Orbitals, bonds and bands

When studying the electronic structure of crystalline solids, physicists tend to think in terms of the mathematical concept of electronic bands, the so-called dispersion 

Let us start by considering the general many-electron problem of Ng valence electrons, which contribute to chemical bonding, and A ion ions, which contain the nuclei and the tightly bound core electrons. The positions of the electrons and ions are given by r, and Rj, respectively, referred to the same arbitrary origin. This problem can be described quantum-mechanically, in the absence of external fields, by the Hamilton operator Ho  

In Eqs. (1.4a), (1.4b) and (1.4c) mg and Mj represent the electron and ion masses, respectively, and V the Laplacian operator (V = V. V = d /dx + d /dy + d /dz ). In Eq. (1.4a) we have inserted a Coulomb term for the electron-electron repulsive interactions and for Vion-ion and Vg-ion. the ion-ion and electron-ion interaction potentials, we leave open their explicit form but we assume that they can be described as sums over two-particle interactions. 

Since electrons are much faster than nuclei, owing to Wg Mj, ions can be considered as fixed and one can thus neglect the //ion-ion contribution (formally Mion-ion Hee, where Vion-ion is a Constant). This hrst approximation, as formulated by N. E. Born and J. R. Oppenheimer, reflects the instantaneous adaptation of electrons to atomic vibrations thus discarding any electron-phonon effects. Electron-phonon interactions can be a-posteriori included as a perturbation of the zero-order Hamiltonian Hq. This is particularly evident in the photoemission spectra of molecules in the gas phase, as already discussed in Section 1.1 for nJ, where the 7T state exhibits several lines separated by a constant quantized energy. 

The second important approximation that enables the resolution of Eq. (1.6) consists in considering that every electron is subject to an effective interaction potential V(ri), which takes into account the full attractive electron-ion interactions as well as somehow a part of the repulsive electron-electron interactions. Ideally we would like to express Hq in the form  

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Orbitals, bonds and bands We should thus solve the Schrodinger equation ... 

As orbitals spread into bands, orbitals oriented for a or a bonds spread into the widest bands. 7t orbitals form narrower bands and 5 bonding orbitals form the narrowest bands. 

The third band system, involving the removal of an electron from the 1 2 orbital, is vibrationally complex, consistent with the orbital being strongly bonding and favouring a linear molecule. Presumably both Vj and V2 are excited but the bands in this system are considerably broadened, making analysis unreliable. 

The structure of CaB contains bonding bands typical of the boron sublattice and capable of accommodating 20 electrons per CaB formula, and separated from antibonding bands by a relatively narrow gap (from 1.5 to 4.4 eV) . The B atoms of the B(, octahedron yield only 18 electrons thus a transfer of two electrons from the metal to the boron sublattice is necessary to stabilize the crystalline framework. The semiconducting properties of M B phases (M = Ca, Sr ", Ba, Eu, Yb ) and the metallic ones of M B or M B5 phases (Y, La, Ce, Pr, Nd ", Gd , Tb , Dy and Th ) are directly explained by this model . The validity of these models may be questionable because of the existence and stability of Na,Ba, Bft solid solutions and of KB, since they prove that the CaB -type structure is still stable when the electron contribution of the inserted atom is less than two . A detailed description of physical properties of hexaborides involves not only the bonding and antibonding B bands, but also bonds originating in the atomic orbitals of the inserted metal . ... 

Gas-surface interactions and reactions on surfaces play a crucial role in many technologically important areas such as corrosion, adhesion, synthesis of new materials, electrochemistry and heterogeneous catalysis. This chapter aims to describe the interaction of gases with metal surfaces in terms of chemical bonding. Molecular orbital and band structure theory are the basic tools for this. We limit ourselves to metals. 

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