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  • 1
    Digitale Medien
    Digitale Medien
    A mathematical investigation of the Car-Parrinello method (1998)
    Springer
    Numerische Mathematik 78 (1998), S. 359-376 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991): 34C15, 34C40, 70F20, 81Q15, 81V55
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract. The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the quantitative influence of the involved parameter $\mu$ , the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order ${\cal O}(\mu^{1/2})$ – provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain non-degeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050316
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  • 2
    Digitale Medien
    Digitale Medien
    Adaptive accuracy control for Car-Parrinello simulations (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 179-186 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991):65M99, 34C15, 34C40, 70F20, 81Q15, 81V55
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary. The Car-Parrinello (CP) approach to ab initio molecular dynamics serves as an approximation to time-dependent Born-Oppenheimer (BO) calculations. It replaces the explicit minimization of the energy functional by a fictitious Newtonian dynamics and therefore introduces an artificial mass parameter $\mu$ which controls the electronic motion. A recent theoretical investigation shows that the CP-error, i.e., the deviation of the CP–solution from the BO-solution decreases like $\mu^{1/2}$ asymptotically. Since the computational effort increases like $\mu^{-1/2}$ , the choice of $\mu$ has to find a compromise between efficiency and accuracy. The asymptotical result is used in this paper to construct an easily implemented algorithm which automatically controls $\mu$ : the parameter $\mu$ is repeatedly adapted during the simulation by choosing $\mu$ as large as possible while pushing an error measure below a user-given tolerance. The performance and reliability of the algorithm is illustrated by a typical example.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050445
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  • 3
    Digitale Medien
    Digitale Medien
    A theoretical model for molecules interacting with intense laser pulses: The Floquet-based quantum-classical Liouville equation (2001)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 115 (2001), S. 5733-5743 
    Details
    ISSN: 1089-7690
    Quelle: AIP Digital Archive
    Thema: Physik , Chemie und Pharmazie
    Notizen: The Floquet-based quantum-classical Liouville equation (F-QCLE) is presented as a novel theoretical model for the interaction of molecules with intense laser pulses. This equation efficiently combines the following two approaches: First, a small but spectroscopically relevant part of the molecule is treated quantum-mechanically while the remaining degrees of freedom are modeled by means of classical molecular dynamics. The corresponding nonadiabatic dynamics is given by the quantum-classical Liouville equation which is a first-order approximation to the partial Wigner transform of full quantum dynamics. Second, the dynamics of the quantum subsystem is described in terms of instantaneous Floquet states thus eliminating highly oscillatory terms from the equations of motion. The resulting F-QCLE is shown to have a well defined adiabatic limit: For infinitely heavy classical particles and for infinitely slow modulation the dynamics adiabatically follows the Floquet quasi-energy surfaces for a strictly time-periodic field. Otherwise, nonadiabtic effects arise both from the motion of the classical particles and from the modulation of the field which is assumed to be much slower than the carrier frequency. A numerical scheme to solve the F-QCLE is based on a Trotter splitting of the time evolution. The simplest implementation can be realized by an ensemble of trajectories stochastically hopping between different Floquet surfaces. As a first application we demonstrate the excellent agreement of quantum-classical and fully quantum-mechanical dynamics for a two-state model of photodissociation of molecular fluorine. In summary, due to the favorable scaling of the numerical effort the F-QCLE provides an efficient tool for the simulation of medium to large molecules interacting with intense fields beyond the perturbative regime. © 2001 American Institute of Physics.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.1398577
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  • 4
    Digitale Medien
    Digitale Medien
    Quantum-classical molecular dynamics as an approximation to full quantum dynamics (1996)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 105 (1996), S. 1074-1083 
    Details
    ISSN: 1089-7690
    Quelle: AIP Digital Archive
    Thema: Physik , Chemie und Pharmazie
    Notizen: 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 wave function and short wave asymptotics for its "classical'' part. Both steps can be rigorously justified under the same smallness assumptions. This throws some light on the time-dependent self-consistent-field method and on mixed quantum-semiclassical models, which also depend on the separation step. On the other hand, the theory leads to a characterization of the critical situations in which the QCMD model is in danger of largely deviating from the solution of full Schrödinger equation. These critical situations are exemplified in an illustrative numerical simulation: the collinear collision of a classical particle with a harmonic quantum oscillator. © 1996 American Institute of Physics.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.471952
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  • 5
    Digitale Medien
    Digitale Medien
    Hybrid Monte Carlo with adaptive temperature in mixed-canonical ensemble: Efficient conformational analysis of RNA (1998)
    New York, NY [u.a.] : Wiley-Blackwell
    Journal of Computational Chemistry 19 (1998), S. 1689-1697 
    Details
    ISSN: 0192-8651
    Schlagwort(e): hybrid Monte Carlo ; generalized ensemble ; reweighting ; n-butane ; triribonucleotide ; Chemistry ; Theoretical, Physical and Computational Chemistry
    Quelle: Wiley InterScience Backfile Collection 1832-2000
    Thema: Chemie und Pharmazie , Informatik
    Notizen: A hybrid Monte Carlo method with adaptive temperature choice is presented that 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 is tested by comparing analytical and numerical results for the small n-butane molecule before simulations are performed for a triribonucleotide. Sampling the complex multiminima energy landscape of this small RNA segment, we observe enforced crossing of energy barriers.   © 1998 John Wiley & Sons, Inc.   J Comput Chem 19: 1689-1697, 1998
    Zusätzliches Material: 8 Ill.
    Materialart: Digitale Medien
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  • 6
    Buch
    Buch
    Metastability and Markov state models in molecular dynamics /; 24 (2013)
    New York, NY :Courant Inst. of Mathematical Sciences [u.a.],
    Details
    Titel: Metastability and Markov state models in molecular dynamics /; 24
    Autor: Schütte, Christof
    Beteiligte Person(en): Sarich, Marco
    Verlag: New York, NY :Courant Inst. of Mathematical Sciences [u.a.],
    Erscheinungsjahr: 2013
    Seiten: IX, 128 S. : , graph. Darst.
    Serie: Courant lecture notes in mathematics 24
    ISBN: 978-0-8218-4359-8
    Materialart: Buch
    Sprache: Englisch
    URL: https://www.zbmath.org/?q=an:06252799
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  • 7
    Buch
    Buch
    ¬A¬ quasiresonant smoothing algorithm for solving large highly oscillatory differential equations from quantum chemistry. Zug. Berlin, Freie Univ., Diss. 1994 (1994)
    Aachen :Shaker,
    Details
    Titel: ¬A¬ quasiresonant smoothing algorithm for solving large highly oscillatory differential equations from quantum chemistry. Zug. Berlin, Freie Univ., Diss. 1994
    Autor: Schütte, Christof
    Verlag: Aachen :Shaker,
    Erscheinungsjahr: 1994
    Seiten: 128 S.
    Serie: Berichte aus der Mathematik
    Materialart: Buch
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  • 8
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    Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of RamanSpectroscopy. (1992)
    Details
    Publikationsdatum: 2014-02-26
    Beschreibung: Models for occupation dynamics in discrete quantum systems lead to large or even infinite systems of ordinary differential equations. Some new mathematical techniques, developed for the simulation of chemical processes, make a numerical solution of countable systems of ordinary differential equations possible. Both, a basic physical concept for the construction of such systems and the structure of the numerical tools for solving them are presented. These conceptual aspects are illustrated by a simulation of an occupation process from spectroscopy. In this example the structures of rotation spectra observed in infrared spectroscopy are explained and some possibilities for an extension of the model are shown.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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  • 9
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    Smoothed Molecular Dynamics for Thermally Embedded Systems (1995)
    Details
    Publikationsdatum: 2014-02-26
    Beschreibung: This paper makes use of statistical mechanics in order to construct effective potentials for Molecular Dynamics for systems with nonstationary thermal embedding. The usual approach requires the computation of a statistical ensemble of trajectories. In the context of the new model the evaluation of only one single trajectory is sufficient for the determination of all interesting quantities, which leads to an enormous reduction of computational effort. This single trajectory is the solution to a corrected Hamiltonian system with a new potential $\tilde{V}$. It turns out that $\tilde{V}$ can be defined as spatial average of the original potential $V$. Therefore, the Hamiltonian dynamics defined by $\tilde{V}$ is smoother than that effected by $V$, i.e. a numerical integration of its evolution in time allows larger stepsizes. Thus, the presented approach introduces a Molecular Dynamics with smoothed trajectories originating from spatial averaging. This is deeply connected to time--averaging in Molecular Dynamics. These two types of {\em smoothed Molecular Dynamics} share advantages (gain in efficiency, reduction of error amplification, increased stability) and problems (necessity of closing relations and adaptive control schemes) which will be explained in detail.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 10
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    Computation of Essential Molecular Dynamics by Subdivision Techniques I: Basic Concept (1996)
    Details
    Publikationsdatum: 2014-02-26
    Beschreibung: The paper presents the concept of a new type of algorithm for the numerical computation of what the authors call the {\em essential dynamics\/} of molecular systems. Mathematically speaking, such systems are described by Hamiltonian differential equations. In the bulk of applications, individual trajectories are of no specific interest. Rather, time averages of physical observables or relaxation times of conformational changes need to be actually computed. In the language of dynamical systems, such information is contained in the natural invariant measure (infinite relaxation time) or in almost invariant sets ("large" finite relaxation times). The paper suggests the direct computation of these objects via eigenmodes of the associated Frobenius-Perron operator by means of a multilevel subdivision algorithm. The advocated approach is different to both Monte-Carlo techniques on the one hand and long term trajectory simulation on the other hand: in our setup long term trajectories are replaced by short term sub-trajectories, Monte-Carlo techniques are just structurally connected via the underlying Frobenius-Perron theory. Numerical experiments with a first version of our suggested algorithm are included to illustrate certain distinguishing properties. A more advanced version of the algorithm will be presented in a second part of this paper.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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