Chemistry Reference
In-Depth Information
independent representatives from among a large set of basis functions is not unique.
This has led to confusion in the literature, and meaningless claims of new methods.
The (H
2
O)
20
dodecahedron serves as a good example.
In 1998, McDonald et al. [25] examined the energy of the 30,026 isomers of
the (H
2
O)
20
dodecahedron with an empirical potential (Fig. 34) and found that the
number of a certain type of H-bond was the most important factor determining
the energy of those clusters. They linked the energy of the dodecahedral (H
2
O)
20
isomers to
n
(2A
→
2A)
, the number of 2A
→
2A bonds, bonds in which both waters
are double-acceptors (2A). In terms of directed graphs, 2A waters on the dodecahe-
dron are vertices with two incoming bonds. Since all water of the dodecahedron is
three-coordinate, this implies that both water molecules of a 2A
→
2A bond have
a dangling hydrogen atom. The maximum number of 2A
2A bonds possible in
the (H
2
O)
20
dodecahedron is 10, an example of which can easily be discerned in
structure (1) of Fig. 33 where the top and bottom pentagons each contain five bonds
with nearest-neighbor dangling hydrogen. These are the least stable dodecahedral
water clusters. The minimum number of 2A
→
2A bonds is 3. These are the most
stable water clusters, an example of which is structure (3) of Fig. 33.
A decade later Kirov et al. [200] proposed a “new discrete model” for the
(H
2
O)
20
dodecahedron as an improvement over previous work. Their model “has
only one preferred H-bond, the one of (t1d)-type”. The designation “t1d” refers
to a trans bond (Fig. 2) with one dangling bond on the donor water molecule. The
number of t1d bonds is easily shown to be given in terms of the number 2A bonds
by the following relation:
→
n
t1d
=
10
−
n
(2A
→
2A)
(47)
(Let
n
2A
be the total number of double-acceptor waters,
n
2A
→
2A
be the number
of bonds in which a 2A donates to another 2A, and
n
2A
→
2D
be the number of
bonds in which 2A donates to a 2D (double-donor water). Since each double
acceptor is a single donor, each 2A is in 1:1 correspondence with a bond that point
from a 2A to either another 2A or a 2D:
n
2A
=
n
2A
→
2A
+
n
2A
→
2D
. Therefore,
n
2A
→
2A
=
n
2A
→
2D
. An H-bond from a 2A to a 2D is what Kirov et al. [200]
call a t1d bond. For polyhedral water clusters with three-coordinate waters,
n
2
A
=
1
n
2A
−
2
(number of vertices), which is 10 for the case of the (H
2
O)
20
dodecahedron.
Hence, we arrive at the conclusion that
n
t1d
=
n
2A
→
2A
.)
McDonald et al. [25] found that
n
(2A
→
2A)
ranged from 3 to 10 with lowest
energy correlated with lowest
n
(2A
→
2A)
. Kirov et al. found in their “new” model
that
n
t1d
ranged from 0 to 7 (Table 3 of [198]) with lowest energy correlated with
highest
n
t1d
, as must occur according to Eq. (47). Clearly, the model of [199] is
merely a linear transformation of the parametrization used by McDonald et al. [25].
In dodecahedral (H
2
O)
20
,n
(2A
→
2A)
, and obviously by Eq. (47)
n
t1d
as well, is
a second-order graph invariant. They are parameters on the lowest level (second-
order graph invariants), of a hierarchy of topological parameters of increasing
10
−
