Computer Animations of Molecular Vibration
Michael McGuan and Robert M. Hanson
Summer Research 2004
Department of Chemistry
St. Olaf College
Northfield, Minnesota 55057
Abstract
Vibrating molecules can be displayed in a web browser with the Java applet Jmol. In order for Jmol to
display a vibrating molecule, it must be given a vector for each atom in the direction that the atom moves
when the molecule is vibrating. To facilitate this process, we have developed a web site that uses a
JavaScript application to allow the user to construct a molecule with simple geometry. The user
communicates with the JavaScript application by entering data into an html page. The JavaScript
application then calculates the vectors on each atom for the selected mode of vibration and displays the
vibrating molecule using Jmol. The web site is available at
www.stolaf.edu/depts/chemistry/mo/struc/vibrate.htm
Degenerate Modes
Types of Vibration
There are two types of molecular vibration. In a stretching vibration, atoms move parallel to their
bond axes and the bond lengths change while the angles generally do not change significantly. In a
bending vibration, atoms move perpendicular to their bond axes and the bond angles change while
bond lengths remain the same.
Sometimes sets of vibrations occur at exactly the same frequency. These vibrations are said to be
degenerate. A degenerate set of n vibrations transforms as one of the n-dimensional representations in
the character table, but individual members of the degenerate set do not transform like any 1-dimensional
representation. When a symmetry operation is performed on one member of a degenerate set, it need not
be converted into itself or minus itself as would be the case for a 1-dimensional mode. Symmetry
operations may turn one mode in a degenerate set into one of the other modes or a linear combination of
the other modes. In order to determine the symmetry of a degenerate set of vibrations, it is necessary to
first represent members of the set by vectors.
Here are two degenerate vibrations for a square
pyramidal molecule with C4v symmetry. A vector has
been assigned to each each vibration.
Goals of the Project
The goal of our project was to create a web page that will display vibration modes for user-created
molecules. These modes are calculated from user input for molecular geometry, bond lengths, and
bond angles.
C 4v
stretching
bending
This is the character table for
the C4v point group.
user
Molecular Symmetry
output
Normal modes of vibration involve highly symmetrical movement of atoms. These modes can be
determined by considering the symmetry of the vibrating molecule. A molecule’s symmetry consists
of a set of operations which, when performed, leave the molecule in an orientation that is
indistinguishable from its original orientation.
html page
Cn: rotate molecule 360/n degrees about axis
effect on
i: center of inversion; sends each point with
coordinates (x,y,z) to (-x,-y,-z)
S4
effect on
Jmol
JavaScript
application
Introduction
Matter is composed of particles called atoms, which combine by forming chemical bonds to make
entities known as molecules. Real bonds are not rigid like those depicted in ball and stick models.
Rather, bonded atoms behave as though they were connected by a spring, and are therefore free to
oscillate in space. This type of motion is called vibration, and it results in stretching of bonds and
deformation of the molecule’s shape without changing the molecule’s center of mass or its angular
momentum. Each mode of vibration occurs at a specific frequency that depends on the masses of the
vibrating atoms and the strength of the bond between them.
Character table for point group D3h
A1’
1
3C 2 ’ s h
2S 3 3s v
1
1
1
1
1
A2’
E’
1
1
-1
1
1
-1
2
-1
0
2
-1
0
A1”
1
1
1
-1
-1
-1
A2”
E”
1
1
-1
-1
-1
1
2
-1
0
-2
1
0
D 3h
E
2C 3
Representations with A or B symmetry are 1-dimensional and always
have characters of 1 or –1. A character of 1 indicates that the
representation is symmetric with respect to the operation. A character of
–1 means that the representation is antisymmetric with respect to the
operation.
Determining the Number of Vibrations
Each atom in a molecule can move independently in the x, y, and z directions, so there are 3n possible
motions for a molecule containing n atoms. Of these, 3 correspond to translation of the entire molecule
in the direction of one of the axes. For nonlinear molecules, an additional 3 motions correspond to
rotation about an axis. In linear molecules, only 2 rotations are possible. The remaining motions are
vibrations, and they are called the normal modes of vibration. 3n-6 normal modes are available to a
nonlinear molecule, while 3n-5 are available to a linear molecule.
This is a 1-dimensional bending vibration mode for a molecule with D3h
symmetry. To determine the symmetry type of this mode, operate on it with each
of the symmetry operations in the D3h point group and see whether the direction of
the vectors is reversed or not.
y
y
z
x
operation
x
E
2C3
3C2’
sh
2S3
3sv
z
A2
1
1
1
-1
-1
B1
1
-1
1
1
-1
B2
E
1
-1
1
-1
1
2
0
-2
0
0
vector
0
1
C4
sv
C2
sd
-1
1
0
0
0
-1
0
0
1
0
0
1
0
0
1
1
1
0
-1
-1
0
1
0
0
1
0
1
-1
0
2
-1
0
0
1
0 -1
0 -1
1
0
-2
0
0
0
1
0
This set of degenerate modes has the same characters as the E representation in the C4v character table.
In general, for an n-dimensional mode, each vibration is assigned an nx1 vector, and each symmetry
operation is represented by an nxn matrix.
Displaying Vibrating Molecules with Jmol
After a molecule’s vibration modes have been determined by a symmetry analysis, these modes can
be displayed in Jmol. First, it is necessary to specify each atom’s location in space with x, y, and z
coordinates. These can be calculated from experimentally determined bond lengths and angles.
Once coordinates have been assigned, a vibration can be added with vectors. These vectors
determine the direction in which atoms move during vibration.
(0,1.6,0)
(0,2.01,0)
2.01 Å
105°
(-0.85,-0.46,-1.47)
(0.19,-1.40,0.33)
(0,0,0)
(0,1.6,0)
(-0.85,-0.46,1.47)
experimental data
(0.19,-1.40,-0.33)
coordinates
vectors
atomic numbers
rotation
character
1
1
-1
-1
-1
translation
y
x
z
1
(-0.38,-1.4,0)
result
z
linear
1
1.76 Å
z
y
1
translation
x
z
1
(1.70,-0.46,0)
y
nonlinear
1
0
0
character
The set of symmetry operations that can be performed on a molecule is called its point group. The
symmetry of a vibration mode is related to the vibrating molecule’s point group. Every point group has a
character table that describes ways in which vibration modes can behave when subjected to the symmetry
operations. The rows in the character table are called representations, and the entries in each cell are
called characters. The dimension of a representation is defined to be the character under the identity
operation E. Vibrations always have the same symmetry as one of the rows of the character table.
A1
1
1
matrix
Symmetry of Vibrations
2s d
1
Sn: rotate molecule 360/n degrees about axis, then reflect through
plane perpendicular to axis
E: do nothing to the molecule
2s v
C2
E
C3
s : reflect molecule through plane
2C 4
The effects of each symmetry operation on the degenerate set can be expressed as a matrix. The first
column of the matrix is the effect of the symmetry operation on the first mode, and the second column is
the effect of the symmetry operation on the second mode. The character of the symmetry operation is
the sum of the diagonal elements in the matrix.
operation
Symmetry Operations
E
vibration
x
rotation
1
Data about molecular structure and
vibration vectors is included in the
html code which invokes the Jmol
applet. Jmol reads this code and uses
it to display a vibrating molecule.
coordinates
vectors
The characters for this vibration match the A2” column of the character table, so this mode
is called A2” bend.