Molecular Symmetry
The symmetry of a molecule can be described by listing all the symmetry elements of
the molecule. A molecule possesses a symmetry element if the application of the
operation generated by the element leaves the molecule in an indistinguishable state.
§9.1 Symmetry elements and operations
A symmetry operation is the operation of actually doing something to a shape so that
the result is indistinguishable from the initial state. A symmetry element is a
geometrical object which is said to generate the operation. The element has the same
symbol as the operation.
(1) an n-fold rotation axis, Cn, which generates proper rotations and n is the order of
the rotation
One symmetry element may generate more than one operation, e.g. a C3 axis
generates two operations termed C3 and C32.
We say that C33, or indeed any Cnn = E, where E is the identity operation, or the
operation of doing nothing.
(2) the identity, E, meaning “do nothing”.
This sounds a bit trivial, but it is necessary to include the identity in the description of
a molecule’s symmetry in order to be able to the theory of Groups.
(3) a plane of symmetry, σ, which generates one operation, that of reflection in the
plane
A symmetry plane must either pass through an atom, or else that type of atom must
occur in pairs, symmetrically either side of the plane.
例 1. 找出水分子 H2O 的 4 個 symmetry elements 以及各自對應的 symmetry
operation。
例 2. 找出下列分子的 symmetry elements 以及各自對應的 symmetry operation;
O
S
Cl
Cl
N
Pyridine
(4) an inversion center, i, which generates the operation of inversion through the
center
An inversion center may be in space in the center of a molecule or at a single atom in
the center of the molecule. If the inversion center is in space, for example, ethane in
the staggered conformation, all atoms must be present in even numbers, spaced either
side of the center.
H
H
H
C
i
C
H
H
H
If the inversion center is at an atom, then that type of atom only must be present in an
odd number.
例 3. 找出下列分子的 symmetry elements 以及各自對應的 symmetry operation;
H
H
C
H
C
H
(5) a rotation-reflection axis, Sn, which generates an improper rotation
The Sn operation is rotation by 1/n of a turn, followed by reflection in a plane
perpendicular to the axis, e.g. ethane in the staggered conformation has an S6 axis.
例 4. 在 ethane 的某一個 H 原子上附加一個“箭頭”，練續進行 6 次 S6，畫
出每一次的結果：
Note that S62 = C3, S63 = I, S64 = C32. In general Snn/2 = I if n is even and n/2 is odd.
For a molecule having Sn with odd n, it may at first seem trivial because the Cn axis
and a perpendicular plane must both be present. Using as the model a flat equilateral
triangle with one vertex “labelled”, we can show that Sn of odd n is rather different
from even n.
In contrast to S6 and C3, applying the operation n times, where n is the order of the
axis, does not bring us back to the identity. Keep going,
For n odd, Sn2n = E. In the above example, S32 = C32, S33 = σ, S34 = C3, S36 = E.
例 5. Do a similar analysis for the symmetry element C6 of benzene, and write the
operations which have the same effect as C6, C62, C63, C64, C65, and C66.
當分子或是離子具有不只一個旋轉軸時，“級數”(order) 最高的旋轉軸 – Cn
with the maximum n – 稱為“主軸”(the principal axis)。和主軸垂直的對稱面標記
h (horizontal) – σh，包含主軸的對稱面標記 v (vertical) – σv；一個對稱面若是包
含主軸，同時平分二個垂直主軸的 C2 之間的夾角，則標記 d(dihedral) – σd。
* We cannot have arbitrary combinations of symmetry operations. For example, a
molecule with a Cn axis can have only mirror planes either containing the axis or
perpendicular to the axis.
例 6. 寫出 XeF4 所有的 symmetry elements：
F
F
Xe
F
C2"
F
C4
C2'
C2
S4
注意！經過原子的 C2 是 C2’，平分鍵角的是 C2”；分子所在的平面因為垂直主
軸是 σh，包含主軸的對稱面，如果包含化學鍵 (C2’) 是 σv (σ)，如果平分化學
鍵的夾角是 σd (σ’)。
XeF4 的 symmetry elements 有 E、C4、C2、S4、i、2C2’、2C2”、σh、2σv、2σd。
例 7. 寫出 NH3、trans-dichloroethene 所有的 symmetry elements。
§9.2 Multiplication table
If two symmetry operations are combined, or performed successively on a molecule,
the result is always the same as the application of one symmetry operation alone. For
example, the complete list of symmetry elements for H2O is {E, C2, σ, σ’}, where σ is
the molecular plane. It is not difficult to verify that
σC2 = σ’, where the operation C2 followed by the operation σ’ gives the same result as
the operation σ’.
O
O
H
H
C2

H
H
σ

O
H
H
─────────── σ’ ───────────
We also find that C2σ = σ’. In this case the two operations C2 and σ commute, σC2 =
C2σ.
The multiplication table for the symmetry operations of H2O is
E
C2
σ
σ'
E
E
C2
σ
σ'
C2
C2
E
σ'
σ
σ
σ
σ'
E
C2
σ'
σ'
σ
C2
E
例. 寫出 NH3、trans-dichloroethene 的 multiplication table。
§9.3 Point groups
(The following scheme is called the Schonflies classification scheme, most favored by
chemists. Crytallographers generally use the International or Hermann-Mauguin
classification.)
寫出下列分子的 symmetry elements：
F
F
HC
CH
B
C
H
F
H
O
H
H
C
H
Cl
C
Cl
N
There are many other examples of several molecules having the same set of symmetry
elements, regardless of the chemical composition of the molecules. It is convenient to
classify all such molecules by a single symbol which summarizes their symmetry.
The symmetry symbol consists of three parts:
capital letter
number
small letter
(i) The number indicates the order of the principal (highest order) axis. This axis
conventionally defines the vertical direction.
(ii) The capital letter is D if an n-fold principal axis is accompanied by n two-fold
axes at right angle to it; otherwise the letter is C.
(iii) The small letter is h if a horizontal symmetry plane is present. If n vertical
symmetry planes are present, the letter is v for a C group but d (= dihedral, because
the n vertical planes lie between the n C2 axes) for a D group. Notice that h takes
precedence over v or d. If no horizontal and vertical planes are present, the small
letter is omitted.
例 1. 分析下列分子的“對稱”，寫出正確的 symmetry symbol：
NH3; cis- and trans-dichloroethrne; benzene; ethane in the staggered conformation;
en
+3
or
Co
en
en
en
;
+
Cl
Co
or cis[Co(en)2Cl2]+ (en = NH2CH2CH2NH2)
Cl
en
O
O
H2N
H
H
H
H
H
or
NH2
H;
* A simple rule to remember is that any n-fold staggered structure (like C2H6,
Mn2(CO)10, etc.) is given the symbol Dnd.
例 2. The above analyses can be applied to simple linear molecules which have an
infinity-fold axis, e.g.
H
C
C
H , D∞h;
H
Cl ,
C∞v.
例 3. Some rather rare molecules possess only two elements of symmetry, and these
are given a special symbol:
(i) E and i only, Ci, e.g. the staggered conformation of an exotic substituted ethane,
F
Cl
H
H
Cl
F
(ii) E and σ only, Cs; e.g. CH2FCl, monosubstituted naphthalene,
Cl
(iii) E and Sn only, Sn.
Many molecules have no symmetry at all, i.e. their only symmetry element is the
identity. Such molecules are given the symbol C1.
The set of symmetry operations of any geometrical shape forms a mathematical group.
The elements or members of a group obey four rules:
(i)
The product of two members of the group, and the square of any member, is also
a member of the group.
(ii) There must be an identity element.
(iii) Combination must be associative, i.e. (AB)C = A(BC).
(iv) Every member must have an inverse which is also a member, i.e. if A is a
member, then A-1 must also be a member, where AA-1 = A-1A = E.
* Notice that it is the set of symmetry operations that form the group.
Since all the symmetry elements of a molecule always pass through one common
point, and the operations generated by the element will leave this point unchanged,
the set of symmetry operations of a molecule is called the point group.
There is a simple way of classifying a molecule into its point group.
【Atkins’ Physical Chemistry (9e) Fig11.7】
例 4. Find out the point group of each of the following:
PF5; POCl3 (note that P=O);
F
H
C
cyclohexane (chair)
cyclohexane (boat)
F
C
H
§9.4 Classes of symmetry operations
Two symmetry operations A and B are in the same class if there is some operation X
such that
XAX-1 = B (X-1 is the inverse of X)
It is said that B is the similarity transform of A, and that A and B are conjugate.
例 1. NH3 has 5 symmetry elements: E, C3, 3σ.
C3 and C32 are in the same class;
3
σ1, σ2, and σ3 are in the same class.
H
The symmetry operations of the C3v point group
N
H
2
H
1
are commonly written in class as:
E, 2C3, 3σ.
例 2. Group the operations of the D3h point group (e.g. BCl3) into classes.
 E, 2C3, 3C2, σh, 2S3, 3σv.
A set of operations are in the same class if they are equivalent operations.