## 2 The Hume-Rothery Electron Concentration Rule  
The cohesive energy of a solid refers to the energy necessary to separate its constituent atoms from each other and bring them to an assembly of neutral free atoms. Therefore, in forming solids, cohesive energy is gained by lowering the total energy relative to that of the free atoms through the formation of different bonding types, namely, ionic, covalent, and metallic. Since ionic bonding originates from electrostatic interaction among charged atoms, it is not strictly necessary to consider the overlap of wave functions between neighboring atoms to evaluate the cohesive energy of these compounds. Conversely, in both covalent and metallic bonding types one must consider the formation of a valence band arising from quantum resonance effects involving the atomic orbitals of condensing atoms. Thus, as atoms get closer and closer to each other, the atomic orbitals in the original assembly of neutral atoms are lifted and split leading to the formation of a valence band upon hybridization. In this picture, the fundamental reason for the existence of the metallic state is that in an isolated metal atom the valence electrons occupy positions close to the upper edge of the potential well, so that the perturbations introduced during the condensation processes by neighboring metal atoms lead to delocalization of the valence electrons [25]. Accordingly, metallic bonding is described by considering that the free electrons charge is uniformly distributed throughout an ordered array of potentials due to positive ions and that the total charge of electrons exactly balances that of the ions. Along with the electrostatic energy among electrons and ions, one must include in the total energy budget the potential energy of electrons moving in the ionic lattice, the electron-electron interactions and the kinetic energy of electrons. Since no more than two electrons can occupy the same quantum state, the available electronic states are progressively filled up to the Fermi energy, \(\mathrm{E_{F}}\), and the resulting kinetic energy of the electronic system systematically increases with the electron concentration. Thus, in the free electron model the average kinetic energy per electron is given by \(<\)T\({}_{\mathrm{e}}\)\(>\) = 3E\({}_{\mathrm{F}}\)/5 = 2903/(r\({}_{\mathrm{s}}\)/a\({}_{0}\))\({}^{2}\) kJ mol\({}^{-1}\), where r\({}_{\mathrm{s}}\) is the Fermi-surface radius and a\({}_{0}\) is the lattice parameter [26; 27]. Hence, since \(<\)T\({}_{\mathrm{e}}\)\(>\) 0, the larger the Fermi-surface size the more favorable the metallic bonding formation. Therefore, a variety of metallic phases are stabilized in nature by mechanisms able to lower the kinetic energy of electrons as much as possible. Of particular interest to us is the so-called Hume-Rothery concentration rule, since it was systematically exploited by [PERSON] and co-workers as a useful guide to synthesize the first generation of high-quality, thermodynamically-stable QCs mentioned in Section 1. This rule accounts for the  
Figure 3: Two examples of different kinds of critical wave functions in one-dimensional Fibonacci QCs: (**a**) Electronic charge distribution for an extended state with E = \(-13/50=-0.26\) and a transmission coefficient T = 1 (reprinted from [PERSON]. _Aperiodic Structures in Condensed Matter: Fundamentals and Applications_; CRC Press: Boca Raton, FL, 2009. With permission from CRC Taylor & Francis group); (**b**) electronic charge distribution for a low diffusivity state with E = \(-0.335440\dots\) and a transmission coefficient T = \(0.4689\dots\)stabilization of certain alloys and compounds with closely related structures that exhibit the same ratio of number of valence electrons to number of atoms (the so-called electron-per-atom ratio, e/a) [27; 28].