4.4 Examples 79
simulations starting from points along the trajectory in the time window are
employed to verify whether the system is in local equilibrium in this region, and
long simulations for a number of temperatures are used to measure the probability
flow from the region and thus the escape time. Unsurprisingly, searching for
locally ergodic regions in this fashion is quite expensive computationally.
Clearly, the concern about the stability of the locally ergodic regions leads to the
study of the barrier structure of the landscape. Quite generally, the procedures to
analyze this structure, e.g. to find saddle points and transition paths and to measure
the probability flow on the landscape, are considerably more involved and often less
robust than the methods employed to determine local minima. Most of the methods
used to gain information about the barriers separating two local minima, or about
the transition regions connecting the various locally ergodic regions, do not rely on
stochastic explorations. Instead, they are essentially deterministic schemes based
on information about the local minima, the gradients, and the curvatures (via the
eigenvectors of the Hessian at the minima or saddle points) [116–141], although
some stochastic procedures such as the threshold algorithm are also employed [39,
40, 112, 50]. For more details, we refer to the literature, e.g., [11].
4.4
Examples
Since the topic of this chapter is structure prediction using simulated
annealing-type methods, all examples presented below employ stochastic single-
and multiwalker algorithms as an essential feature. Moreover, we exclude those
random-walker-based procedures that are discussed elsewhere in this book,
such as basing hopping or random searches. Furthermore, after two decades of
research, there exist very many examples of structure prediction using simulated
annealing; due to lack of space, only a few, all dealing with extended solids, can be
presented in some detail here.
11)
In order to organize the presentation in a sensible fashion, we note that the
studies one finds in the literature under the heading of ‘‘structure prediction’’
of solids can be divided into three different classes: On the one extreme is the
structure determination, where important structural information, typically a unit
cell and its content, is known from the experiment. On the other extreme is the
unrestricted structure prediction, where only the stoichiometry but neither the unit
cell nor the number of formula units/(primitive) unit cell is known. And if we
11) Large finite molecules such as clusters
[142–152], polymers [9, 153–155] or pro-
teins [156–162] constitute a different class
of chemical systems where over the past two
decades much effort has been invested in
the identification of their structures, often
using simulated annealing as the global op-
timization technique. Since these systems
contain a finite number of atoms and lack
the complicating feature of periodic bound-
ary conditions, they have served as a testing
ground not only for simulated annealing
but for many of the global optimization and
structure prediction techniques mentioned
in this book. As this chapter deals with
structure prediction of solids, we refer to
the literature mentioned for more details
and specific examples.