In the molecular orbital theory of H_{2}, 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 zaxis).
Formation of σ molecular orbitals 
s,s overlap 
s,p_{z} overlap 
p_{z},p_{z} 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 
p_{x},p_{x} or p_{y},p_{y} overlap 

Antibonding 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,p_{z} σ 
p_{z},p_{z}
σ 
p_{x},p_{x}
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 O_{2} and F_{2}. Earlier in
the period, the separation of the 2s and 2p orbitals is lower,
and so interaction between the σ orbitals
means that the 2σ_{u} orbital
moves to lower energy, and the 3σ_{g}
orbital moves to higher energy, and hence the 1π_{u}
orbital falls to a lower energy than the 3σ_{g}
orbital. The homonuclear diatomic molecules Li_{2} to
N_{2} 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 3σ_{g}.
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 σ_{g},
whereas the out of phase, antibonding, overlap between s orbitals
gives a σ orbital which is antisymmetric
about inversion, and this is labeled σ_{u}.
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 π_{u},
whereas the out of phase, antibonding, overlap between p orbitals
gives a p orbital which is symmetric about inversion, and this is
labeled π_{g} [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. 4σ_{u}^{*}
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 (c_{A})^{2}, and so when c_{A} and
c_{B} 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.
