An efficient time-splitting method for the ehrenfest dynamics

The Ehrenfest dynamics, representing a quantum-classical mean-field type coupling, is a widely used approximation in quantum molecular dynamics. In this paper, we propose a time-splitting method for an Ehrenfest dynamics, in the form of a nonlinearly coupled Schr ̈odinger-Liouville system. We prove that our splitting scheme is stable uniformly with respect to the semiclassical parameter, and, moreover, that it preserves a discrete semiclassical limit.Thus one can accurately compute physical observables using time steps induced only by the classical Liouville equation, i.e., independent of the small semiclassical parameter - in addition to classical mesh sizes for the Liouville equation. Numerical examples illustrate the validity of our meshing strategy

Additional file 1: of Impact of communities, health, and emotional-related factors on smoking use: comparison of joint modeling of mean and dispersion and Bayes’ hierarchical models on add health survey

Appendix: SAS Code Used to Perform Models. (DOCX 150 kb

Summary statistics of the sample.

<p>Summary statistics of the sample.</p

Results for three-level hierarchical models and joint model.

<p>Results for three-level hierarchical models and joint model.</p

XRD patterns of Fe-precipitates formed at 48 h sealed (a) and unsealed reaction solutions (b).

<p>XRD patterns of Fe-precipitates formed at 48 h sealed (a) and unsealed reaction solutions (b).</p

Changes in the concentrations of DO, Fe²⁺, Fe³⁺, and Fe-precipitates during the jarosite formation process with limiting oxygen supply (■sealed at 0 h, Δsealed at 48 h, ○unsealed).

<p>Changes in the concentrations of DO, Fe²⁺, Fe³⁺, and Fe-precipitates during the jarosite formation process with limiting oxygen supply (■sealed at 0 h, Δsealed at 48 h, ○unsealed).</p

Changes in the concentrations of DO, Fe²⁺, Fe³⁺, and Fe-precipitates during the jarosite formation process with reaction solution loading volume percentages ranging from 20% to 70% (■70% of loading volume, ○40% of loading volume, Δ20% of loading volume).

<p>Changes in the concentrations of DO, Fe²⁺, Fe³⁺, and Fe-precipitates during the jarosite formation process with reaction solution loading volume percentages ranging from 20% to 70% (■70% of loading volume, ○40% of loading volume, Δ20% of loading volume).</p

SEM images of Fe-precipitates formed in the solutions with the loading volume percentage of 20% (a) and 70% (b).

<p>SEM images of Fe-precipitates formed in the solutions with the loading volume percentage of 20% (a) and 70% (b).</p

EDS analysis of Fe-precipitates formed at 48 h sealed and unsealed reaction solutions.

<p>EDS analysis of Fe-precipitates formed at 48 h sealed and unsealed reaction solutions.</p