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Symmetry Elements, Symmetry Operations and Point Groups
Appendix 2 Symmetry Elements, Symmetry Operations and Point Groups S , Rotation of the molecule through an angle 360° jn followed by reflection of all atoms through a plane perpendicu lar to the axis of rotation; the combined operation (which may equally follow the sequence reflection then rotation) is called improper rotation', E, the identity operation which leaves the molecule unchanged.
An object has symmetry when certain parts of it can be interchanged with others without alter ing either the identity or the apparent orientation of the object. For a discrete object such as a molecule 5 elements of symmetry can be envis aged:
n
axis of symmetry, C; plane of symmetry, σ; centre of inversion, i; improper axis of symmetry, 5; and identity, E.
The rotation axis of highest order is called the principal axis of rotation; it is usually placed in the vertical direction and designated the zaxis of the molecule. Planes of reflection which are perpendicular to the principal axis are called horizontal planes (h). Planes of reflection which contain the principal axis are called vertical planes (v), or dihedral planes (d) if they bisect 2 twofold axes. The complete set of symmetry operations that can be performed on a molecule is called the sym metry group or point group of the molecule and the order of the point group is the number of symmetry operations it contains. Table A2.1 lists the various point groups, together with their ele ments of symmetry and with examples of each.
These elements of symmetry are best recognized by performing various symmetry operations, which are geometrically defined ways of exchanging equivalent parts of a molecule. The 5-symmetry operations are: C„, rotation of the molecule about a symmetry axis through an angle of 360°/n\ η is called the order of the rotation (twofold, threefold, etc.); σ reflection of all atoms through a plane of the molecule; i, inversion of all atoms through a point of the molecule; 1290
Symmetry
elements
Table A2.1 Point group C C Ci Ci x
s
c
3
Civ
and operations
1291
Point groups
Elements of symmetry Ε Ε, E, E, E, E,
Examples CHFCIBr S 0 F B r , HOC1, BFCIBr, S O C l , S F N F C H C I B r - C H C I B r (staggered) H 0 , cw-[Co(en) X ] P P h (propeller) H 0 (V-Shaped), H C O (Y-shaped), C1F (T-shaped), S F (see-saw), S i H C l , cw-[Pt(NH ) Cl ], C H C1 G e H C l , PC1 , 0 = P F SF5CI, I F , X e O F [Ni(n -C H )(NO)] [Cr(n .C H )(n -C Me )] NO, HCN, C O S rran,s-N F B(OH)
ο i Ci C Ci, 2σ
2
2
2
2
2
υ
2
2
2
4
3
CIH
C
3h
C*h
D Did D 3
3d
D
2h
E, Ε, Ε, Ε, Ε, Ε, Ε, E, E, E,
C , 3(τ CA, 4σ Cs, 5σ Ce, 6σ C o o , οοσ Ci, OH, i C , σ , i CA, OH, i C , 3C Ci, 2C , 2o , 3
ν
2
6
6
6
4
2
4
4
3
S4
2
3h
4h
3
v
OH,
2
v
Oh,
4
3
2
v
2
6
4
3
8
E, C , 3C , 3o , E, CA, 4 C , 4o ,
2
4
2
4
3
3
D D
6
[Re (^,n -S0 ) ] trischelates [M(chel) ], C H (gauche) B C 1 (vapour, staggered), A s S R W = = W R (staggered) Sg (crown), closo-B 10H10 " B C 1 (planar), B2H6, i r a n i - [ P t ( N H ) C l ] , iran5-[Co(NH ) Cl Br ]-, l A - Q r ^ O b BC1 , P F r B N H , [ R e H ] ~ XeF , PtCU , irani-[Co(NH ) Ηl ] , [ R e C l ] " , closo- 1 , 6 - C B H [Fe(rη -C H ) ] eclipsed, B H " , I F C H , [Cr(n -C H ) ] (eclipsed) ci , c o cyc/o-Cl B N R S i ( S i M e ) , [Pt(PF ) ] S i F , B C 1 , [Ni(CO) ], [ I r ( C O ) ]
E, C , 3C , 3o , i, S6 E, CA, 4 C , 4o , S E, Ci, 2C , 2o , OH, i d
5
6
2
d
2
5
2
2
d
3
4
3
Η
2
2
5
3
6
ν
3
2
6
5
υ
2
2
5
ν
3
2
3
υ
3
2
3
3
3
Civ C$v Csv Cev Coov
5
2
2
2
2
2
S i,
2
3
3
5
3
3
6
9
2 -
SA
+
4
3
4
2
2
2
D Den 5h
E, C5, 5C , 5o , Oh, S$ E, Ce, 6 C , 6o , Oh, i, Se E, COQ, ooC , ooo , i E,S E, 3C , 4 C E, 4C , 6o , 3S E, 4 C , 3C , 3o , i, 4S 2
2
2
d
6
v
h
d
3
h
6
7
6
7
6
7
2
2
4
3
4
2
2
6
2
3
3
4
2
5
6
4
Τ T T
2
5
v
2
SA
8
5
v
4
4
4
4
4
3
4
4
4
4
4
12
3
[Co(N0 )6] ~ (trans N 0 groups eclipsed), [M(n -N0 ) ]«-, [W(NMe ) ]
6
2
2
2
3
O h
h
E, 3CA, 4 C , 6 C , 3o , 6o , i, 3SA, 4 S E, 6 C , 1 0 C , 1 5 C , \5o , i, \2S , \0S 3
5
2
3
h
2
d
v
l0
It is instructive to add to these examples from the numerous instances of point group symme try mentioned throughout the text. In this way a facility will gradually be acquired in discerning the various elements of symmetry present in a molecule. A convenient scheme for identifying the point group symmetry of any given species is set out in the flow chart. Starting at the top of the chart (1)
J. DONOHUE,
tallografiya 26,
Sov. Phys. Crystallogr. 26, 5 1 6 ( 1 9 8 1 ) ; Kris908-9
(1981).
2
6
6
2
2
6
2
each vertical line asks a question: if the answer is "yes" then move to the right, if "no" then move to the left until the correct point group is arrived at. Other similar schemes have been d e v i s e d . ( 2 _ 5 )
2
R. L. CARTER, / . Chem. Educ. 45, 44 (1968). F . A. COTTON, Chemical Applications of Group Theory, 2nd edn., pp. 45-7, Wiley-Interscience, New York, 1971. J. D. DONALDSON and S . D. Ross, Symmetry and Stereo chemistry, pp. 35-49, Intertext Books, London, 1972. J. A. SALTHOUSE and M. J. W A R E , Point Group Character Tables and Related Data, p. 29, Cambridge University Press, 1972. 3
4
5
1
6
2
S F , ΒΟΗ6 (octahedron), CgHg (cubane) Bi Hj 'icosahedron)
6
1292
Appendix
F i g u r e A2.1
2
Point group symmetry flow chart.
An object has symmetry when certain parts of it can be interchanged with others without alter ing either the identity or the apparent orientation of the object. For a discrete object such as a molecule 5 elements of symmetry can be envis aged:
n
axis of symmetry, C; plane of symmetry, σ; centre of inversion, i; improper axis of symmetry, 5; and identity, E.
The rotation axis of highest order is called the principal axis of rotation; it is usually placed in the vertical direction and designated the zaxis of the molecule. Planes of reflection which are perpendicular to the principal axis are called horizontal planes (h). Planes of reflection which contain the principal axis are called vertical planes (v), or dihedral planes (d) if they bisect 2 twofold axes. The complete set of symmetry operations that can be performed on a molecule is called the sym metry group or point group of the molecule and the order of the point group is the number of symmetry operations it contains. Table A2.1 lists the various point groups, together with their ele ments of symmetry and with examples of each.
These elements of symmetry are best recognized by performing various symmetry operations, which are geometrically defined ways of exchanging equivalent parts of a molecule. The 5-symmetry operations are: C„, rotation of the molecule about a symmetry axis through an angle of 360°/n\ η is called the order of the rotation (twofold, threefold, etc.); σ reflection of all atoms through a plane of the molecule; i, inversion of all atoms through a point of the molecule; 1290
Symmetry
elements
Table A2.1 Point group C C Ci Ci x
s
c
3
Civ
and operations
1291
Point groups
Elements of symmetry Ε Ε, E, E, E, E,
Examples CHFCIBr S 0 F B r , HOC1, BFCIBr, S O C l , S F N F C H C I B r - C H C I B r (staggered) H 0 , cw-[Co(en) X ] P P h (propeller) H 0 (V-Shaped), H C O (Y-shaped), C1F (T-shaped), S F (see-saw), S i H C l , cw-[Pt(NH ) Cl ], C H C1 G e H C l , PC1 , 0 = P F SF5CI, I F , X e O F [Ni(n -C H )(NO)] [Cr(n .C H )(n -C Me )] NO, HCN, C O S rran,s-N F B(OH)
ο i Ci C Ci, 2σ
2
2
2
2
2
υ
2
2
2
4
3
CIH
C
3h
C*h
D Did D 3
3d
D
2h
E, Ε, Ε, Ε, Ε, Ε, Ε, E, E, E,
C , 3(τ CA, 4σ Cs, 5σ Ce, 6σ C o o , οοσ Ci, OH, i C , σ , i CA, OH, i C , 3C Ci, 2C , 2o , 3
ν
2
6
6
6
4
2
4
4
3
S4
2
3h
4h
3
v
OH,
2
v
Oh,
4
3
2
v
2
6
4
3
8
E, C , 3C , 3o , E, CA, 4 C , 4o ,
2
4
2
4
3
3
D D
6
[Re (^,n -S0 ) ] trischelates [M(chel) ], C H (gauche) B C 1 (vapour, staggered), A s S R W = = W R (staggered) Sg (crown), closo-B 10H10 " B C 1 (planar), B2H6, i r a n i - [ P t ( N H ) C l ] , iran5-[Co(NH ) Cl Br ]-, l A - Q r ^ O b BC1 , P F r B N H , [ R e H ] ~ XeF , PtCU , irani-[Co(NH ) Ηl ] , [ R e C l ] " , closo- 1 , 6 - C B H [Fe(rη -C H ) ] eclipsed, B H " , I F C H , [Cr(n -C H ) ] (eclipsed) ci , c o cyc/o-Cl B N R S i ( S i M e ) , [Pt(PF ) ] S i F , B C 1 , [Ni(CO) ], [ I r ( C O ) ]
E, C , 3C , 3o , i, S6 E, CA, 4 C , 4o , S E, Ci, 2C , 2o , OH, i d
5
6
2
d
2
5
2
2
d
3
4
3
Η
2
2
5
3
6
ν
3
2
6
5
υ
2
2
5
ν
3
2
3
υ
3
2
3
3
3
Civ C$v Csv Cev Coov
5
2
2
2
2
2
S i,
2
3
3
5
3
3
6
9
2 -
SA
+
4
3
4
2
2
2
D Den 5h
E, C5, 5C , 5o , Oh, S$ E, Ce, 6 C , 6o , Oh, i, Se E, COQ, ooC , ooo , i E,S E, 3C , 4 C E, 4C , 6o , 3S E, 4 C , 3C , 3o , i, 4S 2
2
2
d
6
v
h
d
3
h
6
7
6
7
6
7
2
2
4
3
4
2
2
6
2
3
3
4
2
5
6
4
Τ T T
2
5
v
2
SA
8
5
v
4
4
4
4
4
3
4
4
4
4
4
12
3
[Co(N0 )6] ~ (trans N 0 groups eclipsed), [M(n -N0 ) ]«-, [W(NMe ) ]
6
2
2
2
3
O h
h
E, 3CA, 4 C , 6 C , 3o , 6o , i, 3SA, 4 S E, 6 C , 1 0 C , 1 5 C , \5o , i, \2S , \0S 3
5
2
3
h
2
d
v
l0
It is instructive to add to these examples from the numerous instances of point group symme try mentioned throughout the text. In this way a facility will gradually be acquired in discerning the various elements of symmetry present in a molecule. A convenient scheme for identifying the point group symmetry of any given species is set out in the flow chart. Starting at the top of the chart (1)
J. DONOHUE,
tallografiya 26,
Sov. Phys. Crystallogr. 26, 5 1 6 ( 1 9 8 1 ) ; Kris908-9
(1981).
2
6
6
2
2
6
2
each vertical line asks a question: if the answer is "yes" then move to the right, if "no" then move to the left until the correct point group is arrived at. Other similar schemes have been d e v i s e d . ( 2 _ 5 )
2
R. L. CARTER, / . Chem. Educ. 45, 44 (1968). F . A. COTTON, Chemical Applications of Group Theory, 2nd edn., pp. 45-7, Wiley-Interscience, New York, 1971. J. D. DONALDSON and S . D. Ross, Symmetry and Stereo chemistry, pp. 35-49, Intertext Books, London, 1972. J. A. SALTHOUSE and M. J. W A R E , Point Group Character Tables and Related Data, p. 29, Cambridge University Press, 1972. 3
4
5
1
6
2
S F , ΒΟΗ6 (octahedron), CgHg (cubane) Bi Hj 'icosahedron)
6
1292
Appendix
F i g u r e A2.1
2
Point group symmetry flow chart.