Download Symmetry and Group Theory

III-1
III. Symmetry and Group Theory
Levine – Chap 12
McHale – Appendix C
More focused texts: Cotton – Chap 3
Harris & Bertolucci – Chap 1
A. Categorize Symmetry – operations that put molecule into equivalent state/orientation
[exchange equivalent atoms by rotation/reflection/inversion]
O
Water example H1
H2
rot
o
180
O
H2
H1
H1, H2 indistinguish
 cos  sin    x 
   
Mathematical representation of rotation: 
  sin  cos   y 
1 0  x
- x

     
for rotation about z of = 180o:
 0 -1  y
- y
 x 
 
 y 
Another operation: reflection through (xz) plane: (x,y,z) → (x,-y,z)
Operations (symbols)
Cn – rotation about on axis by 2
n
 – reflection through  plane
 – vertical
h – horizontal -  Cn
d – dihedral – bisect 2 C2 axes
i – inversion: (x,y,z) → (-x,-y,-z)
Sn – improper rotation: combine: ( 2 rotation + h reflection  Sn)
n
z
N
NH3 example: C3 – z-axis 2 = 120º rotation - 2 of them, i.e.  2
3
3
 – contain each N-H bond, exchange other 2
(always get nC for Cn → C3v , if there is a v)
CH4 example: C2 – bisect each CH2 (3 of them x,y,z)
S4 – Same - 90º + flip H’s (6 of them  x,y,z)
C3 – along each C-H bond (8 - exchange 3 H’s  2 )
3
 – contain each CH2 (6 - 2 on each x,y,z)
SF6 example: C4, C2, S4 – each F-S-F bond pair (x,y,z)
‘s on axes (contain xy, yz, xz) and d’s between axes
plus i – all invert through center
H1
y
H3
x H2
H
H
C
H
H
III-2
Categorize these symmetry operations – Point Group
– mathematical entity with defined properties, relations between operations
Molecules that have same group (set of operations) will have common properties
Comes from – [H, sym] = 0 i.e. energy is uncharged if molecules same
Basic operations: Cn, Sn, , n, d, i, + E (do nothing) “identity”
1 – Find all symmmetry elements by inspection (models help)
2 – Use flow chart (handout, e.g. Levine) to determine point group
a. Special → Spherical → full rotation group (atoms)
Cubic → T d, On, In … (2 or more n ≥ 3 Cn)
Linear → C∞, D∞h ( has i)
b. Cn determine group → highest order rotation → it n ≥ 3
This will be the principle axis for symmetrical top
 (if none then C1, Ci, Cs → depend on other elements)
c. If n C2  Cn → then a D group (otherwise C)
if n → Dnh or Cnh
n → Dnd or Cn
none → Dn or Cn
III-3
N
N
C
C
Pt
Practice (Handout)
,
C
,
N
C
, FeCl5Br , FeCl5H2O ,
N
O
FCH3 , B2H6 , BH3 , H2C
CH2 , Fe(Cp)2 –ferrocene
III-4
Value – symmetry characterization can tell if QM integral = 0 or ≠0
How to handle → collection of operations forms a Group
Group Theory can be used to evaluate properties ( =0 or ≠0 ) without knowing function
[note: analogy to angular momentum
→ YJM are symmetry representations of full rotation group]
Read PJS handout (mathematical start—on line, Web Site)
Cotton, Chap 2; Harris & Bertolucci, Chap 1; Douglas & Hollingsworth, Chap 2
B. Group – Definitions: set of elements {} = G under an operation 
Properties:
1) Closure – A  B  G, A, B  G (don’t leave the group)
2) Identity – E  A = A, All A < G, E  G (one does nothing)
3) Reciprocal (Inversion) – A-1  A = E and A-1  G if A  G
(can undo anything, A-1 is in G)
4) Associative – (A  B)  C = A  (B  C)
Note: order maintained but can combine pairs either way
Mathematical examples:
a) {1, -1}  = multiplication
E = 1, self inverse
b) {negative, positive integers}  = addition, E = 0, -N = N-1 (inversion)
c) {E, A, A2}  successive application A,
E = A3 (cyclic), A-1 = A2
Symmetry example (H20) {E, C2, ,   } each is self inverse
Multiplication Table
E
C2
  
E
E
C2
  
When working out keep
track of sign of x,y,z
x  -x
C2 y   y →  
C2
C2

 

 
E
 
 
E

C2

C2
E
Note: a) not require A  B = B  A (commutation) but if true  Abelian Group
b) each element occurs only once in multiplication Table
Ai, AK  G, Ar = AiAK-1  G, ArAK = Ai Ai in column
but h elements Ai and h entries in a column → occur only once
Form of multiplication table will be limited
E A  = {A,E};
h = 2 

 A E
E A B 
h = 3 A B E 
B A A 


cyclic group
A  A  A 2; A 2  A  A 2  E
only one
2
if A  E then B  A  A  B  E
III-5
Construct Multiplication Table NH3 → note self-inversion 
define class x-1 AX  class but C3-1 = C32
NH3 – C3 – (C3, C32) – class
{A,B} class conjugate
(1,2,3) – class
complete set
See Multiplication Table blocks out by classes
Sub Group {E, C3, C32} cyclic – step in itself
Mirrors: self inverse 1-1 = 1
but ij = C3m (rotation)
{E i} also sub group – 3 of them
III-6
C. Representation
Since multiplication tables limited  characteristic of interconversion of elements
– choose simplest one to represent the rest (easier manipulation)
2
1
2
3
4
E
C2
1
1
1
1
1
-1
1
-1

1
1
-1
-1

1
-1
-1
1
1-4 are representation of the group
all have same multiplication table as the 2
Class
X-1AX = B
A, B conjugate
If {A,B} – complete the class
i) every element conjugate to itself X-1XX = X
ii) if A conj B then B conj A X-1AX = B XB = AX XBX-1 = A
iii) If A conj B and C, then B conj C
X-1BX = A = Y-1CY  YX-1BXY-1 = C XY-1 = (XX-1)-1  B conj C
iv) E in class by itself (X-1 EX = E always)
order of the class is integral factor of order of group, h
Representation behaves like vectors
C2: {1 1 1 1}, {1 -1 1 -1}, {1 1 -1 -1}, {1 -1 -1 1}
Note – these are orthogonal (scale pred = 0)
Now any four vector is linear ???? of these
(a b c d) =  c i i
→ i spare space
i
1 0  , 1 0  , 1 0  , 1 0  , etc
Also matrices: 5 
 
 
 

 0 1  0 - 1  0 - 1  0 1
so this is possible to extend to larger and larger reps
5   1 0  not independent, combine i i = 1-4
but note:
  4 
General
similarity transform {E,A,B…} set of matrices
{A-1E Z, Z-1A Z, Z-1B Z, …} = {E´, A´, B´, …}
These also representation of the group (same multiplication table)
A ·B = D
A´B´ = (Z-1A Z)(Z´B Z) = Z-1A B Z
-1
= Z D Z = D´
For any matrix Rep there exist unitary matrix Z or
unitary def: UU+ = E, U-1 = U+, Uij-1 = Uji+
+ – adj out
which does similarly transform of rep to block diagonal
For the smallest blocks → irreducible representation
 a 0 
 1

(blocks do not mix on multiplication:
 0 a 

2
 b 0    a b 0 
 1
   1 1

 0 b   0  a b 


2
2 2
III-7
maintain shape on multiplication (Cotton, Pg 66)
each block has multiplication table of group → rep of group
Grand Orthogonality Theory
 i (R)nm  j (R)nm * 
R



h
ij nm mm
 i j
R → element in the group (symmetry op)
i → irreducible representation (j another one)
n,m → indices of matrix elements in representation
ℓi → order (dimension) of matrix in i
h → order of the group
 i  j 
Ex: C2
4 
1 ij
1 1 = 1+1+1+1 = 4
23 = 1-1-1+1 = 0
R
Impact
a) Products of elements of irreducible rep. restricted
b) Irreducible rep behave as orthogonal and vector ( δ ij )
c) Magnitude normalized to
h
 i for i
d) Elements of matrices orthogonal for different rep.: nn , mm
e) All of this limits number and size irreducible rep.
sum:  i 2  h – clear dimensional construct
– also affects “shape” of rep.
ex: C2 - h = 4 must have E – 1111 ℓi = 1 solution ℓi = 2 (only one), ℓi = 1 (4 of them)
C2 = h = 6 solution: ℓi = 1 (6 of them), ℓi = 1 (-- 2), ℓj = 2 ( – one), only alteration
now need rule to choose → classes
But if ℓi = 1 – 6 vectors must be orthogonal –– can you do it?
Characters → simplify matrix rep by Trace
Trace of matrix invariant to similarity transform
[eg: reducible rep → sum irreducible rep → character same]
i
i (R)  Tr i (R)   i (R) ← diagonal sum of matrix element
 1


Now all elements in class related by similarity transform so have same character
[condense Tables]
GOT recall the mm´ nn´ → only diagonals contribute



*
  i (R)mm    j (R)nn  
R m
 n

h
i j
ij  nm
mn
III-8
since ij

 nn   i
 i j   i
nm
left ij → orthogonal really only i = j contribute
*
 i (R)  j (R)  h ij
R
 characters for h-dimensional orthogonal vector
sort by class
*
 i (CK )  j (CK ) NK  ij h
K
CK – class, Xi (CK) character of ith rep in CK class
NK = number of elements in class
Reduction of the space → reduced dimension = number classes
since all elements of class same character → vectors dependent
or there are not h independent vectors
number classes = number of independent components (character element)
= number of independent vectors can form
number classes = number of irreducible rep → character Tables
Grand Orthogonality Examples
C2
1
2
3
4
E
C2
1
1
1
1
1
-1
1
-1

1
1
-1
-1

e.g.
 i i
R
1
-1
-1
1
 41
2 2  [ (1 1  (-1)(-1)  1 1  (-1)(-1) ]
 4



if i  j
 i j 0
R
try
2 3  [ (1 1  (1)  1  1 (-1)  (-1)(-1)]
Note: the 5 , 6 rep’s are not irreducible
n
  i2  h
C2: h = 4
could have ℓi = 1, n=4 or ℓi = 2, n = 1
i -1
but must have one ℓi = 1 for simple rep: 1,1,1,1
ℓi = 1, only 4 irreducible rep
C3: h = 6
can do: ℓi = 1, n=6
are there 6 independent 6-vectors
n = 5, 4 will not work
ex: n = 5
  i2  5
 i  8
for ℓi = 1
for ℓi = 1, ℓ5 = 1
III-9
n = 3: ℓ1 = ℓ2 = 1, ℓ3 = 2
  i2
 1 1 4  6
n = 8 – ℓi = 1
C4: h = 8
trivial
 2 
N = 5 – ℓi = 1, ℓ5 = 2
C2 Representations
1
2
3
4
E
1
1
1
1
C2
1
-1
1
-1
8
Multiplication Table
(yz) ´(xz)
1
1
1
-1
-1
-1
-1
1
E
C2
E
E
C2
C2
C2
E

´

´
´

(yz) ´(xz)

´
´

E
C2
C2
E
1 0  1 0  1 0  1 0   1 0 
5  

, 
, 
, 
  
 0 1  0 - 1  0 - 1  0 1  0 4 
1 0 
eg. 

 0 - 1
like:
C2

·
1 0 
1 0 

  

 0 - 1
 0 1

´
=
z
O
H1
y
H
H2 Interchange matrix; operate on  1  vector
1 0 


 0 1
E
 0 1


1 0 
C2
 0 1


1 0 
 xz
 H2 
1 0 


 0 1
x
Rotation matrix; operate on  y  vector
z
 
1 0 0 


0 1 0
 0 0 1


E
- 1 0 0


0 - 1 0
0 0 1 


C2
-1 0 0


 0 1 0
 0 0 1



 1 0 0


 0 -1 0
 0 0 1



yz
III-10
 cos  sin 0 


 - sin cos 0 
 0
0
1

 = 180º
Chem 406
We have seen that a set of matrices of dimension n can form a representation of the
group if it has the same multiplication table
Note: if n = 1 have set of integers, these will only work if each i  1
In general can convert this set of matrices to another with similarity transform.
i.e. (E, A, B …) → (E', A', B' …)
where A' = Z-1 A Z
and Z is a unitary matrix
Z Z+ = E (identity matrix)
or Z-1 = Z+ , + → conjugate transpose or adjunct
aside Hermitian: H = H+ ,
When Z is chosen so that (E, A, B …) become “most block diagonal” then the blocks are
irreducible representation and original matrices
 A

 i

A
 , etc.
j
i.e. A  


A 


These matrices can represent symmetry effects on a molecule:
O
H1
 H1 
 as vector
 H2 
H2 interchange of H1, H2 rep 
1 0 

 0 1
E: 
Test
 0 1
 0 1
1 0 
 : 
 ´: 

1 0 
1 0 
 0 1
C2: 
1 0   0 1 1 0 
C2  

  
  
 0 1 1 0   0 1
interchange of x y z (rotation of axes)
x
 y  vector
z
 
III-11
 cos  sin 0 


C2:  - sin cos 0 
 0
0
1

1 0 0 


E:  0 1 0 
 0 0 1


=
 - 1 0 0


 0 - 1 0 
0 0 1 
1



 =  1 

1

1



´ =  - 1 

1

Note – these are all block diagonal if multiplication matrices product block diagonal and
elements on diagonal of product correspond to products of diagonal → each
block is a representation - same multiplication
Examples
z
N
H1
y
H3
x H2
NH3 → C3
Elements:
C3 about z , 3 (N – H)
Operations: E; C3, C32 ; 1, 2, 3
Note: C3-1 = C32
i = i-1
self inverse
Group Theory should reflect similarity of ’s, C3
Multiplication Table
h=6
E
C3
C32
1
2
3
E
C3
E
C3
C3
C32
1
2
3
C32
3
1
2
C32
C32
E
E
C3
2
3
1
1
1
2
3
E
C3
2
2
3
1
C32
C3
3
3
1
2
X-1 A X: E C3 E = C3
C32 E
C3 C32
C3 C3 C32 = C3
C3-1 = C32
1 C3 1 = 1 • 3 = C32
2) {E, 1} also SUBGROUP for each i,
order = 2
E
(self-conjugate)
C32 C3 C3 = C3
Note:
1) {E, C3, C32} form a cyclic group
SUBGROUP, order = 3
is a dvisor of h = 6
Class formed
(C3, C32)
same follow for
III-12
2 C3 2 = 2 • 1
3 C3 3 = 3 • 2
= C32
= C32
i = i-1
(1, 2, 3)
Mullikan notation
Labeling irreducible rep as i is okay but gives no clue as to composureMullikan notation designed to denote dimension and symmetry (sym/anti) with r / t
center open
Dimension: 1D
2D
3D
4D
A or B
E
T (old/phys – F)
U (old/G)
Principal axis
1D – A symmetrical
B anti symmetrical
C2 Principal subscript 1D 1 - symmetrical
(or vertical plane)
2 – anti symmetrical
note ambiguity if 2 classes 
h - horizontal plane superscript
all dimensional
′ – symmetrical
″ – anti symmetrical
inversion center subscript
all
g – symmetrical
u – anti symmetrical
1 1 1 1 1 → A1
2 1-1 1-1 → B1
3 1-1-1 1 → B2
4 1 1 -1 -1 → A2
Note all groups need A total symmetry
Ex. C2 - all 1D
C3
C4
1 - 1 1 1 1 → A1
2 - 1 1 1 1 → A2
3 - 2 - 1 0 → E
A1, A2, B1, B2, E
Special Case – Abelian Groups
Abelian → operations (el of G) commute
Now if AB = BA for all el < G
X-1AX = X-1XA = A
therefore every element is in a class by itself
Thus given our rules: number of class = number of irreducible rep
h dimension abelian group → h rep
from   i  h
 i  1 for all irreducible rep
III-13
So Abelia has h 1-D irreducible rep and momentum
Example – Cyclic Group: E, A, A2… Ah-1
So for 1D
(E)  1
E = Ah
1
( A )  (1 ) h
E A
A1  1 1
A2
Ah-1
1 1 11
Easy solution: p ( A )  e2i p h , p (E)  [e2i p h ] h  e
In this sense A1 = h
p = h all rep e2i = 1
HANDOUT
Example:
C3
C3
C32
C32 = E
1
2
3
e2i/3
e4i/3
e2i = 1
e4i/3
e8i/3
(e2i)2 = 1
e6i/3 = 1
e12i/3 = 1
(e2i)2 = 1
1
1
1
E
1
1
1
C32
C32
1
1
e2i/3
e4i/3
e4i/3
e2i/3 (e2i=1)
so then 3 = A1 =
rearrange
A1
→ clockwise
→ counter
Note: e4i/3 = e-2i/3 = (e2i/3=1) *
A1
E
E
1
1
1
C32
C32
1

*
1
*

The second two representations are basically the same thing but correspond to
rotations clockwise and counter-clockwise – eventually they will behave alike → in
quantum mechanics
→ give the symbol E – degenerate but separable if have ???? of ????
HANDOUT / Example
What do all of these things have to do with molecules?
III-14
Consider:
to describe bonding we will need to discuss
O
H
H
O: 2s, 2pz,
H: 1s1, 1s2
2s :
2pz :
2px :
2py
E
1
C2
1

´
1
1
1
1
1
1
1
-1
-1
1
2px,
2py
i.e. no effect
A1
-B1
(try)
H
H1 + H2 :
1
1
1
1
A
H1 - H2 :
1
-1
-1
1
-B1
–- doesn’t work H1 → H2
or 2p = 2px + 2py + 2pz = 3
= A1 + B1 + B2
aA1 = 1
4
aA2 = 1
4
aB1 = 1
4
aB = 1
2
4
-1
1
1
(3 • 1 + -1 • 1 + 1 • 1 = 4
9
(3 • 1 + -1 • 1 + 1 – -1 + 1 = 1 (3 – 3) = 0
4
(3 • 1 + (-1)(1) + 1 • -1 + 1 = 4 = 1
4
(3 • 1 + (-1)(1)
same
or consider vibration of H2O bond stretches, angle bonds
 =
OH =
OH + OH =
OH - OH =
or
2OH =
E
1
C2
1

´
1
1
A1
1
1
don’t work
A1
-B1
1
1
1
-1
1
-1
2
0
0
2
= 1 (2 • 1 + 2 • 1) = 1
4
aB1 = 1 (2 • 1 + 2 • 1) = 1
4
aA
1
HANDOUT
Cotton, Chap 5
2nd first
Product of two symmetry species transforms as the direct product of their
representation
III-15
1-D – this is easy because rep = character = number
hence 1  2 = X1(R1) X2(R1) , X1(R2) X2(R2) , … , X1(Rh) X2(Rh)
Examples – use C2
A1
A2
B1
B2
yz
E
C2
 xz
 
1
1
1
1
1
1
-1
-1
1
-1
1
-1
1
-1
-1
1
A2  B1 = 1•1, 1• -1, -1•1, -1• -1 = 1, -1, -1, 1 = B2
Rules: Since all 1-D can use shortcut tricks:
A2  B2 = B1
B1  B2 = A2 etc.
A•A=B•B=A
A•B=B•A=B
1•1=2•2=1
2•1=1•2=2
similarly
1 • 1 = 11 • 11 = 1
1 • 11 = 11 • 1 = 11
g•g=u•u=g
u•g=g•u=u
HANDOUT
Degenerate representations not so trivial
Direct product matrix
 a11b11 a11b12  a12b11  a1nb1m 




 a11 a12  a1n   b11 b12  b1m   a11b21




a

b





 21
  21
  


a21b11 a21b12


 
 




a
ann   bm1
bmm   

 n1

a
a
a
b
 n1
nn
nn mm 
n  n
m  m
mn  mn
new rep is much bigger and consists of series of m x n matrices
– can see this will block out
In our rotation
A(R)  B(R) = AB(R)
in general AB will be a reducible rep
n x n times
III-16
ex.
 1
3 
1 0    2
2  ,   1 0 , 
E ~ 
, 



1 
 0 1  3
 0 1

 2
2
E
C3
 x2
In C3

1

0
E•E = 
0

0

XE•E =
0
1
0
0
4
 1

0 0  4

 3
0 0  4
, 
1 0   3

4
0 1  
 3
 4
 3
4
1
4
3
4
 3
4
 3
4
3
4
1
4
 3
4
3 
4 
1 0 0


 3
4  , 0  1 0
0 0  1
 3

4 
0 0 0

1 
4 
1
0

0
0

1 
0
Ex – Cotton, Chap 5
1) Two most commonly used integrals for Quantum Mechanics calculations:
*
*
 i Ĥ  j d  dij and  i  j d  Sij
Rules above say 2nd one Sij = 0 unless i = j

true also 1st one, Hij because Ĥ transforms as A1 so  Ĥ 
j
since XH(R) Xj(R) = Xj(R)
   j 
i.e. character product of rep = product character and character A1 = 1 for all R
or Hij = Hii ij
2) Dipole selection rules – from time dependent perturbation th/semi ???
you learned  i* ˆ  j d  0 for electric dipole allowed transition
n
   qi rˆi sum over charges in system, q’s constant
i 1
 transforms in same way as ̂ = (x, y, z)
For non-zero integral need: i  xyz  j  A
1
or if i  A , common for ground state then i  x, y,z (reducible in general)
1