P. Nettesheim and S. Reich, Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics, in [6], 1999, pp. 412-420.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The classical particles and the wave functions evolve on time scales 1 and , respectively. Clearly, e cient numerical methods for (1) are a prerequisite for the numerical treatment of (2) or related models. In recent years, a variety of numerical methods have been proposed for either (1) or (2) [1,4,10 12,15,16], but all of these methods are based on the assumption of a xed positive . For small , however, they require time steps h . For 0 and h , they do not approximate the adiabatic limit system, which is given by the Born Fock quantum adiabatic theorem in case of (1) or by the ....

P. Nettesheim and S. Reich, Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics, in [6], 1999, pp. 412-420.

....) with fi = 0:7. Preservation of weakly stable limit cycles. We consider the nonlinear oscillator (Van der Pol equation) q 00 = Gammaq (1 Gamma q 2 )q 0 ; 0:01; 3.1) which has a stable limit cycle close to the circle of radius 2. It is known (Stoffer [24] Hairer Lubich [13]) that symmetric or symplectic one step methods give qualitatively correct numerical solutions even when the stepsize is much larger than . Fig. 3 shows the numerical solutions obtained by three different multistep methods. For the strictly stable explicit Adams method (A) the numerical solution ....

....manifold theorem and the finite time estimates between the numerical solution and the solution of the modified equation, it finally follows that the numerical method has a weakly attractive invariant closed curve that is O(e Gammafl =h ) close to the limit cycle of the modified equation. See [13, 24, 25] for more details and for extensions of such a result to the preservation of weakly attractive invariant tori of more general dissipatively perturbed Hamiltonian systems. 4.2. Multistep methods. We now explain the numerical behavior shown in Fig. 3. We consider the symmetric two step scheme yn 1 ....

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P. Nettesheim and S. Reich, Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics, in [3], 1998.

....is 1 16, but it does so in electron ion interactions, with a mass ratio of 1 2000 or less. In the present paper we will not deal with the additional numerical difficulties resulting from a very small mass ratio. Various time integration schemes for the QCMD equations (1. 1) have been proposed in [2, 9, 17, 18, 19, 25]. Starting from the observation that (1.1) is a Hamiltonian system, most of these papers construct symplectic methods for (1.1) This appears promising in view of the known strong results on long time integration by symplectic methods, which are obtained using a backward error analysis that ....

.... grid values of a smooth potential, then an attractive scheme is the Strang splitting exp( Gammai A)b S m =m b ; where, for = m, S = exp( Gamma i 2 V ) exp( Gammai U ) exp( Gamma i 2 V ) Multiple time stepping with Strang splitting has been advocated in the QCMD context in [18]. Its accuracy, or the required step number m, depends strongly on the spatial regularity of the data b. It is very efficient for smooth data, but becomes inaccurate for rough data. It is shown in [13] that, for arbitrary ff 2 [0; 2] k exp( Gammai A)b Gamma S m =m bk C i m j ff kA ....

P. Nettesheim and S. Reich, Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics, in [4], 1999, pp. 412--420.

....space discretization of the Schrodinger equation. A feature in common is the presence of widely different time scales for the quantum and the classical evolution, which leads to particular challenges for the time integration. Various time integration schemes for (1. 1) have been proposed in [1, 6, 12, 13, 14]. So far, no convergence analysis is available which takes the highly oscillatory behaviour of the wave functions into account and therefore assumes no bounds on their derivatives. Such an error analysis is of interest not only from a purely mathematical point of view but it also gives important ....

P. Nettesheim and S. Reich, Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics, in [3], 1998.

....2 subject to the orthogonality constraint ( i ; j ) ij . Again there is a clear separation between fast electronic degrees of freedoms and the slow classical degrees of freedom which can be exploited by our averaging method. As a nal example, we consider quantum classical molecular dynamics [7, 4]. We are especially interested in the case of very large systems, where a full non adiabatic 6 LEIMKUHLER AND REICH quantum mechanical approximation is only needed for small portion of the system such as the quantum mechanical transition of a hydrogen atom. In the simpli ed case of a single ....

P. Nettesheim and S. Reich, Symplectic multiple-time-stepping integrators for quantumclassical molecular dynamics (1998), Lecture Notes in Computational Science and Engineering, Vol. 4, 412-420, Springer.

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