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Lecture 2/11 - uni-wuppertal.de
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Lecture 2/11
Transcript of Lecture 2/11 - uni-wuppertal.de
Lecture 2/11
Permutation symmetry example: PH3
1
2
3
GCNP={E, (12), (13), (23), (123), (132) }
The CNP group has 6 elements – but what are the
cyclic permutations (123) and (132) ?
(12) 1 2 3 2 1 3 =
(123) 1 2 3 2 3 1 =
(132) 1 2 3 3 1 2 =
Transposition
Cyclic permutations
{E, (12), (13), (23), (123), (132)}
are all possible permutations
(123) 1 2 3 2 3 1 =
(132) 1 2 3 3 1 2 =
N-convention
S-convention
In this lecture, we use the
N-convention exclusively
These books only use the
N-convention
Per Jensen and P. R. Bunker: The Symmetry of Molecules, in:
"Encyclopedia of Chemical Physics and Physical Chemistry" (J. H. Moore
and N. D. Spencer, Eds.), IOP Publishing, Bristol, 2001.
(123)(12) 1 2 3 2 1 3 = (123)
= 3 2 1 (13) = 1 2 3
(123)(12) = (13)
Examples of „multiplication“
(12)(123) 1 2 3 2 3 1 = (12)
= 1 3 2 (23) = 1 2 3
(12)(123) = (23)
(12)(12) 1 2 3 2 1 3 = (12)
= 1 2 3 E = 1 2 3
(12)(12) = E; (12) = (12)-1
Examples of „multiplication“
(132)(123) 1 2 3 2 3 1 = (132)
= 1 2 3 E = 1 2 3
(132)(123) = E; (132) = (123)-1
(123) = (132)-1
Multiplication table for the S3 group
Products are Arow Acolumn
E
(123)
(132)
(12)
(23)
(13)
E
E
(123)
(132)
(12)
(23)
(13)
(123)
(123)
(132)
E
(13)
(12)
(23)
(132)
(132)
E
(123)
(23)
(13)
(12)
(12)
(12)
(23)
(13)
E
(123)
(132)
(23)
(23)
(13)
(12)
(132)
E
(123)
(13)
(13)
(12)
(23)
(123)
(132)
E
Group axioms
•All possible products RS = T belong to the group
•Group contains neutral element E (which does nothing)
•Each element has inverse element R1 (R1R =RR1 =E) in the group
•Associative law (AB)C = A(BC) holds, e.g.
(12)(123)(23) = (12) [(123)(23)] = [(12)(123)] (23)
= (12)(12) = (23) (23) = E
since (123)(23) = (12) and (12)(123) = (23)
Examples:
C2v = {E, C2, 1, 2 }
C2v(M) = {E, (12), E*, (12)*}
H2O
S3 = {E, (12), (13), (23), (123), (132) } PH3
Subgroups
A and B are both groups
A = {A1, A2, A3, ......................., An }
B = {A1, A2, A3, ....., Am }
m < n
B A
B is a subgroup of A
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Example: Subgroup of
S3 = {E, (12), (13), (23), (123), (132) }
Example of Abelian group: AB = BA always
S2 = {E, (12) }
Multiplication table:
E (12)
E E (12)
(12) (12) E
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Example: Subgroup of
S3 = {E, (12), (13), (23), (123), (132) }
S3(C)
= {E, (123), (132) }
E (123) (132)
E E (123) (132)
(123) (123) (132) E
(132) (132) E (123)
Multiplication table:
Example of Abelian group: AB = BA always
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Isomorphism
A = {A1, A2, A3, ....., An }
B = {B1, B2, B3, ....., Bn }
The two groups are isomorphic if there exists a one-to-one
correspondence of all elements
Ai Bi for all i
so that
Ai Aj = Ak Bi Bj = Bk
Example: C2v and C2v(M)
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We compare C2v and C2v(M)......
Multiplication table (Rrow Rcolumn)
E C2 yz xy
E E C2 yz xy
C2 C2 E xy yz
yz yz xy E C2
xy xy yz C2 E
E (12) E* (12)*
E E (12) E* (12)*
(12) (12) E (12)* E*
E* E* (12)* E (12)
(12)* (12)* E* (12) E
C2v and C2v(M) are isomorphic!
C2v
C2v(M)
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Homomorphism
A = {A1, A2, A3, ....., An }
B = {B1, B2, B3, .., Bm } m < n
The two groups are homomorphic if there exists a correspondence
of all elements
Ai Bi
so that
Ai Aj = Ak Bi Bj = Bk
Example: C2v(M) and S2 = { E, (12) }
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We compare C2v(M) and S2......
Multiplication table (Rrow Rcolumn)
E (12) E* (12)*
E E (12) E* (12)*
(12) (12) E (12)* E*
E* E* (12)* E (12)
(12)* (12)* E* (12) E
C2v(M)
Correspondence:
E and (12) E
E* and (12)* (12) S2
C2v(M) and S2 are homomorphic!
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Classes
A = {A1, A2, A3, ....., An }
Ai , Ak A
The elements Ai and Ak belong to the same class,
if there exists an element R A such that
Ai = R-1 Ak R
Each element in A belongs to one class only. M
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Example: Class structure of
S3 = {E, (12), (13), (23), (123), (132) }
1. Select element X S3
2. Form R-1 X R for all elements R S3
3. The elements R-1 X R form a class.
1) X = E
R-1 E R = R-1
R = E for all R S3
E forms a class of its own for any group
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Example: Class structure of
S3 = {E, (12), (13), (23), (123), (132) }
2) X = (123)
E-1 (123) E = E (123) E = (123)
(123)-1 (123) (123) = E (123) = (123)
(132)-1 (123) (132) = (123) E = (123)
(12)-1 (123) (12) = (12) (123) (12) = (132)
(23)-1 (123) (23) = (23) (123) (23) = (132)
(13)-1 (123) (13) = (13) (123) (13) = (132)
Form R-1 X R for all elements R S3
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Example: Class structure of
S3 = {E, (12), (13), (23), (123), (132) }
Form R-1 X R for all elements R S3
3) X = (12)
E-1 (12) E = E (12) E = (12)
(123)-1 (12) (123) = (13)
(132)-1 (12) (132) = (23)
(12)-1 (12) (12) = (12)
(23)-1 (12) (23) = (13)
(13)-1 (12) (13) = (23)
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Example: Class structure of
S3 = {E, (12), (13), (23), (123), (132) }
Classes:
{ E }
{ (123), (132) }
{ (12), (13), (23) }
S3 has three classes with 1, 2, and 3
elements, respectively.
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Point groups Example: PH3 at equilibrium
Point group:
C3v = {E, C3, C32, 1, 2, 3 }
Symmetry elements:
C3, 1, 2, 3
C3 Rotation axis
k Reflection plane
Other point groups have rotation-reflection axes also
Point group operations work on points in space P
P‘ = 1 P
„Multiplication“ is possible, e.g.
C32 = 2 1
C32 = 2 1
(Geometrical) symmetry operations in C3v
Reflection
Reflection
Rotation
Multiplication table
Multiplication table for C3v
Point groups systematically
Platonic solids
Ih Ih
Td Oh Oh
tetrahedron cube octahedron
dodecahedron icosahedron buckminsterfullerene
Ih
Point groups systematically
C1 Bromochlorofluoromethane CHBrClF
C2h trans-1,2-dichloroethylene
D6h benzene C6H6
Cs thionyl chloride SOCl2
C3h
boric acid
D2d allene C3H4
Cv hydrogen fluoride HF
C2v
water H2O
Td
methane CH4
Dh carbon dioxide CO2
C3v phosphine PH3
Oh cubane C4H8
C2 hydrogen peroxide H2O2
D3 tris(ethylenediamine)cobalt(III) cation
Ih buckminster- fullerene C60
C3
triphenylphosphine
D2h ethylene C2H2
Source: Wikipedia
Multiplication table for C3v
Multiplication table for the S3 group
Products are Arow Acolumn
E
(123)
(132)
(12)
(23)
(13)
E
E
(123)
(132)
(12)
(23)
(13)
(123)
(123)
(132)
E
(13)
(12)
(23)
(132)
(132)
E
(123)
(23)
(13)
(12)
(12)
(12)
(23)
(13)
E
(123)
(132)
(23)
(23)
(13)
(12)
(132)
E
(123)
(13)
(13)
(12)
(23)
(123)
(132)
E
C3v and S3 are isomorphic!
A matrix group
1 0
0 1
2
1
2
3
2
3
2
1
2
1
2
3
2
3
2
1
1 0
0 1
2
1
2
3
2
3
2
1
2
1
2
3
2
3
2
1
´M1 = E E M4 = (12) 1
´ M5 = (23) 2
´ M6 = (13) 3
´M2 = (123) C3
´M3 = (132) C32
C3v C3v S3 S3
M1
M2
M3
M4
M5
M6
M1
M1
M2
M3
M4
M5
M6
M2
M2
M3
M1
M6
M4
M5
M3
M3
M1
M2
M5
M6
M4
M4
M4
M5
M6
M1
M2
M3
M5
M5
M6
M4
M3
M1
M2
M6
M6
M4
M5
M2
M3
M1
Multiplication table for the matrix group
Products are Mrow Mcolumn
Table is an image of those for C3v and S3
Representations
A = {A1, A2, A3, ....., An }
M = {M1, M2, M3, ....., Mm}
A is a general group
M is a matrix group of quadratic (qq) matrices
A and M are isomorphic (n = m):
M is a faithful, q-dimensional representation of A
A and M are homomorphic (n > m):
M is an unfaithful, q-dimensional representation of A
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Representations
A = {A1, A2, A3, ....., An }
M = {M1, M2, M3, ....., Mm}
Ai () Mi
so that
Ai Aj = Ak () Mi Mj = Mk
The product Mi Mj is a standard matrix product.
If Ai () Mi,, then Ai-1
() Mi-1, the matrix
inverse to Mi,.
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Faithful, two-dimensional representation of
1 0
0 1
2
1
2
3
2
3
2
1
2
1
2
3
2
3
2
1
1 0
0 1
2
1
2
3
2
3
2
1
2
1
2
3
2
3
2
1
´M1 = E E M4 = (12) 1
´ M5 = (23) 2
´ M6 = (13) 3
´M2 = (123) C3
´M3 = (132) C32
C3v C3v S3 S3
M1
M2
M3
M4
M5
M6
M1
M1
M2
M3
M4
M5
M6
M2
M2
M3
M1
M6
M4
M5
M3
M3
M1
M2
M5
M6
M4
M4
M4
M5
M6
M1
M2
M3
M5
M5
M6
M4
M3
M1
M2
M6
M6
M4
M5
M2
M3
M1
Multiplication table for the matrix group
Products are Mrow Mcolumn
Table is an image of those for C3v and S3
C3v, S3, and the matrix group are isomorphic!
