3.3.1 Computational skeletal electron sharing SES and conceptual SEP values

The decisive observation from a conceptual point of view is that the values of deltahedral electron sharing SES₃^Δ(B_n) are very close to the conceptual SEP values (Figure 2), while the best approximation to the 2SEP skeletal electrons counts SEC₃^Δ(B_n) is still missing an amount of electrons currently considered too high for detailed conceptual considerations. Thus, SES₃^Δ(B_n) is the quantum chemically computed quantity, that will be preferentially used to relate G₃^Δ(cluster) values to the Lipscomb picture. Noteworthy, this is not electron counting like the Wade's 2SEP is, but bond counting in the framework of delocalization indices.

3.3.2 Approaching closo styx values from skeletal SES and DR values

For boron hydrides and related systems, the chemical bonding scenario has been characterized as a mixture of 2-center and 3-center bonding as encoded in the styx code.³ In this code, parameter s denotes the number of skeletal 3-center 2-electron BHB bonds, t the number of skeletal 3-center 2-electron B–BB bonds, y the number of skeletal 2-center 2-electron BB bonds, and x the number of extra (besides the one always present exohedral bond per skeletal atom) 2-center 2-electron BH bonds.

A general description of the styx relations of 2- and 3-center bonding with increasing deltahedral cluster size n, is given in Supporting Information.

In order to establish a connection to the cluster G values above, a conceptual quantity has to be defined that has the same kind of principle behavior, namely boundary values 0 and 1 for pure 2- and 3-center bonding, respectively. For this purpose, two rather different conceptual frameworks have been connected. The styx values correspond to a conceptual framework completely different from the one of the G values, that is, t and y are not even conceptually similar to Δ_fluc and Δ_self, respectively. For example, while pure 3-center bonding without 2-center bonding, that is, y = 0 and t = n, is clearly conceivable, the situation Δ_self = 0, and Δ_fluc >0 is not possible. Thus, a one-to-one correspondence cannot even be expected to be found.

The key idea to circumvent this problem is to define from the initial two quantities of each framework, y and t, versus Δ_self and Δ_fluc, two derived quantities within each framework, Γ (concept) and SEP (concept) versus G and SES (and SEC), which can be used for pairwise comparison, Γ versus G, and SEP versus SES (and SEC), between the two frameworks. This will be shown in the following.

Another important difference between the styx picture and the present computational one employing quantities based on delocalization indices is the inclusion of an atomic definition, which leads to incorporation of bond-polarity effects not present in the conceptual styx picture. Effective atomic charges obtained from the actual DI calculations within the QTAIM framework has an effect on the cluster electron count and the cluster effective SES numbers. In order to relate conceptual quantities Γ and SEP to computational ones, this leads also to the additional requirement, that computational SES values are consistent with the corresponding G(cluster) values, that is, the same 2- and 3-center contributions have to be included. This way, to each DR G obtained for a certain 2-, 3-, or multiatomic unit (edge, triangle, cluster) there is also an exactly corresponding “bond count” SES, the former being the ratio, and the latter the sum of Δ_self and Δ_fluc. This is considered as the strength of the DR approach. Moreover, it is possible to relax this complete consistency between G values and associated SES values, by scaling the skeletal electron sharing values to yield skeletal electron counts SEC (Equation 27), which is an approximate procedure. The advantage gained is that SEC does represent an electron count like 2SEP and comparison between them is on the same footing, though the SEC values, while having the correct trend with cluster size, are significantly smaller (Figure 2) than the conceptual ones.

Summarizing, instead of chemically interpreting the initially obtained computational quantities Δ_self and Δ_fluc of the DI calculation, the two derived quantities G and SES will be shown to have a chemical meaning useful to discuss mixed 2- and 3-center bonding scenarios. They can be compared to conceptual Γ and SEPs, thus avoiding comparison of t, y with Δ_self and Δ_fluc.

In detail, the bond contributions used in the specific kind of G value for the borane cluster must be related to the Wade's number of skeletal electron pairs SEP implemented in the styx code, since SEP = t + y for closo-boranes. This has been achieved on the computational side by choosing the deltahedral variants G^Δ(B_n) and the corresponding SES₃^Δ with the same 2- and 3-center DIs included.

Equations (37-40) have a conceptual meaning only if 0 ≤ G^Δ(B_n) ≤ 1, such that values of ${\tilde{t}}^{\mathit{eff}}\left({\mathrm{B}}_n\right)$, ${\tilde{y}}^{\mathit{eff}}\left({\mathrm{B}}_n\right)$ and $t^{\mathit{eff}}\left({\mathrm{B}}_n\right)$, $y^{\mathit{eff}}\left({\mathrm{B}}_n\right)$ are positive.

4.1.1 Cluster-wise bonding

The relevant numerical results are given in Table 2. Comparison of the systematic evolution of the deltahedral boron hydride cluster DRs G^Δ(cluster) with conceptual Γ(n) values (Figure 3) reveals a clear correspondence. The computational curve is always found displaced to higher values with slight variations of the intercurve distances. The positive deviations of G^Δ(cluster) may be related to the electronic underpopulation of the B atoms obtained in all these clusters (Table 2), which has been already mentioned upon comparison of the computational N^eff(B_n) with conception N^el(B_n) values (Figure 2). Already B₄H₄ displays a significant underpopulation of about 0.6 ve, and for the charged species the additional 2-charge of the clusters is not completely transferred to B as well. It leads to an electronic underpopulation of about 0.6–0.7 ve, which is expected to cause an increased 3-center character of deltahedral cluster bonding. This is what is actually observed. The case of B₄H₄ is an important test case for the DR methodology, because it represents the upper boundary case of pure 3-center 2-electron bonding characterized by Γ(4) = 1. The value obtained G^Δ(B₄) = 1.01 demonstrates that the specific setup of the DR formula (Equation 8) conceptually related to the 2-electron system H₃¹⁺ also works for the multi-electron case. The slight overshooting of the conceptual boundary value of 1 is considered as nonsignificant. With this boundary value being fixed, it is obvious that cluster DR values between 0 and 1 conform to mixed 2- and 3-center character of bonding in multi-atomic multi-electronic chemical systems. As the conceptual counterpart C₄H₄ is discussed in the DR framework in Supporting Information.

The 2- and 3-center bond orders ${\tilde{y}}^{\mathrm{eff}}\left({\mathrm{B}}_n\right)\text{and}{\tilde{t}}^{\mathrm{eff}}\left({\mathrm{B}}_n\right)$, respectively, summing up to SES₃^Δ(B_n) are obtained from (Equations 37 and 38). The consistent usage of the same quantities Δ_self(B_n) and Δ_fluc(B_n) for G^Δ(B_n) and SES₃^Δ(B_n) values leads to the characteristic trend of ${\tilde{t}}^{\mathrm{eff}}\left(n\right)\text{and}{\tilde{y}}^{\mathrm{eff}}\left(n\right)$ values, which agrees fairly well with conceptual t(n), y(n) of the styx code (Figure 4A). The tendency of the G^Δ(B_n) values to increased 3-center character is visible in the separate ${\tilde{t}}^{\mathrm{eff}}\left(n\right)$ and ${\tilde{y}}^{\mathrm{eff}}\left(n\right)$ curves, where ${\tilde{t}}^{\mathrm{eff}}\left(n\right)$ is slightly too high, and ${\tilde{t}}^{\mathrm{eff}}\left(n\right)$ too low compared to styx values. Furthermore, the quantities denoted $t^{\mathrm{eff}}\left(n\right)$ and $y^{\mathrm{eff}}\left(n\right)$ (Figure 4B) correspond to electron pair counts (Equations 39 and 40) and have the same meaning as the Lipscomb styx values. They show the correct trend with increasing cluster size, but at too small values for both quantities. This is caused by the electronic underpopulation on the boron atoms compared to the Wade's 2SEP counts (Figure 2), which is directly passed also to SEC₃^Δ(B_n). In the remaining part of the contribution, the focus will be on the DRs and associated SES values, because they are directly related without a further approximation.

4.1.2 Local edge and triangle bonding

For the deltahedral boron hydride clusters n = 4, 6 12 with shapes of platonic solids all skeletal edges and faces are identical. In this case, the cluster-wise DR is identical to the local ones, that is, G^Δ(B_n) = $G^{\mathrm{\Delta }}\left(B,B^{\prime }\right)$ = $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}\left(B,B^{\prime },B^{\prime \prime }\right)$.

The remaining clusters feature skeletal species (vertices) with different number of skeletal connections (vertex degrees), which yields topologically different edges and faces characterized by different average vertex degrees. Within the framework of octet-rule fulfillment in the delocalized TORI (topological octet-rule implementation) framework,¹¹ it is found that locally averaged realizations of the styx code in the skeletal framework are dependent on local topological features as encoded in the average vertex degrees of each deltahedral edge and face. As a consequence, the skeletal edges and faces with higher average vertex degrees are expected to display the higher proportion of 3-center bonding contributions, that is, $G^{\mathrm{\Delta }}\left(B,B^{\prime }\right)$ and $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}\left(B,B^{\prime },B^{\prime \prime }\right)$ values, respectively, than those with lower ones.

The results of this analysis are depicted in Figure 5.

Among the remaining clusters, the cases n = 5, 7 are different from the other ones, because their symmetries allow for only one type of triangle. The presence of two topologically different types of vertices (atomic species), the apical (ap) and the equatorial (eq) ones, leads to two topologically different types of skeletal edges, namely B^eq–B^eq edges between vertices of like vertex degree and those between B^ap–B^eq vertices of different vertex degrees. The other clusters n = 8–11 generally display different types of edges and faces with different edge and vertex compositions.

In the triangular bipyramidal structure of B₅H₅²⁻ (D_3h) with vertex composition D3₂.4₃ (Figure 1) and $\overline{\mathit{vd}}\left(n\right)=3.6$, the two apical and three equatorial atoms display skeletal vertex degrees of 3 and 4, respectively. The additional inclusion of the exo-ligand yields a classical 4-coordination of the former, and a 5-coordination of the latter species. This topological situation yields two types of skeletal connections (edges), the six shorter B^ap–B^eq edges of vertex composition D3.4 (topologically under-connected with respect to $\overline{\mathit{vd}}\left(n\right)=3.6$), and the three longer B^eq–B^eq edges of composition D4₂ (over-connected). In contrast, in B₇H₇²⁻ (D_3h) with composition D5₂.4₅ (Figure 1) and $\overline{\mathit{vd}}\left(n\right)=4.3$, the B^ap–B^eq edges (D4.5) are now overconnected, and B^ap–B^eq (D4₂) edges are under-connected.

Within the framework of octet-rule fulfillment, allowed local realizations of the styx code in the skeletal framework are obtained.^{11, 35} As a consequence, it is expected, that the longer over-connected edges display a higher proportion of 3-center bonding contributions (and smaller 2-center DIs) than the shorter under-connected ones.

Specifically, the under-connected edges B^ap–B^eq in B₅H₅²⁻ ($\overline{\mathit{vd}}$ = 3.5), and B^eq–B^eq in B₇H₇²⁻ ($\overline{\mathit{vd}}$ = 4.0), are expected to display lower (higher) G₂ (2c-DI) values, than the over-connected edges B^eq–B^eq in B₅H₅²⁻ ($\overline{\mathit{vd}}$ = 4), and B^ap–B^eq in B₇H₇²⁻ ($\overline{\mathit{vd}}$ = 4.5).

The DR results are completely consistent with these expectations (Figure 5, top, middle). All deltahedral surface triangles Δ = (B^ap,B^eq,B^eq’) are topologically and symmetrically equivalent, which leads to G^Δ(B_n) = $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}\left(B,B^{\prime },B^{\prime \prime }\right)$.

Compared to the n = 5, 7 clusters, the clusters n = 8–11 with their vertex compositions D4_r.5_s.6_u (Figure 1) additionally display topologically different triangles characterized by different vertex compositions and average vertex degrees $\overline{\mathit{vd}}\left(\mathrm{B},{\mathrm{B}}^{\prime },{\mathrm{B}}^{″}\right)$ (Equation 43). Triangles with lower, higher or approximately equal average vertex degree than the cluster one $\overline{\mathit{vd}}\left(n\right)$, can be topologically classified as over-, under-, or equi-connected, respectively.

As discussed in detail for the n = 5, 7 clusters, these topological details of the polyhedral skeleton types are found to lead to local metric effects also in cluster structures n = 8–11 like edge-length variations, and concomitantly varying degrees of edge- and also triangle-wise electron delocalization, which can be monitored by delocalization indices and DRs. For deltahedral closo-boranes B_nH_n²⁻ with n = 5, 7, 8, 9, 10 it is observed, that edges with lower average vertex degree display smaller edge lengths, higher δ(B,B′) values (Figure 5, top), lower $G^{\mathrm{\Delta }}\left(B,B^{\prime }\right)$ values (Figure 5, middle), and lower $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}\left(B,B^{\prime },B^{\prime \prime }\right)$ values (Figure 5, bottom). In other words, edges and faces composed of atomic species on topologically under-connected sites display a lesser degree of three-center-wise delocalization than those composed of topologically overconnected sites, as generally expected. For clusters with n = 5, 7, 8, 9, 10 this relation is strictly valid. However, the occurrence of one clear exception from this correlation for n = 11 (Figure 5, top, middle bottom) shows, that there are in general additional effects at work that originate not only from average edge or triangle topology.

These types of effects are also responsible for the variations of the G₂^Δ values in crystalline α-Boron (Figure 5, middle), where all edges displaying the same vertex composition D5₂ are topologically similar. The α-Boron crystal structure displays distorted icosahedral B₁₂ clusters with two chemically different B atoms and four chemically different skeletal connections. Nevertheless, they still feature the same D5₁₂ vertex composition as the undistorted B₁₂H₁₂²⁻ cluster; the observable variations of the G₂^Δ values (Figure 5, top) are not caused by topological features of the deltahedron.

For the closo-clusters with only one type of deltahedral triangle (n = 4–7), the local triangle DR value is already given by the global value G^Δ(B_n). This is no longer the case for B_nH_n²⁻ clusters for n = 8–11. It is interesting to analyze the spread of partial DRs $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}{\left(B,B\prime ,B\prime \prime \right)}^{\mathrm{\Delta }i}$ for each type of triangle Δ_i in the less symmetrical clusters n = 8 to 11 and for the n = 12 distorted icosahedron in crystalline α-Boron. In analogy to the behavior found for the edges (Figure 5, top, middle), within each cluster also comparably large differences of local $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}{\left(B,B\prime ,B\prime \prime \right)}^{\mathrm{\Delta }i}$ values can occur (Figure 5, bottom). With the exception of the n = 11 cluster, the sequence of increasing partial $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}{\left(B,B\prime ,B\prime \prime \right)}^{\mathrm{\Delta }i}$ values again follows the sequence of average vertex composition for each triangle type Δi. The exceptional triangle with the lowest $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}{\left(B,B\prime ,B\prime \prime \right)}^{\mathrm{\Delta }i}$ value (Figure 5, bottom) has an exceptional vertex composition of D4.5.6, which is indicated by the average vertex degree $\overline{\mathit{vd}}=5.0$, while all other triangles with the same $\overline{\mathit{vd}}$ have composition D5₃ (indicated by $\overline{\mathit{vd}}=5$).

Especially large $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}{\left(B,B\prime ,B\prime \prime \right)}^{\mathrm{\Delta }i}$ values are found in B₉H₉²⁻ and B₁₁H₁₁²⁻, where they occur for the triangles with rather long edges of 199–199–199 pm (the only D5₃ type triangles in B₉H₉²⁻) and 187–202–202 pm (the only D5₂.6 type triangles in B₁₁H₁₁²⁻). These deltahedral triangles are characterized by small 3c-DIs (≈0.11) and small 2-center DIs (≈0.33) along the edges (Figure 5, top), which is quite different from the situation in B₄H₄ with similarly high $G_{\text{part}}^{\left\{\mathrm{\Delta }\right\}}\left(B,B^{\prime },B^{\prime \prime }\right)$ values, but with large deltahedral 3c-DIs (0.38) and 2c-DIs (0.78) observed. These large triangles appear with only a rather small frequency of 2, and their contribution to the G^Δ(cluster) values amounts to less than 10%. They seem to occur as a compromise for the sake of the overall cluster shape and stability.

Altogether, the local partial DR variations are well balanced, such that the G^Δ(cluster) values resemble fairly well the conceptual Γ(B_n) (Figures 3 and 5) ones, and yield in combination with SES₃^Δ values computational (${\tilde{t}}^{\mathrm{eff}},{\tilde{y}}^{\mathrm{eff}}$) values (Equations 37 and 38) in fair agreement with the conceptual (t, y) ones (Figure 4).