CHAPTER III. Perturbational Molecular Orbital Theory

In principle we can perform some sort of molecular orbital calculation on molecules of almost any complexity. It is, however, often extremely profitable to relate the properties of a complex system to those of a simpler one. To appreciate the orbital structure of complex systems it is much more insightful to start off with the levels of a simpler one and 'switch on' a perturbation. three examples of different types of perturbations which we will use are shown below:

Consider a set of unperturbed (zeroth order in the language of perturbation theory) orbitals, , and a set of associated energies, , which correspond to the solutions for the molecules on the left side. In general -

What perturbation theory does is to express the new (perturbed) energies and wavefunctions in terms of the old (unperturbed) one. This is done in terms of a power series:

and

The superscripts give the order of perturbation, i.e. "0" is the zeroth order (the unperturbed value), "1" is first order, "2" is second order, etc.

The changes in the overlap and resonance integrals are given by the quantities dS_mn and dH_mn so that

Or better in terms of a molecular basis:

and, as before we have the relationship:

We will look at intermolecular perturbations in this Chapter.

A. Intermolecular Perturbation - the nondegenerate case.

Suppose that we have MOs from fragment (or atom) A and MOs from fragment (or atom) B, along with their orbital energies.

Let us assume for simplicity that, when two fragments interact with each other, no geometry change occurs within each fragment so that there is no geometry perturbation to consider within each fragment. To obtain the orbitals of the fragments A and B joined by a single bond, one might break the bond to generate A^. and B^. radicals, A⁺ and B^- ions, or A^- and B⁺ ions and then carry out molecular orbital calculations for these fragments. Therefore, for those atomic orbitals c_m and c_n located on the fragments A and B, respectively, the dH_mn and dS_mn values are simply the H_mn and S_mn values of the composite system AB, respectively and

The subscripts A and B have been added only to help the reader to keep track of which fragment a particular orbital has originated from. So the resultant wavefunction can be written as:

Where (and with no derivation)

The sizes of the corrections to the mixing coefficients are, in general:

So

And the orbital energies are given by:

B. Linear H₃, HF and the three orbital problem

Let us start with the H₃ molecule in a linear geometry.

The energies of y_i, y_k and y_j are given by:

So the resultant orbital interaction picture looks like -

Note that the way that the starting wavefunctions are written:

We will now evaluate the mixing coefficients. Let us start with interaction1:

We will write the first and second order coefficients in a short-hand way with their signs as

So

Or in graphical terms;

Notice that there are no nodes perpendicular to the molecular chain. This is the most bonding MO that is possible

For interaction 2:

There is one node perpendicular to the molecular chain. This is a nonbonding MO - The overlap between adjacent AOs is neither bonding nor antibonding

For interaction 3:

Finally, there are two nodes perpendicular to the internuclear axis. This is the most antibonding MO that is possible.

This is a very general pattern. For the two orbital interaction there is a bonding combination that is stabilized and an antibonding one that is destabilized. For the three orbital pattern there is one MO which is stabilized and is the most bonding combination; there is a destabilized MO which is the most antibonding combination: and there is one MO which is caught in between and is nonbonding.

C. HF and Hybridization

Let us take another example of a three orbital problem and build up the s orbitals of HF. Before this we need to know how to work out hybridization. This is construction of an orbital from two atomic orbitals with different angular quantum numbers. Some examples are given below:

There are many more... Consider the s and p AO of F to be y_i and y_k, respectively. We will interact these two AOs with the H s AO, y_j. This is shown below.

The orbital phases have again been arranged so that

The energies are:

There is one minor complication here. y_i does mix into y_k as we shall see in second order and this creates a third order energy change which is destabilizing. So y_k is stabilized less than what one might expect - it is the "nonbonding" member of the three orbital problem.

The MO shapes are constructed in the following way:

This orbital is obviously bonding.

y_k is:

This is the middle orbital. Notice that the overlap between F p and H s is bonding, but that between F s and H s is antibonding. The orbital is hybridized away from the hydrogen atom - it is essentially nonbonding.

Now finally y_j is:

The overall picture is then:

D. Degenerate Perturbation.

Let us consider an intermolecular perturbation where an orbital of the fragment A is degenerate with an orbital of the fragment B. We arrange the orbital phases of and such that > 0 and thus < 0. The wavefunctions are just linear combinations of the two orbitals, i.e.

And the energy corrections are:

So the result is: