Continuous point groups are symmetry groups described by continuous transformations (rather than discrete ones), where operations like rotations can vary smoothly and form a continuum of symmetries about a fixed point.
Examples include:
-
- SO(3): the group of all possible rotations about a point in three-dimensional space, with no reflections or inversions. It represents full rotational symmetry of a sphere and is a fundamental example of a continuous symmetry group in chemistry, physics and mathematics. This symmetry is a good approximation for spherically symmetric systems such as isolated atoms or idealised spherical tops.
-
: contains all possible rotations about an axis (infinite-fold rotational symmetry) plus an infinite number of vertical mirror planes. Heteronuclear diatomic molecules belong to this group.
-
: includes all the symmetry elements of , along with a horizontal mirror plane, inversion symmetry, and an infinite set of two-fold rotation axes perpendicular to the main axis. Homonuclear diatomic molecules belong to this group.
As the SO(3) group has been covered extensively in a previous article, we shall focus on the and point groups.
point group
To understand how the character table is derived, we must first define the basis functions. In general, the spherical harmonics 
are used because they form a complete basis set for any point group. )
is equivalent to the linear function 
because )
is invariant under all symmetry operations of a point group. Therefore, 
is invariant under all symmetry operations of the point group. Mathematically, this means that it has an eigenvalue of +1 under every operation, for example 
. Hence, transforms according to the totally symmetric irreducible representation 
. The rotation vector (or axial vector) 
, however, transforms according to 
because its curved arrow around the -axis reverses under 
and returns an eigenvalue of -1.
The remaining irreducible representations are doubly-degenerate and are generated from linear combinations of the basis functions and 
. Under a rotation about the -axis by an angle 
, each component transforms as e^{im_l(\phi-\alpha)}=e^{-im_l\alpha}Y_l^{m_l})
:
The character 
corresponding to 
is the trace of this 2×2 matrix. Using Euler’s formula 
, we obtain
=2cos(m_l\alpha))
Although, the quantum numbers 
and 
in are associated with a single electron, they can be replaced by 
and 
for many-electron systems. In such cases, the basis functions are constructed as linear combinations of products of )
. A properly coupled state with definite and projection is an eigenfunction of 
, with 
. Since each component of the linear combination has an azimuthal dependence proportional to 
, the overall function transforms as 
, where 
. Therefore,
=2cos(M_L\phi))
where we have swapped the arbitrary symbol with 
to be consistent with the above character table.
Since 
and 
correspond to the doubly-degenerate states 
and 
, we can rewrite:
=2cos(\vert M_L\vert\phi)=2cos(\Lambda\phi)\;\;\;\;\;\;\;\;130)
where 
is the projection of 
in a many-electron system.
This is why 
takes only non-negative integer values (including 0) and is denoted by special symbols known as molecular term symbols:
It follows that the irreducible representations 
are also labelled as 
respectively. Setting 
in eq130 corresponds to the identity operation 
, which gives =2)
for all doubly-degenerate irreducible representations. Furthermore, a reflection of the basis functions and in a plane containing the -axis yields:
=\Psi_{polar}e^{-iM_L\phi}=\Psi_L^{-M_L})
Thus,
with the character of 
and hence 
being 0 for all doubly-degenerate irreducible representations.
point group
The point group, unlike the point group, includes the inversion symmetry element. Although molecules that belong to this group, such as homonuclear diatomic molecules, are symmetric under inversion, the basis functions (orbitals or rotations) may or may not be. Therefore, every irreducible representation of must be either symmetric or antisymmetric with respect to inversion. These are labelled with the subscripts 
(gerade, German for “even”) or 
(ungerade, German for “odd”).
The characters associated with all and irreducible representations for , 
and 
are the same as those in the point group. Since performing the inversion operation twice returns every point to its original position, we have 
, where 
is the identity matrix. For a one-dimensional representation, this restricts the possible characters of 
to +1 (for a gerade irreducible representation) or -1 (for an ungerade irreducible representation). The matrix representation of in a two-dimensional representation must be:
with 
, =2)
for gerade representations, and 
, =-2)
for ungerade representations.
Since 
and the basis function is invariant under the rotation for one-dimensional irreducible representations, 
. Therefore, the characters )
for and representations are +1 and -1 respectively. Noting that 
, multiplication on the right by 
gives 
. A rotation by 180° about the -axis gives the eigenvalue of -1 for odd states (e.g. 
) and +1 for even states (e.g. 
). Thus, )
is given by ^{\Lambda})
for 
. It follows that for two-dimensional representations:
=\chi(\hat{\sigma}_h)\chi(\hat{C}_{\infty})=\chi(\hat{i})\chi(\hat{C}_2)2cos(\Lambda\phi)=\chi(\hat{i})(-1)^{\Lambda}2cos(\Lambda\phi))
where =+1)
for gerade irreducible representations and -1 for ungerade irreducible representations.
Therefore, for two-dimensional and irreducible representations are ^{\Lambda}2cos(\Lambda\phi))
and ^{\Lambda+1}2cos(\Lambda\phi))
respectively.
Finally, 
denotes a rotation about an axis perpendicular to the -axis. It can be expressed as 
because transforms \rightarrow(x,-y,z))
and then transforms \rightarrow(-x,y,-z))
, which is equivalent to a net rotation of 180° about the 
-axis (see diagram above). Therefore, =\chi(\hat{i})\chi(\hat{\sigma}_v))
, resulting in the corresponding characters in the character table.