Verlet time propagation

Each of these operators is unitary U —t) = U t). Updating a time step with the propagator Uf( At)U At)Uf At) yields the velocity-Verlet algorithm. Concatenating the force operator for successive steps yields the leapfrog algorithm ... 

Fig. 2. The time evolution of the total energy of four water molecules (potential-energy details are given in [48]) as propagated by the symplectic Verlet method (solid) and the nonsymplectic fourth-order Runge-Kutta method (dashed pattern) for Newtonian dynamics at two timestep values.

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

At this point, the kinetic energy at time t + At is available. Figure Ic gives a graphical representation of the steps involved in the velocity Verlet propagation. 

Our AIMD simulations are all-electron and self-consistent at each 0.4 femtoseconds (fs) time step. Variational fitting ensures accurate forces for any finite orbital or fitting basis sets and any finite numerical grid. These forces are used to propagate the nuclear motion according to the velocity Verlet algorithm [22]. The accuracy of these methods is indicated by the fact that during the 500 time-step simulations of methyl iodide dissociation described below, the center of mass moved by less than 10-6 A. 

Nuclear motion can be described by quantum mechanical propagation of the vibrational wave function or classical motion on the time-dependent adiabatic potentials. In our approach, the classical equations of motion are solved by the velocity Verlet algorithm. The typical time increment for integration. At, was 0.5 fs. 

Various numerical methods have been proposed for collisional Hamiltonian systems [136, 176, 184, 263, 348, 352]. Typically, these schemes rely on the Verlet method to propagate the system between collisions, with collisions detected either (i) by checking for overlap at the end of the step, (ii) checking for overlap during the step, or (iii) approximating the time to collision before the step. Collisions lead to momentum exchange between particles according the principle outlined above. 

Computer simulation of the system modeled by Eq. [1] requires some sort of time discretization scheme. The method proposed by Verlet propagates positions by... 

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