Chemistry Reference
In-Depth Information
40
30
20
10
Figure 34.
Energies
of local potential energy min-
ima of the (H
2
O)
20
dode-
camer compared with a graph
invariant prediction to the en-
ergies based on 20 randomly
selected isomers.
0
0
10
20
30
OSS2 energy (kcal mol
-1
)
The plot in Fig. 34 is actually not a least-squares fit of (H
2
O)
20
dodecamer en-
ergies to invariant functions, as was provided for ice structures in Section III.
Rather, the energies of only 20 of the 30026 (H
2
O)
20
isomers, calculated using
the OSS2 empirical potential [197, 198], were used to fit seven invariants. [One
of those invariants was a constant, analogous to the
E
0
term in Eq. (11), so more
precisely an irrelevant overall constant and the six invariant functions were used
to describe the energy of 30,026 isomers in Fig. 34.] The plot in Fig. 34 compares
the prediction from the invariant fit to the energies of only 20 isomers, to the actual
energy calculated using the same model for all 30,026 isomers. The energies were
organized in this way to show that graph invariants furnish a powerful method
to quickly sift through data, and to search for the ground state. With just 20 data
points, candidates for the ground state are identified, and the energy of most struc-
tures is fit to within a small fraction of the total energy range. The small number
of points that lie far from the line in Fig. 34 are quite interesting cases. They are
structures that optimize to a different, sometimes zwitterionic, topology from the
starting configuration [199].
Graph invariants and their higher tensorial extensions provide a complete and
unambiguous method for correlating physical properties with H-bond topology.
Normal H-bonding of neutral molecules constrain the directed graphs describing
H-bond topology to have at most two incoming and two outgoing bonds at each
vertex. Under these constraints, many of the possible invariant combinations are
linearly dependent on others. The minimal number of linearly independent invari-
ants at any given order can be established, but of course the selection of linearly
