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demonstrated by Ercolani
et al.
[10] While the direct counting method is more illustrative
and useful for relatively simple systems, the symmetry number method is particularly
suitable for large supramolecular assemblies. Using the latter method, the overall statisti-
cal factor for equilibrium 1 is calculated as the ratio of total symmetry numbers for react-
ing and produced microspecies according to Equation 3.2.
m
n
M
L
¼
ðs
tot
Þ
ðs
tot
Þ
M;L
m
v
¼
l
=
r
ð
3
:
2
Þ
;
n
M
m
L
n
tot
ðs
Þ
The total symmetry number
s
tot
of a microspecies is the product of the external
s
ext
and
internal
s
mix
produced by the
number of isomeric microspecies contributing to the macrospecies. The graphical applica-
tion of the symmetry numbers method for the formation of mononuclear Cu(I) complexes
is given in Figure 3.1 and gives
s
int
symmetry numbers modulated by the mixing entropy
Cu;L
1;2
12. The counting using the direct method gives
six possibilities to arrange two bidentate ligands in the coordination tetrahedron of Cu(I).
In addition, these two ligands may adopt two possible arrangements (head-head, head-
tail). Since the resulting assembly is achiral, the overall statistical factor is calculated by
multiplying all these possibilities, and
v
¼
Cu;L
1;2
12 is in agreement with the
method of symmetry numbers. The statistical factor translates into the free energy contri-
bution to the overall energetic balance with the expression D
G
M;L
stat
v
¼
2
6
¼
M;L
m
¼
RT
ln
ðv
Þ
.
;
n
3.2.2 Extension to Polynuclear Edifices
The thermodynamic description of intermolecular connections and the chelate effect pre-
sented in the previous section apply also for large supramolecular systems [6]. However,
their appropriate parameterization requires a special consideration of new factors related
to these complexes. The formation of multicomponent assemblies necessarily implies the
presence of two or more
x
-dentate binding sites within the ligand structure, which may
become rather sophisticated. Therefore, the number of possible products increases rapidly
with the number of components and with the complexity of ligands. The binding sites are
interconnected with an organic linker, which plays an important role in the rational pro-
gramming of desired assemblies. A good connector must exhibit a smooth balance
between flexibility and rigidity in order to provide good spatial dispositions for efficient
metal binding. With this difficult task in mind, a number of synthetic challenges must be
resolved in order to prepare a “good” organic receptor. Moreover, the connection of dif-
ferent binding sites into a discrete supramolecular edifice requires intramolecular binding
events (macrocyclization processes), which operate over long distances, compared with
the chelate effect. Particularly in helicates, at least one intramolecular binding takes place
and the magnitude of this interaction is crucial for the stability of the final edifice with
respect to concurrent intermolecular reactions. Let us develop these phenomena in detail.
3.2.2.1 Modelling Intramolecular Interactions
The mechanistic and thermodynamic description of long-distance intramolecular connec-
tions (ring size
6, distinct binding sites) is, in principle, reminiscent of the short-
distance chelate effect discussed for binding
x
-dentate binding sites in mononuclear
complexes (Figure 3.2). The illustration of this macrocyclization reaction is given for the
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