Refine
Year of publication
- 1997 (6) (remove)
Document Type
- ZIB-Report (4)
- Article (2)
Language
- English (6)
Is part of the Bibliography
- no (6)
Institute
- ZIB Allgemein (4)
It was revealed that the QCMD model is of canonical Hamiltonian form with symplectic structure, which implies the conservation of energy. An efficient and reliable integrator for transfering these properties to the discrete solution is the symplectic and explicit {\sc Pickaback} algorithm. The only drawback of this kind of integrator is the small stepsize in time induced by the splitting techniques used to discretize the quantum evolution operator. Recent investigations concerning Krylov iteration techniques result in alternative approaches which overcome this difficulty for a wide range of problems. By using iterative methods in the evaluation of the quantum time propagator, these techniques allow for the stepsize to adapt to the coupling between the classical and the quantum mechanical subsystem. This yields a drastic reduction of the numerical effort. The pros and cons of both approaches as well as the suitable applications are discussed in the last part.
In molecular dynamics applications there is a growing interest in including quantum effects for simulations of larger molecules. This paper is concerned with {\em mixed quantum-classical} models which are currently discussed: the so-called QCMD model with variants and the time-dependent Born-Oppenheimer approximation. All these models are known to approximate the full quantum dynamical evolution---under different assumptions, however. We review the meaning of these assumptions and the scope of the approximation. In particular, we characterize those typical problematic situations where a mixed model might largely deviate from the full quantum evolution. One such situation of specific interest, a non-adiabatic excitation at certain energy level crossings, can promisingly be dealt with by a modification of the QCMD model that we suggest.
\noindent In molecular dynamics applications there is a growing interest in so-called {\em mixed quantum-classical} models. These models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of quantum mechanics. A particularly extensively used model, the QCMD model, consists of a {\em singularly perturbed}\/ Schrödinger equation nonlinearly coupled to a classical Newtonian equation of motion. This paper studies the singular limit of the QCMD model for finite dimensional Hilbert spaces. The main result states that this limit is given by the time-dependent Born-Oppenheimer model of quantum theory---provided the Hamiltonian under consideration has a smooth spectral decomposition. This result is strongly related to the {\em quantum adiabatic theorem}. The proof uses the method of {\em weak convergence} by directly discussing the density matrix instead of the wave functions. This technique avoids the discussion of highly oscillatory phases. On the other hand, the limit of the QCMD model is of a different nature if the spectral decomposition of the Hamiltonian happens not to be smooth. We will present a generic example for which the limit set is not a unique trajectory of a limit dynamical system but rather a {\em funnel} consisting of infinitely many trajectories.
A hybrid Monte Carlo method with adaptive temperature choice is presented, which exactly generates the distribution of a mixed-canonical ensemble composed of two canonical ensembles at low and high temperature. The analysis of resulting Markov chains with the reweighting technique shows an efficient sampling of the canonical distribution at low temperature, whereas the high temperature component facilitates conformational transitions, which allows shorter simulation times. \\The algorithm was tested by comparing analytical and numerical results for the small n-butane molecule before simulations were performed for a triribonucleotide. Sampling the complex multi-minima energy landscape of these small RNA segments, we observed enforced crossing of energy barriers.