Molecular terms refer to MO-LCAO theory of molecular orbitals.
A molecule is said to be homonuclear if it's made of two identical atoms. For this kind of molecules there are special symmetries in the hamiltonian:
- symmetry with respect to the center :
![\left[H,P\right]=0](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/11cf400fb3ee866d48c37f29d6834f550cb8c2e3)
- symmetry with respect to the bound axis, which we choose to be the z axis:
![\left[H,L_{z}\right]=0](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/c0ac7b1348dac73f03429c691430ce75ab68ef87)
- symmetry with respect to every plane orthogonal to the bound axis :
. However
and we can't construct a C.S.C.O of eigenstates common to
. We have to choose
or
, but since
the eigenstates of
with
are also eigenstates of
and we can classify these states with an additional index
A molecular orbital is a linear combination of atomic orbitals of atom
with that of atom
and is described by a greek small letter identifying m-value plus a subscript for the parity,
for even orbitals or
for odd orbitals.
![{\displaystyle {\begin{aligned}m=0\rightarrow \sigma \\m=\pm 1\rightarrow \pi \\m=\pm 2\rightarrow \delta \\\dots \\{\mbox{even orbitals}}\quad \left[\sigma ,\pi ,\dots \right]_{g}\\{\mbox{odd orbitals}}\quad \left[\sigma ,\pi ,\dots \right]_{u}\end{aligned}}}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/d9aa5374e8d7e09602d2fc8c6e98fa06cd6dc5bc)
Furthermore we have to keep in mind that every pair of atomic orbitals gives rise, when combined, to a bonding orbital and an antibonding orbital (denoted with a superscript

).
Once we have found molecular orbitals we fill them with the electrons. The
molecular terms are then given by the assignment of the total angular momentum projection

,a superscript on the left for spin multiplicity and the subscript for the overall parity, plus the

superscript for

terms.
![{\displaystyle ^{2s+1}\left[\Sigma,\Pi,\Delta,\Phi,\dots\right]_{g/u}}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/a5fb17ea649bf41fe85023bffb210a46c5138dbc)
Let us discuss an example, the

molecule.
- Write the expression for each atom electronic configuration:
![C_{\mbox{ground state}} = [He]2s^22p^2](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/0a59dc2fa4539dfad18558ea56ee55907bc91785)
- Since
and parity is governed by
-values we can combine atomic orbitals with
. Furthermore since
we must have
. Thus we are allowed to combine
with
and
with
.


- Put the electrons in the so-found molecular orbitals starting from the lowest energy orbital. There isn't a general rule which regulates the energy scale of molecular orbitals. However for homonuclear molecules experiments show that:
In our example the electronic configuration of
is then
![{\displaystyle [He]1\sigma_g^2 1\sigma_u^2 1\pi_u^4}](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/9cc06eb18a3aa545acba00cbc3e99a96b95096a3)
- Closed shells do not contribute to the molecular terms, so consider just not completely full shells. Find the possible values of
and
looking at the allowed configurations.For each configuration find the corresponding
and,eventually,
indexes with the help of the following rules: 
. The molecular term for
ground state is then
.
Let us discuss the problem of finding molecular terms for the excited configuration of
:
. We neglect the
's orbitals since they're full. Possible configurations are:
So that we have for
the terms (following the order of the image):
. For
there is just one term that is
. Now we add each term corresponding to
possible configurations to the term related to
, and we obtain:

The number of states is easily found considering that each term (except

's) respresents 2 possible values of

so that the total states are

for each non-

term and

for

's terms
Let us verify our result. The configuration

corresponds to

states, which is just the number found.
A molecule is said to be non-homonuclear if it's made of two atoms of different species. For this kind of molecules there are less symmetries than in homonuclear molecules:
- There isn't symmetry with respect to the center :
![\left[H,P\right]\neq 0](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/a53bde3b4b2ae7fad96dc04ae4ba0bde2bc72efc)
- The symmetry with respect to the bound axis, which we choose to be the z axis,is still present:
![\left[H,L_{z}\right]=0](//restbase.wikitolearn.org/en.wikitolearn.org/v1/media/math/render/svg/c0ac7b1348dac73f03429c691430ce75ab68ef87)
- The symmetry with respect to every plane orthogonal to the bound axis is still present and the same considerations hold.
The procedure for finding molecular orbitals and molecular terms is almost the same sketched for homonuclear molecules. The main difference is:
- Since symmetry with respect to the center has been broken there is no supscript
neither in molecular orbitals nor in molecular terms. Furthermore we can combine also atomic orbitals with
if they have
For example
and
can mix together. Because of the energy difference between different n shells and in order to simplify calculations we will not assume
(and other analogue pairs) mixing to be possible if one of these orbitals is completely full.