2.2 Visualizing the Landscape 33
structure into two parts, each one is a tree graph [69] and appropriate names have
previously been suggested for each pattern [49]. The landscape in Figure 2.1a, with
a well-defined global minimum, and relatively small downhill barriers, reminds
us of a ‘‘palm tree,’’ while the larger barriers in Figure 2.1b produce a ‘‘weeping
willow.’’ The topology in Figure 2.1c is qualitatively different, since there are distinct
energy scales for the barriers, which lead to a hierarchical structure. The resulting
pattern is reminiscent of a ‘‘banyan tree,’’ where there are numerous branches
leading to low-lying minima that are separated from the global minimum by high
barriers. These competing minima can be connected to quantitative measures of
‘‘frustration’’ in the energy landscape [70, 71]; an extreme example corresponding
to a glassy landscape will be discussed in Section 4.
Some specific examples are shown in Figure 2.2. The palm tree motif is illustrated
by parts (a)–(d), for (a) an atomic cluster with a Mackay icosahedron [72] as the global
minimum, (b) a self-assembling icosahedral shell [41], (c) a supercell representation
of bulk silicon [64], and (d) a polyalanine peptide. The willow tree pattern revealed
for C
60
in Figure 2.2e is the result of a well-defined global minimum, but large
barriers for interconverting isomers with different σ -bonding frameworks. Finally,
the graph for (H
2
O)
20
in Figure 2.2f has a hierarchical structure, where sets of
minima are disconnected together when certain edges of the graph are removed.
The form of these graphs provides a very convenient way to think about the
likely efficiency of relaxation to the global minimum. A good ‘‘structure-seeking’’
system needs to possess a well-defined free energy minimum, which is kinetically
accessible over the temperature range of interest. The interplay of thermodynamic
and kinetic factors is particularly noteworthy, and cannot be described without a
measure of connectivity between the local minima. It is the palm tree graph that
we associate with structure-seeking properties. However, the willow tree graph also
has a well-defined global minimum, and efficient relaxation is again possible for
a sufficiently high temperature, in good agreement with experimental routes to
buckminsterfullerene [49, 67, 73].
The identification of structure-seeking, self-assembling, or self-organizing sys-
tems with the palm tree pattern is consistent with the ‘‘folding funnel’’ view of
protein folding, described in terms of a set of kinetically convergent pathways [74].
A well-defined free energy minimum exists for such landscapes over a range of
temperature [42], and the lack of competitive structures can be identified with
‘‘minimal frustration’’ [75].
Before comparing glassy and crystalline landscapes in Section 4 it is instructive
to consider a double-funnel landscape, which contains two palm tree features. Here
we consider a cluster of 38 atoms bound by the LJ potential [61], which has served as
a benchmark for global optimization [1, 8–10], thermodynamic sampling [76–79],
and rare event dynamics [1, 20, 36]. The global potential energy minimum is a
truncated octahedron, which is separated by a high barrier from structures based on
incomplete Mackay icosahedra [77]. The octahedral region of configuration space
is favored by potential energy, while the icosahedral region is favored by entropy,
and the change in the global free energy minimum with increasing temperature
provides a finite system analog of a first order phase transition. This transition