In the molecular orbital theory of H2, we consider the molecular orbitals as made up of the symmetric and antisymmetric combination of the individual 1s atomic orbitals on the two atoms. In general, however, there is more than one occupied orbital in the original atoms. The choice of the atomic orbitals needed to describe the molecular orbitals is known as the minimal basis set.

This is generally taken to be the atomic orbitals in the valence shell of the atoms.

Molecular orbitals are formed from the overlap of the basis atomic orbitals. Different atomic orbitals overlap in different ways, and this depends on the symmetry of the atomic orbitals.

σ orbitals

These are formed from the overlap of atomic orbitals which are spherically symmetric about the internuclear axis (this is normally defined as the z-axis).

Formation of σ molecular orbitals
s,s overlap s,pz overlap pz,pz overlap

π orbitals

These are formed from the overlap of atomic orbitals which contain a nodal plane including the internuclear axis (as shown by the arrow in the table):

Formation of π molecular orbitals
px,px or py,py overlap

Anti-bonding orbitals

The diagrams in the tables above show the bonding overlap, when the coefficients in the LCAO equation mean that the overlap is in phase. Antibonding orbitals are formed when the overlap is out of phase (phase is shown in the diagrams in the table below by the hatching).

Antibonding molecular orbitals
s,s σ s,pz σ pz,pz σ px,px π

Molecular Orbital Diagrams

When the atomic orbitals of the same symmetry overlap, a set of molecular orbitals is generated. The energies of the molecular orbitals can be shown relative to the energies of the atomic orbitals in a molecular orbital diagram.

Molecular orbital diagram for a homonuclear diatomic molecule (for period 2 elements)

Bonding orbitals are shown at lower energy than the constituent atomic orbitals and antibonding orbitals are at higher energy than the atomic orbitals. In the species shown, the 2s and 2p orbitals are widely separated in energy, and so the 2s orbitals overlap with each other but not with the 2p orbitals. Similarly, the 2p orbitals overlap with each other, now giving s and p molecular orbitals from the different symmetry types of the p orbitals.

The 1s orbitals are so low in energy that they are not considered in the molecular orbital scheme: the electrons in the 1s orbitals are so tightly bound to the nucleus that they do not contribute to the bonding, and so do not affect the molecular orbital structure. The 1s orbitals are known as core orbitals.

The diagram above is suitable for the homonuclear diatomic molecules O2 and F2. Earlier in the period, the separation of the 2s and 2p orbitals is lower, and so interaction between the σ orbitals means that the 2σu orbital moves to lower energy, and the 3σg orbital moves to higher energy, and hence the 1πuorbital falls to a lower energy than the 3σg orbital. The homonuclear diatomic molecules Li2 to N2 have the molecular orbital diagram in the table below.

Molecular orbital diagram for a homonuclear diatomic molecule (for early period 2 elements)

Nomenclature of Molecular Orbitals

In the discussion of molecular orbitals above, we have seen labels like 3σg. The nomenclature of the molecular orbital to give a term like this follows various rules:

The σ and π labels refer to the symmetry of the wavefunction when viewed along the internuclear axis. Thus, σ orbitals are spherically symmetric along this axis, whereas π orbitals have a nodal plane containing the internuclear axis (in short, they look like atomic s and p orbitals respectively).

The g and u labels refer to the symmetry of the wavefunction with respect to inversion in the center of symmetry. Thus, the in phase, bonding, overlap between s orbitals gives a σ orbital which is symmetric about inversion, and is labeled σg, whereas the out of phase, antibonding, overlap between s orbitals gives a σ orbital which is antisymmetric about inversion, and this is labeled σu.

On the other hand, the in phase, bonding, overlap of the p orbitals gives a p orbital which is anitsymmetric about inversion, and this is labeled πu, whereas the out of phase, antibonding, overlap between p orbitals gives a p orbital which is symmetric about inversion, and this is labeled πg [see the shading in the diagrams above, representing in and out of phase overlap].

Thus g and u do not correspond to bonding and antibonding.

Antibonding orbitals are often denoted with a star, eg. 4σu* might be seen in the diagram above.

The numbering scheme is often confusing as there are a range of numbering schemes in existence. The numbering above follows the rules that they are numbered from the lowest energy, valence level, orbital upwards, and there is a different set of numbering for the orbitals of different symmetry types, ie. the σ and π orbitals are numbered separately. This style of numbering is, however, not unique, and sometimes the core orbitals are included in the numbering scheme.

Heteronuclear diamtomic Molecular Orbitals

Heteronuclear diatomic molecules have similar molecular orbital diagrams, but they take account of the fact that the basis atomic orbitals are different. Therefore, the diagram looks similar but is skewed. This means that the coefficients in the LCAO approximation equation for the atomic orbitals of the two species are not equal.

Wavefunction for a heteronuclear diatomic

The probability of finding an electron in the A atomic orbital contribution of the molecular orbital is given by (cA)2, and so when cA and cB have different magnitudes, the electrons tend to be more associated with one of the atoms in the diatomic than the other. Thus, the electrons become drawn towards the species with the higher coefficient, and the bond is polarised. The limit of polarization is where the electrons become completely associated with one species, and an ionic bond is formed.