1. Physical Chemistry
2019/2/24

2. Chapter IV Molecular Symmetry and
Point Group
Reference Books:
F. Albert, Cotton, Chemical Application of Group Theory, Wiley Press, New York, 1971.
(中译本:群论在化学中的应用,科学出版社,1984)
(2) David M. Bishop, Group Theory and Chemistry, Clarendon Press, Oxford, 1973.
(中译本:群论与化学,高等教育出版社,1984)
2019/2/24

3. Symmetry is all around us and is a fundamental property of nature.
2019/2/24

4. 2019/2/24

5. Motivational Factors The nature and degeneracy of vibrations.
The legitimate AO combinations for MOs.
The appearances and absences of lines in a molecule’s spectrum.
The polarity and chirality of a molecule.
2019/2/24

6. §4-1. Symmetry Elements and Operations
The term symmetry is derived from the Greek word “symmetria” which means “measured together”.
We require a precise method to describe how an object or molecule is symmetric.
2019/2/24

7. Symmetry Operation Symmetry Element
A symmetry operation is a movement of a body such that, after the movement has been carried out, every point of the body is coincident with an equivalent point (or perhaps the same point) of the body in its original orientation.
Symmetry Element
A symmetry element is a geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations may be carried out.
2019/2/24

8. §4-1-1. Symmetry Elements and Operations Required in Specifying Molecular Symmetry
2019/2/24

9. 1. The Identity Operation E Ê
No matter how asymmetrical a molecule is, it must have an identity operation, E
The symbol “E” comes from the German, “eigen,” meaning “the same”
2019/2/24

10. 2. Proper Axes and Proper Rotations Cn
An n-fold rotation is symbolized by the element Cn, and represents n–1 rotational operations about the axis.
2019/2/24

11. The operation C1 is merely E.
Molecules may have many rotation axes. But the axis with the highest n is designated as the principal axis
The operation C1 is merely E.
2019/2/24

12. 3. Symmetry Planes and Reflections
There are 3 types of planes:
Vertical, horizontal and dihedral
2019/2/24

13. Vertical v
If the reflection plane contains the Principle Axis, it is called a “vertical plane.”
2019/2/24

14. Horizontal plane h
If the reflection plane is perpendicular to the Principle Axis, it is called a “horizontal plane.” h C
A molecule can have only one h
2019/2/24

15. Dihedral d
Vertical planes which bisect the angles between adjacent pairs of C2 axes perpendicular to the principle axis
2019/2/24

16. 4. Improper Axes and Improper Rotations
Sn=Cn h
Rotations by 2/n followed by reflection in a plane  to the Sn axis.
2019/2/24

17. i 5. The Inversion Center i = S2 = C2h=hC2 (x, y, z)  (-x, -y, -z)
2019/2/24

18. §4-1-2. Multiplication Table Multiplication Table of H2O
C2v E C2 sxz syz
E E C2 sxz syz
C2 C2 E syz sxz
sxz sxz syz E C2
syz syz sxz C2 E
Operate first
Operate
second
2019/2/24

19. §4-1-3. General Relations Among Symmetry Elements and Operations
1. Products
(1) The product of two reflections, intersecting at an angle of = 2/2n, is a rotation by 2 about the axis defined by the line of intersection
2019/2/24

20. (2) When there is a rotation axis, Cn, and a plane containing it, there must be n such planes separated by angles of 2/2n;
(3)The product of two C2 rotations about axes which intersect at an angle  is a rotation by 2 about an axis perpendicular to the plane of the C2 axes;
(4) A proper rotation axis of even order and a perpendicular reflection plane generate an inversion center.
2019/2/24

21. 2. Commutation
The identity operation and the inversion with any operations;
Two rotations about the same axis;
Reflections through planes perpendicular to each other;
Two C2 rotations about perpendicular axes;
Rotation and reflection in a plane perpendicular to the rotation axis.
2019/2/24

22. §4-2. Molecular Point Group
§ Definitions and Theorems of Group Theory
1. Definitions
A group is a collection of elements which are interrelated according to certain rules.
2019/2/24

23. (3) The associative law of multiplication must hold; A(BC)=(AB)C
The product of any two elements in the group must be an element in the group;
AB=C
(2) One element in the group must commute with all others and leave them unchanged; E----the identity element EX=XE=X
(3) The associative law of multiplication must hold; A(BC)=(AB)C
(4) Every element must have a reciprocal, which is also an element of the group AA-1= A-1A= E
2019/2/24

24. (ABC·····XY)-1 = Y-1X-1····C-1B-1A-1
2. Theorems of Group
(1) For a certain element, there is only one reciprocal in the group
(2)There is only one identity element in one group.
(3)The reciprocal of a product of two or more elements is equal to the product of the reciprocals, in reverse order.
(ABC·····XY)-1 = Y-1X-1····C-1B-1A-1
(4) If A1, A2 and A3····· are group elements, their product, says B, must be a group element, A1A2A3····· =B
2019/2/24

25. 3. Some Important Conceptions
Order----the number of elements in a finite group
Finite groups; Infinite groups
Subgroup--- the smaller groups,whose elements are taken from the larger group
{1，-1，i，-i }
{1，-1 }
矩阵乘法
2019/2/24

26. Class---A complete set of elements which are conjugate to one another
Similarity Transform
A, B and X are elements of a group, if
B=X-1AX
We say B is conjugate with A.
Class---A complete set of elements which are conjugate to one another
2019/2/24

27. §4-2-2. Molecular Point Groups
Point group-----All symmetry elements in a molecule intersect at a common point, which is not shifted by any of the symmetry operations.
Schoenflies Symbols
2019/2/24

28. Cn groups---only one Cn Axis
n Cn symmetry operations, g=n C1
CFClBrI
2019/2/24

29. C2 (E, C2)
2019/2/24

30. C3, (E, C3, C32) C3
2019/2/24

31. Cn add a horizontal plane h g=2n n=1, C1h= Cs
2. Cnh groups:
Cn add a horizontal plane h
g=2n
n=1, C1h= Cs
Cn h =Sn
2019/2/24

32. Cs
H2TiO
HOCl
CH3OH
2019/2/24

33. C2h (E, C2, h, i)
Trans-C2H2Cl2 
2019/2/24

34. C3h (E, C3, C32, h, S3, S32)
B(OH)3, planar
2019/2/24

35. Cn add a vertical plane v
3. Cnv groups: g=2n
Cn add a vertical plane v
C2v (E, C2, 1, 2)
H2O
2019/2/24

36. C3v (E, 2C3, 3v)
staggered-C2H3F3 C3
NH3
2019/2/24

37. C4v
C6v
OXeF4
2019/2/24

38. AB type of diatomic molecules
Cv : C+v
AB type of diatomic molecules C
v
2019/2/24

39. 4. Sn groups--- with only one Sn Axis
When n is odd, Sn = Cnh
When n is even, the group is called Sn and consists of n elements
S2=Ci, S4, S6
2019/2/24

40. trans-C2H2F2Cl2Br2   i Ci
2019/2/24

41. S4
2019/2/24

42. Cn axis add n C2 axes perpendicular to Cn (g=2n)
5. Dn groups
Cn axis add n C2 axes perpendicular to Cn (g=2n) D3
2019/2/24

43. 6. Dnh groups Dn group + h g=4n nC2 Cn, h Cn h =Sn C2 h = n v
2019/2/24

44. D2h E, 3C2, s2=i, h, 2v
ethylene
B4(CO)2
2019/2/24

45. D3h
Ph(Ph)3
2019/2/24

46. D4h
PtCl4 2-
CAl4-
Mn2(CO)10
2019/2/24

47. D6h
D5h
2019/2/24

48. 2019/2/24

49. A2 type of diatomic molecules
Dh : C v +h
A2 type of diatomic molecules h C
2019/2/24

50. 7. Dnd groups
Dn + d
d Cn  n d
d C2  S2n
g=4n
2019/2/24

51. D2d (E, 2S4, C2, 2C2’, 2d)
2019/2/24

52. D2d
2019/2/24

53. D3d
C2H6
2019/2/24

54. D4d
2019/2/24

55. 8. T, Th, Td
Td — 4C3 , 3C2, 6d ;
g =24 ?
2019/2/24

56. C3
CCl4
C10H16 (adamantance) ?
2019/2/24

57. Th, T, T
h =12 Th
h =24
2019/2/24

58. 9. O, Oh
Oh— 4C3 , 3C4, i ; g =48
UF6
2019/2/24

59. ?
C8H8 (Cubane)
UF6
2019/2/24

60. O
h =24
2019/2/24

61. 10. I, Ih
Ih — E, 6C5 , 10C3, 15C2 , i ; g =120
C60
C180
2019/2/24

62. §4-2-3. A systematic procedure for symmetry classification of molecules
1. Determine whether the molecule belongs to one of the “special” groups:
Cv , Dh , Td , Oh , Ih
2. No proper or improper rotation axes:
C1, Cs, Ci
3. Only Sn (n even) axis:
S4, S6, S8….
2019/2/24

63. 4. One Cn axis, with no C2’s  Cn
(1) if there are no symmetry elements except the Cn axis, the group is Cn
(2) if there are n vertical planes, the group is Cnv
(3) if there is a horizontal plane, the group is Cnh
2019/2/24

64. (2) If there is a horizontal plane, the group is Dnh
5. If in addition to the principal Cn axis, there are n C2 axes lying in a plane  the Cn axis, the molecule belongs to D group:
(1) If there are no symmetry elements besides Cn and C2, the group is Dn
(2) If there is a horizontal plane, the group is Dnh
(3) If there is no h, but a set of d planes, the group is Dnd
2019/2/24

65. 2019/2/24

66. Illustrative Examples
Example 1. H2O2
2019/2/24

67. A. The staggered configuration
Example 2. Ferrocene
A. The staggered configuration
2019/2/24

68. B. The eclipsed configuration
2019/2/24

69. Problem: 一个立方体，顶点上放置八个完全相同的小球，当依次取走一个、两个、三个或四个球时，余下的球构成的图形属于什么点群？
2019/2/24

70. §4-2-4 Symmetry-based Molecule Properties
Dipole Moment
2019/2/24

71. A molecule has a Cn axis,the dipole moment should be along the axes.
A moloecule has a symmetry plane and reflection,the dipole moment should be along the plane.
2019/2/24

72. A molecule with two or more Cn rotations about perpendicular axes will show no dipole moment.
e.g. Dn Dnd Dnh
A molecule with a Cn(or Sn) rotation and a plane perpendicular to it will show no dipole moment.
e.g. Cnh Sn
2019/2/24

73. Only molecules with Cn,Cnv and Cs point groups may have Dipole moment.
2019/2/24

74. Optical Rotation
A molecule with any symmetry planes, improper axes or inversion center will show no optical rotation.
Only molecules with Dn,O, T and I point groups may have optical rotation.
2019/2/24

75. Exercises
1. What group is obtained by adding to or deleting the indicated symmetry operation from each of the following groups?
C3 plus i, C3 plus S6, C3v plus i,
D3d minus S6, S6 minus i, Td plus i
2. What is the symmetry of a octahedral UF6 molecule when one or more F is removed:
(1) 1; (2) 5; (3) 2,3; (4) 1,3
(5) 5,6; (6)1,2,3 5 4 1 U 3 2 6
2019/2/24

76. §4-3. Representations of Groups
§ Matrix notation for geometric transformations
(x, y, z)  (x’, y’, z’)
The symmetry operations can be described by matrix
2019/2/24

77. A. The identity
B. Inversion
2019/2/24

78. C. Reflections
2019/2/24

79. D. Proper Rotation
 is the angle between r and z axis
2019/2/24

80. 2019/2/24

81. E. Improper Rotation
2019/2/24

82. x, y, z coordinates, z as the rotation axis, C3v operations:
2019/2/24

83. §4-3-2. Representations and Characters 1. Representations of Groups
1. Representations of Groups
A set of matrices, each corresponding to a single operation in the group, that can be combined among themselves in a manner parallel to the way in which the group elements combine.
2019/2/24

84. X A(R) X-1 = B(R) A (R) ---A representation of the group
Then, the new set of matrices B(R) is also a representation of the group.
A and B equivalent representations
2019/2/24

85. Representation of C3v point group
(X, Y, Z)
2019/2/24

86. Representation of C3v point group
2019/2/24

87. X A(R) X-1 = B(R) A (R) is a representation of the point group
B(R) is also the representation of the point group
A 和 B equivalent representation of the point group
2019/2/24

88. The sum of the diagonal elements of a square matrix : 
2. Characters
The sum of the diagonal elements of a square matrix : 
2019/2/24

89. Therefore, we have: (2) Conjugate matrices have identical characters
(1) The characters of AB and BA are equal
(2) Conjugate matrices have identical characters
Therefore, we have:
The matrices of the same class of group elements have equal characters
b. Two equivalent representations have the same characters
2019/2/24

90. 3. Reducible and irreducible representations
All the matrices of a representation of a group can make the same similarity transformation to be block-factored matrices. We call the set of matrices as a reducible representation, otherwise call as an irreducible representation
2019/2/24

91. reducible representation
2019/2/24

92. 4. Character Tables
2019/2/24

93. §4-3-3. The “Great Orthogonality Theorem” and Its Consequences
2019/2/24

94. 1. The Great Orthogonality Theorem
order
Symmetry operation
Irreducible
representation
dimension
2019/2/24

95. 2019/2/24

96. 2. Important rules about irreducible representations and their characters
(1) The vectors whose components are the characters of two different irreducible representations are orthogonal
2019/2/24

97. If i=j, we have:
The sum of the squares of the characters in any irreducible representation equals g.
2019/2/24

98. (2) The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group
2019/2/24

99. Since i(E), the character of the representation of E in the ith irreducible representation is equal to the order of the representation, this rule can also be written as:
2019/2/24

100. (3) The number of irreducible representations of a group is equal to the number of classes in the group
2019/2/24

101. (4) The number of times the ith irreducible representation occurs in a reducible representation can be determined by:
2019/2/24

102. §4-4. Chemical Applications of Group Theory
§ Wave functions as bases for irreducible representations
The eigenfunctions for a molecule are bases for irreducible representations of the symmetry group to which the molecule belongs
2019/2/24

103. The representations are one-dimensional
If  is nondegenerate:
The representations are one-dimensional
K-fold degenerate eigenfunctions are a k-dimensional representation for the group
2019/2/24

104. The same irreducible representation
§4-4-2.Symmetry-adapted linear combinations(SALCS)
The same irreducible representation
Symmetry matching
LCAO-MO
2019/2/24

105. H2O: C2v
(YZ)
2019/2/24

106. The irreducible representation of O atom:
2s∈A pz∈A px∈B py ∈B2
The representation of a single H atom:
E C2 σv σv’
is not the irreducible representation
2019/2/24

107. Symmetry matching Combine two H atoms together: E C2 σv σv’ 1 1 1 1
O 2s 2pz
O 2py
Symmetry matching
2019/2/24

108. Taking energy into consideration LCAO-MO
2019/2/24

109. SALCs refer to one or more sets of orthonormal functions, which are generally either atomic orbitals or internal coordinations of a molecule, and to make orthonormal linear combinations of them in such a way that the combinations form bases for irreducible representations of the symmetry group of the molecule.
2019/2/24

110. §4-4-3. Projection operators
2019/2/24

111. The procedure to construct SALCs
(1) Identify the molecule point group;
(2) Use the orbitals as the basis for a representation;
(3) Reduce to its irreducible components;
(4) Projections operate
2019/2/24

112. Illustrative Examples
Example 1. H2O , C2v group
Use the two H 1s orbitals as the basis for a representation: E C2 σv
σv’ 2
Reduce this to its irreducible components:
2019/2/24

113. 2019/2/24

114. NH3 Γ 3 1 C s E 2 C 3 A 1 1 1 z x + y , z A 1 1 - 1 R E 2 - 1 ( x , yv 3 v A 1 1 1 z x 2 + y 2 , z 2 1 A 1 1 - 1 2 R z E 2 - 1 2 2 ( x , y ) , ( R , R ) ( x - y , x y ) ( x z , y z ) x y Γ 3 1
2019/2/24

115. Smith method
2019/2/24

116. 1 -1 2 -2 3 -3
(X, Y, Z )
Base
2019/2/24

117. 2019/2/24

118. § The direct product
R---an operation in the symmetry group of a molecule
1, 2 --- two sets of functions which are bases for representations of the group
2019/2/24

119. The characters of the representation of a direct product are equal to the products of the characters of the representations based on the individual sets of functions.
2019/2/24

120. §4-4-5. Identifying non zero matrix elements
The integral may be nonzero only if i and j belong to the same irreducible representation of the molecular point group.
2019/2/24

121. Spectral transition probabilities
An electric dipole transition will be allowed with x, y, or z polarization if the direct product of the representations of the two states concerned is or contains the irreducible representation to which x, y, or z, respectively, belongs.
2019/2/24

122. 对C4v群 问使积分 不为零时 应属于哪些不可约表示？ C4v E 2C4 C2 2σv 2σd A1 1 A2 1 B1 B2 2
2019/2/24