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PAPER No.13:Applications of molecular symmetry and group theory MODULE No.18 : Great Orthogonality Theorem and its applications part-2 Character table

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Subject Chemistry

Paper No and Title 13, Applications of molecular symmetry and group theory

Module No and Title 18, Great Orthogonality Theorem and its applications: Part-II Character table

Module Tag CHE_P13_M18

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TABLE OF CONTENTS

1. Learning Outcomes 2. Introduction 3. What is a character table?

3.1 Different sections of character table 4. Notations followed for irreducible representations

4.1 C3v character table and meaning of various terms and parts 5. Summary

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1. Learning Outcomes

After studying this module, you shall be able to:

• Know the applications of rules/theorems derived from GOT. • Learn about character table of a point group. • Know the notations followed for the irreducible representations in character table. • Know example for explanation. • Explain C3v character table. • Know about binary and ternary functions basis.

2. Introduction

You already know about the applications of GOT along with the rules /theorems derived from GOT. These rules have been explained by taking suitable examples. In this module we will earn how to use these rules derived from GOT in construction of character tables of few point group. Character tables are characteristics of point groups and are very useful in dealing with molecular spectroscopy and other related fields. The character tables for each point groups, which are generally encountered, are given in the end of a book on group theory. Great Orthogonality Theorem can be utilized to find (i) total numbers of irreducible representations of a point group and (ii) characters of each of these irreducible representations. Before construction of a character table let us see what it means and what the meaning of various terms used in this table.

3. What is a character table?

The character table of a point group is unique and this is same for all molecules which belong to the same point group.For examples molecules H2O, SO2, pyridine. which belong to C2v point group will have same character table of C2v point group. The general format of a character table is given as:

A B

C D e F

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3.1 Different sections of character table This table is divided into a to f sections ie in six sections (i) Section (a) lists the Schoenflies symbol of point group C2v, C2h, D3 etc. (ii) Section (b) lists the symmetry operations of point group class wise. (iii) Section (c) lists the conventional symbols/notations for various irreducible representations ie Mulliken’s symbols for irreducible representations are used. (iv) Section (d) gives the characters, of all irreducible representations for a symmetry operation in each

class of symmetry operations. (v) Section (e) lists the bases sets of representation for each of the irreducible representation. These may be X, Y, Z axes; Tx, Ty, Tz translational vectors; Rx, Ry. Rz rotational vectors: sets of orbitals; bond vectors.

(vi) Section (f) shows that how the binary functions of x, y, z form the basis for representations for some of the irreducible representations. Binary and ternary combinations can be x2, y2, z2; xy, xz, yz; x2-y2, x2+y2

, x3 and so on. These are important in dealing with symmetries of d, f -- orbitals and in Raman spectroscopy.

Mulliken’s symbols used for irreducible representations have definite meanings. These will be

mentioned here with out going into the source of these. 4. Notations followed for irreducible representations Section (c) of the character table gives the R. S. Mulliken’s symbols / notations ie names for irreducible

representations. These are assigned as: (i) All one dimensional irreducible representations are labeled as A or B as shown for C2v point group

character table as A1, A2, B1, B2 (ii) All two dimensional irreducible representations are labeled as E (one should not confuse with the symbol for identity symbol). (iii) All three dimensional representations are labeled as T or F. Usually T is used in electronic

transitions problems and F is used in vibrational problems. These are important in dealing with higher order representations which are represented by G (four dimensional) and H (five dimensional) and so on.

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(iv) For point group C∞v and D∞h a different notation is used. One dimensional irreducible representation

is represented by the symbol ‘∑’, two dimensional irreducible representation is represented as ‘∏’ .In addition to these general symbols, most Mulliken’s symbols have subscripts and /or superscripts.

For one dimensional irreducible representations subscripts and superscripts have well defined meaning and can be easily worked out.

(v) A one dimensional irreducible representation is given the symbol A if it is symmetric with respect to rotation about the highest order proper axis (principal axis) or improper axis . Symmetric means χ = +1. If the irreducible representation is antisymmetric (χ = -1) with respect to principal axis of rotation Cn it is lableled as B.

C2v E C2

(z)

v(xz) v(yz)

A1 +1

A2 +1

B1 -1

B2 -1

Here C2 axis is the only rotation axis. Characters for irreducible representations A1 and A2 ie A type irreducible representations is χ =+1 and Characters for irreducible representations B1 andB2 ie B type reducible representations is χ =-1 .

For C1, Cs, Ci which do not have Cn axis symbol A is used for one dimensional irreducible representation.

Cs E h

A' +1 +1

A'' +1 -1

Ci E I

Ag +1 +1

Au +1 -1

C1 E

A +1

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(vi a) Subscript 1 is given to one dimensional irreducible representation A or B if it is symmetric with respect to rotation about C2 axis which is perpendicular to Cn axis (principal axis).In the absence of such C2 axis symmetric with respect to σv is considered ie symbols A1 orB1 are assigned. If irreducible representation is antisymmetric with respect to C2┴ Cn (principal axis) or σv the symbols used are A2 or B2.

C2

v

E C2

(z)

v(xz) v(yz)

A1 +1

A2 -1

B1 +1

B2 - -1

For D2 ,D2h point groups where there are three C2 axes and these fall in different class ,the irreducible

representation in which χ for each C2 axis is +1 is labeled as A while other one dimensional irreducible representations is labeled as B. For D2d point group the character of S2n determines the symbol for one dimensional irreducible representation.

D2 E C2 (z) C2 (y) C2 (x)

A +1 +1 +1 +1

B1 +1 +1 -1 -1

B2 +1 -1 +1 -1

B3 +1 -1 -1 +1

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(vi b) For linear molecules the symmetric or antisymmetric in above rule (II a) are indicated by + or - subscript ie ∑+ or∑-

instead of A1 or A2.

C v E 2C ...

v

A1= + +1 ... A2= - +1 ... E1= +2 ... E2= +2 ... E3= +2 ... ... ... ...

... ...

En +2 ... (vii) Subscript g (gerade) or u (ungerade) are given respectively to irreducible representations if these

are symmetric or anti symmetric with respect to centre of inversion ‘i’ ie symbols are Ag, Bg, Au, Bu A1u, B1u etc.

Point groups Cnh (n is even),Dnh (n is even), Dnd (n is odd), Oh, D∞h have centre of inversion i. C2h E C2 (z) i h

Ag +1 +1 +1 +1

Bg +1 -1 +1 -1

Au +1 +1 -1 -1

Bu +1 -1 -1 +1

(viii) Superscripts single prime (′) or double prime(″) are given to irreducible representations which are symmetric or antisymmetric with respect to σh. Point groups Cnh (n is odd), Dnh( n is odd) do not have i but have σh only. C3h E C3(z) (C3)2 h S3 (S3)5

A' +1

E' +2

D2d E 2S4 C2 (z) 2C'2 2 d

A1 +1 +1 +1 +1 +1

A2 +1 +1 +1 -1 -1

B1 +1 -1 +1 +1 -1

B2 +1 -1 +1 -1 +1

E +2 0 -2 0 0

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A'' -1

E''

-2 *

In C3h character table symbol A’ is given when character for h is +1 and A” is give when character

for h is -1 The rules (iii) and (iv) are equally applicable to one -, two-, three- dimensional irreducible representations (ix) If one dimensional irreducible representation has complex characters e, f, g then there must be another equally acceptable irreducible representation with characters e*, f*, g*.These are generally bracketed together and labeled as E. 4.1 C3v character table and meaning of various terms and parts Let us now write down complete character table for C3v point group and explain it in detail. Sections a to f together with explanation is given below in character table of C3v point group

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Section (a) lists C3v a symbol for point group Section (b) lists six symmetry operations of C3v point group, which have been divided into three classes i.e. in E, 2C3, 3σv classes. Total symmetry operations=6 which is h the order of the group. Section( c) tells that as there are only three classes in the group ,there will be only three irreducible representations namely A1,A2 (one dimensional), and E(two dimensional).A1 and A2 are one dimensional irreducible representations ie characters for all matrices of symmetry operations equals to χ (R)= +1 or -1 only. The A1, irreducible representation is symmetric with respect to σv, because C2┴ to C3 is not there. A2, antisymmetric with respect to σv. These A1 and A2 are respectively obtained when Tz or Z and Rz are taken as basis for representation. E is a two dimensional irreducible representation, in which (x,y) or (Tx,Ty) or (RX, Ry) have been used as basis sets for representation Section (d) of the character table lists the characters of matrices for symmetry operations of a particular irreducible representation. These can be obtained by making use of rules of Great Orthogonality Theorem. Section (e) lists the basis sets for representations of irreducible representations (given in part c). For example the basis for A1 is z or Tz and basis for A2 is R z and so on. Let us try to find the basis sets for these representations and we take NH3 molecule which belongs to C3v point group. Let us take z-vector (unit vector along Z-axis) and Rz vector (rotational). These are shown below.

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NH

HH

z

σv

x

y

Let us take z-vector and see the effect of symmetry operations of C3v point group on this E(z)=z ie χ = +1;C3(z)=z ie χ = +1; σv(z)=z ie χ = +1 Similarly let us see the effect of symmetry

operations of C3v point group on Rz rotational vector. These effects are given as E(Rz)=Rz ie χ = +1;C3(Rz)=z ie χ = +1; σv(Rz)=Rz ie χ = -1. These effects can be tabulated as:

C3v E 2C3 3σv A1 1 1 1 z-unit vector A2 1 1 -1 Rz vector Z- unit vector gives +1characterfor all symmetry operations and this corresponds to A1 irreducible

representation. Rz- unit vector gives +1characterfor for symmetry operations E and C3 and -1character for σv these

characters corresponds to A2 irreducible representation. Let us now take x and y unit vectors and see the effect of symmetry operations E, C3, and σv on these

unit vectors.

E(x) → x i.e. χ = +1;E(y) → y ie χ = +1 ie matrix for E is 1 00 1 and trace of this matrix is +2.

σv(x,y) →contributes χ = 0 for each σv as these belong to same class. The matrices corresponding to three σv,s are given as:

-1 0 0 1

σv'

1/2 √3/2 −√3/2 -1/2

σv"

1/2 √3/2 √3/2 -1/2

σv'"

, ,

Trace in each case is =0 Effect of C3

1 on x,y unit vectors. The matrix for this is

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.

-1/2 -√3/2 √3/2 -1/2

and trace for this is -1

Thus for x,y unit vectors the effect of symmetry operations E C31 and σv gives the characters as

+2 -1 0 respectively. These characters correspond to E irreducible representation in C3v point group table

C3v E 2C3 3σv A1 1 1 1 z-unit vector A2 1 1 -1 Rz vector E 2 -1 0 (x,y)or (Rx,Ry)

The binary or ternary functions x2, y2, z2; xy, xz, yz; x2-y2,x2+y2, x3 as basis for irreducible

representations can be worked by taking direct product of basis functions x,y,z and corresponding irreducible representations.

Let us take z2 function as basis and find irreducible representation for

C3v E 2C3 3σv A1 1 1 1 z-unit vector A1 1 1 1 z-unit vector A1.A1

1 1 1 z2

Similarly other binary and ternary functions can be taken as basis and corresponding irreducible representation can be obtained.

4. Summary

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• Applications of rules/theorems derived from GOT have been explained. • What is character table of a point group is explained. • Notations followed for the irreducible representations of the character table explained.

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• Examples were given. • Explanation for C3v character table is given in detail. • Binary or ternary functions as basis for irreducible representations are explained.

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