Humboldt State University ® Department of Chemistry

Richard A. Paselk

Chem 110	General Chemistry	Fall 2003
Lecture Notes::Lec 20_17 October	© R. Paselk 2003
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Hybrid Atomic Orbitals

We've reviewed atomic orbitals and classical bonding theory, now our question is how can we best understand bonding in molecules, including their shapes etc., in light of modern theory (quantum mechanics)?

We need to keep in mind that our modern picture of simple molecules must be consistent with the classical picture, since it gave us good approximations to observation!

However, when we look at the atomic orbitals for the valence electrons they are generally not arranged in a way that would give the shapes predicted by VSEPR Theory. Thus, the four valence orbitals of atomic carbon are the spherical 2s orbital and the three mutually perpendicular 2p orbitals, while VSEPR predicts that carbon surrounded by four hydrogens will be tetrahedral in shape. [overhead]

So what do we do? Recall that the specific shapes of the orbitals result from the interactions of the electrons with a central positive charge (and each other), so we might expect they would change shape if exposed to an external charge (like a second atom).

One way to model this new situation then is to assume that all four of the atomic orbitals are perturbed into a new configuration. If we assume they all have the same energy (required if they are to form a symmetrical set around the carbon nucleus, for example), then we can assume they each have the average energy of the original four orbitals. We can now come up with a new orbital set by adding the orbitals together, and keeping in mind that we must end up with the same number of orbitals as we started with. If we make this calculation we find there are now four equivalent orbitals arrayed in a tetrahedral geometry, just as we predicted with VSEPR Theory - ta da! [overhead]

Notice, that with this Hybrid Orbital Theory we are looking at individual atoms, not molecules. All of our calculations and predictions are for the atoms. We now make molecules by overlapping the new hybrid orbitals with other hybrid orbitals or with atomic orbitals of other atoms to make a molecule.

Let's look now at the examples and illustrations in your text , noting single and multiple bonds etc.

• Tetrahedral Electronic Geometry = sp³. Four orbitals (s + 3 p's) combined. (Note the sum of "exponents" = number of orbitals)
  • Methane (CH₄) - tetrahedral molecule. [overhead]
  • Ammonia (NH₃) - trigonal bipyramidal molecule.
• Trigonal Planar Electronic Geometry = sp². Three orbitals (s + 2 p's) combined, one p orbital left as is.
  • Ethylene (H₂CCH₂) - each carbon has trigonal planar geometry. [overheads]
• Linear Electronic Geometry = sp¹, or sp. Two orbitals (s + p) combined, two p orbitals left as is. [overhead]

Note we get two basic bond types when we overlap orbitals:

1. Sigma (s) bonds: These are cylindrically symmetrical around the axis connecting the bonded atoms. Single bonds are always sigma bonds, and in a multiply bonded system the "first" or "central" bond is a sigma bond. [overhead]
2. Pi (p) bonds: these are made up of two lobes with planar symmetry round a plane though the nuclei of the two bonded atoms. The "second" and "third" bond of multiply bonded atoms are pi bonds. For systems with two pi bonds the bond panes are perpendicular to each other. [overhead]

The hybrid atomic orbital model is a localized electron model - the quantum calculations are looking at the atoms individually. The hybrid orbital model is particularly useful to us at this time because it gives nice pictures of two aspects of bonding:

• Molecular shape - look at sp, sp², sp³, dsp³, and d²sp³ (overhead, text figure)
• Multiple bond formation - sigma and pi bonds.

However, the localized electron, hybrid orbital theory does not do well in other areas:

• Since the electrons are assumed to be localized, resonance must be invoked to explain partial bonds etc.
• It gives no direct information about bond energies since it is not calculating the way electrons are shared.
• It doesn't work well for unpaired electrons in molecules.

In the hybrid orbital model described we look at the atoms individually in creating the orbitals, then we allow them to overlap to give bonds. Of course in a real molecule nature does not distinguish between atoms and orbitals in this way, in fact when atoms form a bond new orbitals are formed based on the entire molecule. Now I want to introduce some of the concepts involved in this molecular orbital. picture.

Molecular Orbitals

Molecular Orbital Model of Bonding: As with atoms, we will begin with the simplest system, in this case the dihydrogen molecule, H₂. (Strictly speaking, the simplest molecule is the dihydrogen molecular ion, H₂⁺, with a single electron.)

As I noted in the beginning of our discussion of modern bonding, orbitals are conserved, so if we add two hydrogen atoms, H_a & H_b together, the two 1s orbitals should give us two molecular orbitals, MO₁ and MO₂:

Note that one orbital will have a lower energy and the second a higher energy as expected from the approximate conservation of orbital energies we noted earlier. And when we add and subtract the two atomic orbitals they give molecular orbitals of quite different shapes. (overhead, text figure)

The molecular orbitals resulting from this combination are symmetrical along the atomic axis between the bonded atoms, and as before are referred to as sigma (s) molecular orbitals. The two orbitals, however have much different properties.

• The ground level (lower energy orbital) is a bonding orbital and called simply a sigma orbital. The electron density for this orbital is largely distributed between the atoms.
• The high energy orbital actually has most of the electron density not between the nuclei, so the nuclei and electrons will repel each other, and no bond is formed. This orbital is referred to as an antibonding orbital and given the designation sigma star (s*).
• Note that if electrons occupy both the bonding and antibonding orbitals there will be no net bond formed!

Bond Order = (#bonding electrons - # antibonding electrons)/2. Divide by two to get "classical" two electron bond. Bond order gives a measure of bond strength in units of an electron-pair bond.

• If we look at hydrogen, H₂, both electrons go into the ground state (lowest energy) MO, giving a bond order of one, so H₂ has a single bond.
• If we look at the next possible homonuclear diatomic molecule, He₂, the four electrons will first fill the lowest energy MO, but the next two go into the higher energy, antibonding MO. The bond order is then 0, and theory predicts no bonding and no He₂ molecule.

So far we've looked only at atoms with s-electrons and s-orbitals. What happens when we have p-electrons? The first element with p-electrons is boron, with a valence electronic configuration of 2s²2p¹. So what happens if we combine two boron atoms and calculate the new energy levels for the potential molecule?

• First let's look at a simple calculation assuming the s and p orbitals do not interact with each other.
  • Because the s and p orbitals are of significantly different energies we'll get two distinct sets of MO's. (text figure 9.36)
  • As expected the 2s orbitals will combine to give sigma MO's with a splitting just like we saw for hydrogen.
  • The p-orbitals will be a bit more complex. (overhead, figures 9.33 and 9.34)
    • One set, call them the p_x orbitals will overlap with cylindrical symmetry about the axis connecting the nuclei giving a set of sigma orbitals.
    • The other two sets have planar symmetry and give pi orbitals.
  • With this calculation the energy diagram shows, starting at the lowest energy, a s_2s, a s_2s*, a s_2p, two p orbitals of equal energy (y, and z), two p* orbitals of equal energy, and a s_2p* orbital.
  • Filling from the bottom with the six electrons of the two boron atoms we should see two bonds and one antibond giving a total of one bond, which is what we observe. However, diboron is paramagnetic, which is not at all expected from our filling diagram. What's wrong? Our model is too simple.
• Recall from our earlier discussion that when we look at a molecule the electrons of that molecule are now just that - they belong to the molecule. In our first calculation above we assumed we could treat the electron energy levels the same as we did for the atoms. But when we put the two atoms together as a molecule we shouldn't be surprised that the energy levels are more complex - the valence s and p orbitals of the atoms are all considered together and a whole new set is calculated. Your author refers to this as "mixing" the s and p orbitals, but really there are no s or p valence orbitals in the molecule.)
  • So when we calculate the orbitals fresh, assuming the molecule we of course still get the same number of orbitals, and even the same types, but the energy levels differ. (text figure 9.38)
    • The s_2s, and s_2s* orbitals remain separate from the other orbitals, but s_2s* energy is lowered.
    • The order of the s_2p and p_2p orbitals is reversed, with the two p_2p at a lower energy. The order of the excited (*) orbitals remains the same.
    • As a result, filling from the lowest energy, we see a s_2p bond, a s_2p* antibond, and a p_2p bond made up of two unpaired electrons in different p_2p orbitals. Thus the diboron molecule is predicted to be paramagnetic, as observed.

© R A Paselk

Last modified 17 October 2003