Single switch surface hopping for molecular dynamics with transitions

A trajectory surface hopping algorithm is proposed, which stems from a mathematically rigorous analysis of propagation through conical intersections of potential energy surfaces. Since nonadiabatic transitions are only performed when a classical trajectory attains one of its local minimal surface gaps, the algorithm is called single switch surface hopping. Numerical experiments for a two mode Jahn–Teller system are presented, which illustrate the asymptotic justification of the method as well as its good performance in the physically relevant parameter range

Trigonometric pulse envelopes for laser-induced quantum dynamics

We relate powers of trigonometric functions to Gaussians by proving that properly truncated cosn functions converge to a Gaussian as n tends to infinity. For an application, we analyse the laser-induced population transfer |X1Σ+ → |A1Πx in a two-level model system of aluminium monochloride (AlCl) with fixed nuclei. We apply linearly x-polarized ultraviolet laser pulses with a trigonometric envelope function, whose square has full width at half-maximum of 2.5 fs and 5.0 fs. Studying population dynamics and optimized laser parameters, we find that the optimal field amplitude for trigonometric pulses with n = 20 and n = 1000 has a relative difference of 1%, which is below experimental resolution

Propagation through conical crossings: An asymptotic semigroup

We consider the standard model problem for a conical intersection of electronic surfaces in molecular dynamics. Our main result is the construction of a semi-group in order to approximate the Wigner function associated with the solution of the Schrödinger equation at leading order in the semiclassical parameter. The semigroup stems from an underlying Markov process that combines deterministic transport along classical trajectories within the electronic surfaces and random jumps between the surfaces near the crossing. Our semigroup can be viewed as a rigorous mathematical counterpart of so-called trajectory surface hopping algorithms, which are of major importance in molecular simulations in chemical physics. The key point of our analysis, the incorporation of the nonadiabatic transitions, is based on the Landau-Zener type formula of Fermanian-Kammerer and Gérard[10] for the propagation of two-scale Wigner measures through conical crossings

Propagation through Generic Level Crossings: A Surface Hopping Semigroup

We construct a surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels. The underlying time-dependent Schrödinger equation has a matrix-valued potential, whose eigenvalue surfaces have a generic intersection of codimension two, three, or five in Hagedorn's classification. Using microlocal normal forms reminiscent of the Landau–Zener problem, we prove convergence to the true solution with an error of the order $\varepsilon^{1/8}$, where $\varepsilon$ is the semiclassical parameter. We present numerical experiments for an algorithmic realization of the semigroup illustrating the convergence of the algorithm

Single switch surface hopping for a model of pyrazine

The single switch trajectory surface hopping algorithm is tested for numerical simulations of a two-state three-mode model for the internal conversion of pyrazine through a conical intersection of potential energy surfaces. The algorithm is compared to two other surface hopping approaches, namely, Tully’s method of the fewest switches [J. Chem. Phys. 93, 1061 (1990)] and the method by Voronin et al. [J. Phys. Chem. A 102, 6057 (1998)] . The single switch algorithm achieves the most accurate results. Replacing its deterministic nonadiabatic branching condition by a probabilistic accept-reject criterion, one obtains the method of Voronin et al. without momentum adjustment. This probabilistic version of the single switch approach outperforms the considered algorithms in terms of accuracy, memory requirement, and runtime

Wigner measures and codimension two crossings

This article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels. Codimension two crossings are the simplest energy level crossings, which affect the Born–Oppenheimer approximation in the zeroth order term. The model we study is a two-level Schrödinger equation with a Laplacian as kinetic operator and a matrix-valued linear potential, whose eigenvalues cross, if the two nucleonic coordinates equal zero. We discuss the case of well-localized initial data and obtain a description of the wavefunction’s two-scaled Wigner measure and of the weak limit of its position density, which is valid globally in time

A fluoroplanigraphy system for rapid presentation of single plane body sections

Fluoroplanigraphic system for rapid presentation of single plane body sections with reduced X ray exposure to patient