TY - GEN A1 - Schütte, Christof A1 - Nettesheim, Peter T1 - Non-Adiabatic Effects in Quantum-Classical Molecular Dynamics N2 - 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 {\em 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 {\em 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. T3 - ZIB-Report - SC-98-38 KW - quantum-classical molecular dynamics KW - non-adiabatic processes KW - Schrödinger equation KW - highly oscillatory phase KW - adiabatic limit KW - quantum adiabati Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3817 ER - TY - GEN A1 - Bornemann, Folkmar A. A1 - Nettesheim, Peter A1 - Schütte, Christof T1 - Quantum-Classical Molecular Dynamics as an Approximation to Full Quantum Dynamics N2 - This paper presents a mathematical derivation of a model for quantum-classical molecular dynamics (QCMD) as a {\em 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. T3 - ZIB-Report - SC-95-26 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-1922 ER - TY - GEN A1 - Nettesheim, Peter A1 - Schütte, Christof T1 - Numerical Integrators for Quantum-Classical Molecular Dynamics N2 - 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. T3 - ZIB-Report - SC-97-42 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3111 ER - TY - GEN A1 - Nettesheim, Peter T1 - Second Order Transitions in Quantum-Classical Molecular Dynamics N2 - 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 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 [{\em 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. T3 - ZIB-Report - SC-98-37 KW - quantum-classical molecular dynamics KW - mixed quantum-classical models KW - asymptotics KW - quantum populations Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3805 ER - TY - GEN A1 - Nettesheim, Peter T1 - An Explicit and Symplectic Integrator for Quantum-Classical Molecular Dynamics N2 - This paper presents an explicit and symplectic integrator called PICKABACK for quantum-classical molecular dynamics. This integration scheme is time reversible and unitary in the quantum part. We use the Lie formalism in order to construct a formal evolution operator which then is split using the Strang splitting yielding the symplectic discretization PICHABACK. Finally the new method is compared with a hybrid method in application to two examples: a collinear collision with a quantum oscillator and additionally a photodissociation process of a collinear ArHCI-molecule. T3 - ZIB-Report - SC-95-40 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-2062 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 - Nettesheim, Peter A1 - Reich, Sebastian T1 - Symplectic Multiple-Time-Stepping Integrators for Quantum-Classical Molecular Dynamics N2 - The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. T3 - ZIB-Report - SC-97-56 Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3250 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 - TY - CHAP A1 - Nettesheim, Peter A1 - Schütte, Christof ED - Deuflhard, Peter ED - Hermans, J. ED - Leimkuhler, Benedict ED - Marks, A. ED - Reich, Sebastian ED - Skeel, R. T1 - Numerical Integrators for Quantum-Classical Molecular Dynamics T2 - Computational Molecular Dynamics Y1 - 1999 UR - http://publications.imp.fu-berlin.de/137/ VL - 4 SP - 396 EP - 411 PB - Springer ER - TY - CHAP A1 - Schütte, Christof A1 - Nettesheim, Peter ED - Keil, F. ED - Mackens, W. ED - Voss, H. ED - Werther, J. T1 - Nonadiabatic Effects in Quantum-Classical Molecular Dynamics T2 - Scientific Computing in Chemical Engineering II Y1 - 1999 UR - http://publications.imp.fu-berlin.de/136/ SP - 42 EP - 56 PB - Springer ER -