120A lecture 5 - point groups

120A lecture 5 - point groups - Welcome to:! Chemistry...

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Unformatted text preview: Welcome to:! Chemistry 120A! Lecture 5! Point Groups and ! Character Tables! Prof. Joshua Figueroa ! Symmetry Elements! Axis of Rotation, Cn! Planes of Reflection, σ The Inversion Center, i! Improper Rotation Axes, Sn! The Identity, E! Properties of an ! object or molecule! Symmetry Operations! Symmetry operations are carried out! with respect to symmetry elements! C2 Point Groups! In Chemistry, a Point Group represents the full set of symmetry elements and symmetry opera7ons for a given molecule For any molecule, its unique combina7on of symmetry elements and opera7ons results in a unique point group. A molecule can not belong to more than one point group. Molecules that have dis7nctly different shapes, but possess iden7cal Sets of symmetry elements and opera7ons are said to belong to the same point group Point groups are governed by the mathema7c principles of Group Theory. The four defining characteris7cs of a group must hold for all symmetry (point) groups Group Theory: Defining Characteristics of Groups! 1  ­ The product of any two elements in the group and the square of each element must be an element in the group 2  ­ One element in the group must commute with all others and leave them unchanged 3  ­ The associa?ve law of mul?plica?on must hold 4  ­ Every element must have a reciprocal, which is also an element of the group Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. EXAMPLES: Oh – Octahedral point group (Contains all symmetry elements and opera1ons of a perfect octahedron) C∞v – “C infinity v” (Designates an infinite number of σv planes parallel to an infinite Cn rota1on axis) Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. EXAMPLES: Oh – Octahedral point group (Contains all symmetry elements and opera1ons of a perfect octahedron) C∞v – “C infinity v” (Designates an infinite number of σv planes parallel to an infinite Cn rota1on axis) C O Infinite σv’s Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. EXAMPLES: Oh – Octahedral point group (Contains all symmetry elements and opera1ons of a perfect octahedron) C∞v – “C infinity v” (Designates an infinite number of σv planes parallel to an infinite Cn rota1on axis) D∞h – “D infinity h” (Designates that a single σd is perpendicular to an infinite number of σv planes, which are parallel to an infinite Cn rota1on axis) Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. EXAMPLES: Oh – Octahedral point group (Contains all symmetry elements and opera1ons of a perfect octahedron) C∞v – “C infinity v” (Designates an infinite number of σv planes parallel to an infinite Cn rota1on axis) D∞h – “D infinity h” (Designates that a single σd is perpendicular to an infinite number of σv planes, which are parallel to an infinite Cn rota1on axis) O O Infinite σv’s One σh Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. EXAMPLES: Oh – Octahedral point group (Contains all symmetry elements and opera1ons of a perfect octahedron) C∞v – “C infinity v” (Designates an infinite number of σv planes parallel to an infinite Cn rota1on axis) D∞h – “D infinity h” (Designates that a single σd is perpendicular to an infinite number of σv planes, which are parallel to an infinite Cn rota1on axis) O O D designates ‘dihedral’ – That there Is either an axis of rota7on or a mirror plane perpendicular to the highest order axis of rota7on Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis C! 3 N H Three σv’s H H Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis C! 3 N H Three σv’s H H D3h – Designates that there are n two fold rota1on axis perpendicular to a three ­fold primary rota1on axis AND a σh plane perpendicular to the primary rota1on axis Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis C! 3 N H Three σv’s H H D3h – Designates that there are n two fold rota1on axis perpendicular to a three ­fold primary rota1on axis AND a σh plane perpendicular to the primary rota1on axis F F B F Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis C! 3 N H Three σv’s H H D3h – Designates that there are n two fold rota1on axis perpendicular to a three ­fold primary rota1on axis AND a σh plane perpendicular to the primary rota1on axis There are three C2’s F F B C 3! F C 2! Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis C! 3 N H Three σv’s H H D3h – Designates that there are n two fold rota1on axis perpendicular to a three ­fold primary rota1on axis AND a σh plane perpendicular to the primary rota1on axis There are three C2’s F F B C 3! F C 2! σh Point Groups! Point groups are designated by specific unique labels that describe the totality of the symmetry elements and opera?ons of the groups. More EXAMPLES: C3v – Designates that there is a three ­fold primary rota1on axis and n σv planes parallel to that axis C! 3 N H Three σv’s H H D3h – Designates that there are n two fold rota1on axis perpendicular to a three ­fold primary rota1on axis AND a σh plane perpendicular to the primary rota1on axis σv There are three C2’s There are three σv’s F F B C 3! F C 2! σh Determining Molecular! Point Groups! Low Symmetry Groups (No sym, mirror (s) or inversion (i) only) C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih High Symmetry Groups C or S2n Groups D Groups Dnh Dnd Dn Cnh Cnv S2n Cn Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N O D Groups C or S2n Groups A – Oh C – C∞v B – C2v Dnh Dnd Dn Cnh Cnv S2n Cn D – D∞h Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N O D Groups C or S2n Groups A – Oh C – C∞v B – C2v Dnh Dnd Dn Cnh Cnv S2n Cn D – D∞h Determining Molecular! Point Groups! EXAMPLES H H H C H C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih D Groups C or S2n Groups A – Td C – C3v B – D3d Dnh Dnd Dn Cnh Cnv S2n Cn D – Oh Determining Molecular! Point Groups! EXAMPLES H H H C H C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih D Groups C or S2n Groups A – Td C – C3v B – D3d Dnh Dnd Dn Cnh Cnv S2n Cn D – Oh Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih O H H D Groups C or S2n Groups A – D2d C – Cs B – C2h Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih O H H D Groups C or S2n Groups A – D2d C – Cs B – C2h Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N H H H D Groups C or S2n Groups A – Td C – D3h B – C3V Dnh Dnd Dn Cnh Cnv S2n Cn D – D3d Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N H H H D Groups C or S2n Groups A – Td C – D3h B – C3V Dnh Dnd Dn Cnh Cnv S2n Cn D – D3d Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih F F B F D Groups C or S2n Groups A – Td C – D3h B – C3V Dnh Dnd Dn Cnh Cnv S2n Cn D – D3d Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih F F B F D Groups C or S2n Groups A – Td C – D3h B – C3V Dnh Dnd Dn Cnh Cnv S2n Cn D – D3d Determining Molecular! Point Groups! EXAMPLES H H H B H B H H C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih D Groups C or S2n Groups A – Cs C – D2d B – Ci Dnh Dnd Dn Cnh Cnv S2n Cn D – D2h Determining Molecular! Point Groups! EXAMPLES H H H B H B H H C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih D Groups C or S2n Groups A – Cs C – D2d B – Ci Dnh Dnd Dn Cnh Cnv S2n Cn D – D2h Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih H3N Cl Pt NH3 Cl D Groups C or S2n Groups A – Cs C – Oh B – C4h Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih H3N Cl Pt NH3 Cl D Groups C or S2n Groups A – Cs C – Oh B – C4h Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih NH3 H3N H3N Mo CO CO CO D Groups C or S2n Groups A – D3h C – Oh B – C3v Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih NH3 H3N H3N Mo CO CO CO D Groups C or S2n Groups A – D3h C – Oh B – C3v Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES CO H3N H3N Mo CO NH3 CO C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih D Groups C or S2n Groups A – D3h C – Oh B – C3v Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v Determining Molecular! Point Groups! EXAMPLES CO H3N H3N Mo CO NH3 CO C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih D Groups C or S2n Groups A – D3h C – Oh B – C3v Dnh Dnd Dn Cnh Cnv S2n Cn D – C2v ...
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