Molecular_orbitals

The stationary states electrons can have within molecules.

It's next to impossible to find out what the orbitals of a molecule are directly. Instead, one approximates the molecular orbitals as linear combinations of some basis for the electron's state space, usually what each atom's orbitals would be if it was on its own. Some qualitative rules:

• There must be as many molecular orbitals as there were basis orbitals.
• Basis orbitals mix more (ie contribute more to the same molecular orbitals) when they are closer in energy.
• Molecular symmetries map stationary states to stationary states, so any collection of degenerate molecular orbitals must transform according to some representation of the symmetry group. As a result, basis orbitals that transform according to different representations don't mix.

As a simple example consider H₂, with the atoms labelled H' and H". The lowest energy atomic orbitals, 1s' and 1s", don't transform according to the symmetries of the molecule. However, the following linear combinations do:

1s' - 1s" Antisymmetric combination: negated by reflection, unchanged by other ops

1s' + 1s" Symmetric combination: unchanged by all symmetry ops

Since these are of very different energy than all the other atomic orbitals, we would expect these two combinations to be close approximations to the lowest two molecular orbitals. In general, the symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. Since the H₂ molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (and thence more stable) than two free hydrogen atoms.