In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements. The entries consist of characters, the traces of the matrices representing group elements of the column's class in the given row's group representation. In chemistry, crystallography, and spectroscopy, character tables of point groups are used to classify e.g. molecular vibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks on physical chemistry, quantum chemistry, spectroscopy and inorganic chemistry devote a chapter to the use of symmetry group character tables.[1] [2] [3] [4] [5] [6]
G
G
G
\rho(g)=1
g\inG
Here is the character table of
C3=\langleu\rangle
| ( 1 | ( u | ( u2 | |
1 | 1 | 1 | 1 |
\chi1 | 1 | \omega | \omega2 |
\chi2 | 1 | \omega2 | \omega |
where ω is a primitive cube root of unity. The character table for general cyclic groups is (a scalar multiple of) the DFT matrix.
Another example is the character table of
S3
| (1) | (12) | (123) | |
| χtriv | 1 | 1 | 1 |
| χsgn | 1 | −1 | 1 |
| χstand | 2 | 0 | −1 |
where (12) represents the conjugacy class consisting of (12), (13), (23), while (123) represents the conjugacy class consisting of (123), (132). To learn more about character table of symmetric groups, see http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_groups.
The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1×1 matrices containing the entry 1). Further, the character table is always square because (1) irreducible characters are pairwise orthogonal, and (2) no other non-trivial class function is orthogonal to every character. (A class function is one that is constant on conjugacy classes.) This is tied to the important fact that the irreducible representations of a finite group G are in bijection with its conjugacy classes. This bijection also follows by showing that the class sums form a basis for the center of the group algebra of G, which has dimension equal to the number of irreducible representations of G.
See main article: Schur orthogonality relations. The space of complex-valued class functions of a finite group G has a natural inner product:
\left\langle\alpha,\beta\right\rangle:=
| 1 | |
| \left|G\right| |
\sumg\alpha(g)\overline{\beta(g)}
where
\overline{\beta(g)}
\beta
g
\left\langle\chii,\chij\right\rangle=\begin{cases}0&ifi\nej,\ 1&ifi=j.\end{cases}
For
g,h\inG
| \sum | |
| \chii |
\chii(g)\overline{\chii(h)}=\begin{cases}\left|CG(g)\right|,&ifg,hareconjugate\ 0&otherwise.\end{cases}
where the sum is over all of the irreducible characters
\chii
\left|CG(g)\right|
g
For an arbitrary character
\chii
\left\langle\chii,\chii\right\rangle=1
The orthogonality relations can aid many computations including:
V=\left\langle\chi,\chii\right\rangle
\left|G\right|=\left|Cl(g)\right|*
| \sum | |
| \chii |
\chii(g)\overline{\chii(g)}
If the irreducible representation V is non-trivial, then
\sumg\chi(g)=0.
More specifically, consider the regular representation which is the permutation obtained from a finite group G acting on (the free vector space spanned by) itself. The characters of this representation are
\chi(e)=\left|G\right|
\chi(g)=0
g
Vi
\left\langle\chireg,\chii\right\rangle=
| 1 | |
| \left|G\right| |
\sumg\chii(g)\overline{\chireg(g)}=
| 1 | |
| \left|G\right| |
\chii(1)\overline{\chireg(1)}=\operatorname{dim}Vi
Then decomposing the regular representations as a sum of irreducible representations of G, we get
Vreg=oplus
| \operatorname{dim | |
| V | |
| i |
Vi}
|G|=\operatorname{dim}Vreg=\sum(\operatorname{dim}
| 2 | |
| V | |
| i) |
over all irreducible representations
Vi
10=12+12+22+22
Complex conjugation acts on the character table: since the complex conjugate of a representation is again a representation, the same is true for characters, and thus a character that takes on non-real complex values has a conjugate character.
Certain properties of the group G can be deduced from its character table:
|G| x |G|
The character table does not in general determine the group up to isomorphism: for example, the quaternion group and the dihedral group of order 8 have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative by E. C. Dade.
The linear representations of are themselves a group under the tensor product, since the tensor product of vector spaces is again . That is, if
\rho1:G\toV1
\rho2:G\toV2
\rho1 ⊗ \rho2(g)=(\rho1(g) ⊗ \rho2(g))
[\chi1*\chi2](g)=\chi1(g)\chi2(g)
The outer automorphism group acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphism
g\mapstog-1
C3
u\mapstou2,u2\mapstou,
\chi1
\chi2
\omega
\omega2
Formally, if
\phi\colonG\toG
\rho\colonG\to\operatorname{GL}
\rho\phi:=g\mapsto\rho(\phi(g))
\phi=\phia
Aut
Out
This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.
To find the total number of vibrational modes of a water molecule, the irreducible representation Γirreducible needs to calculate from the character table of a water molecule first.
Water (
C2v
C2v
A1 | 1 | 1 | 1 | 1 | z | x2,y2,z2 | |
A2 | 1 | 1 | −1 | −1 | Rz | xy | |
B1 | 1 | −1 | 1 | −1 | Ry,x | xz | |
B2 | 1 | −1 | −1 | 1 | Rx,y | yz |
Functions:
x
y
z
Rx
Ry
Rz
x2+y2
x2-y2
x2
y2
z2
xy
yz
zx
When determining the characters for a representation, assign
1
0
-1
\Gammareducible
x,y,z
Unless otherwise stated, for the identity operation
E
3
\sigma
1
i
-3
Cn
Sn
Cn=2\cos\theta+1
Sn=2\cos\theta-1
\theta=
| 360 | |
| n |
A simplified version of above statements is summarized in the table below
| Operation | Contributionper unshifted atom | |
|---|---|---|
E | 3 | |
C2 | −1 | |
C3 | 0 | |
C4 | 1 | |
C6 | 2 | |
\sigmaxy/yz/zx | 1 | |
i | −3 | |
S3 | −2 | |
S4 | −1 | |
S6 | 0 |
\Gammareducible
=
x
| Number of unshifted atom(s) | 3 | 1 | 3 | 1 | |
| Contribution per unshifted atom | 3 | −1 | 1 | 1 | |
\Gammared | 9 | −1 | 3 | 1 |
From the above discussion, a new character table for a water molecule (
C2v
A1 | 1 | 1 | 1 | 1 | |
A2 | 1 | 1 | −1 | −1 | |
B1 | 1 | −1 | 1 | −1 | |
B2 | 1 | −1 | −1 | 1 | |
\Gammared | 9 | −1 | 3 | 1 |
\Gammared
N=
| 1 | |
| h |
\sumx
| x | |
| (X | |
| i |
x
| x | |
| X | |
| r x |
nx)
where,
h=
| x | |
| X | |
| i |
=
\Gammareducible
| x | |
| X | |
| r |
=
nx=
So,
| N | |
| A1 |
=
| 1 | |
| 4 |
[(9 x 1 x 1)+((-1) x 1 x 1)+(3 x 1 x 1)+(1 x 1 x 1)]=3
| N | |
| A2 |
=
| 1 | |
| 4 |
[(9 x 1 x 1+((-1) x 1 x 1)+(3 x (-1) x 1)+(1 x (-1) x 1)]=1
| N | |
| B1 |
=
| 1 | |
| 4 |
[(9 x 1 x 1)+((-1) x (-1) x 1)+(3 x 1 x 1)+(1 x (-1) x 1)]=3
| N | |
| B2 |
=
| 1 | |
| 4 |
[(9 x 1 x 1)+((-1) x (-1) x 1)+(3 x (-1) x 1)+(1 x 1 x 1)]=2
So, the reduced representation for all motions of water molecule will be
\Gammairreducible=3A1+A2+3B1+2B2
Translational motion will correspond with the reducible representations in the character table, which have
x
y
z
A1 | z | |
A2 | ||
B1 | x | |
B2 | y |
B1
B2
A1
x
y
z
\Gammatranslational=A1+B1+B2
Rotational motion will correspond with the reducible representations in the character table, which have
Rx
Ry
Rz
A1 | ||
A2 | Rz | |
B1 | Ry | |
B2 | Rx |
B2
B1
A2
x
y
z
\Gammarotational=A2+B1+B2
Total vibrational mode,
\Gammavibrational=\Gammairreducible-\Gammatranslational-\Gammarotational
=(3A1+A2+3B1+2B2)-(A1+B1+B2)-(A2+B1+B2)
=2A1+B1
So, total
2+1=3
2A1
1B1
There is some rules to be IR active or Raman active for a particular mode.
x
y
z
x2+y2
x2-y2
x2
y2
z2
xy
yz
xz
x
y
z
As the vibrational modes for water molecule
\Gammavibrational
x
y
z
Similar rules will apply for rest of the irreducible representations
\Gammairreducible,\Gammatranslational,\Gammarotational
Ethylene is a member of the D2h point group, which has eight Mulliken symbols in the first column. Besides, the ethylene molecule contains six atoms, each with an x, y, and z axis. So, the molecule has a total of 18 axes.
For vibrational modes of the molecule, it is necessary to calculate the irreducible representation Γirreducible. Also, the irreducible representation is related with the reducible representation.
Here is another method to calculate the representation calculation. It is necessary to find the change of x, y and z axes. If the atom changes the place after the operation, there is no contribution to the Γreducible. If the atom keeps the same place after the operation, then check the axis, if the axis keeps same direction, the contribution to the Γreducible.is 1; if the axis reverses to the opposite direction, the contribution to the Γreducible.is -1; if the axis rotates at a certain angle θ, the contribution is cos θ. After calculating all axes of all atoms, there is the value of the reducible representation Γreducible for this operation.In this case, ethylene is the D2h point group with eight symmetry operations in the first line, each operation provides the different Γreducible.
E: Identity Symmetry. All atoms remain in their original positions, so they all have the same x, y, and z axes. The 18 axes remain in the same position, each contributing one to the reducible. The reducible number for E is 18.
C2(x), C2(y): As the molecule rotates along the x or y axis, each atom moves and contributes zero to the reducible. The overall Γreducible for C2(x) and C2(y) are 0.
C2(z): The molecule rotates along the z axis, with only two carbon atoms remaining in the same position. The x and y axes of each carbon atom reverse to the opposite place, but z axis keeps the same direction, contributing negative one of each atom. The overall Γreducible is -2.
i: The molecule is inverse through the center. Since all atoms move places, the overall Γreducible for i is 0.
σ(xy): The molecule flips across the xy plane. The overall Γreducible for σ(xy) is 0, as all atoms move places.
σ(xz): The molecule flips across the xz plane, but two carbon atoms remain in the same place. The x and z axes remain unchanged, each contributing to a single reducible number. However, the y axis reverses and contributes to negative one Γreducible. So, each carbon contributes one Γreducible, the overall Γreducible is 2.
σ(yz): It is different from other operations. All six atoms maintain their original positions. The y and z axes remain the same, but the x axis reverses, resulting in one Γreducible for each atom. The total Γreducible is 6.
New character table for ethylene
\Gammared
| E | C2(x) | C2(y) | C2(z) | i | σ(xy) | σ(xz) | σ(yz) | ||
| Γreducible | 18 | 0 | 0 | -2 | 0 | 0 | 2 | 6 |
| N | |
| Ag |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x 1)+0+0+(2 x 1 x 1)+(6 x 1 x 1)]=3
| N | |
| B1g |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x 1)+0+0+(2 x 1 x (-1))+(6 x 1 x (-1))]=1
| N | |
| B2g |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x (-1))+0+0+(2 x 1 x 1)+(6 x 1 x (-1))]=2
| N | |
| B3g |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x (-1))+0+0+(2 x 1 x (-1))+(6 x 1 x 1)]=3
| N | |
| Au |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x 1)+0+0+(2 x 1 x (-1))+(6 x 1 x (-1))]=1
| N | |
| B1u |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x 1)+0+0+(2 x 1 x 1)+(6 x 1 x 1)]=3
| N | |
| B2u |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x (-1))+0+0+(2 x 1 x (-1))+(6 x 1 x 1)]=3
| N | |
| B3u |
=
| 1 | |
| 8 |
[(18 x 1 x 1)+0+0+((-2) x 1 x (-1))+0+0+(2 x 1 x 1)+(6 x 1 x (-1))]=2
Γirreducible = 3Ag+1B1g+2B2g+3B3g+1Au+3B1u+3B2u+2B3u
Translational motion has x, y and z functions in “linear functions, roatations”. So, Γtrans = 1B1u+1B2u+1B3u
Rotational motion has Rx, Ry and Rz functions in “linear functions, roatations”. So, Γrot = 1B1g+1B2g+1B3g
Vibrational motio: Γvib = Γirreducible-Γtrans-Γrot = 3Ag+1B2g+2B3g+1Au+2B1u+2B2u+1B3u
The final step is to determine which vibrations are IR or Raman active. This means that the symmetry operation can be detected using the infrared or Raman spectrum.
First, for IR to work, they must have x, y, and z functions in "linear functions, rotations". In Γvib, only 2B1u+2B2u+1B3u are IR active.
To be Raman active, "quadratic functions" must include x2, y2, z2,xy, xz, yz, x2+y2 or x2-y2 functions. In Γvib, only 3Ag+1B2g+2B3g are Raman active.
