Local Electronic Structure in AlN Studied by Single-Crystal 27Al and 14N NMR and DFT Calculations.

Both the chemical shift and quadrupole coupling tensors for 14 N and 27 Al in the wurtzite structure of aluminum nitride have been determined to high precision by single-crystal NMR spectroscopy. A homoepitaxially grown AlN single crystal with known morphology was used, which allowed for optical alignment of the crystal on the goniometer axis. From the analysis of the rotation patterns of 14 N ( I = 1 ) and 27 Al ( I = 5 / 2 ), the quadrupolar coupling constants were determined to χ ( 14 N ) = ( 8 . 19 ± 0 . 02 ) kHz, and χ ( 27 Al ) = ( 1 . 914 ± 0 . 001 ) MHz. The chemical shift parameters obtained from the data fit were δ i s o = - ( 292 . 6 ± 0 . 6 ) ppm and δ Δ = - ( 1 . 9 ± 1 . 1 ) ppm for 14 N, and (after correcting for the second-order quadrupolar shift) δ i s o = ( 113 . 6 ± 0 . 3 ) ppm and δ Δ = ( 12 . 7 ± 0 . 6 ) ppm for 27 Al. DFT calculations of the NMR parameters for non-optimized crystal geometries of AlN generally did not match the experimental values, whereas optimized geometries came close for 27 Al with χ ¯ calc = ( 1 . 791 ± 0 . 003 ) MHz, but not for 14 N with χ ¯ calc = - ( 19 . 5 ± 3 . 3 ) kHz.

Variational approach to rare event simulation using least-squares regression.

We propose an adaptive importance sampling scheme for the simulation of rare events when the underlying dynamics is given by diffusion. The scheme is based on a Gibbs variational principle that is used to determine the optimal (i.e., zero-variance) change of measure and exploits the fact that the latter can be rephrased as a stochastic optimal control problem. The control problem can be solved by a stochastic approximation algorithm, using the Feynman-Kac representation of the associated dynamic programming equations, and we discuss numerical aspects for high-dimensional problems along with simple toy examples.

Effective dynamics along given reaction coordinates, and reaction rate theory.

In molecular dynamics and related fields one considers dynamical descriptions of complex systems in full (atomic) detail. In order to reduce the overwhelming complexity of realistic systems (high dimension, large timescale spread, limited computational resources) the projection of the full dynamics onto some reaction coordinates is examined in order to extract statistical information like free energies or reaction rates. In this context, the effective dynamics that is induced by the full dynamics on the reaction coordinate space has attracted considerable attention in the literature. In this article, we contribute to this discussion: we first show that if we start with an ergodic diffusion process whose invariant measure is unique then these properties are inherited by the effective dynamics. Then, we give equations for the effective dynamics, discuss whether the dominant timescales and reaction rates inferred from the effective dynamics are accurate approximations of such quantities for the full dynamics, and compare our findings to results from approaches like Mori-Zwanzig, averaging, or homogenization. Finally, by discussing the algorithmic realization of the effective dynamics, we demonstrate that recent algorithmic techniques like the "equation-free" approach and the "heterogeneous multiscale method" can be seen as special cases of our approach.

Dimension reduction by balanced truncation: application to light-induced control of open quantum systems.

In linear control, balanced truncation is known as a powerful technique to reduce the state-space dimension of a system. Its basic principle is to identify a subspace of jointly easily controllable and observable states and then to restrict the dynamics to this subspace without changing the overall response of the system. This work deals with a first application of balanced truncation to the control of open quantum systems which are modeled by the Liouville-von Neumann equation within the Lindblad formalism. Generalization of the linear theory has been proposed to cope with the bilinear terms arising from the coupling between the control field and the quantum system. As an example we choose the dissipative quantum dynamics of a particle in an asymmetric double well potential driven by an external control field, monitoring population transfer between the potential wells as a control target. The accuracy of dimension reduction is investigated by comparing the populations obtained for the truncated system versus those for the original system. The dimension of the model system can be reduced very efficiently where the degree of reduction depends on temperature and relaxation rate.

Communications: On the linear response of mechanical systems with constraints.

We revisit the problem of the linear response of a constrained mechanical system. In doing so, we show that the standard expressions of Green and Kubo carry over to the constrained case without any alteration. The argument is based on the appropriate definition of constrained expectations by means of which Liouville's theorem and the Green-Kubo relations naturally follow.

Fast simulation of polymer chains.

We propose an algorithm for the fast and efficient simulation of polymers represented by chains of hard spheres. The particles are linked by holonomic bond constraints. While the motion of the polymers is free (i.e., no collisions occur) the equations of motion can be easily integrated using a collocation-based partitioned Gauss-Runge-Kutta method. The method is reversible, symplectic, and preserves energy. Moreover the numerical scheme allows the integration using much longer time steps than any explicit integrator such as the popular Verlet method. If polymers collide the point of impact can be determined to arbitrary precision by simple nested intervals. Once the collision point is known the impulsive contribution can be computed analytically. We illustrate our approach by means of a suitable numerical example.

Data-based parameter estimation of generalized multidimensional Langevin processes.

The generalized Langevin equation is useful for modeling a wide range of physical processes. Unfortunately its parameters, especially the memory function, are difficult to determine for nontrivial processes. We establish relations between a time-discrete generalized Langevin model and discrete multivariate autoregressive (AR) or autoregressive moving average models (ARMA). This allows a wide range of discrete linear methods known from time series analysis to be applied. In particular, the determination of the memory function via the order of the respective AR or ARMA model is addressed. The method is illustrated on a one-dimensional test system and subsequently applied to the molecular dynamics time series of a biomolecule that exhibits an interesting relationship between the solvent method used, the respective molecular conformation, and the depth of the memory.