Fully adaptive propagation of the quantum-classical Liouville equation

Abstract

In mixed quantum-classical molecular dynamics few but important degrees of freedom of a dynamical system are modeled quantum-mechanically while the remaining ones are treated within the classical approximation. Rothe methods established in the theory of partial differential equations are used to control both temporal and spatial discretization errors on grounds of a global tolerance criterion. The TRAIL (trapezoidal rule for adaptive integration of Liouville dynamics) scheme [I. Horenko and M. Weiser, J. Comput. Chem. 24, 1921 (2003)] has been extended to account for nonadiabatic effects in molecular dynamics described by the quantum-classical Liouville equation. In the context of particle methods, the quality of the spatial approximation of the phase-space distributions is maximized while the numerical condition of the least-squares problem for the parameters of particles is minimized. The resulting dynamical scheme is based on a simultaneous propagation of moving particles (Gaussian and Dirac deltalike trajectories) in phase space employing a fully adaptive strategy to upgrade Dirac to Gaussian particles and, vice versa, downgrading Gaussians to Dirac-type trajectories. This allows for the combination of Monte-Carlo-based strategies for the sampling of densities and coherences in multidimensional problems with deterministic treatment of nonadiabatic effects. Numerical examples demonstrate the application of the method to spin-boson systems in different dimensionality. Nonadiabatic effects occurring at conical intersections are treated in the diabatic representation. By decreasing the global tolerance, the numerical solution obtained from the TRAIL scheme are shown to converge towards exact results.

Adaptive Gaussian particle method for the solution of the Fokker‐Planck equation

“…The most simple form would be an identity matrix multiplied by a single standard deviation λ i , as proposed in [16]. More parameters may be introduced into the shape matrices, e.g.…”

“…Hence the overall residual will be much larger than the predicted one. With a new Gaussian at the sample point with the largest local residual, additional sample points in its vicinity are proposed in [16]. This, however, leads to additional Gaussians and sample points in regions, where the measured residual is largest, leaving even larger residuals in other regions undetected.…”

“…In [15] and [16] a mechanism was proposed to increase and decrease the number of basis functions according to a predetermined tolerance and hence to adapt the decomposition to nonlinearities in the ODE. An implementation was the TRAIL (Trapezoid Rule for Adaptive Integration of Liouville dynamics) algorithm, which was first applied to problems of molecular dynamics.…”

“…The aim is to minimize it until a prescribed toleranceˆ spatial is met by choosing a suitable parameter vector p. New Gaussians are added if no combination of the components of p fulfils ini <ˆ spatial . For this optimization process a Gauß-Newton method is proposed in [16] or a Nelder-Mead scheme (cf. [20]) in [14].…”

“…Prior to use (16) for the first time after the initialization, the sample points have to be redistributed, since their location is optimized for the residual with respect to the initial density and not to the residual due to (1).…”