Chapter 03 Group Theory

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Group Theory		� PAGE �25�
Chapter 3 - Group Theory
A Group is a collection of elements which is:
i)	closed under some single-valued associative binary operation
ii)	contains a single element satisfying the identity law
iii)	and has a reciprocal element for each element in the group
Collection: a specified # of elements (finite or infinite)
Elements: the consitituents of the group (i.e., symmetry operations)
Binary Operation: the combination of two elements of a group to yield another element in the group. The combination may be mathematical (addition, subtraction, etc.) or qualitative as in the successive application of two symmetry operations on an object. 
Single-valued: the combination of two elements yields a unique result
Closed: the combination of any two group elements must always yield another element belonging to the group. 
Associative: the associative law of combination must hold for the group.
				(AB)C = A(BC)
In general, however, elements of a group do not have to commute (but they can):
				AB ≠ BA
Identity Law: there must be an element in the group which when combined with any element in the group will leave them unchanged. This element is called the identity or unit element and it commutes with all elements of the group. It is given the symbol E.
			EA = A		AE = A		EE = E
Reciprocal Element: for each element A in a group there must be an element called the reciprocal, A1, such that the following holds:
				AA1 = A1A = E
In general, group multiplication is not commutative, i.e., AB ≠ BA. However, it can be and a group in which multiplication is completely commutative is called an Abelian Group. 
�Group Multiplication Table (matrix operations)
	G3
	E
	A
	B
	E
	E
	A
	B
	A
	A
	B
	E
	B
	B
	E
	A
Each row and each column in a group multiplication table lists each of the group elements ONCE and ONLY ONCE. It therefore follows that no two columns or rows may be identical! 
Consider a “real” C3 table using symmetry elements:
	C3
	E
	C3
	C32
	E
	E
	C3
	C32
	C3
	C3
	C32
	E
	C32
	C32
	E
	C3
�Consider the two different ways we can set up a 4 × 4 table:
	G41
	E
	A
	B
	C
	E
	E
	A
	B
	C
	A
	A
	E
	C
	B
	B
	B
	C
	E
	A
	C
	C
	B
	A
	E
Note that each element times itself generates E. 
	G42
	E
	A
	B
	C
	E
	E
	A
	B
	C
	A
	A
	B
	C
	E
	B
	B
	C
	E
	A
	C
	C
	E
	A
	B
Note that this group table above is cyclic, that is, the group is generated by one element:
	A = A
	
	A3 = C
	A2 = B
	
	A4 = E
Note that the G3 (C3) example above was also cyclic. 
There is only one group combination possible for the G5 group, which turns out to be cyclic as well:
	G5
	E
	A
	B
	C
	D
	E
	E
	A
	B
	C
	D
	A
	A
	B
	C
	D
	E
	B
	B
	C
	D
	E
	A
	C
	C
	D
	E
	A
	B
	D
	D
	E
	A
	B
	C
Note the diagonal lining up of the elements in cyclic groups (symmetry of a matrix sort). 
�SubGroups
A subgroup is a self-contained group of elements residing within a larger group.
	G6
	E
	A
	B
	C
	D
	F
	E
	E
	A
	B
	C
	D
	F
	A
	A
	E
	D
	F
	B
	C
	B
	B
	F
	E
	D
	C
	A
	C
	C
	D
	F
	E
	A
	B
	D
	D
	C
	A
	B
	F
	E
	F
	F
	B
	C
	A
	E
	D
	G3
	E
	D
	F
	E
	E
	D
	F
	D
	D
	F
	E
	F
	F
	E
	D
�Classes
Assume that A and X are elements of a group and we perform the following operation:
X1AX = B
Where B is another element in the group. B is then called the similarity transform of A by X. If this relationship holds, then A and B are said to be conjugate.
The following is true for elements that are related by similarity transforms:
1) Every element is conjugate with itself
A = X1AX
	(X may be equal to the identity element E)
2) If A is conjugate with B, then B is conjugate with A
	Thus, if we have:
X1AX = B
	Then there must exist another element Y such that:
Y1BY = A
3)	Finally, if A is conjugate to both B and C, then B and C must also be conjugate to each other.
A group of elements that are conjugate to one another is called a Class of Elements. 
To determine which elements group together to form a class you have to work out all the similarity transforms for each element in the group. Those sets of elements that transform into one another are then in the same class. 
Consider the C3v symmetry point group “matrix”:
	C3v
	E
	C3
	C32
	v1
	v2
	v3
	E
	E
	C3
	C32
	v1
	v2
	v3
	C3
	C3
	C32
	E
	v2
	v3
	v1
	C32
	C32
	E
	C3
	v3
	v1
	v2
	v1
	v1
	v2
	v3
	E
	C3
	C32
	v2
	v2
	v3
	v1
	C32
	E
	C3
	v3
	v3
	v1
	v2
	C3
	C32
	E
Lets determine the classes of symmetry operations for this point group. Lets start with the similarity transforms for the vertical mirror planes:
v1v1v11 = v1
v2v1v21 = v3
v3v1v31 = v2
�Let’s see how this works graphically:
v2v1v21 = v3
�v3v1v31 = v2
�C3v1C31 = v3
If we continue these similarity transforms we find that the various symmetry operations for C3v break down into the following classes:
				E
				C3, C32
				v1, v2 , v3 
If we examine the character tables in Cotton we find that the symmetry operations are listed and grouped together in these very same classes:
Corollary: the orders of all the classes must be integral factors of the order of a group. 
Order of a point group = # of symmetry operations
�Matrix Operations
Consider the following matrix:
	a11
	a12
	a13
	a14
	. . .
	a1n
	a21
	a22
	a23
	a24
	. . .
	a2n
	a31
	a32
	a33
	a34
	. . .
	a3n
	
	
	
	
	. . .
	
	an1
	an2
	an2
	an2
	. . .
	amn
Character: sum of diagonal elements
In order to multiply two matrices they must be conformable, i.e., to multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B. 
(a11 × b11) + (a12 × b21) = c11
	3 × 2			 2 × 4		 =		3 × 4
�The symmetry operations can all be represented mathematically as 3 × 3 square matrices. 
To carry out the symmetry operation, you multiply the symmetry operation matrix times the coordinates you want to transform. The x, y, z coordinates are written in vector format as a 3 × 1 matrix:
For example, the inversion operation take the general coordinates x, y, z to x, y, z. In matrix terms we would write:
x(new) = (1)(x) + (0)(y) + (0)(z)
y(new) = (0)(x) + (1)(y) + (0)(z)
z(new) = (0)(x) + (0)(y) + (1)(z)
Symmetry Operation Matrices:
Cn × h = Sn
					Cn 				 h
�Group Representations
The set of four matrices that describe all of the possible symmetry operations in the C2v point group that can act on a point with coordinates x, y, z is called the total representation of the C2v group.
	 E			 C2			 xz		 yz
Note that each of these matrices is block diagonalized, i.e., the total matrix can be broken up into blocks of smaller matrices that have no off-diagonal elements between blocks. 
These block diagonalized matrices can be broken down, or reduced into simpler one-dimensional representations of the 3-dimensional matrix. 
If we consider symmetry operations on a point that only has an x coordinate (e.g., x, 0, 0), then only the first row of our total representation is required:
	C2v
	E
	C2
	xz
	yz
	
	
	1
	1
	1
	1
	x
We can do a similar breakdown of the y and z coordinates to setup a table:
	C2v
	E
	C2
	xz
	yz
	
	
	1
	1
	1
	1
	x
	
	1
	1
	1
	1
	y
	
	1
	1
	1
	1
	z
These three 1-dimensional representations are as simple as we can get and are called irreducible representations. 
There is oneadditional irreducible representation in the C2v point group. Consider a rotation Rz :
The identity operation and the C2 rotation operations leave the direction of the rotation Rz unchanged. The mirror planes, however, reverse the direction of the rotation (clockwise to counter-clockwise), so the irreducible representation can be written as:
	C2v
	E
	C2
	xz
	yz
	
	
	1
	1
	1
	1
	Rz
4 Classes of symmetry operations = 
4 Irreducible representations!!
Now lets consider a case where we have a 2-dimensional irreducible representation. Consider the matrices for C3v
	 E			 		C3			 	 v
In this case the matrices block diagonalize to give two reduced matrices. One that is 1-dimensional for the z coordinate, and the other that is 2-dimensional relating the x and y coordinates. 
Multidimensional matrices are represented by their characters (trace), which is the sum of the diagonal elements. 
Since cos(120º) = 0.50, we can write out the irreducible representations for the 1- (z) and 2-dimensional “degenerate” x and y representations: 
	C3v
	E
	2C3
	v
	
	
	1
	1
	1
	z
	
	2
	1
	0
	x,y
As with the C2v example, we have another irreducible representation (3 symmetry classes = 3 irreducible representations) based on the Rz rotation axis. This generates the full group representation table:
	C3v
	E
	2C3
	v
	
	
	1
	1
	1
	z
	
	2
	1
	0
	x,y
	
	1
	1
	1
	Rz
�Character Tables
 Schoenflies symmetry symbol
Mulliken Symbol Notation
A or B: 1-dimensional representations
E : 2-dimensional representations
T : 3-dimensional representations
A = symmetric with respect to rotation by the Cn axis
B = anti-symmetric w/respect to rotation by Cn axis
Symmetric = + (positive) character
Anti-symmetric =  (negative) character
�Subscripts 1 and 2 associated with A and B symbols indicate whether a C2 axis ( to the principle axis produces a symmetric (1) or anti-symmetric (2) result. 
If C2 axes are absent, then it refers to the effect of vertical mirror planes (e.g., C3v) 
Primes and double primes indicate representations that are symmetric ( ( ) or anti-symmetric ( ( ) with respect to a h mirror plane. They are NOT used when one has an inversion center present (e.g., D2nh or C2nh). 
In groups with an inversion center, the subscript “g” (“gerade” meaning even) represents a Mulliken symbol that is symmetric with respect to inversion. The symbol “u” (“ungerade” meaning uneven) indicates that it is anti-symmetric. 
The use of numerical subscripts on E and T symbols follow some fairly complicated rules that will not be discussed here. Consider them to be somewhat arbitrary. 
Square and Binary Products
These are higher order “combinations” or products of the primary x, y, and z axes. 
The Great Orthogonality Theorem
i (R)mn 	The element in the mth row and nth column of the matrix corresponding to the operation R in the ith irreducible representation i. 
i (R)mn*	complex conjugate used when imaginary or complex #’s are present (otherwise ignored)
h 	the order of the group
li	the dimension of the ith representation
(A = 1, B = 1, E = 2, T = 3)
	delta functions, = 1 when i = j, m = m’, or n = n’; = 0 otherwise
The different irreducible representations may be thought of as a series of orthonormal vectors in h-space, where h is the order of the group. 
�Because of the presence of the delta functions, the equation = 0 unless i = j, m = m’, or n = n’. Therefore, there is only one case that will play a direct role in our chemical applications:
�Five “Rules” about Irreducible Representations:
The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order, h, of a group.
	For example, consider the D3h point group:
l(A1’)2 + l(A2’)2 + l(E’)2 + l(A1”)2 + l(A2”)2 + l(E”)2
(1)2 + (1)2 + (2)2 + (1)2 + (1)2 + (2)2 = 12
The sum of the squares of the characters in any irreducible representation is also equal to the order of the group h. 
		
	For example, for the E’ representation in D3h:
(E)2 + 2(C3)2 + 3(C2)2 + (h)2 + 2(S3)2 + 3(h)2
(2)2 + 2(-1)2 + 3(0)2 + (2)2 + 2(-1)2 + 3(0)2 = 12
The vectors whose components are the characters of two different irreducible representations are orthogonal.
�� EMBED Equation.DSMT4 
For example, multiply out the A2’ and E’ representations in D3h: 
1(1)(2) + 2(1)(-1) + 3(-1)(0) + 1(1)(2) + 2(1)(-1) + 3(-1)(0)
2 + (-2) + 0 + 2 + (-2) + 0 = 0
In a given representation the characters of all matrices belonging to operations in the same class are identical.
The number of irreducible representations in a group is equal to the number of classes in the group. 
Row
Row
Column
�
Abelian group
(column) × (row)
Column
E
i
(xy)
(xz)
(yz)
Cn
Sn
�
Characters of the irreducible representations
Mulliken
symbols
x, y, z
Rx, Ry, Rz
Squares & binary products of the coordinates
if i ≠ j
if m ≠ m’
 n ≠ n’
h = 12 (order of group)
g = # of symmetry operations R in a class
Dimensions:
A or B = 1
E = 2
T = 3
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