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Group Theory solutions

Chemistry Group Theory
Course

General Chemistry II (CHEM 208)

6 Documents
Students shared 6 documents in this course
Academic year: 2023/2024
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The College of William & Mary

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Chem 312 Group Theory  Symmetry Operations and Point Groups

Symmetry Element: a geometric entity with respect to which a symmetry operation may be carried out.

Symmetry Operation: a rearrangement of a body after which it appears unchanged. After the operation, every point in the body remains coincident with an equivalent point before the operation. There are five symmetry operations; each is based on the presence of a symmetry element, such as a point, line or plane.

Element Symbol Operation E =

identity E identity E

reflection plane  reflection )

inversion center i inversion (i  i)

proper rotation axis Cn rotation (Cn)n

improper rotation axis Sn improper rotation

(Sn)n (even n) (Sn)2n (odd n)

The reflection plane and inversion center each generate only a single symmetry element. That is, if applied twice, they generate E and if applied thrice, they regenerate themselves. Higher rotation axes have intermediate steps before producing E. For instance, one operation of NH 3 is rotation through 2/3. This is called a C 3 rotation. The molecule may also be rotated by 4/3, (C 32 ) without reproducing the original orientation of the molecule. It takes C 33 to give an operation equivalent to E.

An improper rotation involves the combination of a rotation and a reflection through a plane that is normal to the rotation axis. The rotation and the reflection always commute, so they can be carried out in either order. When n has an odd value, both parts of Sn  the rotation axis (Cn) and reflection plane () must exist independently as symmetry elements of the molecule. However when n is even, Cn/2 must still exist, but Cn and  may not exist independently. Thus even-valued Sn axes are more important than odd-valued, since they sometimes create a higher order axis than Cn alone. It should be noted that S 1 is simply a mirror operation and therefore is , and S 2 involves a 180° rotation plus an orthogonal mirror operation and therefore is the inversion center, i.

Symmetry operations are best illustrated by their applications. Some of these are shown on the next page, as well as in your text.

Symmetry Point Group: A mathematical group of symmetry elements. Each point group obeys the laws governing groups, and can be characterized by its collection of symmetry elements. Each point group produces a group multiplication table.

A. Point Groups Derived from Zero or One Symmetry Elements

1.) C 1 ; elements: E 2.) Cs; elements: E,  3.) Ci; elements: E, i 4.) Cn; elements: E, Cn 5.) Sn; elements: E, Sn, Cn/

Sa

C

Sb

O

Sb

C

Sa

O

C

Pd

C

C C

Nc

aN Nb

dN

i

C

Pd

C

C C

Na

cN Nd

bN

i

Brx

V

Bry

aCl

Clc

Clb

C 31

Brx

V

Bry

cCl

Clb

Cla

C 32

Brx

V

Bry

bCl

Cla

Clc

Cn

Nx

aCl

bCl

S 2

Sn Ny

Cld

Clc

Ny

dCl

cCl

Nx

Cla

Clb

2 2

i) C2v ii) C3h iii) Td iv) C 2 v) C3v vi) D4h vii) C2h viii) D2d

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Group Theory solutions

Course: General Chemistry II (CHEM 208)

6 Documents
Students shared 6 documents in this course

University: The College of William & Mary

Was this document helpful?
Chem 312
Group Theory Symmetry Operations and Point Groups
Symmetry Element: a geometric entity with respect to which a symmetry operation may be carried
out.
Symmetry Operation: a rearrangement of a body after which it appears unchanged. After the
operation, every point in the body remains coincident with an equivalent point before the operation.
There are five symmetry operations; each is based on the presence of a symmetry element, such as a
point, line or plane.
Element
Symbol
Operation
E =
identity
E
identity
E
reflection plane
reflection
)
inversion center
i
inversion
(i i)
proper rotation axis
Cn
rotation
(Cn)n
improper rotation axis
Sn
improper
rotation
(Sn)n (even n)
(Sn)2n (odd n)
The reflection plane and inversion center each generate only a single symmetry element. That
is, if applied twice, they generate E and if applied thrice, they regenerate themselves. Higher rotation
axes have intermediate steps before producing E. For instance, one operation of NH3 is rotation
through 2/3. This is called a C3 rotation. The molecule may also be rotated by 4/3, (C32) without
reproducing the original orientation of the molecule. It takes C33 to give an operation equivalent to
E.
An improper rotation involves the combination of a rotation and a reflection through a plane
that is normal to the rotation axis. The rotation and the reflection always commute, so they can be
carried out in either order. When n has an odd value, both parts of Sn the rotation axis (Cn) and
reflection plane () must exist independently as symmetry elements of the molecule. However when
n is even, Cn/2 must still exist, but Cn and may not exist independently. Thus even-valued Sn axes
are more important than odd-valued, since they sometimes create a higher order axis than Cn alone. It
should be noted that S1 is simply a mirror operation and therefore is , and S2 involves a 180°
rotation plus an orthogonal mirror operation and therefore is the inversion center, i.
Symmetry operations are best illustrated by their applications. Some of these are shown on
the next page, as well as in your text.

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