Closely spaced molecular orbitals

As an illustration, picture a magnesium metal crystal, which has an hep structure. Since each magnesium atom has one 35 and three 3p valence atomic orbitals, a crystal with n magnesium atoms has available n is) and 3n 3p) orbitals to form the molecular orbitals, as illustrated in Fig. 10.20. Note that the core electrons are localized, as shown by their presence in the energy "well" around each magnesium atom. Flowever, the valence electrons occupy closely spaced molecular orbitals, which are only partially filled. 

Conduction bands closely spaced molecular orbitals with empty electron spaces... 

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... 

Recall that in the MO model for the gaseous Li2 molecule (Section 9.3), two widely spaced molecular orbital energy levels (bonding and antibonding) result when two identical atomic orbitals interact. Flowever, when many metal atoms interact, as in a metal crystal, the large number of resulting molecular orbitals become more closely spaced and finally form a virtual continuum of levels, called bands, as shown in Fig. 10.19. 

For C70, molecular orbital calculations [60] reveal a large number of closely-spaced orbitals both above and below the HOMO-LUMO gap [60]. The large number of orbitals makes it difficult to assign particular groups of transitions to structure observed in the solution spectra of C70. UV-visible solution spectra for higher fullerenes (C n = 76,78,82,84,90,96) have also been reported [37, 39, 72]. 

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). 

For the purpose of this presentation we limit ourselves to closed-shell systems, i.e. those systems consisting of an even number, 2N, of electrons which doubly occupy N space orbitals uy,u2,...uN> each with a and 3 spin. The appropriate molecular orbital wavefunction is then given by... 

The electronic properties of solids can be described by various theories which complement each other. For example band theory is suited for the analysis of the effect of a crystal lattice on the energy of the electrons. When the isolated atoms, which are characterized by filled or vacant orbitals, are assembled into a lattice containing ca. 5 x 1022 atoms cm 3, new molecular orbitals form (Bard, 1980). These orbitals are so closely spaced that they form essentially continuous bands the filled bonding orbitals form the valence band (vb) and the vacant antibonding orbitals form the conduction band (cb) (Fig. 10.5). These bands are separated by a forbidden region or band gap of energy Eg (eV). 

The number No of occupied valence SCF orbitals in a molecule is typically less than the total number Nmb of orbitals in the minimal valence basis sets of all atoms. The full valence MCSCF wavefunction is the optimal expansion in terms of all configurations that can be generated from N b molecular orbitals. Closely related is the full MCSCF wavefunction of all configurations that can be generated from Ne orbitals, where Nc is the number of valence electrons, i.e. each occupied valence orbital has a correlating orbital, as first postulated by Boys (48) and also presumed in perfect pairing models (49,50), We shall call these two types of frill spaces FORS 1 and FORS 2. In both, the inner shell remains closed. 

In general, a molecular-centered basis set is not suitable for constructing a function which does not approach spherical symmetry and have most of its structure close to the origin. For example, an extensive linear combination of molecule-centered atomiclike orbitals would be needed to construct the nodes in a b2g molecular orbital of benzene. Also, because the interference effects are specifically characteristic of the interplay between electron wavelength and the set of internuclear spacings, a molecule-centered basis set will not adequately describe interference effects. 

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