Molecular Orbital Theory More Complex Examples

Because we are really only interested in the bonds involving the B-H-B bridges in diborane, we can therefore focus on just that portion of the molecule in our MO analysis. We begin the problem by forming four equivalent sp hybrid orbitals on the 

TABLE 10.17 Representations for the H and B orbitals involved in the B-H-B portion of diborane. 

Using the projection operator, the mathematical forms of the six SALCs (after normalization) are given by Equations (I0.50)-(I0.55)  

AOs on the two H atoms have the same energy before we combine them to make SALCs, but they have different energies after taking the linear combinations. Their Mulliken symbols are also shown in parentheses on the right-hand side of the MO diagram. 

The shapes of the bonding 0 g and b3 MOs clearly show that the electron pairs occupying these MOs are delocalized over all three nuclei. Because of the small size of the H atoms, there also exists a certain amount of direct B-B bonding in diborane that helps to stabilize the molecule. The MO diagram is also consistent with the PES spectrum of diborane, which shows six peaks of comparable intensity at 11.8, 13.3, 13.9, 14.7, 16.1, and 21.4 eV. Four of these peaks are due to ionization from the four terminal B-H bonding MOs (not shown in the figure), while the two peaks at 13.3 and 14.7 eV correspond with the and 0 g MOs, respectively, on the B-H-B bridges. 

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MOLECULAR ORBITAL THEORY MORE COMPLEX EXAMPLES... 

In this section we start, as in valence-bond theory, with a simple molecule, H2, and in the following sections extend the same principles to more complex molecules and solids. In every case, molecular orbitals are built by adding together—the technical term is superimposing—atomic orbitals belonging to the valence shells of the atoms in the molecule. For example, a molecular orbital for Fi2 is... 

The use of computational chemistry to address issues relative to process design was discussed in an article. The need for efficient software for massively parallel architectures was described. Methods to predict the electronic structure of molecules are described for the molecular orbital and density functional theory approaches. Two examples of electronic stracture calculations are given. The first shows that one can now make extremely accurate predictions of the thermochemistry of small molecules if one carefully considers all of the details such as zero-point energies, core-valence corrections, and relativistic corrections. The second example shows how more approximate computational methods, still based on high level electronic structure calculations, can be used to address a complex waste processing problem at a nuclear production facility (Dixon and Feller, 1999). 

A theory which shows greater applicability to bonding in cluster compounds is the Polyhedral Skeletal Electron Pair Theory (PSEPT) which allows the probable structure to be deduced from the total number of skeletal bond pairs (400). Molecular orbital calculations show that a closed polyhedron with n vertex atoms is held together by a total of (n + 1) skeletal bond pairs. A nido polyhedron, with one vertex vacant, is held together by (n + 2) skeletal bond pairs, and an arachno polyhedron, with two vacant vertices, by (n + 3) skeletal bond pairs. Further, more open structures are obtainable by adding additional pairs of electrons. This discussion of these polyhedral shapes is normally confined to metal atoms, but it is possible to consider an alkyne, RC=CR, either as an external ligand or as a source of two skeletal CR units. So that, for example, the cluster skeleton in the complex Co4(CO)10(RCCR), shown in Fig. 16, may be considered as a nido trigonal bipyramid (a butterfly cluster) with a coordinated alkyne or as a closo octahedron with two carbon atoms in the core. 

F. Fenske. We demonstrate for transition metal complexes that the non-empirical Fenske-Hall (FH) approach provides qualitative results that are quite similar to the more rigorous treatment given by density functional theory (DFT) and are quite different from Hartree-Fock-Roothaan (HFR) calculations which have no electron correlation. For example, the highest occupied molecular orbital of ferrocene is metal based for both DFT and FH while it is ligand (cyclopentadienyl) based for HFR. In the doublet (S = 1/2) cluster, Cp2Ni2(pi-S)2(MnCO)3, the unpaired electron is delocalized over the complex in agreement with the DFT and FH results, but localized on Mn in the HFR calculation. A brief description of the theory of FH calculations is used to rationalize the origin of its similarity to DFT. 

Application of valence bond theory to more complex molecules usually proceeds by writing as many plausible Lewis structures as possible which correspond to the correct molecular connectivity. Valence bond theory assumes that the actual molecule is a hybrid of these canonical forms. A mathematical description of the molecule, the molecular wave function, is given by the sum of the products of the individual wave functions and weighting factors proportional to the contribution of the canonical forms to the overall structure. As a simple example, the hydrogen chloride molecule would be considered to be a hybrid of the limiting canonical forms H—Cl, H Cr, and H C1. The mathematical treatment of molecular structure in terms of valence bond theory can be expanded to encompass more complex molecules. However, as the number of atoms and electrons increases, the mathematical expression of the structure, the wave function, rapidly becomes complex. For this reason, qualitative concepts which arise from the valence bond treatment of simple molecules have been applied to larger molecules. The key ideas that are used to adapt the concepts of valence bond theory to complex molecules are hybridization and resonance. In this qualitative form, valence bond theory describes molecules in terms of orbitals which are mainly localized between two atoms. The shapes of these orbitals are assumed to be similar to those of orbitals described by more quantitative treatment of simpler molecules. 

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