Unformatted text preview: Welcome to:!
Chemistry 120A!
Lecture 5!
Point Groups! Prof. Joshua Figueroa ! Symmetry Elements!
Axis of Rotation, Cn!
Planes of Reﬂection, σ The Inversion Center, i! All Molecules have the Iden1ty Element Improper Rotation Axes, Sn!
The Identity, E! Rota%on by 360˚ about an arbitrary axis returns an equivalent conﬁgura%on This axis is referred to as the iden%ty symmetry element, E Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E σ Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E σ Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E σ Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E C2 Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E C2 Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E C3 Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E C3 Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E C32 Symmetry Elements vs Symmetry Operations!
Symmetry opera1ons are carried out with respect to symmetry elements EXAMPLES: 1
A mirror plane, σ, generates a single reﬂec1on opera1on 2
Two consecu1ve reﬂec1ons with respect to a given σ is equivalent to E 3
A C2 axis generates a single two
fold rota1on opera1on 4
A C3 axis generates two opera1ons: rota1on by 2π/3 and rota1on by 4π/3 These opera1ons are called C3 and C32 , respec1vely 5 – Note that C33 = E C33 Symmetry Elements vs Symmetry Operations! C3 How many C3’s are there? There are only 4 unique C3 axes, but there are 8 diﬀerent C3 opera1ons (4 C3’s and 4C32’s) Symmetry Elements!
Axis of Rotation, Cn! Properties of an !
object or molecule! Planes of Reﬂection, σ The Inversion Center, i!
Improper Rotation Axes, Sn! Symmetry Operations! The Identity, E! 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 opera1ons for a given molecule For any molecule, its unique combina1on of symmetry elements and opera1ons results in a unique point group. A molecule can not belong to more than one point group. Molecules that have dis1nctly diﬀerent shapes, but possess iden1cal Sets of symmetry elements and opera1ons are said to belong to the same point group Point groups are governed by the mathema1c principles of Group Theory. The four deﬁning characteris1cs 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 opera7ons of a perfect octahedron) C∞v – “C inﬁnity v” (Designates an inﬁnite number of σv planes parallel to an inﬁnite Cn rota7on 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 opera7ons of a perfect octahedron) C∞v – “C inﬁnity v” (Designates an inﬁnite number of σv planes parallel to an inﬁnite Cn rota7on 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 opera7ons of a perfect octahedron) C∞v – “C inﬁnity v” (Designates an inﬁnite number of σv planes parallel to an inﬁnite Cn rota7on 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 rota7on 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 opera7ons of a perfect octahedron) C∞v – “C inﬁnity v” (Designates an inﬁnite number of σv planes parallel to an inﬁnite Cn rota7on 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 rota7on 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 opera7ons of a perfect octahedron) C∞v – “C inﬁnity v” (Designates an inﬁnite number of σv planes parallel to an inﬁnite Cn rota7on 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 rota7on axis) O O D designates ‘dihedral’ – That there Is either an axis of rota1on or a mirror plane perpendicular to the highest order axis of rota1on 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 rota7on 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 rota7on 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 rota7on 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 rota7on axis perpendicular to a three
fold primary rota7on axis AND a σh plane perpendicular to the primary rota7on 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 rota7on 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 rota7on axis perpendicular to a three
fold primary rota7on axis AND a σh plane perpendicular to the primary rota7on 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 rota7on 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 rota7on axis perpendicular to a three
fold primary rota7on axis AND a σh plane perpendicular to the primary rota7on axis There are three C2’s F B F
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 rota7on 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 rota7on axis perpendicular to a three
fold primary rota7on axis AND a σh plane perpendicular to the primary rota7on axis There are three C2’s F B F
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 rota7on 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 rota7on axis perpendicular to a three
fold primary rota7on axis AND a σh plane perpendicular to the primary rota7on axis σv There are three C2’s There are three σv’s F B F
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 Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N O
C or S2n Groups D Groups A – Oh B – C2v C – C∞v D – D∞h Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N O
C or S2n Groups D Groups A – Oh B – C2v C – C∞v D – D∞h Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih H
H
H C
H A – Td B – D3d C – C3v D – Oh C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih H
H
H C
H A – Td B – D3d C – C3v D – Oh C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih O
H H A – D2d B – C2h C – Cs D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih O
H H A – D2d B – C2h C – Cs D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N
H H
H A – Td B – C3V C – D3h D – D3d C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih N
H H
H A – Td B – C3V C – D3h D – D3d C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih F B F F
C or S2n Groups D Groups A – Td B – C3V C – D3h D – D3d Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih F B F F
C or S2n Groups D Groups A – Td B – C3V C – D3h D – D3d Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih H
H B H B
H A – Cs B – Ci C – D2d D – D2h H
H C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih H
H B H B
H A – Cs B – Ci C – D2d D – D2h H
H C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih H3N
Cl Pt NH3
Cl A – Cs B – C4h C – Oh D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih H3N
Cl Pt NH3
Cl A – Cs B – C4h C – Oh D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih Co
C or S2n Groups D Groups A – Cs B – C4v C – D2d D – D4h Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups! EXAMPLES C1, Cs, Ci Td , Oh , C∞v , D∞h , Ih Co
C or S2n Groups D Groups A – Cs B – C4v C – D2d D – D4h Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih NH3
H3N
H3N Mo
CO CO
CO A – D3h B – C3v C – Oh D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih NH3
H3N
H3N Mo
CO CO
CO A – D3h B – C3v C – Oh D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih CO
H3N
H3N Mo
CO NH3
CO A – D3h B – C3v C – Oh D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn Determining Molecular!
Point Groups!
C1, Cs, Ci EXAMPLES Td , Oh , C∞v , D∞h , Ih CO
H3N
H3N Mo
CO NH3
CO A – D3h B – C3v C – Oh D – C2v C or S2n Groups D Groups Dnh Cnh Dnd Dn Cnv S2n Cn ...
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 Spring '09
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