Point group F Point groups have symmetry about a single point at Point group (point symmetryAll molecules characterized by 32 different combinations of t Symmetry elements are geometric entities about hich a symmetry operation can be performed. In symmetry elements int group, all symmetry elements must pass hrough the center of mass (the point). A symmetry There are two naming systems commonly used opentical to the initial object. object n describing symmetry elements tical tool 1. The Schoenflies notation used extensively by lize and simplify many spectroscopists s in chemistry. A group consists of a set of 2. The Hermann-Mauguin or international ns) that completely describe the notation preferred by crystallographers All Combinations of point Symmetry Elements Are Not Possible Deduction of 32 Point gro The allowed combinations of point symmetry rOtation: 1, 2-,- 4- 6-fold= 5 point group Point Symmetry Elements gRotation-inversion: 2,3,4,6 Compatible With 3D Translations →5 point group among32 COmbinations of rotation: 222. 223 224. 226. 23 nat ry Elemont symbol 432=6 point group among 32 cOmbinations of rotation and an inversion or a Rotoinversion Axis n-1, 2, 3,4,E mirror:15 point group among 32 Mirror I point group among 32 Deduction of 32 Point Grou Schoenflies Symbols 1、2、3、4、6 fold rotation axis exist in crystal, which re marked as Cr Ca Ca C and ce tThe combination of rotation axis can deduce ich only 、23432 which 4 Cn: cyclic, the po the order of the rotation ax g D,: dihedral, the group point which generated onsists of only rotation axis from the combination of 2-fold axis. n is the have in order of the main rotation axis 9T: tetrahedral he combination of rotation axi *0: octahedral with higher order L6L2 3L4L, 3L4L26L2 L 2L9 Point Groups Point groups have symmetry about a single point at the center of mass of the system. Symmetry elements are geometric entities about which a symmetry operation can be performed. In a point group, all symmetry elements must pass through the center of mass (the point). A symmetry operation is the action that produces an object identical to the initial object. Group theory is a very powerful mathematical tool that allows us to rationalize and simplify many problems in chemistry. A group consists of a set of symmetry elements (and associated symmetry operations) that completely describe the symmetry of an object. Point Group Point group (point symmetry)¾¾All molecules characterized by 32 different combinations of symmetry elements There are two naming systems commonly used in describing symmetry elements 1. The Schoenflies notation used extensively by spectroscopists 2. The Hermann-Mauguinor international notation preferred by crystallographers All Combinations of Point Symmetry Elements Are Not Possible The allowed combinations of point symmetry elements are called point groups Point Symmetry Elements Compatible With 3D Translations Deduction of 32 Point Groups Rotation: 1-, 2-, 3-, 4-, 6-foldÞ 5 point group among 32 Rotation-inversion: Þ 5 point group among 32 Combinations of rotation: 222, 223, 224, 226, 23, 432 Þ 6 point group among 32 Combinations of rotation and an inversion or a mirror:15 point group among 32 mirror: 1 point group among 32 Deduction of 32 Point Groups 3L 3L44L36L2 24L3 L66L2 L22L2 1、2、3、4、6 fold rotation axis exist in crystal, which are marked as C1、C2、C3、C4 and C6 . The combination of rotation axis can deduce：222、223、 224、226、23、432，which are marked as D2、D3、D4、 D6、T and O. The above 11 point groups consists of only rotation axis and do not have inversion axis. Schoenflies Symbols vCn : cyclic, the point group which only one rotation axis,n is the order of the rotation axis. vDn : dihedral, the group point which generated from the combination of 2-fold axis, n is the order of the main rotation axis). vT: tetrahedral vO: octahedral The combination of rotation axis with higher order