Consider the effect of an operation, *R*, on a state, Φ_{a}.
The
character of the operation is given by

A state function can be expressed as a linear combination of microstates:

so the character of the operation on the state function can be written in terms of microstates as

Each microstate,

where the molecular orbitals in the microstate consist of a selection of the M.O.s in the active space. Before we continue, let us examine this idea:

Consider a full set of M.O.s:

Let the active space be the M.O.s from 8 to 11. Then microstates containing two electrons would be:

.

These microstates could be represented by M.O. orbital occupancies.
1100
1010
1001
0110
0101
0011.

Remember that the M.O.s here can be of either α
or β
spin.
To continue, we need to evaluate
.
This can be expressed
in terms of M.O.s as:

For convenience, we will represent the integral
by
.
This integral can be described as "The integral over M.O. *
ψ _{k}*
in microstate

Using this abbreviation,
can be written as:

Although it is not immediately obvious, the right-hand term is a determinant, of order

For our purposes, solution of the determinant is best done explicitly. To see
why, note that the number of M.O.s involved in the C.I. (the active space) is
very small. Because of this, the number of electrons, *N*, in the Slater
determinants is also small; *N* has a maximum value of 20. Next, use can be
made of the fact that no point-group operation can mix α
and β electrons. This allows the integral to be split into two parts, each of which
has a maximum value of *N*=10. Finally, remember that *N* is the number of
electrons, not M.O.s, used in the active space. A system of *N* electrons has
the same symmetry as a system in which all the M.O.s which were occupied were
replaced with all the M.O.s which were not occupied (the positron
equivalent) . (This assumes that if every M.O. were
occupied, then the state of the system would be totally symmetric.) Using
this fact, we can replace the *N* occupied M.O.s with *N*' unoccupied M.O.s, if
*N*' < *N*.

When these three points are considered, we see that *N* has a maximum value of
5 (for a system of 10 M.O.s). Each case can be considered separately.

- For
*N*= 1:

or

- For
*N*=2:

or

or

- For
*N*=3:

=

or

=

For higher numbers of electrons, the associated determinant is solved using standard methods.

The total character,
,
is obtained by multiplying the
characters for the
and
parts together:

If the positron equivalent is taken for only one set of electrons, e.g. either the α or the β set, but not both, then the character has to be multiplied by the determinant of the M.O. transform.

These expressions can then be used in

to give the expectation value for the state. Finally, if the state is degenerate, the character is given by summing the components of the state.

For the atom, the Russell-Saunders coupling scheme can be reproduced.
States allowed are *S*, *P*, *D*, *F*, *G*, *H*, *I*, *K*, *L*, and *M*. This
set is more than sufficient to allow all possible Russell-Saunders states
spanned by a basis set of *s*, *p*, and *d* orbitals to be represented. The
highest angular momentum achievable with such a basis set is 8, i.e. *L*. For
simpler atoms (ones with only a *s*-*p* basis set) the allowed states are
*p*^{0},*p*^{6}: ^{1}*S*_{g}, * p*^{1},*p*^{5}: ^{2}*P*_{u}, * p*^{2},*p*^{4}:
* ^{1}S_{g}+^{3}P_{g}+^{1}D_{g}* ,

For the axial infinite groups, allowed states are: Σ,
Π, Δ,
θ,
and Γ.
Even quite simple systems can achieve quite high angular
momentum, thus acetylene, with a `C.I.=4` (the
HOMO *π*
and LUMO *π**)
will contain a * ^{1}Γ_{g}*
state, i.e., the
angular momentum will be 4.