*4.3. Natural Bond Order Correlations*

The distended shapes of windowpane clusters provide clear evidence of the severe effects of "ring strain" in altering the network O−H···O bonds from the idealized geometries of isolated H-bonds in binary complexes. Nevertheless, one expects that network H-bonds should continue to exhibit the robust correlations with NBO/NRT measures of bond order and charge transfer that were previously demonstrated for free binary H-bonded species [63]. We now turn to examining the supramolecular extension of such correlations for the classical bond order–bond length (BOBL) relationships that have long been fruitfully employed in the integer (single-, double-, triple-, etc., bond) range of covalent bonding in molecules [42,43].

A simple example of such BOBL correlations is illustrated in Figure 9 for the 1,4,4W9 windowpane cluster of Figure 2. For each O···H−O linkage, the total *b*O···<sup>O</sup> bond order is obtained as the sum of *b*O···<sup>H</sup> (major) and "long-bond" [64] *b*OˆO (minor) contributions,

$$b\_{\rm O\cdots O} = b\_{\rm O\cdots H} + b\_{\rm CO} \tag{3}$$

with sub-integer values ranging from 0.02 to 0.18 in this simple cluster. As shown in the right panel, the BOBL correlation is of excellent quality (Pearson correlation coefficient χ ≈ −0.97), and the least-squares regression line (shown in the inset) allows close prediction of *R*O···<sup>O</sup> distances to near the 0.01Å level(!), despite the fact that NBO/NRT descriptors receive *no* input from real-space molecular geometry or spatial distribution of electron density. Thus, the resonance–covalency concepts underlying NRT bond order evaluations appear to extend seamlessly into this *sub*-integer range of weak H-bonding in clusters, practically as well as the familiar *supra*-integer range of strong covalent bonding and resonance in molecules.

**Figure 9.** Calculated NRT bond orders *<sup>b</sup>*O···<sup>O</sup> of the 1,4,4W9 windowpane cluster (**left panel**), showing their excellent BOBL correlation (Pearson χ = −0.973) with optimized *R*O···<sup>O</sup> bond lengths (**right panel**).

More complex three-dimensional structures of windowpane clusters obstruct clear visual representation of all relevant *b*O···<sup>O</sup> bond orders and tend to show additional effects of ring strain. Comprehensive listings of *b*O···<sup>O</sup> bond orders and *R*O···<sup>O</sup> distances (Å) for all H-bonds in all clusters (keyed to the atom numberings of Figures 2 and 3) are presented as tables in SI, as exemplified for the 4,4,6W14 cluster in Table 2. In this case, the *<sup>b</sup>*O···O-*R*O···<sup>O</sup> correlation is found to be weaker, but still of reasonably high quality (χ ≈ −0.91), reflecting the heterogeneities of higher-coordination motifs.


**Table 2.** NRT bond orders *<sup>b</sup>*ij and bond lengths *<sup>R</sup>*ij (Å) for all O(*i*)···O(*j*) H-bonds of the 4,4,6W14 cluster (with atom numberings as shown in Figure 3). (See SI for similar tables for all clusters of the present work).

It is also of interest to examine the global BOBL correlations for *all* windowpane clusters of the present work, covering ca. 250 individual *b*O···O-*R*O···<sup>O</sup> H-bonded pairs in a broad variety of coupled coordination motifs. Figure 10 displays the BOBL scatter plot, least-squares regression line, and Pearson correlation coefficient for this entire data set of hydrogen bonds, showing the strong correlation (χ = −0.90) that persists in spite of increasingly heterogeneous cluster topologies.

**Figure 10.** Scatter plot, least-squares regression line, and Pearson correlation coefficient (χ) for *b*ij-*R*ij BOBL correlations of all (~250) H-bonds in the clusters of Figures 2 and 3.

The degraded accuracy of the linear least-squares regression fit in Figure 10 (compared, e.g., to that in Figure 9) can be primarily attributed to the upward deviations from linearity that are evident near *b*ij → 0. However, it is important to recognize that these deviations are *required* on physical grounds, because intermonomer separation should asymptotically *diverge* (*R*ij → ∞) as bond order vanishes (*b*ij → 0). Indeed, only the higher-order connectivity of the H-bond network prevents such asymptotic dissociation when any single H-bond is severed, so the proper appearance of such nonlinearity in the *b*ij → 0 limit serves to further confirm the resonance–covalent nature of H-bonding even in this range of interaction strengths near the limit of chemical interest.
