Molecular orbitals 2

MO diagram of HF:

Generally the more similar in energy two wavefunctions or orbitals are, the stronger they overlap. In MO diagrams, the closer the energy level of a molecular orbital with an atom's atomic orbital, the more likely the electron in that orbital is found on the parent atom. So from above we can predict that electrons in HF are far more likely to be found on the flourine atom - which agrees from what we would predict from knowing about electronegativity.

It also tells us that an electron promoted into the sigma antibonding orbital will be more likely to be found on the hydrogen atom.

The H1s orbital overlaps with the FP_z and F2s orbital. Three atomics orbitals of sigma symmetry therefore overlap to produce three molecular orbitals of sigma symmetry.

The 1s orbital on flourine also possesses sigma symmetry, but since it is not a valance orbital it is dismissed as negligible and not used in the calculation.

Flourine also has two p orbitals which produce two pi molecular orbitals. Hydrogen has no orbitals of the right symmetry to overlap with them. The two pi molecular orbitals are an example of "non-bonding orbitals".

The textbook writes that u and g notation are not used for molecular orbitals in heteronuclear diatomics, I don't have a good explanation for that because I don't know much about symmetry, I assume it's difficult to apply operations to a molecule when electron clouds are not evenly spread out between atoms as in homonuclear diatomics.

The bond order (b) is calculated by (1/2)(electrons in bonding orbitals - electrons in antibonding orbitals). Non-bonding orbitals are ignored. Applying that to the above diagram we get 1. This agrees with the single bond/one shared electron pair from lewis diagrams.

Here is the MO diagram of oxygen again:

This should give a bond order of two, giving the same prediction of a double bond that we would expect from lewis diagrams. You should also be able to tell that the O₂^- ion would have a bond order of 1.5, since the extra electron would only have an antibonding MO to slot into.

As a general rule, bond length decreases with bond order. So adding an extra electron to O₂ would make the molecule longer.

Gerade and ungerade

The symmetry notation u and g are sometimes used when describing molecular orbitals. This refers to the operation of inversion, which requires starting at an arbitrary point in the orbital, traveling straight through the center, and then continuing outwards an equal distance from the center. The orbital is designated g (for gerade, even) if the phase is the same, and u (for ungerade, uneven) if the phase changes sign.

Symmetry symbols can be used as a way to distinguish different orbitals in MO diagrams:

Wavefunctions, bonds and antibonds

Wave functions look like this:

1s

2s

2p

The wave function squared, or ψ², represents the probability of finding that electron in that point in space. Which look like this:

Wave functions can have positive and negative phases. Overlap of opposite phases will cancel out. Physicists sometimes use analogies with macroscopic waves such as water waves, where peaks and troughs will reinforce their own kind, but will cancel eachother out.

Constructive (bonding) overlap of 1s orbitals:

Destructive (antibonding) overlap of 1s orbitals:

A node is a point of 0% probability of finding the electron

I don't think there exists a satisfying explanation why electrons act like this - why they act like waves with different phases and so on. All we know is that there is a mathematical model which describes them - called a wavefunction because it's similar to the mathematical models describing classic waves - and that it fits well with experiment. 

Perhaps when we use computers to upgrade our brains it will be as simple as "a bucket of sand and bucket-shaped hole will cancel out" is to us - and we'd be able to rest knowing that the universe acts in a nice and appropriate way. 

In drawings or 3d models, phases are usually shown as shades, different colors, or + and - symbols. Bonding orbitals are considered "in-phase overlap":

1s - 1s molecular bonding orbital

2p - 2p molecular bonding orbital

Anti-bonding orbitals are considered to be "out of phase overlap", producing these abominations:

1s - 1s molecular antibonding orbital

2p - 2p molecular antibonding orbital

D orbitals

Here are the shapes of the S, P and D atomic orbitals:

It is necessary to learn the differences between the d orbitals, since it comes up when describing bonds. Convention is to take the z axis as the internuclear axis, which I will follow. This means:

d_z²: Cylindrical along the internuclear axis, so can contribute to sigma orbitals

d_xz and d_xy: These look like p orbitals along the internuclear axis, so can contribute to pi orbitals

d_yz and d_x²-_y²: These have no equivalent to what we've seen before. Four lobes from each atomic orbital can overlap to form weak delta (δ) molecular orbitals:

Molecular orbital diagrams

Molecular wavefunctions are formed by taking the product of all the individual electron wavefunctions found in both atoms. This can be denoted as ψ = ψ(1)ψ(2)...ψ(N_e) where N_e is the total number of electrons.

While we need computers to calculate these molecular wavefunctions, we can use rules to predict their shape:

1. N atomic orbitals combine to produce N molecular orbitals.
2. The general pattern is one orbital above (in energy) the parent atomic orbitals, one below, and the rest lying in between.

For example, here is the MO diagram of H₂:

So two atomic orbitals (1s from both hydrogen atoms) produce two molecular orbitals, denoted σ and  σ*. We have two total electrons, so these occupy the lower-energy σ bonding orbital. The empty σ* orbital in the diagram is called an anti-bonding orbital, because it is higher in energy then the parent atoms. Having two electrons in the σ* would pretty much cancel out the extra stability gained from filling the σ orbital. This can be observed in an MO diagram of He₂:

This matches the real world, we don't observe He₂.

A pair of electrons being in a σ molecular orbital is the same thing as a "σ bond" from A-level chemistry. The sigma symbol actually means "cylindrical symmetry around the internuclear axis". We know that bonds between s or sp-hybridized orbitals are sigma, while bonds between p orbitals perpendicular to the internuclear axis are called π. This definition means that bonds between "head on" p orbitals are also called sigma.

Sigma bonds

Pi bond

Here is the MO diagram of O₂:

This predicts that oxygen has two unpaired electrons, which isn't predicted by using lewis diagrams. Experiments easily confirm that these two unpaired electrons exist.

Unpaired electrons (or "free radicals") in A-level chemistry are typically taught as being extremely reactive. This is mostly true for other molecules. Wikipedia writes "with some exceptions, unpaired electrons cause radicals to be highly chemically reactive." I don't know if there are any tricks to help spot the exceptions.

Here is the MO diagram of F₂:

Six 2p orbitals (three from each atom) combine to produce one sigma bonding orbital, two pi bonding orbitals, and their corresponding antibonding orbitals. Some intuition about why this happens can be found by considering how the p orbitals are perpendicular to eachother.

Only two p orbitals can combine "head-on", the other four have to form perpendicular pi bonds, since they are pointing in the wrong direction to form sigma bonds.

Finally, MO diagrams can occasionally have non-bonding orbitals, which are at the same energy level as their parent atoms. "Lone pairs" from A-level chemistry are contained in these.

Resonance structures

When drawing Lewis diagrams, showing resonance structures can more accurately describe what experiments tell us about chemical bonds. For example, the acetate anion has the following resonance:



A double-bond and single-bond have different lengths. The two oxygen bond lengths in this anion are detected experimentally as halfway between the lengths of single and double bonds. So the true nature of a molecule is a blend of their resonance structures.

A structure with multiple resonances is also lower in energy then any single contributing structure. So the negative energy of the above anion is even lower then summing the individual bonds would suggest.

Resonance structures are usually depicted as having the same energy as each other. Structures slightly higher in energy can also contribute, but won't do so as much. Chemists usually have some intuition about what are the stable and therefore most contributing resonance structures. Low-energy structures usually have:

Atoms obeying the octet rule
Negative charge located on electronegative atoms, positive charge on the electropositive atoms
More bonds
No charges
If charge is necessary, have it spread out across the molecule

Consider these negligible resonances for the acetate ion.

This one has both more charge and less bonds, making it too unstable to have a significant contribution.

This has an extra bond, but the negative charge is on an less electronegative element, and most importantly it violates the octet rule for carbon. We never usually see carbon break the octet rule, so this structure would immediately look wrong to a chemist. But it might work if another atom was in place of carbon.

Also consider Furan:
 


Positive charge on the oxygen is not going to be very stable, making the first structure the main contributor. But we can expect experiments to show all the single bonds as having slightly double-bond character.

Sidenotes

It is worth memorizing Avogadro's number, 6.022 x 10²³

It is also worth memorizing a selection of chemistry jokes. In my experience, non-chemists will occasionally ask if you know any. My best ones are below:

How many moles in a guacamole?
Avocardo's number.

Why do chemistry professors like to teach about ammonia? Because it's basic material.

Argon walks into a bar. The bartender says "Get out! We don't serve your noble gases here." Argon doesn't react. 

Which dissolves better in water, a brown bear or a white bear? A white bear, because he's polar.

Also learn the conversion factor from Celsius to kelvin, +273.15. There will be Americans at university, so impress them by knowing how to convert to Celsius from Fahrenheit. You do it roughly by subtracting 32 and dividing by 2.