Simulations of long time scale dynamics using the dimer method
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Simulations of long time scale dynamics using the dimer method a a;b Graeme Henkelman and Hannes J onsson
a Department of Chemistry, Box 351700, University of Washington, Seattle, WA 98195-1700, USA b Faculty of Science, VR-II, University of Iceland, 107 Reykjav k, Iceland May 27, 2001
Abstract
We have carried out long time scale simulations where the \dimer method" [G. Henkelman and H. Jonsson, J. Chem. Phys. 111, 7010 (1999)] is used to nd the mechanism and estimate the rate of transitions within harmonic transition state theory and time is evolved by using the kinetic Monte Carlo method. Unlike traditional applications of kinetic Monte Carlo, the atoms are not assigned to lattice sites and a list of all possible transitions does not need to be speci ed beforehand. Rather, the relevant transitions are found on the y during the simulation. An application to the di usion and island formation of Al adatoms on an Al(100) surface is presented.
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INTRODUCTION
One of the greatest challenges in computational studies of atomic systems is the simulation of long time scale evolution. While it is relatively straightforward to iteratively solve Newton's equations, the time scale that can be simulated that way, even when the simplest interaction potentials are used, is only on the order of nanoseconds for a typical system size and a weeks worth of CPU time on a modern computer. In chemistry and materials processing, most interesting transitions are thermally activated and take place on the time scale of microseconds or even seconds. The disparity of time scales is huge and it is clearly necessary to develop di erent methods for simulating time evolution. Fortunately, there is often a separation of time scales. Vibrational motion of atoms occurs on the short time scale of femtoseconds. For a typical chemical reaction or di usion event, there are on the order of 10 vibrations before there is a suÆciently large uctuation of thermal energy in the right degree of freedom for a transition to take place. Instead of following each vibration and waiting for these rare events, one can use transition state theory (TST) [1, 2, 3, 4, 5, 6] to calculate the average amount of time necessary for the system to make a transition. In order to calculate a rate, a bottleneck through which the system must pass in order to make the transition must be identi ed. This is the so-called transition state. For solid state systems, it is often possible to assume that the system is harmonic near the energy minimum representing the initial state and near a saddle point on the energy surface in the transition state. In this case, there is a simple form of TST referred to as harmonic transition state theory (hTST), in which the rate of a transition, k, can be directly related to properties of the initial state energy minimum and the saddle point [7, 8], B : k = ze (1) 10
hTST
3N
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AA8.1.1
(E z
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ln Rate
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Figure 1: The temperature dependence of the rate of two di eren
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