Verlet approximation

The velocity Verlet algorithm may be derived by considering a standard approximate decomposition of the Liouville operator which preserves reversibility and is symplectic (which implies that volume in phase space is conserved). This approach [47] has had several beneficial consequences. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

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. 

We assume that A is a symmetric and positive semi-definite matrix. The case of interest is when the largest eigenvalue of A is significantly larger than the norm of the derivative of the nonlinear force f. A may be a constant matrix, or else A = A(y) is assumed to be slowly changing along solution trajectories, in which case A will be evaluated at the current averaged position in the numerical schemes below. In the standard Verlet scheme, which yields approximations y to y nAt) via... 

This is the Verlet algorithm for solving Newton s equation numerically. Notice that the term involving the change in acceleration (b) disappears, i.e. the equation is correct to third order in At. At the initial point the previous positions are not available, but may be estimated from a first-order approximation of eq. (16.29). 

The first term m V is a function of x only. Let us assume that we are using the velocity Verlet time integrator, which is the most common. In that case, x is computed with local accuracy 0 dtA) and global accuracy 0(df2), and the velocity v at half-steps is computed with accuracy Oidt2 ) if the following approximation is used ... 

Before introducing a more accurate approximation for v, we recall the basic velocity Verlet algorithm... 

This is known as the Verlet algorithm. Provided that the time step, At, is sufficiently small, this algorithm gives an accurate approximation to the true trajectory defined by Eqs. (9.5). 

Our primary goal was the simulation of entire atomic systems, thus made of electrons and nuclei. As mentioned earlier (see Sect. 2.3), in a large class of systems (e.g. not too high temperature) one can decouple the motion of nuclei and electrons within the Born-Oppenheimer approximation. The previous section was then devoted to the Density Functional Theory solution of the electronic structure problem at fixed ionic positions. By computing the Hellmann-Feynman forces (11) we can now propagate the dynamics of an ensemble of (classical) nuclei as described in Sect. 2.3, using e.g. the velocity verlet algorithm [117]. 

L. Verlet (1967) Computer experiments on classical fluids I. Thermodynamics properties of Lermard Jones molecules. Phys. Rev. 98, p. 159 D. Chandler (1978) Statistical mechanics of isomerization dynamics in liquids and transition-state approximation. J. Chem. Phys. 68, pp. 2959-2970... 

Hansen and Verlet [156] observed an invariance of the intermediate-range (at and beyond two molecular diameters) form of the radial distribution function at freezing, and from this postulated that the first peak in the structure factor of the liquid is a constant on the freezing curve, and approximately equal to the hard-sphere value of 2.85. They demonstrated the rule by application to the Lennard-Jones system. Hansen and Schiff [157] subsequently examined g r) of soft spheres in some detail. They found that, although the location and magnitude of first peak of g r) at crystallization is quite sensitive to the intermolecular potential, beyond the first peak the form of g(r) is nearly invariant with softness. This observation is consistent with the Hansen-Verlet rule, and indeed Hansen and Schiff find that the first peak in the structure factor S k) at melting varies only between 2.85 n = 8) to 2.57 (at n= ), with a maximum of 3.05 at n = 12. 

---