The procedure for working out a molecular orbital of a general diatomic molecule is quite simple. We construct molecular orbitals using the available orbitals on the atoms. We then fill up the molecular orbitals, starting with the lowest in energy, until all the electrons in the species have been assigned to an orbital.
The filling of the orbitals obeys the Pauli exclusion principle, so a maximum of two electrons can be accommodated in each molecular orbital. Hund’s rule of maximum multiplicity is also obeyed – this means that if there are degenerate orbitals (orditals of the same energy) available, electrons occupy them singly with parallel spins before they start to pair up.
To a good approximation, we may consider only the orbitals in the valence shell of the two atoms – the others are too different in energy to be involved in molecular orbital formation (at least at the basic level we are treating the problem).
The molecule H2 is formed from two H atoms, each of which contributes a hydrogen 1s orbital to the formation of molecular orbitals. As we have seen, this produces a bonding 1σ orbital, and an antibonding 1σ* orbital. This is represented diagrammatically as follows:
|The vertical axis gives the relative energies of the orbitals, both atomic and molecular (represented by the horizontal lines). The central set of orbitals are the molecular orbitals, and are labeled for clarity. Note the antibonding orbital is more antibonding than the bonding orbital is bonding (it is displaced further from the energy level of the atomic orbitals). The bonding orbital is occupied by two electrons (represented by the two arrows; the direction of the arrow indicates the spin of the electron, so these two are spin paired, in accordance with the Pauli principle.) The grey lines connect the molecular orbitals to the atomic orbitals that were involved in their formation.
We shall encounter some slightly more complex examples of these diagrams, but the points established above will remain true throughout.
The ground state configuration of H2 is thus 1σ2, indicating that the molecule is bound and stable.
We now consider a hypothetical molecule He2. This would also have two molecular orbitals formed from the overlap of 1s orbitals on the atoms, giving a molecular orbital diagram of the same appearance as the one above. However, since each He atom has two electrons in its outer shell, there are four electrons to be accommodated in the molecular orbitals. Thus both the bonding 1σ and antibonding 1σ* orbitals are fully occupied – we may say that the molecule contains one bond and one antibond. Since the antibonding orbitals are more antibonding than the bonding orbitals are bonding, the molecule as a whole is classified as unbound – unstable with respect to its constituent atoms. Thus we can conclude that the He2 molecule will not exist.
It is appropriate at this point to make a few remarks about the bond order, b, of a molecule. This is defined as follows:
where n is the number of electrons in bonding orbitals, and n* is the number of electrons in antibonding orbitals.
It should thus be apparent that each pair of electrons in a bonding orbital contributes one to the bond order, and each pair of electrons in a an antibonding orbital contributes minus one.
In the case of H2, the bond order is one, which corresponds to the single sigma bond in the molecule. In the case of He2, the bond order is zero, indicating that there is no bond (and thus no such molecule).
The bond order is a useful quantity, about which we can make several statements:
Molecules with a bond order of zero are unbound, and exist as their constituent atoms rather than as a molecule.
Between two atoms A and B of given elements (which can be the same), the greater the bond order, the shorter and stronger the bond. i.e. an O-O bond, with a bond order of one, is longer and weaker than the O=O double bond in an oxygen molecule, which has a bond order of two.
To extend our consideration of molecular orbitals to the homonuclear diatomic molecules of the second row elements (Li to Ne), we must introduce a new principle of MO theory:
All orbitals of the correct symmetry contribute to a given molecular orbital.
So σ molecular orbitals are formed from all the orbitals that have cylindrical symmetry about the internuclear axis (which in this case happen to be the 2s and 2pz orbitals).
However, for the elements O, F and Ne, the energy difference between the 2s and 2pz orbitals is so great that they do not mix to an appreciable extent, and so it is reasonable to make the approximation that they do not in fact contribute to the same orbitals. Thus for the elements O to Ne, which we shall consider initially, the effects of this new principle are not observed. The MO diagram for these elements looks like this:
The 2s orbitals on each atom overlap in precisely the same way as the 1s orbitals of H and He did, giving rise to two s molecular orbitals, one bonding and one antibonding.
The 2pz orbitals, which are directed along the internuclear axis, overlap very strongly, again giving rise to a bonding and an antibonding orbital.
The 2px and 2py orbitals lie perpendicular to the internuclear axis, so may overlap with each other side-on to form π molecular orbitals. The interference between atomic orbitals may again be constructive or destructive, so again bonding and antibonding molecular orbitals are formed. The px orbitals overlap giving rise to one bonding π orbital and one antibonding π orbital, as do the py orbitals. Since the px and py orbitals were degenerate on the two atoms, it should be logical that the two bonding orbitals are degenerate, as are the two antibonding orbitals.
It is now possible to obtain the ground state electronic configurations of the diatomic molecules of the second period from O2 to Ne2 by inserting the correct number of electrons into the molecular orbital diagram above. However, there is little to be gained by going through the exercise in full, so we shall pick out some of the most important and interesting results:
O2 is predicted to contain unpaired spins. Each oxygen atom contributes six valence electrons, giving a total of twelve to occupy the MO’s. The first four fill the orbitals produced by overlap of the 2s atomic orbitals. The next six fill the three bonding orbitals produced from the overlap of the 2p orbitals, which leaves two electrons to be accommodated. These must go into the 1π* orbitals. However, since there are two degenerate orbitals here, the two electrons occupy them with parallel spins (in accord with Hund’s rule of maximum multiplicity). These unpaired spins mean that the O2 molecule is predicted to be paramagnetic, which has been observed experimentally.
This is one of the great successes of MO theory – VB theory is unable to explain why the oxygen molecule is paramagnetic.
The molecule Ne2 is predicted to be unbound (bond order of 0), and has indeed never been observed.
For the elements Li to N, the molecular orbital diagram is not the same as that above. The energy gap between the s and p orbitals is lower for these elements, so the 2s and 2pz orbitals do all contribute to all of the σ molecular orbitals. This leads to an alteration of the order of the molecular orbitals, so that the diagram looks like this:
From this diagram, we can obtain the ground state electronic configurations for all the diatomic molecules of the elements from Li to N. This leads to various results, eg:
Li2 is predicted to be bound, and has indeed been observed in the gas phase.
Be2 is predicted to be unbound, and has never been observed.
In addition, the bond orders predicted by MO theory for the diatomic molecules of ALL the second row elements are in agreement with those in Lewis structures of the molecules (e.g. N2 is predicted a bond order of three, and in its Lewis structure it has a triple bond.)