Traversing the Potential Energy Surface

The eigenvector-following technique[88] can find transition states on a PES by maximizing the energy along a chosen eigenvector of the Hessian whilst simultaneously minimizing the energy along all other eigenvectors. Once a transition state is located, the minima it connects can be found by taking small initial steps away from the stationary point along the eigenvector of the Hessian corresponding to the unique negative eigenvalue (in both the positive and negative directions) and then performing minimizations.

The technique we use to comprehensively search the PES is simple. From a starting minimum we search for transition states along the n_ev eigenvectors with the lowest eigenvalues in both positive and negative directions, and add the new minima that are connected by these transition states to our set of minima. We then repeat this process for all minima until until no new minima are found. The larger the value of n_ev (maximum 3N-6), the more comprehensive the search will be. However, even using the maximum value of n_ev there is no guarantee that this approach will find all the minima on the PES. The method has been previously used to generate large samples of minima and transition states for small LJ clusters[162].

The second method we use involves a Monte Carlo walk between connected minima. First, from a starting minimum we randomly choose an eigenvector along which to search for a transition state; this choice is weighted towards those with lower eigenvalues, because a search along an eigenvector with a low eigenvalue generally converges to a transition state more rapidly. Once a transition state is found, we identify the minima it connects. Eigenvector-following occasionally finds a transition state unconnected to the original minimum, and so we check that the original minimum is at one end of the rearrangement pathway.

Having found a connected minimum, a decision whether to accept the step must then be taken. Here, we choose a Metropolis step criterion, which accepts moves with a probability of min$[1,\exp(-\Delta E/kT)]$,where $\Delta E$ is the change in energy associated with the step[284]. There are many other possible step criteria that could be used. For example, the probability of acceptance could be related to the transition rate as calculated using RRKM theory, and so could be used to simulate the dynamics of the real cluster (assuming the density of states of the minimum and transition state could be calculated accurately). Once the decision to accept the step has been made the whole process is repeated, and a new transition state search is started. A record is kept of each transition state search, so that if the same step is attempted the search does not need to be repeated.

Techniques that search the PES by taking steps in configuration or phase space are hindered at low temperatures by the time scale separation between interwell and intrawell motion; this leads to trapping in local minima. Our method avoids this problem by removing the vibrational motion altogether.

To analyse the behaviour of our method further, the concept of a basin introduced by Kunz and Berry is useful[253,254]. These authors defined a basin as a set of connected minima that belong to sequences of minima which are monotonically decreasing in energy and which lead down to the lowest energy minimum in the basin. (This definition overlaps somewhat with the concept of a funnel that is used in the protein folding literature and which we discussed in Chapter 4.) The global minimum is at the bottom of the primary basin. Basins have a similar role in this `minima space' to that played by minima in configuration space. However, there are far fewer basins than minima on the PES. The time scale for interbasin flow is likely to be slower than for intrabasin flow between minima [253,254]. Therefore, sufficiently deep secondary basins (basins that do not end at the global minimum) can act as traps (§5.4.4).

If a Monte Carlo walk in the minima space is performed at zero temperature, the system is `quenched' to the bottom of a basin. If the number of basins on the PES is not too large, we might expect that a quench from a random configuration might lead to the global minimum, especially as it is likely that the primary basin would have the largest catchment area. This suggests that these quenches may be a useful global optimization tool. If a low rather than a zero temperature is used, the method would be able to escape from shallow basins, whilst not significantly compromising the favourability of downhill moves.