Molecular Orbital Theory pi Orbitals

TABLE 10.13 Symmetric and antisymmetric components of the direct product e x e in the point group. 

In all of the polyatomic MO diagrams that we have examined up until this point, the ligands have always been H atoms. Because the SALCs derived from Is AOs always 

FIGURE 10.36 The effects of the two F molecular vibrations on the trigonal planar geometry. These are the only possible Jahn-Teller distortions for the D, 

Solution. The AOs for C will lie higher in energy than those for O because of the smaller effective nuclear change on the carbon nucleus. The symmetries of the C AOs in the point group are as follows 2s = (Tg , 2p =, and 2p,  

By this point in time, you should not even need to follow the formal procedure of the projection operator method to determine the symmetries and shapes of the SALCs. The Tg+ SALC must be totally symmetric to all of the symmetry operations. The 7 SALC must be antisymmetric with respect to all of the inversion, S, and C2 operations. The jt SALCs will have a nodal plane containing the intemuclear axis, with symmetric with respect to inversion and antisymmetric to inversion. The shapes of the SALCs can be seen in the one-electron MO diagram, where the energies and shapes of the MOs were calculated using Wavefunction s Spartan Student Edition, version 5.0. 

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A covalent bond is formed when an electron pair is shared between atoms. According to valence bond theory, electron sharing occurs by overlap of two atomic orbitals. According to molecular orbital (MO) theory, bonds result from the mathematical combination of atomic orbitals to give molecular orbitals, which belong to the entire molecule. Bonds that have a circular cross-section and are formed by head-on interaction are called sigma (cr) bonds bonds formed by sideways interaction ot p orbitals are called pi (77-) bonds. 

In the early years of quantum theory, Hiickel developed a remarkably simple form of MO theory that retains great influence on the concepts of organic chemistry to this day. The Hiickel molecular orbital (HMO) picture for a planar conjugated pi network is based on the assumption of a minimal basis of orthonormal p-type AOs pr and an effective pi-Hamiltonian h(ctT) with matrix elements... 

The basis of the VSEPR theory is that the shape of a molecule (or the geometry around any particular atom connected to at least two other atoms) is assumed to be dependent upon the minimization of the repulsive forces operating between the pairs of sigma (a) valence electrons. This is an important restriction. Any pi (7t) or delta (8) pairs are discounted in arriving at a decision about the molecular shape. The terms sigma , pi and delta refer to the type of overlap undertaken by the contributory atomic orbitals in producing the molecular orbitals, and are referred to by their Greek-letter symbols in the remainder of the book. 

It is apparent that the molecular orbital theory is a very useful method of classifying the ground and excited states of small molecules. The transition metal complexes occupy a special place here, and the last chapter is devoted entirely to this subject. We believe that modem inorganic chemists should be acquainted with the methods of the theory, and that they will find approximate one-electron calculations as helpful as the organic chemists have found simple Hiickel calculations. For this reason, we have included a calculation of the permanganate ion in Chapter 8. On the other hand, we have not considered conjugated pi systems because they are excellently discussed in a number of books. 

We start with some biographical notes on Erich Huckel, in the context of which we also mention the merits of Otto Schmidt, the inventor of the free-electron model. The basic assumptions behind the HMO (Huckel Molecular Orbital) model are discussed, and those aspects of this model are reviewed that make it still a powerful tool in Theoretical Chemistry. We ask whether HMO should be regarded as semiempirical or parameter-free. We present closed solutions for special classes of molecules, review the important concept of alternant hydrocarbons and point out how useful perturbation theory within the HMO model is. We then come to bond alternation and the question whether the pi or the sigma bonds are responsible for bond delocalization in benzene and related molecules. Mobius hydrocarbons and diamagnetic ring currents are other topics. We come to optimistic conclusions as to the further role of the HMO model, not as an approximation for the solution of the Schrodinger equation, but as a way towards the understanding of some aspects of the Chemical Bond. 

In Fig. 1 we show the correlation between E and experimental heats of formation for the (complete) set of C22H14 benzenoid isomers. For comparison we also present some recent data for the same set of compounds, obtained by a semiempirical MNDO method [21] and by the MMX/PI version of molecular mechanics calculations [22], The only conclusion we wish to draw from Fig. 1 is that HMO theory is capable of reproducing the experimental enthalpies of benzenoid hydrocarbons with an accuracy which is not much worse than that of the much more sophisticated (and highly parametrized) molecular orbital and molecular mechanics approaches. 

A more quantitative approach, using molecular orbital theory and the known frequencies of the molecular bending modes, to show that the pi bonding term is proportional to the number of vacancies in the t%g metal -orbitals has also been made (70). The populations of the three halogen /(-orbitals, and their variance with bending vibrations (assumed to be harmonic), were calculated by means of molecular orbital... 

MOLECULAR ORBITAL THEORY FOR CYCLIC CONJUGATED PI SYSTEMS... 

Section 12.2 Molecular Orbital Theory for Linear Pi Systems... 

Section 12.3 Molecular Orbital Theory For Cyclic Conjugated Pi Systems... 

We can now state that each carbon-to-carbon linkage in benzene contains a sigma bond and a partial pi bond. The bond order between any two adjacent carbon atoms is therefore between 1 and 2. Thus molecular orbital theory offers an alternative to the resonance approach, which is based on valence bond theory. (The resonance structures of benzene are shown on p. 349.)... 

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. 

In this section, we focus on the mode by which orbitals overlap—end to end or side to side—to see the detailed makeup of covalent bonds. These two modes give rise to the two types of covalent bonds—sigma bonds and pi bonds. WeTl use valence bond theory to describe the two types here, but they are essential features of molecular orbital theory as well. 

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