Crystalline solids molecular orbitals

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

The molecular orbital treatment of a crystalline solid considers the outer electrons as belonging to the crystal as a whole (10,11). Sommer-feld s early free electron theory of metals neglected the field resulting... 

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

A crystalline solid can be considered as a huge, single molecule subsequently, the electronic wave functions of this giant molecule can be constructed with the help of the molecular orbital (MO) methodology [19]. That is, the electrons are introduced into crystal orbitals, which are extended along the entire crystal, where each crystal orbital can accommodate two electrons with opposite spins. A good approximation for the construction of a crystal MO is the linear combination of atomic orbitals (LCAO) method, where the MOs are constructed as a LCAO of the atoms composing the crystal [19]. 

Polymer Conformation and Crystallinity. Beyond the stereoregularity and tacticity, the geometrical conformation of the polymer chain in the solid material could influence its electronic structure, through a modification of its valence band molecular orbitals. Indeed, a few years ago, very characteristic band structures were calculated for T, G, TG, and TGTG polyethylenes ( ). More recently. Extended Huckel crystal orbital calculations showed that for isotactic polypropylene, a zig-zag planar or a helical conformation resulted in significant changes in the theoretical valence band spectra, supporting the idea that conformation effects could be detected experimentally by the XPS method ( ). 

Hartree-Fock calculations on molecules commonly exploit the symmetry of the molecular point group to simplify calculations such studies on perfectly ordered bulk crystalline solids are possible if one exploits the translational symmetry of the crystalline lattice (see Ashcroft and Mermin, 1976) as well as the local symmetry of the unit cell. From orbitals centered on various nuclei within the unit cell of the crystal Bloch orbitals are generated, as given by the formula (in one dimension) ... 

Figure 2 shows the energy-level diagram of the molecular orbitals of the 1-D silicon clusters (SiH2)nH2 with silicon 3d orbitals. The levels shown by broken lines are unoccupied. These unoccupied levels correspond to the conduction band in crystalline silicon. The occupied levels shown by solid lines around — 17 to —13eV and —11 to —9eV are the valence orbitals mainly localized on silicon 3s and 3p orbitals, respectively. The unoccupied levels are the... 

Solid-state chemistry uses the same principles for bonding as those for molecules. The differences from molecular bonding come from the magnitude of the molecules in the solid state. In many cases, a macroscopic crystal can reasonably be described as a single molecule, with molecular orbitals extending throughout. This description leads to significant differences in the molecular orbitals and behavior of solids compared with those of small molecules. There are two major classifications of solid materials crystals and amorphous materials. Our attention in this chapter is on crystalline solids composed of atoms or ions. 

Generally, as we shall see in this Chapter, there is only one conformation of a molecule in any one crystal structure. One of the most common questions asked by solution chemists about the results of a crystal structure analysis is how can one be sure that the solid-state conformation is the same as that observed in solution The conformation found for a flexible molecule in the crystalline state is that of one of the various conformers found in solution. This has been verified by other physical methods such as nuclear magnetic resonance. If, however, a molecule is found to have the same conformation in several different crystal structures, it is reasonable to assume that this conformation has a low (although not necessarily the lowest) energy. This assumption can often be tested by calculation (by ah initio molecular orbital calculations, for example) of the appropriate theoretical potential energy curve. 

The methylidyne cubane [Cp Ti(/x-CH)]4 (Scheme 202) is obtained as a dark brown crystalline solid by thermolysis of Cp TiMe in toluene with methane elimination. This transformation was monitored by NMR and no intermediates are observed. The signals assignable to the methylidyne groups appear as singlets at 6 17.75 in the NMR and at 6 490.8 in the 13C NMR spectra. In order to analyze the interaction between the /i3-ligand and the titanium centers, extended Huckel molecular orbital calculations have been carried out. In contrast, the thermolysis of the trinuclear oxo alkyls [Cp Ti(/i-0)(CH2R)]3 (R = H, Me) affords the //3-alkylidyne derivatives [Cp Ti(//-0)]3-(/x3-CR).505-507... 

---