2. Commutation s The identity operation and the inversion with any operations Two rotations about the same axis .o Reflections through planes perpendicular to each other w Two C2 rotations about perpendicular axes s Rotation and reflection in a plane perpendicular to the rotation axis 20218/21 Chemistry Departme nt of Fudan University
Physical ChemistryI Chemistry Department of Fudan University Chapter IV Molecular Symmetry and Point Group 2021/8/21 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
84-2 Molecular Point Group 84-2-1. Definitions and Theorems of Group Theory 1。 Definitions A group is a collection of elements Which are interrelated according to certain rules 20218/21 Chemistry Departme nt of Fudan University
Physical ChemistryI Chemistry Department of Fudan University Chapter IV Molecular Symmetry and Point Group 2021/8/21 22 §4-2. Molecular Point Group §4-2-1. Definitions and Theorems of Group Theory 1. Definitions A group is a collection of elements which are interrelated according to certain rules
Phydical Chemiatry I Chapter IV Molecular Symmetry and Poirmt Group (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 EXXE=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=A-1A=E 20218/21 Chemistry Departme nt of Fudan University
Physical ChemistryI Chemistry Department of Fudan University Chapter IV Molecular Symmetry and Point Group 2021/8/21 23 (1)The product of any two elements in the group must be an element in the group; AB=C (3) The associative law of multiplication must hold; A(BC)=(AB)C (2) One element in the group must commute with all others and leave them unchanged; E----the identity element EX=XE=X (4) Every element must have a reciprocal, which is also an element of the group AA-1= A-1A= E
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. The reciprocal of a product of two or more elements is equal to the product of the reciprocals, in reverse order (ABC…Xy=FX1…CBA (4)If A, A, and A3".are group elements, their product, says B, must be a group element, AjA2a3"=B 20218/21 Chemistry Departme nt of Fudan University
Physical ChemistryI Chemistry Department of Fudan University Chapter IV Molecular Symmetry and Point Group 2021/8/21 24 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
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 0 矩阵乘法 20218/21 Chemistry Departme nt of Fudan University
Physical ChemistryI Chemistry Department of Fudan University Chapter IV Molecular Symmetry and Point Group 2021/8/21 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 } 0 1 1 0 − − 0 1 1 0 − 1 0 0 1 − 1 0 0 1 矩阵乘法