TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - Adaptive Accuracy Control for Microcanonical Car-Parrinello Simulations N2 - 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 {\em decreases} like $\mu^{1/2}$ asymptotically. Since the computational effort {\em 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. T3 - ZIB-Report - SC-96-20 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2311 ER - TY - GEN A1 - Deuflhard, Peter A1 - Dellnitz, Michael A1 - Junge, Oliver A1 - Schütte, Christof T1 - Computation of Essential Molecular Dynamics by Subdivision Techniques I: Basic Concept N2 - 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. T3 - ZIB-Report - SC-96-45 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2553 ER - TY - GEN A1 - Nettesheim, Peter A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - Chebyshev-Approximation for Wavepacket-Dynamics: better than expected N2 - The aim of this work is to study the accuracy and stability of the Chebyshev--approximation method as a time--discretization for wavepacket dynamics. For this frequently used discretization we introduce estimates of the approximation and round--off error. These estimates mathematically confirm the stability of the Chebyshev--approximation with respect to round--off errors, especially for very large stepsizes. But the results also disclose threads to the stability due to large spatial dimensions. All theoretical statements are illustrated by numerical simulations of an analytically solvable example, the harmonic quantum oszillator. T3 - ZIB-Report - SC-96-47 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2575 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Schütte, Christof T1 - A Mathematical Investigation of the Car-Parrinello Method N2 - 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 {\em 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 ${\rm O}(\mu^{1/2})$ --- 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. Finally, we show how to use the gained quantitative insight for an automatic control of the unphysical ``fake'' kinetic energy of the method. T3 - ZIB-Report - SC-96-19 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2302 ER - TY - GEN A1 - Schütte, Christof A1 - Bornemann, Folkmar A. T1 - Homogenization Approach to Smoothed Molecular Dynamics N2 - {\footnotesize In classical Molecular Dynamics a molecular system is modelled by classical Hamiltonian equations of motion. The potential part of the corresponding energy function of the system includes contributions of several types of atomic interaction. Among these, some interactions represent the bond structure of the molecule. Particularly these interactions lead to extremely stiff potentials which force the solution of the equations of motion to oscillate on a very small time scale. There is a strong need for eliminating the smallest time scales because they are a severe restriction for numerical long-term simulations of macromolecules. This leads to the idea of just freezing the high frequency degrees of freedom (bond stretching and bond angles) via increasing the stiffness of the strong part of the potential to infinity. However, the naive way of doing this via holonomic constraints mistakenly ignores the energy contribution of the fast oscillations. The paper presents a mathematically rigorous discussion of the limit situation of infinite stiffness. It is demonstrated that the average of the limit solution indeed obeys a constrained Hamiltonian system but with a {\em corrected soft potential}. An explicit formula for the additive potential correction is given via a careful inspection of the limit energy of the fast oscillations. Unfortunately, the theory is valid only as long as the system does not run into certain resonances of the fast motions. Behind those resonances, there is no unique limit solution but a kind of choatic scenario for which the notion ``Takens chaos'' was coined. For demonstrating the relevance of this observation for MD, the theory is applied to a realistic, but still simple system: a single butan molecule. The appearance of ``Takens chaos'' in smoothed MD is illustrated and the consequences are discussed.} T3 - ZIB-Report - SC-96-31 Y1 - 1996 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2410 ER - TY - JOUR A1 - Nettesheim, Peter A1 - Bornemann, Folkmar A. A1 - Schmidt, Burkhard A1 - Schütte, Christof T1 - An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics JF - Chem. Phys. Lett. Y1 - 1996 UR - http://publications.imp.fu-berlin.de/101/ U6 - https://doi.org/10.1016/0009-2614(96)00471-X VL - 256 IS - 6 SP - 581 EP - 588 ER - TY - JOUR A1 - Bornemann, Folkmar A. A1 - Nettesheim, Peter A1 - Schütte, Christof T1 - Quantum-classical molecular dynamics as an approximation to full quantum dynamics JF - J. Chem. Phys. Y1 - 1996 UR - http://publications.imp.fu-berlin.de/103/ U6 - https://doi.org/10.1063/1.471952 VL - 105 IS - 3 SP - 1074 EP - 1083 ER - TY - JOUR A1 - Nettesheim, Peter A1 - Huisinga, Wilhelm A1 - Schütte, Christof T1 - Chebyshev-Approximation for Wavepacket-Dynamics JF - preprint Y1 - 1996 UR - http://publications.imp.fu-berlin.de/102/ ER -