## 7 From Metallic to Covalent Bonding Styles  
Some time ago, it was pointed out that the presence of a pronounced hollow feature in the DOS, due to hybridization of a narrow d band with a broad sp band, may be regarded as a common feature in transition-metal aluminides. Whether these hollows become actual gaps, as it was predicted for RuAl\({}_{2}\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**QC System** & \(\varrho\)**(4 K) (mTcm)** & **R** \\ \hline AlLiCu & 0.87 & 1.1 \\ AlCuFe & 2.6–11 & 1.2–2.5 \\ AlPdMn & 1.5–10 & 1.1–2.5 \\ AlCuRu & 1.7–51 & 1.2–4.2 \\ AlCuOs & 77–143 & 4.0–4.5 \\ AlPdRe & 2–2000 & 1.8–280 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Chemical trends in the low-temperature electrical resistivity (\(\varrho\)) and the electrical resistivity ratio (R) defined as R = \(\varrho\)(4 K)/\(\varrho\)(300 K) of Al-based i-QCs [57; 69; 70; 71; 72; 73; 74; 75]. Figure 9: **(a)** Room temperature electronic resistivity values retrieved from the literature [14; 76; 77] as a function of the e/a ratio for i-Al\({}_{6}\)Cu\({}_{35<x}\)Ru\({}_{x}\) QCs. **(b)** Occupied state distribution in i-Al\({}_{65}\)Cu\({}_{20}\)Ru\({}_{15}\): Al 3p (dotted line), Al 3s,d (starred line), Cu 3d (line with crosses), and Ru 4d (line with triangles) (courtesy [PERSON]). FeAl\({}_{2}\), and Fe\({}_{2}\)NbAl alloys [18; 81], will depend on fine details of the electronic structure, including the relative positions of the atomic levels, the size of the atoms, the crystal structure, and the number of valence electrons. These results inspired the search for semiconducting QCs, and the electronic structure of several hypothetical approximants corresponding to Al-TM i-QCs was studied by using density functional calculations. In this way, the possible existence of semiconducting approximants and even band insulators in Al-based QCs was put forward [82; 83; 84]. These phases should exhibit a QP chemical bond network extending over long distances, which could only be established on an underlying high structural QP arrangement of atoms throughout the space. Now, since the main building blocks of i-QCs are assumed to be hierarchically-arranged atomic clusters one must consider bonds among atoms inside clusters along with cluster-cluster interactions giving rise to the extended chemical network. The bonding nature of icosahedral clusters of the group 13 elements was earlier investigated to understand the relation between properties of clusters and their own aggregates. By means of molecular orbital quantum calculations it was found that Al\({}_{12}\) and Al\({}_{13}\) atomic icosahedra (the latter with one atom located at the center) exhibited covalent and metallic-type bonding, respectively (see Figure 10a, top panel). Thus, atomic occupation of the icosahedron center induces a metallic-covalent bonding conversion phenomenon in these monoatomic clusters [85]. When going from a metallic to a covalent bonding scenario one may expect to observe some general features in the atomic arrangement of the system, such as a smaller coordination number around most atoms, a sharper directionality of the electronic charge distribution among them, as well as shorter interatomic distances. Indeed, it is well known that different bonding styles (i.e., ionic, covalent, metallic, Van der Waals) are characterized by different values of their corresponding effective radii. Accordingly, it was found that the distance between neighboring Al atoms in Al-based i-QCs is 0.24 nm, approximately 10% shorter than that of fcc-type Al elemen  
Figure 10: (**a**) Electronic charge equidensity surfaces (0.35 e/Å\({}^{3}\)) of the Mackay icosahedral cluster first (top panel) and second (bottom panel) shells in the \(\alpha\)-AlReSi approximant crystal; (**b**) section contour map of charge density of \(\alpha\)-AlReSi in the range 0.00–1.20 e/Å\({}^{3}\), with a step of 0.10 e/Å\({}^{3}\).