In quantum chemistry, **molecular orbitals** are the statistical 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 were on its own. Some qualitative rules:

- There must be as many molecular orbitals as there were basis orbitals.
- Basis orbitals mix more (i.e., 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 do not mix.

_{2}, with the two atoms labelled H' and H". The lowest-energy atomic orbitals, 1s' and 1s", do not transform according to the symmetries of the molecule. However, the following linear combinations do:

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

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

Because these have 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. Because the H_{2} molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (and hence more stable) than two free hydrogen atoms. This is called a covalent bond.

On the other hand, consider a hypothetical molecule of H_{3}, with the atoms labelled H, H', H". Then we would expect three low-energy combinations:

1s + 1s" - 1s' Symmetric 1s - 1s" Antisymmetric 1s + 1s' + 1s" SymmetricThe bonding and antibonding orbitals end up with roughly the same energy, but now there is a third orbital between them, of roughly the same energy as one of the basis orbitals. Because the bonding electron can hold only two electrons, the third electron has to go into the middle level, so there is no energy advantage for the molecule to stay together except under high pressure. Under those conditions, we can expect to see an effectively infinite number of hydrogens packed together, so the molecular orbitals form continuous bands.

Now let's move to larger atoms. Considering a hypothetical molecule of He_{2}, we find that both the bonding and antibonding orbital are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H_{2} + 2 He, so the molecule exists only a short while. In general, we find that atoms such as He that have completely full energy shells rarely bond with other atoms. (In fact there is not a single stable molecule containing He, Ne or Ar.)

The same sort of thing applies for the lower-energy shells in larger molecules: although they mix with other orbitals, there is no real energy advantage gained as a result, so they can be neglected from consideration. Molecular structure relies on the outermost (valence) electrons of the atoms, which are usually of comparable energy. When there is a fair difference between them, though, one finds that one atom's orbitals contribute almost entirely to the bonding orbitals, and the other's almost entirely to the antibonding orbitals. Thus, the situation is effectively that some electrons have been transferred from one atom to the other. This is called a (mostly) ionic bond.

Every molecular orbital actually covers the whole molecule; they are not localized to particular bonds. This actually happens whether the atoms have an energy advantage to grouping or not—strictly speaking, there is mixing between orbitals of atoms light-years away from each other, and although the resulting orbitals do not have energy different from those of the atomic orbitals, the electron density is always high near all the nuclei. This is a reflection of the fact that all electrons are identical, so there is no real way to distinguish the electrons of the two separated atoms. To make up for this, we often take linear combinations of molecular orbitals so that electron density is localized around atoms, and between them (hybridized bonds), but it should be remembered that these are not stationary states, so although they are useful in treating electron density they have no real meaning in terms of energy.

The Linear Combination of Atomic Orbitals approximation for molecular orbitals was introduced in 1929 by Sir John Lennard-Jones. His ground-breaking paper showed how to derive the electronic structure of the fluorine and oxygen molecules from quantum principles. This quantitative approach to molecular orbital theory represents the dawn of modern quantum chemistry.

*See also:*atomic orbital