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### Orbital Energy Level Diagrams

The idea here is that symmetry orbitals of the same symmetry mix and push one another apart.

This is understood by considering a two level system. We have two orbitals,  with energies , of the same symmetry which interact through a perturbation. We solve the secular equations to obtain the new orbitals . The secular equations are

The interaction is responsible for  not being zero. We assume the interaction does not change the diagonal elements, thus . The solutions of the secular equations are
 (50)
If, as usual, the perturbation is much smaller than the energy difference, then we obtain
 (51)
or
 (52) (53)
Thus the lower level is pushed down, and the upper level pushed up viz

This two level picture is important for understanding HOMO LUMO interactions of molecules (the HOMO is the highest occupied molecular orbital of molecule A, and the LUMO is the lowest unoccupied orbital of molecule B; these orbitals are obtained by solving the secular equations of each molecule. Now the molecules interact and the two level picture gives

The magnitude of the interaction will depend upon the magnitude of F12, which itself is determined by their overlap and relative size. The larger coefficients in the molecular orbitals indicate where most of the electrons are (eg on the O in a carbonyl HOMO and on the C in a carbonyl LUMO)

In H2O the A1, B2 combinations of 1sA,1sB will mix with 2pzO,2pyO. To construct the diagram we use the fact that the 2s atomic orbital is lower in energy than the 2p orbitals (why?), and that the 1s H orbitals are highest in energy. Thus we get the following orbital energy diagram

Using an aufbau process to fill the orbitals with electrons, we therefore predict that the ground state electronic configuration is 2a121b223a121b12, where we have used the small letter convention, and remembered that the inner electrons occupy 1a1.
The diagram for NH3 may similarly be constructed:

The ground state configuration is 2a121e43a12.
The diagram for SF6 without and with the inclusion of 3d orbitals is shown

Without the 3d orbitals, the 1a1g and 1t1u are bonding, and the 1eg combination of the 2pF orbitals is nonbonding. The addition of the 3dS orbitals lowers the 1eg orbital energy. Group theory cannot say whether the effect is small or large; in this case the effect is significant and it is important to include `d basis functions' on S to correctly describe the bonding situation.

Next: Walsh Diagrams Up: Contents - Previous: To Determine Symmetry Orbitals
Nicholas Handy

1998-09-23