Ellis A.M., Feher M., Wright T. Electronic and Photoelectron Spectroscopy: Fundamentals and Case Studies

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The principles of point group symmetry and group theory

255

block-diagonalized into a 2×2,a5×5, and another 2×2 matrix:







. 00000

.0 0

. 00000

.0 0

. a

.0 0

. a

.0 0

. a

.0 0

. a

.0 0

. a

.0 0

. 00000

. a

. 00000

. a







If no transformation exists that brings the representation to a block-diagonal form, it is said

to be irreducible.

Unlike the arbitrary matrix representations above, irreducible representations are unique:

they are the simplest representations of the symmetry group. It is, however, often rather

difﬁcult to ﬁnd the appropriate similarity transformation to bring the matrix representation

to an irreducible form. Luckily, as will be shown later, it is usually sufﬁcient to use the

characters of the representation, where the character is deﬁned as the trace of the corre-

sponding matrix (i.e. the sum of its diagonal elements). Dealing with characters is much

simpler than dealing with matrix representations, and these can be collected together into

tables for general use.

There are ﬁve important rules that form the basis of the derivation of character tables.

The reader who is only interested in the use of character tables can simply skip these.

(i) The number of irreducible representations of a group equals the number of classes in

the group.

(ii) The order of the group, h,isdetermined by the dimension of its irreducible represen-

tations, i.e.

h =



(D.4)

where l

is the dimension of the ith irreducible representation.

(iii) The sum of the squares of the characters in any irreducible representation is equal to

the order of the group,

h =



[χ

(R)]

(D.5)

256 Appendix D

where χ

(R)isthe character (trace of the matrix) representing the Rth symmetry oper-

ation in the ith irreducible representation.

(iv) In any irreducible representation the characters that belong to symmetry operations in

the same class are identical.

(v) The following expression holds for the characters of two different irreducible repre-

sentations (orthogonality relation):



(R)χ

(R) = 0, where i = j (D.6)

D.5 Character tables

The properties of point groups can be summarized using character tables. Character tables

list the possible symmetry operations for a given point group along with the irreducible

representations and their characters. The character tables for the most important point

groups can be found at the end of this appendix.

The way character tables are arranged is illustrated below for the C

point group.

E 2C

3σ

11 1z x

+ y

, z

11−1 R

E 2 −10(x, y), (R

, R

)(x

− y

, xy), (xz, yz)

This character table consists of four sections, separated above by double lines (optional). In

the leftmost column, beneath the point group symbol, are the irreducible representations.

These are sometimes also called symmetry species, or simply the symmetry. The uppermost

irreducible representation is always the totally symmetric one, for which all characters are

equal to 1. These characters can be seen to the right of the A

label.

Conventionally, the symbols for irreducible representations are determined in the fol-

lowing way. One-dimensional representations are marked with the letters A or B,two-

dimensional representations by E, three-dimensional ones usually with the letter T.

For

one-dimensional representations, the letter A is used when the character for the rotation

around the principal axis is +1 (i.e. when it is symmetric for this transformation) and B

when this character is −1. The symmetry with respect to the rotation around the axis per-

pendicular to the principal axis (or in its absence reﬂection in the σ

plane) is shown as a

subscript, 1 for the symmetric and 2 for the antisymmetric representation. Reﬂections in

the σ

plane are designated with a prime (symmetric) or double prime (antisymmetric). The

subscripts g and u denote the symmetric or antisymmetric nature of the representation with

respect to inversion.

Exceptions to this are the linear molecule point groups D

∞h

and C

∞v

,where labels such as σ and π are preferred

over A and E. This is discussed again later in the appendix.

The principles of point group symmetry and group theory

257

The second section of the C

character table gives the character for each irreducible rep-

resentation and for each class of symmetry operations. It is useful to note that the character

for the identity operation is equal to the dimension of the irreducible representation.

The ﬁnal two columns provide information about the symmetries of cartesian vectors

(x, y, z),

products of these vectors, and rotations about the cartesian axes (R

, R

, and R

This information is useful for determining spectroscopic selection rules. For example, the z

coordinate axis in the C

point group transforms like the totally symmetric (A

)irreducible

representation, because it is unaffected by the operations of the group. R

also appears on

its own and transforms as the A

irreducible representation. In contrast to these, x and y

(and similarly R

and R

) jointly form a representation. This arises because, after the C

operation is performed, the resulting vector will contain both x and y components. As a

result, x and y are inseparable in this respect, and so they jointly form a representation and

transform as the E irreducible representation.

D.6 Reducible representations, direct products, and direct product tables

There are many occasions in spectroscopy when it is necessary to multiply irreducible

representations, or in the language of group theory, calculate their direct products. The

direct product is obtained by multiplying the characters for each symmetry element. The

resulting representations are often reducible. It can be proved that the number of times (a

)

an irreducible representation occurs within a reducible one can be determined using the

following formula:



red

(R)χ

(R) (D.7)

where χ

red

(R)isthe character of the reducible representation corresponding to operation

R, and χ

(R)isthe character of the irreducible representation. The summation is over all

symmetry operations and h is the order of the group.

This rule can be illustrated by determining the direct product of the E species with itself

within the C

point group, i.e. E ⊗ E.Asthe characters for the E species are 2, −1, and 0,

the characters of the direct product will be  =4, 1 and 0. This is a reducible representation

that can be decomposed to irreducible representations using the formula above, yielding

[1(1)(4) + 2(1)(1) +3(1)(0)] = 1

[1(1)(4) + 2(1)(1) +3(−1)(0)] = 1

[1(2)(4) + 2(−1)(1) +3(0)(0)] = 1

The symmetry of a cartesian vector is the same as the symmetry of the corresponding cartesian axis. For example,

the x axis has both positive and negative regions and any rotation about this axis will leave these unmoved. On the

other hand, a C

rotation about an axis perpendicular to the x axis and passing through the origin will transform x

into −x, and vice versa. In other words, the x axis in this instance will be antisymmetric with respect to C

. Thus

symmetry operations can be applied to cartesian vectors in a manner identical to their application to molecules.

258 Appendix D

Table D.4 Direct product table for point groups C

, C

, D

, C

, D

, S

, D

+ [A

] + E

+ B

+ E

+ [A

] + E

Table D.5 Direct product table for point groups T, T

,O,O

E A

+ [A

] + ET

+ T

+ E + [T

] + T

+ E + T

+ T

+ E + [T

] + T

Table D.6 Direct product table for point groups C

∞v

and D

∞h

−

Π∆



−



−





Π 

+ [

−

] + + 

∆ 

+ [

−

] + 

Note that the number of symmetry operations in each class needs to be considered, and

these are the ﬁrst numbers in each term inside the square brackets. The result is that the

direct product E ⊗ E can be reduced to A

+ A

+ E. That this ﬁnding is correct can be

conﬁrmed by adding up the characters of the three irreducible representations, which will

yield the original reducible representation.

Fortunately, it is not necessary to use (D.7)every time direct products of irreducible

representations are required. Instead, direct product tables are available, which allow the

task to be carried out quickly and easily. Direct product tables often prove themselves to be

just as useful in spectroscopy as character tables, and it is important to be comfortable with

their use. Three direct product tables are, D.1, D.2, and D.3, are shown above, covering a

wide range of point groups.

Interpretation and use of the direct product tables requires a little care. First, notice that

each table applies to a number of different point groups. In some cases the irreducible

representations in the table do not correspond exactly to those of one of the listed point

The principles of point group symmetry and group theory

259

groups. For example, the four irreducible representations of the C

point group are A

, A

, and B

. None of these appears in Table D.4, and yet this is the direct product table

that is supposed to apply to the C

point group. To ﬁnd the direct products we do the

following. First, if the irreducible representations of the point group have no numerical

subscripts, the corresponding subscripts in the direct product table are ignored. Second, if

the irreducible representations have u or g subscripts, or they have a





superscript, the

following additional product rules apply:

Forgand u subscripts: 

⊗ 

= 

, 

⊗ 

= 

, 

⊗ 

= 

, and



⊗ 

= 

For



and



superscripts: 



⊗ 



= 



, 



⊗ 



= 



, 



⊗ 



= 



, and





⊗ 



= 



Thus, for example, if the direct product B

⊗ A

is required for the C

point group, the

above rules show that B

⊗ A

= B

.Asanother example, the E

⊗ E

direct product in

the C

point group is found to be A

+ [A

] + E

. The signiﬁcance of the square bracket

around A

will be seen later.

It is sometimes necessary to extend the concept of direct products to a higher number of

terms. As an example, a triple direct product can be calculated by taking the direct product

of any pair of representations, and then using the result to calculate its direct product with

the third. This operation is commutative, so the order of multiplication does not matter.

Triple direct products are particularly useful in the discussion of spectroscopic selection

rules (see Section 7.1.2).

There are certain simple rules regarding direct products that are helpful to remember

and which can readily be checked by consulting the direct product tables.

The direct product of the totally symmetric irreducible representation with a non-

totally symmetric representation gives the non-totally symmetric representation (as all

characters of the totally symmetric species are 1).

The direct product of any one-dimensional irreducible representation with itself gives

the totally symmetric representation.

The direct product of a higher-dimensional species with itself will be reducible and

always includes the totally symmetric irreducible representation.

D.7 Cyclic and linear groups

The discussion above shows how to interpret, and use, the character tables for most point

groups. However, there are two types of groups that are a little more complicated. One of

these falls into the category of the so-called cyclic groups. They are called cyclic because all

their symmetry elements can be generated from different powers of one of their elements.

Cyclic groups can easily be recognized from their character tables, as the characters of

two-dimensional species contain a function and its complex conjugate. Examples include

the groups C

, C

, and many others. The other category that presents difﬁculties at

ﬁrst sight is the linear molecule point groups C

∞v

and D

∞h

260 Appendix D

Consider the C

point group as an example of a cyclic group. The character table for this

group is as follows:

ε = exp(2πi/3)

A 11 1z, R

+ y

, z

∗

(x , y)(R

, R

)

− y

, xy)

(yz, xz)

In this table, the symbol ε stands for the quantity exp(2πi/3), where i =

√

−1, and ε

∗

the complex conjugate of ε, i.e. exp(−2πi/3). It can be shown that, for the purposes of

many physical applications, the two rows belonging to the E representation can be added,

so that the resulting row only contains real numbers. When this is done the following table

is obtained:

A 11 1 z, R

+ y

, z

E 22cos 2π/32cos 2π/3(x, y), (R

, R

)(x

− y

, xy), (xz, yz)

which can be used like any other character table. As an example, we can try to reduce

the direct product E ⊗ E.Asthe characters for the E representation are 2, 2 cos 2π/3, and

2 cos 2π/3, the characters of the direct product will be  =4, 4 cos

2π/3, and 4 cos

2π/3.

It can be shown by applying well-known trigonometric relationships, namely sin

x + cos

= 1 and cos 2x = cos

x − sin

x, that the characters of the direct product are equal to

 = 4, 2 + 2 cos 2π/3, and 2 + 2 cos 2π/3. It is easy to see that this is simply the sum of

three species, i.e. E ⊗ E = 2A + E.

The character tables for linear molecules, C

∞ v

and D

∞ h

, are also somewhat peculiar at

ﬁrst sight. These two groups differ in the existence of the centre of symmetry as a symmetry

element. As an example, the character table for the point group C

∞ v

is shown below.

∞v

E 2C

∞

... ∞σ

≡ 

11 ... 1 z x

+ y

, z

≡ 

−

11 ... −1 R

≡  22cos φ ... 0 (x, y), (R

, R

)(xz, yz)

≡  22cos 2φ ... 0 (x

− y

, xy)

≡  22cos 3φ ... 0

... ... ... ... ...

First, there is an inﬁnite number of classes because rotation about any angle φ about the C

∞

axis is a symmetry operation and each of these C

∞

elements belongs to a different class.

Similarly, there is an inﬁnite number of σ

planes. The consequence of an inﬁnite number

of symmetry elements is that there is also an inﬁnite number of irreducible representations.

The labelling of these is often slightly confusing. On the one hand, they are sometimes

named according to the conventions described above, i.e. A

, A

, E

, etc. More usually

The principles of point group symmetry and group theory

261

they are labelled according to the convention introduced in electronic structure theory to

describe electronic states of linear molecules, namely , , , , etc. As described in

Sections 4.2.2 and 4.2.3,inelectronic states these labels correspond to different values of

the angular momentum quantum number .

The direct products of irreducible representations in linear groups can be calculated in a

similar manner to other point groups. Taking 

−

⊗  as an example, the characters of the

direct product are  = 2, 2 cos φ,...,0,i.e. 

−

⊗  = .Trigonometric relationships

need to be invoked for the direct products of two- or higher-dimensional representations.

Forexample, the characters of the direct product  ⊗  are  = 4, 4 cos

φ ...,0.Using

the above relationships it can be shown that 4 cos

φ = 2 + 2 cos 2φ, and hence  ⊗  =

 + 

+ 

−

.Inpractice such manipulations are not necessary and direct products can

be obtained simply by inspecting Table D.6.

D.8 Symmetrized and antisymmetrized products

In the description of spectroscopic states it is sometimes necessary to invoke the sym-

metrized and antisymmetrized product of two functions, instead of simply taking their

product. For functions f

and f

, the symmetrized product is

+ f

), whereas the anti-

symmetrized product is

−f

). It can be proved that both of these products are reducible

representations of the point group. In many examples, the antisymmetrized product simply

vanishes.

Symmetrized and antisymmetrized products have special importance when the electronic

state is derived for two electrons. The resulting electronic state can be obtained from the

direct product of the symmetry species of the molecular orbitals. Careful consideration of

the Pauli principle is required if the electrons reside in degenerate orbitals and this is a topic

considered in more detail in the next appendix.Indirect product tables antisymmetrized

direct products are displayed in square brackets.