![Page 1: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/1.jpg)
Chemical Applications of Group Theory
![Page 2: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/2.jpg)
• We have learnt the point group theory of molecular symmetry. We shall learn how to use this theory in our chemical research.
![Page 3: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/3.jpg)
1. Representation of groups
1.1 Matrix representation and reducible representation
![Page 4: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/4.jpg)
1.2 Reducing of representations
• Suppose that we have a set of n-dimensional matrices, A, B,
C, … , which form a representation of a group. These n-D
matrices themselves constitute a matrix group.
• If we make the same similarity transformation on each matrix,
we obtain a new set of matrices,
• This new set of matrices is also a representation of the group.
• If A’ is a blocked-factored matrix, then it is easy to prove that
B’,C’… also are blocked-factored matrices.
... CC ;BB AA 111 '';'
,.....','
4
3
2
1
4
3
2
1
B
B
B
B
B
A
A
A
A
A
A1,A2,A3… are n1,n2,n3…-D submatrices with n= n1 + n2 + n3 + ….
![Page 5: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/5.jpg)
• Furthermore, it is also provable that the various sets of
submatrices
{A1,B1,C1…}, {A2,B2,C2…}, {A3,B3,C3…}, {A4,B4,C4…},
are in themselves representations of the group.
• We then call the set of matrices A,B,C, … a reducible
representation of the group.
• If it is not possible to find a similarity transformation to reduce
a representation in the above manner, the representation is
said to be irreducible.
• The irreducible representations of a group is of fundamental
importance.
![Page 6: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/6.jpg)
2. Character Tables of Point Groups
Bases
![Page 7: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/7.jpg)
2.1 Construction of Character Table
![Page 8: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/8.jpg)
![Page 9: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/9.jpg)
These translation vectors constitute a set of bases of C2v group.
![Page 10: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/10.jpg)
![Page 11: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/11.jpg)
![Page 12: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/12.jpg)
2.2 symmetry species: Mulliken symbols
• All 1-D irreducible reps. are labeled by either A or B, 2-D irreducible rep. by E, 3-D irreducible rep. by T and so on.
• A: symmetric with respect to Cn rotation, i.e., (Cn)=1.
• B: asymmetric with respect to Cn rotation, i.e., (Cn)=-1.
• Subscriptions 1 or 2 designates those symmetric or asymmetric with respect to a C2 or a sv .
• Subscripts g or u for universal parity or disparity.
• Superscripts ‘ or ‘’ designates those symmetric or asymmetric with respect to sh
![Page 13: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/13.jpg)
2.3 Symmetry of molecular properties
![Page 14: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/14.jpg)
![Page 15: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/15.jpg)
![Page 16: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/16.jpg)
![Page 17: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/17.jpg)
![Page 18: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/18.jpg)
• This formula was derived from the “Great orthorgonality theorem”.
![Page 19: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/19.jpg)
![Page 20: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/20.jpg)
![Page 21: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/21.jpg)
![Page 22: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/22.jpg)
![Page 23: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/23.jpg)
![Page 24: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/24.jpg)
![Page 25: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/25.jpg)
![Page 26: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/26.jpg)
3. Symmetry adapted Linear Combinations of AOs–
A convenient way to construct MOs
Symmetry of wavefunctions of a molecule
Symmetry Operation on Schrödinger Eq.
Ĥy = Ey RĤy = REy R
RĤR-1Ry = ERy
ĤRy = ERy Ry is also an eigenvector of
Ĥ, with the same eigenvalue
E as y corresponds to.
![Page 27: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/27.jpg)
Symmetry of wavefunctions
If eigenvalue E is not degenerate, then
Ry = cy
Which means y is a basis for some 1-D I.R..
![Page 28: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/28.jpg)
Symmetry of wavefunctions
If eigenstate E is l-fold degenerate, then
Which means set {y1, y2, …, yl} is a basis for this l-D I.R.
E {y1, y2, …, yl}
11 12 1 1
21 22 2 2
1
1 2
l
ll
i ij j
j
l l ll l
r r r
r r rR r
r r r
y
yy y
y
![Page 29: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/29.jpg)
Symmetry of wavefunctions
In LCAO approach of MO theory, we have
So we need to solve the secular equation
),,2,1(,01
nrESHc irir
n
i
i
11 11 12 12 1 1 1
21 21 22 22 2 2 2
1 1 2 2
0
n n
n n
n n n n nn nn n
H ES H ES H ES c
H ES H ES H ES c
H ES H ES H ES c
i
iic
![Page 30: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/30.jpg)
Symmetry of wavefunctions
We can transform the original bases of AO’s into Linear
Combinations of AO’s which have the same properties of
some I.R.s (Symmetry Adapted Linear Combination),
therefore the Secular Equation is block diagonalized.
1 1
2 2
n n
y
y
y
AO SALC
?
![Page 31: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/31.jpg)
Symmetry of wavefunctions
H-ES
1
2
...
...
0
0
Block Diagonalized
AO SALC
n n
n2 n2
n1 n1
1 1 2 2n n
![Page 32: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/32.jpg)
Symmetry of wavefunctions
Step 1 Reduce the REP spanned by the AO basis.
Obtain the ni’s for each I.R.
Step 2 Construct ni SALCs for each I.R..
Step 3 Transform the hamiltonian matrix into block
diagonalized form, and solve it.
![Page 33: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/33.jpg)
Example 1:
• Neglecting the O 1s orbital/electrons.
H2O LCAO with the following atomic orbitals
O: 2s; 2px, 2py, 2pz
2H: 1sa, 1sb
Symmetry: C2v
![Page 34: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/34.jpg)
H2O
C2v I C2 sv sv'
A1
A2
B1
B2
z
xy
x,xz
y,yz
1 1 1 1
1 1 -1 -1
1 -1 1 -1
1 -1 -1 1
Character Table
Symmetry of AOs
from Oxygen
A1 : 2s, 2pz
B1 : 2px
B2 : 2py
![Page 35: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/35.jpg)
Symmetry Reduction of 2-D Bases Set {1sa, 1sb}
C2v I C2 sv sv'
A1
A2
B1
B2
z
xy
x,xz
y,yz
(R) 2 0 0 2
1 1 1 1
1 1 -1 -1
1 -1 1 -1
1 -1 -1 1
1
1(1 2 1 0 1 0 1 2) 1
4An
2
1(1 2 1 0 1 0 1 2) 0
4An
1
1(1 2 1 0 1 0 1 2) 0
4Bn
2
1(1 2 1 0 1 0 1 2) 0
4Bn
1 2A B Two 1-D REPs
![Page 36: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/36.jpg)
SALC - Symmetry Adapted Linear Combination of AOs
For this trivial problem, it is very simple, we
intuitively determined the new basis as
A1:
A B
11s + 1s
2
A B
11s - 1s
2
+ +
+ - B2:
![Page 37: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/37.jpg)
H2O
A1 symmetry AOs or SALCs for LCAO
A1 : O2s, O2pz
1A
Too complicated for analysis!
![Page 38: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/38.jpg)
H2O- Hybridization of Oxygen’s 2s and 2pz
+
-
2s
2pz
h
h´ Hybridization
![Page 39: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/39.jpg)
H2O- Hybridization of Oxygen’s 2s and 2pz
h h´ 1A
1
*' Ah d Small, not effective bonding
1
*
Ah d Large, effective bonding
![Page 40: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/40.jpg)
H2O- Chemical Bonding in A1 REP
h
h´
1A
+ + +
+ + -
1 1A Ah
1 1A Ah
0 1
Non-bonding
![Page 41: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/41.jpg)
H2O- Chemical Bonding in B2 REP
2py
- +
+
- +
-
- +
- +
+ -
0 1
22 Byp
2B
22 Byp
![Page 42: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/42.jpg)
H2O- Summary on Chemical Bonding
1a
O 2H H2O
1s
2s
2p
h, h´
b1, b2 1s(a,b) 2b
1a1
2a1
3a1
4a1
1b2
2b2
1b1
![Page 43: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/43.jpg)
H2O- Summary on Molecular Energy Level Sequences
+ + +
2a1 3a1
+ + -
4a1
- +
- +
1b2
+ -
+ -
2b2
1b1
![Page 44: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/44.jpg)
Use Projection operator to construct SALCs
• Definition:
for j-th ir. rep. of the point group. This equation was derived from the “great orthogonality theorem”.
• A non-normalized SALC can be constructed from bases set by the formula:
i
R
jj
i
jj
SALC RRh
lP ˆ)(ˆ
![Page 45: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/45.jpg)
Example: p-MOs of C3H3 (D3h)
• = A2" + E"
• However, it is more convenient to
reduce its symmetry to D3. Then we
have
= A2 + E
![Page 46: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/46.jpg)
)(3
1)(
6
1
)(6
1
ˆ)ˆ(6
1ˆ
321321321
1
"
21
'
2121
2
31
1
31
1122
CCCCCE
RRPR
AA
)(3
1
ˆ
321
122
AA
PA
)2(3
1)2(
6
2
ˆ)ˆ(6
1ˆ
3211
2
31
1
31
11
CCE
RRPR
EE
)2(6
1
ˆ
321
11
EE PA
Using orthogonality and normalization, we have )(2
1322 E
p 3 0 -1 p = A2 + E
![Page 47: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/47.jpg)
Another way to derive the third MO:
)2(6
11321
1
3 EC
1. Find an operation to covert the second wavefunction into an
nonequivalent one (not +/- of the original one). A C3 operation
works well. Then we have
2. A linear combination of this new one and the original one gives
rise to
)(2
1
)(2
3
)2)(2
1()2(
322
32
32113211
1
3
E
EE AC
![Page 48: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/48.jpg)
A general simplification • Reducing the symmetry to C3 subgroup.
p 3 0 0 p = A + E
3211
2
31
1
3111ˆ)ˆ(ˆ CCERRP
R
AA
3211
2
31
1
311
)1(
1
)1( **ˆ)ˆ(ˆ CCRRPR
EE
3211
)2(
1
)2( *ˆ)ˆ(ˆ R
EE RRP
![Page 49: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/49.jpg)
)2(6
1))
3
2cos(2)
3
2cos(22(
)ˆˆ(
321321
1
)2(
1
)1(
1
p
p
A
PPA EEE
)(3
1)(
ˆ
321321
11
A
PA AA
)(2
1))
3
2sin(2)
3
2sin(2(
)ˆˆ(
3232
1
)2(
1
)1(
2
p
p
Ai
PPA EEE
![Page 50: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/50.jpg)
Example 2: C6H6 D6h C6
p 6 0 0 0 0 0 p = A + B + E1 + E2
)(6
1
ˆ)ˆ(ˆ
654321654321
1
5
61
4
61
3
61
2
61
1
6111
A
R
AACCCCCERRP
)(6
1
ˆ)ˆ(ˆ
654321654321
1
5
61
4
61
3
61
2
61
1
6111
B
R
BB CCCCCERRP
![Page 51: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/51.jpg)
C6H6 D6h C6
6 0 0 0 0 0 = A + B + E1 + E2
654321
1
5
61
4
61
3
61
2
61
1
611
)1(
1
)1(
**
**ˆ)ˆ(ˆ 11
CCCCCERRPR
EE
654321
1
5
61
4
61
3
61
2
61
1
611
)2(
1
)2(
**
**ˆ)ˆ(ˆ 11
CCCCCERRPR
EE
![Page 52: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/52.jpg)
)22(12
1
))3
cos()3
cos()3
cos()3
cos((2)ˆˆ()1(
654321
6543211
)2(
1
)1( 111
p
p
p
p
APPAEEE
)(2
1
))3
sin()3
sin()3
sin()3
sin((2)ˆˆ()2(
6532
65321
)2(
1
)1( 111
p
p
p
p
AiPPAEEE
Similarly, we have
)(2
1/)()2(
)22(12
1)()1(
653265
65432165
2
2
iA
A
E
E
![Page 53: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/53.jpg)
CO2 has 3 modes of vibration
Vibrational spectroscopy
O=C=O O=C=O O=C=O
Infra-red inactive -
no dipole change IR active IR active
H2O has 3 modes of vibration
IR
active
IR active IR active
H O
H H O
H H O
H
Number of active modes tells us about symmetry
![Page 54: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/54.jpg)
Molecular vibrations - number of modes
4 atoms - can move independently in x, y, z directions
x
y z
x
y z
x
y z
x
y z
3N degrees of freedom for a N-atom molecule.
If atoms fixed, there are: 3 translational degrees
3 rotational degrees
and the rest (3N-6) are vibrational modes
![Page 55: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/55.jpg)
No. of modes of each symmetry species
Example - SiH2Cl2 Point group C2v
Character table
C2v E C2 sv(xz) sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz Si
Cl2
H1
Cl1
H2
z
x
y
Draw x, y and z vectors on all atoms
Count +1, -1, 0 if vector transforms to itself, minus
itself, or moves
Perform symmetry operations
![Page 56: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/56.jpg)
Character table
C2v E C2 sv(xz) sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Si
Cl2
H1
Cl1
H2
z
x
y
Operation E
Si atom x transforms into Si x count +1
y transforms into Si y count +1
z transforms into Si z count +1
total +3
Same for other 4 atoms grand total +15
![Page 57: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/57.jpg)
Character table
C2v E C2 sv(xz) sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Si
Cl2
H1
Cl1
H2
z
x
y
Operation C2 Si atom x transforms into Si -x count -1
y transforms into Si -y count -1
z transforms into Si z count +1
total -1
H1 and H2 move - swap places count 0
Cl1 and Cl2 swap places count 0
grand total -1
![Page 58: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/58.jpg)
Character table
C2v E C2 sv(xz) sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Si
Cl2
H1
Cl1
H2
z
x
y
Operation sv(xz) Si atom x transforms into Si x count +1
y transforms into Si -y count -1
z transforms into Si z count +1
total +1
H1 and H2 also lie in xz plane, and behave as Si count +1 each
Cl1 and Cl2 swap places count 0
grand total +3
![Page 59: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/59.jpg)
Character table
C2v E C2 sv(xz) sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Si
Cl2
H1
Cl1
H2
z
x
y
Operation sv(yz) Si atom x transforms into Si -x count -1
y transforms into Si y count +1
z transforms into Si z count +1
total +1
H1 and H2 swap places count 0
Cl1 and Cl2 also lie in yz plane, and behave as Si count +1 each
grand total +3
![Page 60: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/60.jpg)
No. of modes of each symmetry species
Example - SiH2Cl2 Point group C2v
Overall we have:
E C2 sv(xz) sv(yz)
+15 -1 +3 +3
This is the reducible representation of the set
of 3N (=15) atomic displacement vectors
We reduce it to the irreducible representations,
using a formula
![Page 61: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/61.jpg)
Character table
C2v 1E 1C2 1sv(xz) 1sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Formula is )(.)(.1
RRgh
aR
Ri
Reducible representation 15 -1 3 3
No. of A1 motions = 1/4 [1.15.1 + 1.(-1).1 + 1.3.1 + 1.3.1] = 5
Reduce the reducible representation
![Page 62: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/62.jpg)
Character table
C2v 1E 1C2 1sv(xz) 1sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Formula is )(.)(.1
RRgh
aR
Ri
Reducible representation 15 -1 3 3
No. of A1 motions = 1/4 [1.15.1 + 1.(-1).1 + 1.3.1 + 1.3.1] = 5
No. of A2 motions = 1/4 [1.15.1 + 1.(-1).1 + 1.3.(-1) + 1.3.(-1)] = 2
![Page 63: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/63.jpg)
Character table
C2v 1E 1C2 1sv(xz) 1sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Formula is )(.)(.1
RRgh
aR
Ri
Reducible representation 15 -1 3 3
No. of A1 motions = 1/4 [1.15.1 + 1.(-1).1 + 1.3.1 + 1.3.1] = 5
No. of A2 motions = 1/4 [1.15.1 + 1.(-1).1 + 1.3.(-1) + 1.3.(-1)] = 2
No. of B1 motions = 1/4 [1.15.1 + 1.(-1).(-1) + 1.3.1 + 1.3.(-1)] = 4
No. of B2 motions = 1/4 [1.15.1 + 1.(-1).(-1) + 1.3.(-1) + 1.3.1] = 4
![Page 64: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/64.jpg)
Symmetry species of all motions are:-
5A1 + 2A2 + 4B1 + 4B2 - the irreducible representation
Translations, rotations, vibrations
3 of these are translations of the whole molecule
3 are rotations
Symmetry species of translations are given by
vectors (x, y, z) in the character table
Symmetry species of rotations are given by Rx,
Ry and Rz in the character table
![Page 65: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/65.jpg)
Symmetry species of all motions are:-
5A1 + 2A2 + 4B1 + 4B2
Character table
C2v 1E 1C2 1sv(xz) 1sv(yz) h = 4
A1 +1 +1 +1 +1 z x2, y2, z2
A2 +1 +1 -1 -1 Rz xy
B1 +1 -1 +1 -1 x, Ry xz
B2 +1 -1 -1 +1 y, Rx yz
Translations are:- A1 + B1 + B2
Rotations are:- A2 + B1 + B2
- so vibrations are:- 4A1 + A2 + 2B1 + 2B2
Translations, rotations, vibrations
![Page 66: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/66.jpg)
Symmetry species of vibrations
are:- 4A1 + A2 + 2B1 + 2B2
Vibrational modes of SiH2Cl2
What does each of these modes look like?
2 rules
(i) there is 1 stretching vibration per bond
(ii) must treat symmetry-related atoms together
![Page 67: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/67.jpg)
Vibrational modes of SiH2Cl2
2 rules
(i) there is 1 stretching vibration per bond
(ii) we must treat symmetry-related atoms together
We therefore have:-
two stretching modes of the SiCl2 group
two of the SiH2 group
The remaining five modes must be deformations
(angle bending vibrations)
![Page 68: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/68.jpg)
Vibrational modes of SiH2Cl2
We therefore have:-
two stretching modes of the SiCl2 group
We can stretch the two Si-Cl bonds
together in phase
or together out of phase
![Page 69: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/69.jpg)
Is vibration symmetrical with
respect to each symmetry
operation?
- if yes +1, if no -1
From the character table,
this belongs to the symmetry
species A1
We call the mode of vibration
sym SiCl2
E C2 sxz syz
+1 +1 +1 +1
x
z
y
![Page 70: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/70.jpg)
Is vibration symmetrical
with respect to each
symmetry operation?
- if yes +1, if no -1
E C2 sxz syz
From the character table,
this belongs to the
symmetry species B2
We call the mode of
vibration asym SiCl2
+1 -1 -1 +1
x
z
y
![Page 71: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/71.jpg)
Vibrational modes of SiH2Cl2
We therefore have:-
two stretching modes of the SiCl2 group
We can stretch the two Si-H bonds
together in phase
or together out of phase
and two stretching modes of the SiH2 group
![Page 72: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/72.jpg)
x
z
y
From the character table, this
belongs to the symmetry
species A1
We call the mode of vibration
sym SiH2
E C2 sxz syz
+1 +1 +1 +1
![Page 73: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/73.jpg)
From the character table, this
belongs to the symmetry
species B1
We call the mode of vibration
asym SiH2
E C2 sxz syz
+1 -1 +1 -1
x
z
y
![Page 74: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/74.jpg)
Vibrational modes of SiH2Cl2
We now have:-
two stretching modes of the SiCl2 group
two of the SiH2 group
The remaining five modes must be deformations
(angle bending vibrations)
As with stretches, we must treat symmetry-
related atoms together
![Page 75: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/75.jpg)
From the character table, this
belongs to the symmetry
species A1
We call the mode of vibration
sym SiCl2 (or SiCl2 scissors)
E C2 sxz syz
+1 +1 +1 +1
x
z
y
![Page 76: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/76.jpg)
From the character table, this
belongs to the symmetry
species A1
We call the mode of vibration
sym SiH2 (or SiH2 scissors)
+1 +1 +1 +1
E C2 sxz syz
x
z
y
![Page 77: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/77.jpg)
From the character table, this
belongs to the symmetry
species B1
We call the mode of vibration
SiH2 (or SiH2 wag)
E C2 sxz syz
+1 -1 +1 -1
x
z
y
![Page 78: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/78.jpg)
From the character table, this
belongs to the symmetry
species B2
We call the mode of vibration
SiH2 (or SiH2 rock)
+1 -1 -1 +1
E C2 sxz syz
x
z
y
![Page 79: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/79.jpg)
y
x
From the character table, this
belongs to the symmetry
species A2
We call the mode of vibration
SiH2 (or SiH2 twist)
E C2 sxz syz
+1 +1 -1 -1
![Page 80: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/80.jpg)
Vibrational modes of SiH2Cl2
Overall, we now have:-
two stretching modes of the SiCl2 group
A1 + B2
two of the SiH2 group
A1 + B1
five deformation modes
2A1 + A2 + B1 + B2
Together, these account for all the modes we
expect:
4A1 + A2 + 2B1 + 2B2
![Page 81: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/81.jpg)
![Page 82: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/82.jpg)
![Page 83: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/83.jpg)
![Page 84: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/84.jpg)
![Page 85: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/85.jpg)
![Page 86: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/86.jpg)
![Page 87: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/87.jpg)
Good Luck In the Final Exam!
![Page 88: Chemical Applications of Group Theory - Xiamen Universitypcss.xmu.edu.cn/old/users/xlu/group/courses/structurechem/chapter... · Chemical Applications of Group Theory • We have](https://reader034.vdocuments.us/reader034/viewer/2022042502/5b3b1f387f8b9a4b0a8e9aed/html5/thumbnails/88.jpg)
Final Exam
• Content: Chapters 5-9
• Time: June 13, 8:00-10:00
• Venue: 群贤二-102
• Tools: 科学计算器、笔等普通文具
• 考前答疑: June 10-12,
嘉锡楼316(圆弧形办公室)