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C734b Irreducible Representations and Character Tables
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Irreducible Representations Irreducible Representations
and Character Tablesand Character Tables
C734b
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Irreducible Representations Irreducible Representations
Suppose {Γ(A), Γ(B), …} for an l-dimensional matrix representation of G. Then if S is any non-singular lxl matrix (det S ≠ 0) the {Γ’(A), Γ’(B), …} also form a l-dimensional representation of G where
( A ) S ( A ) S
1
−
similarity transform
Note:Note: ( A ) ( B ) S ( A ) SS ( B ) S S ( A ) ( B ) S
Γ' Γ = −^1 Γ^ −^1 Γ = −^1 Γ Γ
S ( AB ) S '( AB )
⇒ {Γ’(A),^ Γ’(B), …} is also a representation of G
Two representations related by a similarity transformation are said to be equivalentequivalent.
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Suppose that Γ^1 and Γ^2 are matrix representations of G with dimensions l 1 and l 2 , respectively, and that for every operation A of G a (l 1 + l 2 )-dimensional matrix is defined by:
( )
( )
( ) ⎟
A
A
A 2
1
( ) ( )
( )
( )
( )
( ) ⎟
B
B
A
A
A B 2
1
2
1
( ) ( ) ( ) ( )
( ) ( )
( AB )
AB
AB
A B
A B
1 2 2
1 1
⇒ {Γ(A),^ Γ(B),…} as defined also form a representation of G This representation of G is called the direct sum of Γ^1 and Γ^2
1 2 ⇒Γ=Γ ⊕Γ
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Alternatively, we can regard Γ as reduced into Γ^1 and Γ^2
A representation of G is reducible if it can transformed by a similarity transformation into an equivalent representation, each matrix which has the same block diagonal form. Then each of the smaller representations Γ^1 , Γ^2 , Γ^3 etc are also representations of G
A representation that can not be reduced any further is called an irreducible representation , IR
of fundamental importance
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Rule 1.) can be written as:
[ E ] h i
∑ i =
χ( )^2
since χi (E), the character of the representation of E in the ith^ IR = order of the representation.
Rule 2.)Rule 2.)^ The sum of the squares of the characters in any IR = h
[ R ] h R
∑ i =
2
χ ( ) ( simple test of irreducibility )
Rule 3.)Rule 3.) the vectors whose components are the characters of two different IRs are orthogonal:
∑ (^ R )^ j ( R )=^0 R
χ i χ
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Rule 4.):Rule 4.): In a given reducible or irreducible representation the character of all matrices belong to the same class are identical.
Rule 5.): Rule 5.): The number of IRs = number of classes in a group.
From rules 2.) and 3.): (^) j ij R
∑ χ i^ ( R )χ( R )= h^ δ
Denote the number of elements in the mth^ class by gm , the number in the nth class by gn, etc. and let there be k classes.
Then: (^) p ij
k
p
∑ χ^ i ( Rp )χ j ( Rp ) g = h^ δ
Here Rp is any one of the operations in the pth^ class.
This means the k χi (Rp) quantities in the Γi IR behave like components of a k-dimensional vector which is orthogonal to the k-1 other vectors.
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Example:Example: C3v {E, C 3 +, C 3 - , σa, σb, σ 3 }
defined w.r.t. the xz plane
y
x
a
b
c
+π/
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Matrix representations Matrix representations
( E )
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
− −
=
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎟ ⎠
⎜ ⎞ ⎝
⎟ ⎛ ⎠
⎜ ⎞ ⎝
⎟ ⎠
⎜ ⎞ ⎝
⎟ − ⎛ ⎠
⎜ ⎞ ⎝
⎛
Γ + = 0 0 1
0 2
1 2
3
0 2
3 2
1
0 0 1
0 3
cos^2 3
sin^2
0 3
sin^2 3
cos^2
( 3 ) π^ π
π π
C
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Lastly:
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛ −
=
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
⎟ ⎠
⎜ ⎞ ⎝
⎟ − ⎛ ⎠
⎜ ⎞ ⎝
⎛
⎟ ⎠
⎜ ⎞ ⎝
⎟ ⎛ ⎠
⎜ ⎞ ⎝
⎛
= 0 0 1
23 21 0
21 23 0
0 0 1
sin^23 cos^230
0 3
sin^2 3
cos^2 π π
π π
σ (^) C
Each matrix is block diagonal into two representations by inspection:
1 2 Γ ⊕Γ
Χ(Γ^1 ) = {E, C 3 +, C 3 - , σA, σB, σC } ={2, -1, -1, 0, 0, 0 } = 1 st^ class 2 nd^ class 3 rd^ class
Χ(Γ^2 ) = {1 1, 1 1, 1, 1}
Point 4
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i) i) Are Γ^1 and Γ^2 irreducible? Yes if (^ ) point1.)
2
R h
R
∑χ i =
h = 6
Γ^1 : (2) 2 + (-1)^2 +(-1) 2 + 0 2 + 0^2 +0^2
irreducibleirreducible
Γ^2 : (1) 2 + (1)^2 + (1) 2 + (1)^2 + (1) 2 + (1)^2 = 6 irreducibleirreducible
ii)ii) Is χ(Γ^1 ) orthogonal to χ(Γ^2 )? point 3.)
(2)(1) + (-1)(1) +(-1)(1) +(0)(1) + (0)(1) + (0)(1) = 2 – 2 = 0 yesyes
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iii)iii) Are Γ^1 and Γ^2 the only irreducible representations?
No! No! There are 3 classes: E, {C 3 +, C 3 - }, {σA,^ σB,^ σC }
∴There must be one more.^ point 5.)
Let that representation be Γ^3
Ql^21 + l^22 +l^23 = 6 point1.)
and l^1 =^2 ,^ l 2 =^16411
2
⇒ l 3 = − − =
∴ χ(E) for^ Γ
(^3) =1; the IR is 1-dimensional point 1.)
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∴ ∑χ( Γ^1 )χ(Γ^3 )= 0
R
and (^ ) ( )^0
Γ^2 Γ^3 =
R
∴ (1)(2)(1) + (2)(-1)χ^3 (C^3 ) +(3)(0)χ^3 (σ) = 0^ (i)
and (1)(1)(1) + (2)(1)χ 3 (C 3 ) + (3)(1)χ 3 (σ) = 0 (ii)
in class
∴ From (i): -2χ^3 (C^3 ) = -2^ ⇒^ χ(C^3 ) = 1
∴ From (ii): 1+(2)(1)(1) + 3χ 3 (σ) = 0^ ⇒^3 χ 3 (σ) = -
∴ Χ^3 (σ) = -
Note:Note: (^ )^6 IR point 2.)
Γ^3 = ⇒
R
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Character TablesCharacter Tables
Tabulation by class the characters of the IRs of a point group
The Schonflies symbol is in the upper left-hand corner
Each column is headed by the number of elements in class x symbol for that element. For example 2C 3 for {C 3 +, C 3 - } in C3v
Each row ≡ Γ label for the IR given by Mulliken notation: a)a)^ 1-D IRs symmetric to Cn rotation; that is,^ χ(Cn) = +1:^ A otherwise if χ(Cn) = -1: B b)b) Subscripts 1 or 2 are used depending on whether the IR is symmetric or anti-symmetric, χ = +1 or -1, to a perpendicular C 2 axis or σv c)c)^ Prime or double prime superscripts to indicate IRs which are symmetric or anti-symmetric to σh (if it exists). d)d)^ g or u subscripts depending if IR is symmetric or anti-symmetric with respect to I (if it exists). g ≡ gerade and u ≡ ungerade.
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e)e) 2-D or 3-D IRs are labelled by E and T , respectively. (Don’t confuse this with R = E or T groups).
Right-hand side of character table tells how components of
r = e ˆ 1 x + e ˆ 2 y + e ˆ 3 z
r
or how quadratic functions, xy, z^2 , etc. transform.
These will be useful down the road for understanding the IRs for p and d orbitals.
Rx, R (^) y, Rz tell how rotations about x, y, and z, transform, respectively.
The notation for the IRs of the axial groups C∞v and D∞h is different.
IRs are classified according to the magnitude of the z-component of angular momentum Lz along the symmetry axis, z ≡ Λ
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...
0 1 2 3 ...
Σ Π Δ Φ
Λ = Lz =
All IRs are 2-D except Σ. Subscripts g and u are the same but + or – superscripts are used on Σ if χ(σv) = +1 or -1, respectively.
For L (^) z. 0, χ(C 2 `) and χ(σv) = 0.
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Should we care? Yes because many important chemistry items share the same symmetry properties
What else behaves symmetryWhat else behaves symmetry--wise as x, y, z?wise as x, y, z?
The p-orbitals: px, p (^) y, p (^) z
The components of the dipole moment: μˆ^ =− er^ :μ x =− ex μ y =− ey μ z =− ez
r
The dipole moment governs the strongest single-photon absorption and emission transitions. Their IRs will help understand electronic spectroscopy
Bond lengths also behave as x, y, z. Their IRs come up in infrared spectroscopy
Translations of molecules behave as x, y, z
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What else behaves symmetry- What else behaves symmetry-wise as binary properties of x, y, z?wise as binary properties of x, y, z?
The d-orbitals: d^ z 2 dx (^2) − y 2 dxy dxz dyz
Components of the electric quadrupole, Q
Q (^) ij rirj r ij d r
2 3 ≡ (^) ∫ρ ( 3 − δ )
Important for weak electric quadrupole transitions; important in solid state NMR
Binary products also show up as the operator for two-photon transitions in molecules
Components of the polarizability tensor α. Their IRs come up in Raman spectroscopy
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What else behaves symmetry- What else behaves symmetry-wise as Rwise as R xx RR yy RR zz ??
The components of orbital angular momentum: Lx, L (^) y, L (^) z
The components of the magnetic dipole: m BL
r^ r
Here μB is the Bohr magnetron constant. Important for weak magnetic dipole transitions. Also important in NMR.
What about sWhat about s--orbitals?orbitals?
s-orbitals look the same regardless of symmetry operation. Hence their IR is the totally symmetric IR of the point group under discussion. This is always true and therefore, s is not labeled in the character table.
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What about fWhat about f--orbitals?orbitals?
There are seven: f^ z^3 fxz^2 fyz^2 fxyz fz ( x^2 − y^2 ) fx ( x^2 − 3 y^2 ) fy ( 3 x^2 − y^2 )
Their IRs are not listed in character tables but if you know how x, y, z, and the binary operators behave you can deduce the triple products by taking direct products of the appropriate IRs.
What about electron spin? What about electron spin?
Like every thing to do with electron spin, the behavior is weird. We’ll consider this later.
Point:Point:^ You can learn a lot from a character table without doing a single calculation!
