using five approximate treatments, i.e., the usual TDSCF, the three TDSCF2 forms, and the TDSCF2-CI one. The quantities for comparison with the exact results include the decay P(t) of the initial state, the time dependencies of the energies e(i) of individual modes, and the overlap S(t) of the corresponding approximate wave function with the exact one. We find, indeed, that explicit inclusion of pair correlations improves the description of the quantum dynamics of the system.

DOI
10.1063/1.465669

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We discuss a generalized dynamic mean-field method combining the advantages of explicit pair correlations and of configuration interaction. The approximate dynamical method, which we call time-dependent self-consistent-field configuration interaction (TDSCF2-CI), is constructed by including N(N−1)/2 TDSCF2 configurations. In each configuration a given pair of N coupled modes is directly (not in the mean-field approach) correlated; the N(N−1)/2 configurations include all such choices of pairs. As such, it has both the usual advantages of TDSCF and improvements due to some inclusion of correlations (exact results for any two-mode problem, improved descriptions of dynamical corrections, and greater accuracy). A three-mode model Hamiltonian is analyzed using five approximate treatments, i.e., the usual TDSCF, the three TDSCF2 forms, and the TDSCF2-CI one. The quantities for comparison with the exact results include the decay P(t) of the initial state, the time dependencies of the energies e(i) of individual modes, and the overlap S(t) of the corresponding approximate wave function with the exact one. We find, indeed, that explicit inclusion of pair correlations improves the description of the quantum dynamics of the system.
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A temperature-dependent Schrödinger equation is proposed for the study of quantum dynamics in systems which interact with an external bath. The derivation is based on a time-dependent self consistent field (TDSCF) approximation for a system-bath wave function. As previously shown by Miller, this approximation leads to a quantum mechanical analog to the classical generalized Langevin equation. By replacing the time-evolution of the quantum mechanical bath observables with the corresponding classical trajectories, the fluctuating force and the nonlinear friction kernel in the Langevin-Schrödinger equation become temperature-dependent where the fluctuations intensity is proportional to T. Application of the new equation to the spin-boson model shows agreement with numerically exact path integral calculations for T>0. We relate the success of the TDSCF approximation in this case to the bath-induced noise which diminishes the importance of quantum mechanical system-bath correlations.
Typical (sub)system-bath quantum dynamical problems are often investigated by means of (approximate) reduced equations of motion. Wavepacket approaches to the dynamics of the whole system have gained momentum in recent years and there is hope that properly designed approximations to the wavefunction will allow one to correctly describe the subsystem evolution. The continuous-configuration time-dependent self-consistent field (CC-TDSCF) and local coherent-state approximation (LCSA) methods, for instance, use a simple Hartree product of bath single-particle-functions for each discrete variable representation (DVR) state introduced in the Hilbert space of the subsystem. Here we focus on the above two methods and replace the DVR states with the eigenstates of the subsystem Hamiltonian, i.e., we adopt an energy-local representation for the subsystem. We find that stable and semiquantitative results are obtained for a number of dissipative problems, at the same (small) computational cost of the original methods. Furthermore, we find that both methods give very similar results, thus suggesting that coherent-states are well suited to describe (local) bath states. As a whole, present results highlight the importance of the system basis-set in the selected-multiconfiguration expansion of the wavefunction. They suggest that accurate and yet computationally cheap methods may be simply obtained from CC-TDSCF/LCSA by letting the subsystem states be variationally optimized.
We present the concept, derivation, and implementation of dynamical configuration interaction, a quantum embedding theory that combines Green's function methodology with the many-body wave function. In a strongly correlated active space, we use full configuration interaction (CI) to describe static correlation exactly. We add energy-dependent corrections to the CI Hamiltonian which, in principle, include all remaining correlations derived from the bath space surrounding the active space. Next, we replace the exact Hamiltonian in the bath with one of excitations defined over a correlated ground state. This transformation is naturally suited to the methodology of many-body Green's functions. In this space, we use a modified GWGWGW/Bethe-Salpeter equation procedure to calculate excitation energies. Combined with an estimate of the ground-state energy in the bath, we can efficiently compute the energy-dependent corrections, which correlate the full set of orbitals, for very low computational cost. We present dimer dissociation curves for H2{\mathrm{H}}_{2}H2​ and N2{\mathrm{N}}_{2}N2​ in good agreement with exact results. Additionally, excited states of N2{\mathrm{N}}_{2}N2​ and C2{\mathrm{C}}_{2}C2​ are in excellent agreement with benchmark theory and experiment. By combining the strengths of two disciplines, we achieve a balanced description of static and dynamic correlation in a fully ab initio, systematically improvable framework.
A configuration-interaction time-dependent density functional theory (CI-TDDFT) for nuclear dynamics is developed. In this framework, the correlated nuclear many-body wave function is expanded in terms of time-dependent many-particle configurations built from a common set of orthonormal single-particle states. The equations of motion for both the expansion coefficients and the single-particle states are derived self-consistently using the Dirac-Frenkel time-dependent variational principle. This formulation extends conventional time-dependent density functional theory (TDDFT) by incorporating configuration mixing and beyond-mean-field correlations, while preserving energy and particle-number conservation. As an illustrative application, the method is implemented using the relativistic point-coupling functional PC-PK1 in the particle-hole channel and a monopole pairing interaction in the particle-particle channel, and is applied to the study of isoscalar giant monopole resonance in 58^{58}58Ni and 60^{60}60Ni. Numerical tests show that both the total energy and particle number are conserved, with relative deviations within 4×10−44\times 10^{-4}4×10−4 during the time evolution. Compared with conventional TDDFT, CI-TDDFT yields broader strength distributions for giant monopole resonances while keeping the main peak positions close to those from TDDFT. This broadening is associated with configuration mixing in the valence space and suggests a coupling of the monopole oscillation to additional collective degrees of freedom. These results demonstrate the potential of CI-TDDFT as a quantum, microscopic beyond-mean-field framework for nuclear dynamics.
A general hierarchy of approximations to the many-body Schr\"odinger equation is presented which reduces to the time-dependent mean-field approximation in lowest order and provides systematic corrections in subsequent orders. The theory is applied to two interacting systems described by the Lipkin model Hamiltonian. Comparison of the results of the lowest two orders of approximation with the exact solution demonstrates the practicality of the method and its potential for generalizing nuclear dynamics beyond the mean field theory.NUCLEAR STRUCTURE Time-dependent approximation to nuclear dynamics. Corrections to time-dependent mean-field theory. Solution of Lipkin model for two interacting systems.