The MO Approximation

For unsubstituted aromatic hydrocarbons all the carbon atoms are assigned the same Coulomb integral (a) and all C—C bonds are assigned the same resonance Integral (/3). 

The resonance integral of the 7r-bond between the heteroatom and carbon is another possible parameter in the treatment of heteroatomic molecules. However, for nitrogen compounds more detailed calculations have suggested that this resonance integral is similar to that for a C—C bond and moreover the relative values of the reactivity Indices at different positions are not very sensitive to change in this parameter. 

Theoretical reactivity indices of heteroaromatic systems distinguish reactivity toward electrophilic, nucleophilic and homolytic reactions. 

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To make the connection with MO theory, consider the SOPP approximation of valence bond theory and write the ground state wave function as in Eq. (4). For the MO approximation and are the same and this leads to... 

Eq. (5) where the are localized molecular orbitals (LMOs) which are related to the familiar CMOs by a unitary transformation. Because of the orthogonality of molecular orbitals the autocorrelation function for the MO approximation is simply the sum of the time-dependent probability amplitudes for the hole to be in the various LMOs,... 

Syntheses and properties of cyclobutadiene-transition metal complexes have been discussed in detail by Maitlis 167). Brown 168), and others 169) have reviewed the metal-ligand bond in terms of the MO approximation. The main bonding in these complexes is due to an overlap of the two degenerate nonbonding cyclobutadiene orbitals with spd hybrid metal atomic orbitals. 

This is true only for the MO approximation. Ab initio VB calculations by Palke (1986) on ethylene show the equivalent banana bond description of the double bond in terms of equivalent nonorthogonal hybrids to be more stable than the a-n description. 

One of the deficiencies of the MO methods, especially the simple ones, is that they tend to exaggerate uneven distribution of electrons in a molecule and thus make it more polar (with a higher dipole moment) than it actually is. The result is that dipole moments which are based on charge densities obtained from eigenfunctions of the MO approximations are usually considerably higher than the actual experimental dipole moments. Two old, well-known examples of theoretical dipole moments obtained by the HMO method are fiilvene (11) whose calculated dipole moment is 4.7 D [72-74] and the experimental value is 1.2 D [68], and azulene (15), with a calculated dipole moment of 6.9 D [72] and the experimental value of 1.0 D [68]. 

The MO approximation puts the electrons of a molecule in molecular orbitals, which extend over the whole molecule. As an approximation to the molecular orbitals, we usually use linear combinations of atomic orbitals. The VB method puts the electrons of a molecule in atomic orbitals and constructs the molecular wave function by allowing for exchange of the valence electron pairs between the atomic orbitals of the bonding atoms. We compared the two methods for H2. We now consider other homonuclear diatomic molecules. 

In the MO approach, H 2 has the ground-state configuration (o-gls) (o- ls). With no net bonding electrons, no bonding is predicted, in agreement with the VB method. The MO approximation to the wave function is... 

The MO approximation to the ground state of water is a Slater determinant of the form... 

The MO approximation represents a rough approximation to reality. So is Koopmans s theorem, which proves to be poorly satisfied for most molecules. But these approximations are often used for practical purposes. This is illustmted by a certain quantitative relationship, derived by Grochala et al. ... 

Unfortunately, VB wave functions are hard to use for molecules larger than H2. Slater determinants are easier to handle and generally give a rather good approximation at the equilibrium positions of the nuclei. We therefore primarily use the MO approximation as a basis for our discussions. 

The notation just introduced is rather more than a convenient shorthand for specifying which orbitals are occupied and by how many electrons. It expresses the fact that the MO approximation to the molecular wave function is a product of one-electron wave functions, i.e. orbitals, each taken to a power equal to the number of electrons occupying it. We recall that the irreducible representation of a product of coordinates is the direct product of their irreps extending the same idea to the product of orbitals, we see that the irrep of an electron configuration is simply the direct product of the irreps of its occupied... 

In the present case, we assume that we know the total energy of the solvent and solute molecules when they are isolated. The interaction energy can then be found by a straightforward classical electrostatic calculation. In the MO approximation, this is particularly simple because we regard... 

Longuet-Higgins and Abrahamson and has been rather generally adopted. This is based on the idea that if a molecule has symmetry and if the symmetry is conserved during a reaction, the wave function will tend to retain similar symmetry. Thus if the molecule has a plane of symmetry, and if the wave function is initially symmetric with respect to reflection in that plane, and if the plane of symmetry is retained throughout a reaction, then the wavefunction must remain symmetric. In the MO approximation, it can be shown that similar restrictions apply to MOs. This has led to the interpretation of pericyclic reactions in terms of the conservation of orbital symmetry during them. 

The MO approximation is based on the assumption that an individual spin orbital (pioc or (piP (where cpi is the function of space coordinates and a (m = 1/2) or P(nis= — 1/2) are the functions of spin coordinates) corresponds to each electron. The full wave function of a many-electron system ip in the Hartree-Fock approximation is written as a Slater determinant whose form provides for the property of antisymmetry of ip, required by the Pauli principle, with respect to the pairwise permutation of any electron... 

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