CHEM 511 Chapter 3 page 1 of 12

Chapter 3

Introduction to Molecular Symmetry

This chapter will deal with the symmetry characteristics of individual molecules, i.e., how

molecules can be rotated or imaged along certain axes and be indistinguishable form a non-

rotated/imaged molecule.

Symmetry Operation: an operation performed on an object which leaves it in a configuration

that is indistinguishable from, and superimposable on, the original configuration.

EX. BF3

Rotations (Cn)

If the object can be rotated about an axis, we say there is rotational symmetry

EX. BF3—this can be rotated by ⅓ of a circle, so it has C3 symmetry

EX. PtCl42-

EX. CCl4

The axis with the highest n is called the principal axis.

EX. BF3

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Mirror Planes (σ)

A plane of symmetry (mirror plane) exists if reflection of all parts of a molecule through this

plane results in an indistinguishable configuration

EX. BF3

A σd occurs if a σ both contains the primary axis and bisects two adjacent 2 fold axes.

Center of Inversion (i)

If reflection of all parts of an object through the center of the object produces an

indistinguishable configuration, the object has a center of symmetry aka center of inversion

EX. Which of the following have an i?

BF3

C6H6

CO2

H2O

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Improper axis of rotation (Sn) If you rotate an object about an axis (a Cn operation), then follow it by a reflection through a

plane perpendicular to the axis—and the molecule comes back indistinguishable, you have an Sn

operation.

EX. BF3

Does [PtCl6]2- have an Sn operation?

Identity Operator (E)

This is, in essence, a 360º rotation—leaving the molecule unchanged. At the very least, ALL

molecules have this type of symmetry element.

EX. Can you think of a molecule that has NO Cn, Sn, i, or σ(h, v, d)?

Linking operations together If an object has a Cn, then if you perform Cn n-times, you rotate the object back to the original

position.

EX. BF3

Similarly, combining a C3 with a σh is an S3 operation: S3 = C3 × σh

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Assigning point groups

Sometimes (particularly in spectroscopy and MO theory) we need to know all of the symmetry

operations that pertain to a certain chemical. Rather than try to figure these out for every species,

a flow chart allows us to assign a “point group” which will characterize all of the symmetry

elements (Figure 3.10). Note: it isn’t necessary to find ALL symmetry elements when assigning

a point group. Additional flow charts are at the back of this chapter notes.

Assign point groups to the following:

BF3

CHBrClF

H2O

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HCN

acetylene

Special point groups (Td, Oh, Ih)

Certain molecular shapes have a high degree of symmetry and have special designations:

Td = tetrahedral

Oh = octahedral

Ih = icosahedral

Character tables The appendix has a list of common character tables—these will show all of the symmetry

elements for a particular point group

EX.

D4h E 2C4 C2 2C2’ 2C2” i 2S4 σh 2σv 2σd

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The left column has the symmetry labels:

A and B refer to singly degenerate modes

E refers to doubly degenerate modes

T refers to triply degenerate modes

Relating symmetry to vibrational spectroscopy For molecules to give rise to strong absorptions in the IR spectrum, they must change the

molecule’s dipole moment

An additional technique called Raman spectroscopy will give rise to strong signals if there is a

change in the polarizability of the molecule

Rule of mutual exclusion: for centrosymmetric molecules, IR active modes will NOT be Raman

active and vice versa

For linear molecules, the number of fundamental vibrational modes is: 3n-5 (n = # of atoms)

For nonlinear molecules, the number of fundamental vibrational modes is: 3n-6 (n = # of atoms)

EX. How many vibrational modes are in CO2? H2O?

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So how do character tables help us with this?

For H2O (C2v) the character table is

Consider the symmetry operations on H2O—how many bonds remain unchanged after each operation?

Looking at the C2v character table, which rows add up to give the reducible representation?

For the A1 we note all values are “1”—this means the molecule is left unchanged with this

operation. In B2, some values are -1, which means the molecule is reversed. Use this approach

for the stretching or bending modes.

How are the vibrational modes of the symmetric stretch affected by the symmetry operations in C2v?

Asymmetric stretch affected by symmetry operations?

Recall that H2O will have 3 modes—we’ve described two stretching, so the last one must be the

scissoring (bending).

If the symmetry label (A1, B2, etc) has an x, y, or z in the character table, the mode is IR

active

If the symmetry label has a product term (z2, x2, xy, etc) in the character table, the mode

is Raman active

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XY3 molecules with D3h symmetry Consider SO3: how many vibrational modes will it have?

What values do we find for the reducible representation:

D3h E C3 C2 σh S3 σv

Which rows sum up to be this reducible representation?

Are these IR-active, Raman-active, both, or neither?

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XY3 molecules with C3v symmetry Consider NH3: How many vibrational modes would it have?

What values do we find for the reducible representation:

C3v E C3 σv

Which rows sum up to be this reducible representation?

Are these IR-active, Raman-active, both, or neither?

Your text goes over several other point groups and their respective IR-active bands: Point

Group

Example

molecule

shape Degrees of

freedom

Do some modes represent an

unchanged dipole?

C2v ClF3

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Point

Group

Example

molecule

shape Degrees of

freedom

Do some modes represent an

unchanged dipole?

D4h PtCl42-

Td CCl4

Oh SF6

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Chirality

To determine if a molecule is chiral, what two characteristics (operations) do we seek?

Note that an improper axis of rotation is compatible with these and a better way to determine if a

molecule is nonsuperimposable on its mirror image is to look for an Sn operation. Chiral

molecules lack an Sn.

Ex. Determine the point group of Δ-trioxalatoferrate(III)

Note this molecule of tetrafluorospiropentane: Does it have an i? a σ? is it chiral?

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