Stepsize

Robert D. Skeel and Jeffrey J. Biesiadecki. Symplectic integration with variable stepsize. Ann. Num. Math., 1 191-198, 1994. 

Fig. 1. Comparison of two different dynamical simulations for the Butane molecule Verlet discretization with stepsize r = O.OOSfs. Initial spatial deviation 10 A. Left Evolutions of the total length (=distance between the first and the last carbon atom) of the molecule (in A). Right Spatial deviation (in A) of the two trajectories versus time.

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

Essential Dynamics In most applications details of individual MD trajectories are of only minor interest. An illustrative example due to Grubmuller [10] is documented in Figure 3. It describes the dynamics of a polymer chain of 100 CH2 groups. Possible stepsizes for numerical integration are confined... 

Fig. 2. Left Time average (over T = 200ps) of the molecular length of Butane versus discretization stepsize r for the Verlet discretization. Right Zoom of the asymptotic domain (r < 10 fs) and quadratic fit.

Covering of Energy Cells Assume that the energy cells under consideration are compact sets and the stepsize r is fixed. We want to construct a collection B of boxes in phase space such that the union Q of these subsets is a covering of the energy cell we focus on. To this end, consider... 

Fig. 6. The density of the invariant measure of the potential Vi for total energy F = 4.5. Results of the subdivision approach (left) and a direct simulation with about 4.5 million steps for stepsize t = 1/30 (right).

In all ealculations done so far a fixed stepsize r = 0.1 has been used. Hence an application of formula (9) leads to the following table concerning flip- flop probabiliti< s Ix tween different conformations. 

Fig. 2. Top left short time interval results for the SPL method with stepsize At = 0.05 (100 steps) top right corresponding energy variation. Center and Bottom equivalent diagrams for RK4a (At = 0.1) and RK4b (At = 0.1).

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

The control scheme tries to choose the stepsize t so that 111 11 = TOL in some adequate norm. In case of a tolerance exceeding error, i.e., for Herll > TOL, one reduces the stepsize according to... 

When considering the construction of exactly symmetric schemes, we are obstructed by the requirement to find exactly symmetric approximations to exp(—ir/f/(2fi,)). But it is known [10], that the usual stepsize control mechanism destroys the reversibility of the discrete solution. Since we are applying this mechanism, we now may use approximations to exp —iTH/ 2h)) which are not precisely symmetric, i.e., we are free to take advantage of the superior efficiency of iterative methods for evaluating the matrix exponential. In the following, we will compare three different approaches. 

In [13], an efficient residual error estimation scheme has been introduced for controlling the quality of the approximation. This gives us a stopping criterion for the iteration guaranteeing that the quality of the approximation fits to the accuracy requirements of the stepsize control. 

The stepsize controlling adaptive QCMD integrators presented in the previous section differ only with respect to the approximation of the quantum propagation. We herein compare three of these integrators, all of them... 

To begin with, we compare the stepsizes used in the simulations (Fig. 3). As pointed out before, it seems to be unreasonable to equip the Pickaback scheme with a stepsize control, because, as we indeed observe in Fig. 3, the stepsize never increases above a given level. This level depends solely on the eigenvalues of the quantum Hamiltonian. When analyzing the other integrators, we observe that the stepsize control just adapts to the dynamical behavior of the classical subsystem. The internal (quantal) dynamics of the Hydrogen-Chlorine subsystem does not lead to stepsize reductions. 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

Fig. 3. Stepsize r used in the simulation of the collinear photo dissociation of ArHCl the adaptive Verlet-baaed exponential integrator using the Lanczos iteration (dash-dotted line) for the quantum propagation, and a stepsize controlling scheme based on PICKABACK (solid line). For a better understanding we have added horizontal lines marking the collisions (same tolerance TOL). We observe that the quantal H-Cl collision does not lead to any significant stepsize restrictions.

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. 

The effeet of integration method and stepsize must be eheeked for every application where temperature runaway is possible. Those will be mostly oxidations, but other reaetions ean be very exothermie, too. During the 1973/74 oil erisis, when synthetie natural gas projeets were in vogue, one of the CO hydrogenation teehnologies was found to be very exothermie and prone to runaway also. 

Run your study at the Hartree-Fock level, using the 6-31+G(d) basis set. Use a step size of 0.2 amu bohr for the IRC calculation (i.e., include IRC=(RCFC, StepSize=20) in the route section). You will also find the ColcFC option helpful in the geometry optimizations. 

MaxPoints option 174 RCFC option 176 Restart option 174 StepSize option 200 isodensity surface 238... 

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