Chapter VI Diatomics

A. Hybridization - the mixing of two AOs on the same atom with different angular quantum numbers.

AOs mix like vectors -

f µ c1px + c2py

The first two examples have cl = c2 > 0 and cl = -c2 > 0, respectively. In the second a z2-x2 AO is generated. All this amounts to is changing the coordinate system which is arbitrary anyway. Hybridization, on the other hand creates different orbitals centered on the same atom. Again it is a consequence of vector addition-

Some examples are:

B. Homonuclear Diatomics

Let us first just use degenerate perturbation theory to construct the MOs and associated energies.

CASE A

What is not quite right - or at least we have to consider here is that while the <s|p> integral is zero when the AOs are on the same atom (they are orthogonal), this is not the case when they are on different atoms -

Notice that we have assigned phases with the AOs so that

Recall from perturbation theory:

Thus the energy and wavefunction corrections become

So the resultant orbitals can be drawn as-

CASE B

Contour plots of the MOs for N2 are shown on the next page. The hybridization is very clear here.

Obviously there must be some mixing which leads to hybridizationin each case, but the essential difference between case A and case B is the position of the 2sg+ MO relative to pu. The changes as a function of what molecule one is talking about. The problem to be considered is the magnitude of

Molecule electronic configuration state re(Å) B.O.

Li2 ()2 2.67 1

Be2 ()2()2 2.45 0

B2 ()2()2()2 1.59 1

C2 ()2()2()4 1.24 2

N2 ()2()2()4()2 1.11 3

O2 ···()2()4()2 1.21 2

F2 ···()2()4()4 1.42 1

Ne2 ···()2()4()4()2 3.1 0

Na2 ()2 3.08 1

Mg2 ()2()2 3.89 0

Al2 ()2()2()1()1 2.72 1

Si2 ()2()2()2()2 2.25 2

P2 ()2()2()2()4 1.89 3

S2 ···()2()4()2 1.89 2

Cl2 ···()2()4()4 1.99 1

Ar2 ···()2()4()4()2 3.76 0

In the first row there is an inversion of the order - one goes from case B to case A on going from N2 to O2. One reason is that the atomic s/p energy gap increases on going from left to right in the Periodic Table.

Think why this should be true...

With a larger gap the energy dominator for the equation on the previous page is expected to be larger so the second order correction will be smaller. For the second row things are different. We always have case A from Al2 onward. The difference here lies primarily in the p-p p type overlap. For the first row the p AOs are contracted very much compared to the second row so p type overlap for the second row and beyond is quite weak. Therefore the pu - pg splitting is much smaller for the second row and consequently pu itself lies at a higher energy.

Returning to the diatomic splitting patterns in case A and case B one can also tell something about the strength of the A-A bond. One seemingly innocent way of doing this is to use bond order. The bond order is defined as the number of bonding MOs minus the number of antibonding ones. This has been listed in the previous table. The A-A bond length roughly scales with the bond order. Remember, however, that as one moves from right to left in the Periodic Table, the element becomes more electronegative and the covalent radius will decrease (the AOs become more contracted). The actual bond strengths for the dimers are -

B. VB versus MO models and through-bond coupling

Let us go back to N2. We would write this in a valence bond perspective as:

Or we could take a more elegant approach by having sp hybrids at each N atom and constructing an orbital interaction diagram:

But this is not at all like we came up with in case B (or case A) from the LCAO approach. Note that 1sg+ must be the s MO and the 1su+ and 2sg+ must be the lone pairs in the delocalized LCAO approach. Which is correect? Photoelectron spectroscopy provides the answer. The experiment involves photoionization of gas phase molecules.

And by using Koopmans' theorem:

ei = hn - KE

where hn is the energy of the laser source (normally 21.1 eV)

The energies of the MOs can be experimentally determined. There is one minor complication. The ionizations are not discrete lines because there is vibrational fine structure.

So for H2 we get:

The PE spectrum for N2 is

The 1sg+ level is at lower potential energy than 21.4 eV so a more energetic source must be used to see this ionization. Clearly the VB model is wrong. The two lone pair orbital are separated by more than 2 eVs! How is this possible? Where did the VB model go wrong?? The VB model neglects that the s and the lone pair orbitals are not orthogonal. The can and will interact with each other. This is called through - bond coupling:

There are many instances of where through - bond conjugation has been experimentally verified.

C. Electronegativity Perturbation Theory

This is a real time - saving feature since we can exploit symmetry to build MOs and energies for a very symmetrical molecule and then use this form of perturbation theory to construct the MOs and energies for a less symmetrical molecule. The master equations utilize the fact that when an atom - say A - is convert to another - say B - more or less electronegative one. Then the Coulomb integral for all AOs on B will change by some amount, i.e. <ca|Heff|ca> = Haa + da. Thus, da> 0 when the atom B is has become less electronegative than A or da < 0 when atom B is more electronegative than A for all AOs. We disregard any changes (and there certainly will be) in the overlap integrals and bond lengths. The equations for this brand of perturbation theory are:

Here cai and caj are coefficients for the ith and jth MOs, respectively, for ca, the AO that is undergoing the electronegativity change. As a very simple example let's see what happens when we go from H2 to H-He+. First we can easily work out what happens from an orbital interaction diagram:

Now let us take the results from H2 and use them with electronegativity perturbation theory to get those for H-He+. In this case da is (-) since He is much more electronegative than H. The energy for s is then given by:

Both corrections are stabilizing. Now for s*-

So s* is stabilized less than s isgoing from H2 to H-He+. The wavefunctions are simple to figure out from the orbital interaction picture. The electronegativity perturbation

D. The MOs of CO

A much harder problem to do is to construct the MOs of CO from the AOs of C and O atoms and try to work out the orbital interactions. The p pattern is still easy but the s system is quite difficult to see. An alternative is to use the A2 homonuclear diatomic MOs for C22- and then use electronegativity perturbation theory to get at CO, i.e.

Let's first tackle the p system with the knowledge that since O is more electronegative than C, da < 0:

So the result is:

We have to be careful with the s system There are four MOs here and in principle all can mix with each other. But the situation is more simple. Let us start with the highest MO 2su+ mixing into the lowest MO, 1sg+ -

But now 1su+ lies much closer in energy. It will mix into 1sg+ as :

Work through for yourself how 2sg+ mixes into 1sg+ and what mixes into 1su+. The bottom line is that for the s and p AO coefficients to have the same sign in the mixing they must have the same hybridization. So 2sg+ mixes only with 1su+ and 1sg+ with 2su+. The net result of this perturbation treatment is then relatively simple two orbital mixings -

Countor plots show these features nicely:

E. M-CO interactions

How does a transition metal bond to CO? Is it through the oxygen or through the carbon end? A transition metal will always have a acceptor orbital (empty MO) to interact with the HOMO of CO - 3s. It will also have two donor orbitals (filled MOs) to interact with the LUMO of CO - 2p. The interactions are then:

It is easy to see that the overlap between these fragment orbitals is always larger for the M-CO case. Since the three interactions are two orbital - two electron ones, they are stabilizing. In fact they are the primary mode of bonding in transition metal carbonyl complexes. This treatment is called the Dewar - Chatt - Duncanson model of bonding. We will see this feature again and again in later material.