On The Singular Limit Of The Quantum Classical Molecular Dynamics Model
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Author | : Folkmar A. Bornemann |
Publisher | : |
Total Pages | : 19 |
Release | : 1997 |
Genre | : Molecular dynamics |
ISBN | : |
Download On the Singular Limit of the Quantum Classical Molecular Dynamics Model Book in PDF, Epub and Kindle
Abstract: "In molecular dynamics applications there is a growing interest in so-called 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 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 quantum adiabatic theorem. The proof uses the method of 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 funnel consisting of infinitely many trajectories."
Author | : Christian Lubich |
Publisher | : European Mathematical Society |
Total Pages | : 164 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 9783037190678 |
Download From Quantum to Classical Molecular Dynamics Book in PDF, Epub and Kindle
Quantum dynamics of molecules poses a variety of computational challenges that are presently at the forefront of research efforts in numerical analysis in a number of application areas: high-dimensional partial differential equations, multiple scales, highly oscillatory solutions, and geometric structures such as symplecticity and reversibility that are favourably preserved in discretizations. This text addresses such problems in quantum mechanics from the viewpoint of numerical analysis, illustrating them to a large extent on intermediate models between the Schrodinger equation of full many-body quantum dynamics and the Newtonian equations of classical molecular dynamics. The fruitful interplay between quantum dynamics and numerical analysis is emphasized.
Author | : Folkmar A. Bornemann |
Publisher | : |
Total Pages | : 0 |
Release | : 1995 |
Genre | : Molecular dynamics |
ISBN | : |
Download Quantum-classical Molecular Dynamics as an Approximation to Full Quantum Dynamics Book in PDF, Epub and Kindle
Abstract: "This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a partial classical limit of the full Schrödinger equation. This limit is achieved in two steps: separation of the full wavefunction and short wave asymptotics for its 'classical' part. Both steps can be rigorously justified under certain smallness assumptions. Moreover, the results imply that neither the time-dependent self-consistent field method nor mixed quantum-semi-classical models lead to better approximations than QCMD since they depend on the separation step, too. On the other hand, the theory leads to a characterization of the critical situations in which the models are in danger of largely deviating from the solution of the full Schrödinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of an Argon atom with a harmonic quantum oscillator."
Author | : Christof Schütte |
Publisher | : |
Total Pages | : 15 |
Release | : 1997 |
Genre | : Mechanics |
ISBN | : |
Download Approximation Properties and Limits of the Quantum Classical Molecular Dynamics Model Book in PDF, Epub and Kindle
Abstract: "In molecular dynamics applications there is a growing interest in including quantum effects for simulations of larger molecules. This paper is concerned with 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."
Author | : Peter Nettesheim |
Publisher | : |
Total Pages | : 15 |
Release | : 1998 |
Genre | : Molecular dynamics |
ISBN | : |
Download Second Order Transitions in Quantum Classical Molecular Dynamics Book in PDF, Epub and Kindle
Abstract: "Mixed quantum-classical models have attracted considerable interest due to the expectation that they correctly describe non-adiabatic processes of full quantum dynamics. One of these models, the so-called QCMD model, represents most degrees of freedom of the molecular system by the means of classical mechanics but an important, small portion of the system is modeled by a quantum wavefunction: the wavefunction is nonlinearly coupled to the classical motion via a singularly perturbed Schrödinger equation. In extension to the analysis given by F.A. Bornemann [Homogenization in Time of Singularly Perturbed Mechanical Systems, Lecture Notes in Mathematics, no. 1687, 1998, Springer, Berlin], the article presents an asymptotic expansion up to second order in the perturbation parameter. This result allows for the construction of new models and numerical integration schemes."
Author | : Karen Drukker |
Publisher | : |
Total Pages | : 156 |
Release | : 1998 |
Genre | : Atomic structure |
ISBN | : |
Download Molecular dynamics of mixed quantum-classical systems Book in PDF, Epub and Kindle
Author | : Christof Schütte |
Publisher | : |
Total Pages | : 15 |
Release | : 1998 |
Genre | : Born-Oppenheimer approximation |
ISBN | : |
Download Non-adiabatic Effects in Quantum Classical Molecular Dynamic Book in PDF, Epub and Kindle
Abstract: "In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The article is concerned with the so-called QCMD model. This model describes most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. We review the conditions under which the QCMD model is known to approximate the full quantum dynamical evolution of the system. In most quantum-classical simulations the Born-Oppenheimer model (BO) is used. In this model, the wavefunction is adiabatically coupled to the classical motion which leads to serious approximation deficiencies with respect to non-adiabatic effects in the fully quantum dynamical description of the system. In contrast to the BO model, the QCMD model does include non-adiabatic processes, e.g., transitions between the energy levels of the quantum system. It is demonstrated that, in mildly non-adiabatic scenarios, so-called surface hopping extensions of QCMD simulations yield good approximations of the non-adiabatic effects in full quantum dynamics. The algorithmic strategy of such extensions of QCMD is explained and the crucial steps of its realization are discussed with special emphasis on the numerical problems caused by highly oscillatory phase effects."
Author | : Christof Schütte |
Publisher | : |
Total Pages | : 19 |
Release | : 2000 |
Genre | : |
ISBN | : |
Download Non-adiabatic Effects in Quantum-classical Molecular Dynamics Book in PDF, Epub and Kindle
Author | : Frerich Keil |
Publisher | : Springer Science & Business Media |
Total Pages | : 457 |
Release | : 2012-12-06 |
Genre | : Science |
ISBN | : 3642601855 |
Download Scientific Computing in Chemical Engineering II Book in PDF, Epub and Kindle
The application of modern methods in numerical mathematics on problems in chemical engineering is essential for designing, analyzing and running chemical processes and even entire plants. Scientific Computing in Chemical Engineering II gives the state of the art from the point of view of numerical mathematicians as well as that of engineers. The present volume as part of a two-volume edition covers topics such as the simulation of reactive flows, reaction engineering, reaction diffusion problems, and molecular properties. The volume is aimed at scientists, practitioners and graduate students in chemical engineering, industrial engineering and numerical mathematics.
Author | : Ali M. Nassimi |
Publisher | : |
Total Pages | : 260 |
Release | : 2011 |
Genre | : |
ISBN | : |
Download A Study of Quantum-classical Dynamics in the Mapping Basis Book in PDF, Epub and Kindle