[RowanF]

From what I've read, I gather that atomic orbitals are always supposed to combine to form an equal number of molecular orbitals. I'm having a lot of trouble interpreting what this really means though. For example, to form H2, a 1s atomic orbital combines with a 1s atomic orbital from a second hydrogen atom. This can lead either to a bonding orbital or an anti-bonding orbital. The bonding orbital forms as it is lower in energy.

What I don't understand is how the number of molecular orbitals combine equally with respect to the original number of atomic orbitals. On the diagram, you mark two electrons of opposite spin in the bonding square, leaving the higher energy anti-bonding square empty. So does this single square represent two orbitals? I often hear it being referred to in the singular, as if we ended up with only one MO after combining two AOs.

Thanks for any help.

[Borek]

Empty orbital is still an orbital.

[Enthalpy]

Any wave function, including orbitals, can host two electrons with opposite spins. This is a property of fermions: they must be in different states, but the spin orientation counts as a part of the state, and since electrons can have two spin orientations, two fit in the same otherwise identical state, which is usually worded as "same state" or "same orbital".

Note A: beware that any detail of the state counts. 2p orbitals can be oriented along x, y or z. So while each orbital 2p_x, 2p_y and 2p_z can host two electrons with opposed spin, together they host 6 electrons. This is often worded loosely as "the 2p orbital can host 6 electrons".

Note B: molecular orbitals exist as soon as atoms come close to an other, whether the molecule is energetically favourable or not. I proposed it as a possible process for sonoluminescence, where pressure would create excimers or exciplexes that emit light when less strong pressure permits them to split. Temporary molecules, unfavourable due to anti-bonding orbitals, could also result from mere pressure during sonication, and be reaction intermediates. It could also happen in detonations.

Each individual hydrogen atom could host two electrons as a 1s orbital, but there is only one electron. An H₂ molecule can host two electrons as the bonding orbital and two as the antibonding. As there are only two electrons, they can occupy both the more favourable bonding orbital.

It is a very efficient reason for atoms to make molecules and explains why lone atoms are extremely rare on Earth - from H to H₂ releases more heat than burning H₂ to H₂O. This spin pairing is not a force between the electrons, it's just the possibility electrons have to rearrange on more favourable molecular orbitals. In a first approximation, valid if the atoms aren't too close, the bonding and anti-bonding orbitals have energies symmetrical below and above the energy of the initial atomic orbitals, so if electrons have to fill the anti-bonding orbital too, the molecule isn't energetically favourable. So helium doesn't make He₂ molecules naturally, but hydrogen does make H₂.

A molecule can have some electrons on anti-bonding orbitals. For instance, N₂ puts all 2p electrons on bonding orbitals and is chemically quite inert, but O₂ has two electrons more that must occupy anti-bonding orbitals. O₂ is energetically much more favourable than 2O, but is reactive. F₂, with two electrons more than O₂, is even more reactive. Ne₂ doesn't form at all usually.

Older models tell that N₂ has a triple bond and O₂ a double bond. Molecular orbitals tell that three 2p electron pairs in O₂ are on bonding molecular orbitals and one pair on an anti-bonding one, so the formation of O₂ releases about as much energy as two bonding orbitals.

[Mitch]

Each square in the middle of the diagram represents one molecular orbital, it isn't just a square.

[Corribus]

Analogy:

Imagine you and your brother each have a hamster with its own big cage. Your hamster's cage is about as nice as your brother's hamsters cage, but they are nice in different ways. One day you get together with your brother and say, "Hey wouldn't our hamsters be more happy in one single cage?" Your brother agrees. The two of you then consider which cage to put the hamsters in. They are both the same size, but yours has a nice wheel, whereas your brother's has a good sleeping place. Yours has sturdy walls, but your brother's has a nicer door. Etc. It's hard to decide which is the better cage. Your brother floats an idea: what if you deconstruct the two cages and take the better parts from each one and build a really nice cage to put them in? With the leftover parts, a second cage can be built for emergencies, like if you need to separate the hamsters when they are sick. So you do just this and end up with two new cages, one that is much nicer than either cage you started with, and one much less nice, which will remain empty but can still hold two hamsters if you needed it to.

Combining atomic orbitals to form molecular orbitals is like deconstructing and rebuilding hamster cages. You always maintain the same number of total orbitals (cage components) and can build the same number of orbitals (cages) from them, but rather than having n amount of so-so orbitals (cages), you get some lower energy orbitals (better cages) and some higher energy orbitals (worse cages). The average energy of the orbital (cage quality) is more or less the average of the orbital energies (cage qualities) that you started with. Your electrons (hamsters) prefer the lower energy orbitals (nicer cages), but they have to share, leaving the higher energy orbitals (worse cages) empty. The higher energy orbitals (worse cages) can still hold electrons (hamsters), but the electrons (hamsters) don't really like to be in them if they don't have to.

At the risk of confusing you, it's good to keep in mind that most atomic and molecular orbitals don't really exist. They are mathematical functions, based on approximations, to describe where electrons tend to be in space given other systemic parameters. The LCAO-MO (linear combination of atomic orbitals to form molecular orbitals) approach is ONE such model of the electronic structure of molecules, and that is what you are learning now. It is based on the idea that the orbital formed in a molecule is APPROXIMATELY equivalent to a situation where you add atomic orbitals together in linear fashion. But some theories don't even involve orbitals at all.