Orbitals in solids

In a lattice, the set of atomic orbitals are ideal for the formation of Bloch functions. A band can be obtained by combining the atomic orbital functions. This model can be said to be composed of the orbital functions of the solid. Eq. (2.15) describes the combination for the one-dimensional case 

If one looks at a chain of atoms (in one dimension) the region between = 0 and n/a determines the smallest cell of the reciprocal space. This is called a Brillouin zone. 

Each kj. represents one orbital of the sohd. If the length of the chain reaches typical numbers of atoms in macroscopic dimensions, a very large number of crystal orbitals are formed. On the energy scale a single orbital is no longer distinguishable a band is formed. A similar result was obtained with the nearly free electron model. 

A short description of molecular orbitals in sohds will be given in the following text. As was aheady mentioned, a comprehensive treatment of the orbital concept of bonding in sohds was presented by R. Hoffmann.  

---

It is known that two dominant inter-site interactions between the orbitals in solids are the superexchange-type and the cooperative Jahn-Teller type interactions. The former is attributed to the virtual exchange of an electron under the strong on-site Coulomb interaction [8-11], The explicit form of this interaction is given by... 

Nonbonding 4/electrons have been predicted and found spectroscopically to have comparable binding energies than outer bonding orbitals in solids. It is this fact combined with the knowledge that the 4/orbitals have a mean radius of about 0.3 A, i.e. smaller than that of the 5p and 5s electrons, and are localized, that makes the RE so unique in the periodic table. In contrast both 3d and 5/orbitals (20) are known to show significant hybridization with outer valence states. 

The generalisation of this kind of procedure is the generation of so-called Bloch orbitals in solid-state theory, where translational symmetry will generate one symmetry orbital from amy orbital in the unit cell auid the variation problem is solved in terms of orbitals of the same symmetry type generated from different orbitals in the same unit cell. 

S. Svensson, N. Mftrtensson, E. Basilier, P.A. Malmquist, U. Gelius, K. Siegbahn Core and valence orbitals in solid and gaseous mercury by means of ESCA. J. Electr. Spectrosc. Rel. Phenom. 9, 51 (1976)... 

For solids, the number of atoms and therefore the number of orbitals goes to infinity. The valence orbitals develop into continuous energy bands. The details of the electronic structure of a crystalline solid are described in terms of the band structure. The energy up to which the bands are occupied is the Fermi energy (Ef) already mentioned in Section I.E. When the Fermi energy lies within a band, the solid is a metal when it lies in a gap between two bands, the solid is a semiconductor (small gap) or an insulator (large gap). As in molecules, the core orbitals in solids still behave like those in atoms. 

Electronic Properties When molecular orbitals are formed from N atoms, atomic orbital combined to form N molecular orbitals. In solids, N is very large, resulting in a large number of orbitals [40]. The overlap of a large number of orbitals leads to closely spaced molecular orbitals which form a virtually continuous band (Fig. 3) [41]. The overlap of the highest occupied molecular orbitals (HOMO) results in the formation of a valence band and a conduction band is formed from... 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

The orbitals and orbital energies produced by an atomic HF-Xa calculation differ in several ways from those produced by standard HF calculations. First of all, the Koopmans theorem is not valid and so the orbital energies do not give a direct estimate of the ionization energy. A key difference between standard HF and HF-Xa theories is the way we eoneeive the occupation number u. In standard HF theory, we deal with doubly oecupied, singly occupied and virtual orbitals for which v = 2, 1 and 0 respectively. In solid-state theory, it is eonventional to think about the oecupation number as a continuous variable that can take any value between 0 and 2. 

These reactions show sulfur in the role of an oxidizing agent. The properties of compounds such as ZnS suggest they contain the sulfide ion, S-2. The formation of this ion again can be expected on the basis of the fact that the neutral sulfur atom has two electrons less than enough to fill the valence orbitals. Acquisition of two electrons completely fills the low energy valence orbitals and solid ionic compounds can be formed. 

Conjugated polymers are generally poor conductors unless they have been doped (oxidized or reduced) to generate mobile charge carriers. This can be explained by the schematic band diagrams shown in Fig. I.23 Polymerization causes the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the monomer to split into n and n bands. In solid-state terminology these are the valence and conduction bands, respectively. In the neutral forms shown in Structures 1-4, the valence band is filled, the conduction band is empty, and the band gap (Eg) is typically 2-3 eV.24 There is therefore little intrinsic conductivity. 

Electrical conduction in metals can be explained in terms of molecular orbitals that spread throughout the solid. We have already seen that, when N atomic orbitals merge together in a molecule, they form N molecular orbitals. The same is true of a metal but, for a metal, N is enormous (about 1023 for 10 g of copper, for example). Instead of the few molecular orbitals with widely spaced energies typical of small molecules, the huge number of molecular orbitals in a metal are so close together in energy that they form a nearly continuous band (Fig. 3.43). 

Bonding in solids may be described in terms of bands of molecular orbitals. In metals, the conduction bands are incompletely filled orbitals that allow electrons to flow. In insulators, the valence bands are full and the large band gap prevents the promotion of electrons to empty orbitals. 

There are two possible structures for simple alkyl radicals. They might have sp bonding, in which case the structure would be planar, with the odd electron in ap orbital, or the bonding might be sp, which would make the structure pyramidal and place the odd electron in an sp orbital. The ESR spectra of CHs and other simple alkyl radicals as well as other evidence indicate that these radicals have planar structures.This is in accord with the known loss of optical activity when a free radical is generated at a chiral carbon. In addition, electronic spectra of the CH3 and CD3 radicals (generated by flash photolysis) in the gas phase have definitely established that under these conditions the radicals are planar or near planar. The IR spectra of CH3 trapped in solid argon led to a similar conclusion. " °... 

The method involves the irradiation of a sample with polychromatic X-rays (synchrotron radiation) which inter alia promote electrons from the innermost Is level of the sulfur atom to the lowest unoccupied molecular orbitals. In the present case these are the S-S antibonding ct -MOs. The intensity of the absorption lines resulting from these electronic excitations are proportional to the number of such bonds in the molecule. Therefore, the spectra of sulfur compounds show significant differences in the positions and/or the relative intensities of the absorption lines [215, 220, 221]. In principle, solid, liquid and gaseous samples can be measured. 

Figure 3. Molecular-orbital diagrams as obtained by the ROHF method. Dashed lines indicate MOs dominated by the metal d-orbitals, the solid lines stand for doubly occupied or virtual ligand orbitals. Orbitals which are close in energy are presented as degenerate the average deviation from degeneracy is approximately 0.01 a.u. In the case of a septet state (S=3), the singly occupied open-shell orbitals come from a separate Fock operator and their orbital energies do not relate to ionization potentials as do the doubly occupied MOs (i.e. Koopmann s approximation). For these reasons, the open-shell orbitals appear well below the doubly occupied metal orbitals. Doubly occupying these gives rise to excited states, see text.

Fig. 10.2 Crystal Orbital Overlap Population (COOP) and Densities of States (DOS) plots for SrCa2ln2Ce (a) COOP plots of the In-In (solid) and In-Ge (dashed) interactions (b) DOS plots of the total DOS (dotted), ln-5py lone pair (dashed), and ln-5px p-states (solid).

As described in Section 10-, the bonding in solid metals comes from electrons in highly delocalized valence orbitals. There are so many such orbitals that they form energy bands, giving the valence electrons high mobility. Consequently, each metal atom can be viewed as a cation embedded in a sea of mobile valence electrons. The properties of metals can be explained on the basis of this picture. Section 10- describes the most obvious of these properties, electrical conductivity. 

The description derived above gives useful insight into the general characteristics of the band structure in solids. In reality, band structure is far more complex than suggested by Fig. 6.16, as a result of the inclusion of three dimensions, and due to the presence of many types of orbitals that form bands. The detailed electronic structure determines the physical and chemical properties of the solids, in particular whether a solid is a conductor, semiconductor, or insulator (Fig. 6.17). 

Molecular Orbital Theory and Chemical Bonding in Solids... 

MOLECULAR ORBITAL THEORY AND CHEMICAL BONDING IN SOLIDS... 

---