Two-orbital Systems

As a simple introduction to PMO theory suppose we consider the bond formation between the two-atom, two-orbital system shown in Figure 4.8. The energy gained on forming the bond A—B is given by the second-order perturbation equation... 

In Table 7 the six-membered monocyclic dienes are represented by the conjugated 1,3-cyclohexadiene and its isomer 1,4-cyclohexadiene. 1,3-Cyclohexadiene has a nonplanar equilibrium conformation that is primarily influenced by three factors -electron interaction (optimal for a planar conformation) angle strain and torsion strain (both optimal for a planar conformation). The reduced overlap between the two --orbital systems is, for the observed C=C—C=C angle of 18°, estimated at ca 10% and should therefore not influence the conjugation stabilization drastically, compared to a conformation with coplanar C=C bonds. 

We derived the 2 x 2 matrices of Eqs. 4.55 starting with a two-orbital system. These results can be generalized to n orbitals ... 

So we get the same energy levels as from matrix diagonalization ( f 2 = 1.414). To find the coefficients we substitute the energy levels into the secular equations for the propenyl system these are, projecting from the secular equations for a two-orbital system, Eqs. 4.49 ... 

Knowing the VB wave function for a 3e bond (Eq. 3.47), we can construct the wave function for structure 1 by considering this electronic distribution as made of two independent three-electron/two-orbital systems orthogonal to... 

Figure 4.1S. The n MOs and n energy levels for a two-/ -orbital system in the SHM. The MOs are composed of the basis functions (two p AOs) and the eigenvectors, while the energies of the MOs follow from the eigenvalues (Eq. (4.66)). The paired arrows represent a pair of electrons of opposite spin (in the electronic ground state of the neutral ethene molecule V l is occupied and Vt2 is empty).

Fig. 20-9. A hypothetical two-orbital system in which two possible distributions of two electrons and the resulting total energies are as shown.

Next, we consider the two-electron states and energies. This is essential since the ultimate objective is to find the exact two-electron eigenstates and eigenvalues of the two-electron two-orbital system. This can be done only because the model is so very simple that the solution requires a diagonalization of matrices no larger than 3x3. 

The manner in which Cl corrects the deficiency of the monodeterminantal MO wavefunction is often illustrated by reference to the simple example of a two electron-two orbital system, e.g., H2, pi ethylene, etc. Elementary discussions of this type can be found in a number of elementary texts. See, inter alia Borden, W.T. "Modern Molecular Orbital Theory for Organic Chemists" Prentice-Hall Englewood Cliffs, NJ, 1975. The following are some of the methodologies used for obtaining correlated wavefunctions ... 

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