The geometrical model for obtaining the D3h group’s
characters of irreducible representation and symmetry
types
Tadeusz Bancewicz
Adrian Kaminski a)
Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
Abstract
The intention of this work is to present a visual method which
serves to obtain the characters of irreducible representations
(CIR) and symmetry types of point group D3h. The used method
is based on transformations of geometrical figures, which have
the same symmetrical operations as a given molecule. The
problems – which constitute one of the subjects of Molecular
Spectroscopy explorations with Group Theory application – require,
in this approach, only elementary calculus and simple geometry
and seems to be convenient for introductory chemistry and
physics. The author prepared software and computer simulations
in order to show the simplicity of the method. These teaching
aids are presented and described in this paper. The elaboration
also contains the mathematical insight into the applied method.
There are calculated active vibrations and frequencies for the D3h
group, too. Results are strictly related to the geometrical model.
I. INTRODUCTION
Atomic cores in a molecule oscillate around a position of equilibrium.
These complicated motions can be decomposed into components receiving
normal vibrations, which are often used in Molecular Physics by Group
Theory to determine – among other things – dislocation and amplitudes of
nuclei in a molecule. The aforementioned Theory applies symmetry
operations to a given molecule to determine CIR and simultaneously, kind
of oscillation, whose effects are visible (if active) on a screen as a form
of a spectrum. Symmetry is essential for our consideration too. There
exists the definition of symmetry given by Weyl1 and contained in “The
Feynman Lectures of Physics”2 as well as equivalent definition which
refers to the symmetry operation. It says: “The symmetry operation to a
molecule is such a transformation, as a result of which the particle takes
a new position indistinguishable from the initial one.”3 The definition is
1
adopted for exploring vibrations of flat molecules which belong to D3h
point group like, for instance, SO3 and ion NO3¯. The point group is
realized by a set of all symmetry operations to a given molecule and an
action which relies on composing (i.e. making in turn) the operations. But
instead of exploring directly a molecule, we use a method which is –
previously used for C2v and C4v point groups3 – applied to a geometrical
figure which has the same symmetry operations as a given molecule. The
shape of the figure corresponds to the shape of the potential well created
by the molecule’s atomic cores. The author expanded the method on the
D3h group. Transforming the figure through all symmetry operations
enable us to obtain CIR and the type of vibration in a very simple and
elegant way. Furthermore these methods seem to be transparent and
comprehensible, especially applicable for introductory chemistry and
physics courses at university. The paper is organized as follows:
Section II. presents software and computer simulations of the discussed
molecule’s oscillations, as well as transformations of some figures which
symmetry operations correspond with the ones of a given molecule.
Animations and software are available at <www.matphys.kki.pl> (also see
Ref.4)
Section III. deals with finding CIR on geometrical figure transformations
through symmetry operations.
Section IV. gives mathematical insight of applied method as well as
calculations of active vibrations and frequencies in IR.
II. COMPUTER SIMULATIONS AND SOFTWARE
There is no other better way to attract someone’s attention and make him
(or her) interested in presented topic than visualization and possibility of
interactive work within a considered problem. Therefore the author created
a number of computer simulations (animations) and software 4 to give the
reader an opportunity of elaborating the topic by him or herself.
Basically, there are two groups of animations – the first one uses a simple
geometrical figure and its transformations through all symmetry operations
which are assigned to a given molecule. There is a chance to discover
CIR by just conversing the figure and observing results. It is possible to
open the simulations in most of Media Players. Figures are mostly
coloured to draw attention and minimize any mistakes during observation.
Results serve as data for software.
The second group constitutes stereo forms of molecules together with
their oscillations what enables to observe deflections of the atoms of
molecules and simultaneously notice directions of their dislocations. In
2
addition, there exists the opportunity to predict, whether or not, a
vibration is active in IR. In other words, whether it is observable as a
spectrum, and if so, what kind of spectrum that would be – parallel or
perpendicular bands. Animated oscillations are arranged in types and are
compatible to software results.
The software uses symmetry operations and found – through application of
simulations – characters of irreducible representations, to get the number
and symmetry type of a given oscillation ( more details on symmetry type
you can find at < www.matphys.kki.pl >). Symmetry operations – necessary
for the program – are subject to be found either by using the examined
molecule or geometrical figure to which it corresponds.
Adopted conditions for the program are standard ones. This means that
the lengths of bonds were measured at determined (standard) temperature
and pressure.
III. CHARACTERS OF IRREDUCIBLE REPRESENTATIONS
It is well known that CIR enables one to determine the type of
oscillations of a given molecule. The approach here is based on applying
a geometrical figures which have the same symmetry operations as an
examined molecule. The figure which is taken for finding irreducible
representations, reflects the atomic cores of a molecule and illustrates an
energetic level. Applied signs (+), (–), describe the quantum state of a
molecule. They constitute the representation of a wave function.
Symmetry, which often constitutes the connector between physical laws,5
is here the basis for understanding various kinds of molecular spectra.
The outline of the first figure is used and remains unchangeable for the
next ones. Alterations may occur as division of the first figure. The
reader can find the similar approach in Siegel’s article,6 where the
discussed field of exploration was described by a strict geometrical
display. This way allows one to find all CIR of considered molecules.
Explorations are oriented on D3h point group and molecules which belong
to it like NO3¯ and SO3.
A. Non-degenerate vibrations
An equilateral triangle is used as a figure to be transformed because it
has the same shape as the molecule. Symmetry operations, which can be
assigned to both NO3¯ (SO3) and the equilateral triangle are as follows:
E, 2C3, 3C2, σh, 2S3, 3σv. For more details see Ref.7
In order to get CIR of A1’ symmetry type we take aforementioned
triangle to all symmetry operations. We use triangle’s outline for
transformations. In computer simulations, the triangle is coloured, for
3
better visualization. No matter how one transforms the triangle one gets
the same shape (Fig. 1(a)).
(a)


Front
Back

 
 
(b)
(c)

 
 

 
 
Back
Front
(d)
Fig.1. Figures which serve to obtain CIR of non-degenerated vibrations:
(a) it serves to obtain A1’ symmetry type
(b) two plane - sites of the same figure (2 subfigures) enable one
to obtain A2’’ type
(c) triangle is divided by its height and this gives us A2’
(d) two subfigures of the same triangle but with opposite signs
on both its plane – sites. It serves to obtain A1’’ type
If a figure comes into itself (after accomplishing an operation), then we
ascribe it 1. Conversely if a figure is passing into the opposite one as a
result of the sign’s change, then we ascribe it -1. Hence, one receives A1’
symmetry type (Table 1).4 In Fig. 1(b) we have the same triangle but with
different signs on each plane-site. Additionally, there is no division of the
triangle’s side. This figure gives us A2” symmetry type (Table 1), which
is visible in the spectrum as a parallel band.4
Tab.1. Table of characters of NO3¯
A1’
A1’’
A2’
A2’’
E
2C3 3C2 σh
2S3 3σv
1
1
1
1
1
1
1
1
1
-1
-1
-1
1
1
1
1
-1
-1
1
-1
1
-1
-1
1
4
E’
E”
2
-1
0
2
-1
0
2
-1
0
-2
1
0
A new form of the triangle is shown in Fig. 1(c). The triangle’s heights
divide each its sides into two parts. The front sign corresponds to the
rear one. Transformations give A2’ symmetry type (Table 1), of inactive
vibrations.4
In Fig. 1(d) we have the same figure as at A2’ but with opposite signs on
both triangle’s plane – sites. If we make transformations then we obtain
CIR of A1’’ symmetry type.
B. Degenerated vibrations
Let us divide an equilateral triangle into two equal parts. As a result we
obtain two right triangles. There are 3 ways in which such a triangle can
be obtained depending on which one of the triangle’s heights we choose.
Therefore, we may obtain either a or b or c (Fig. 2(i)) or if the triangle
is reflected with respect to the horizontal axis then we obtain r, s, t (Fig.
2(i)). The only assumption is that we reflect the first triangle a receiving
r, and the successive triangles s and t we receive as a result of
revolution of the triangle r at 120˚ and 240˚ the counterclockwise
direction with respect to the axis which is perpendicular to the triangle’s
plane and passes through its center. In this way, one obtains two groups
of triangles a, b, c and r, s, t from which each next one is turned by 120˚
with respect to the previous one. One may perform the reflection triangle
a onto r not necessarily with respect to horizontal axis but an arbitrary
one which is parallel to any triangle’s side. We choose one triangle from
each group e.g. a and r, but equally we may choose any other two
triangles. These are so called Introductory Figures (IF). They receive
numbers which are twice as high as numbers due to figures obtained as a
result of transformation (Eq. 1). Next, we make transformations due to
symmetry operation of NO3¯ (SO3). Each IF is transformed separately and
numerical results are added. We ascribe numbers as follows: IF are
ascribed with the number 1, their oppositions (marked by “prim”) or
figures of opposite signs on each triangle’s plane – sites are ascribed to
-1. The other figures are ascribed to -1/2 and their oppositions (marked
by “prim”) 1/2 (Eq. 1). Figures receive the above numbers after
accomplishing an operation. For instance, we want to find character for
C3. If our IF are a and r then revolution a by 120˚ gives b which
receives number -1/2 and revolution r at the same angle gives s
5
a
  1,
r
a' 
  1,
r' 
b
c 
1
 ,
s
2
t 
b' 
c'  1

s'  2
t' 
(1)
which receives the -1/2. We add up the numbers and receive -1 or the
number which is character for C3.
 
a
r
 
 
a
r
 



b
s


Front

c
t





b
s



(i)
c
t


 
a
r
 



b
s



c
t
(ii)


Back
Fig.2. Figures which serve to obtain degenerated triangles:
(i) two groups of triangles (a, b, c) and (r ,s, t) which enable E’ type to be found.
(ii) the same two groups as at (i) but with a different sign on each surface
of the same triangle. It serves to obtain CIR of E” symmetry type.
In case of operation C2 and σv we make not one but 3 transformations of
each IF due to their being 3 axes (C2) and 3 planes of reflection (σv).
We add up results of the transformations separately in each group and
then add up results of both groups. For instance, we want to receive a
character for C2. Let us assume a and r as IF. We transform a with
respect to the axis which contains the triangle’s height and passes through
its bottom side. The result is a’ which is ascribed with number -1. Then,
we transform a with respect to successive axis, for example, the one
which contains the triangle’s height and passes through the its left side.
The result is b’ which receives 1/2 according to Eq. 1. Lastly figure a is
transformed with respect to the third axis which brings it to c’ and
number 1/2. We add up the numbers receiving 0. We do the same with
the second IF or r. The result in the second group is 0. Finally, we add
6
up the results of both groups of C2 operation which receives 0 or CIR.
All the above brings us to the possibility of obtaining CIR for E’
symmetry type (Table 1) which gives doubled degenerated oscillations of
perpendicular bands spectrum, available at simulations.4
Fig. 2(ii) presents the same two groups of triangle as those serving to
find E’ but with a different sign on each surface of the same triangle.
Analogically we can obtain E” symmetry type. The only difference is at
3C2 and 3σv where we subtract the results received in each group. All
types discovered show strict compatibility with those of Carter,8 which
joins molecular symmetry with group theory, and the elaboration by
Cornwell.9
IV. MATHEMATICAL INSIGHT OF THE APPLIED METHOD.
ACTIVE VIBRATIONS IN IR.
A. The mathematical inspection:
We place the NO3¯ (SO3) molecule in plane XY (Fig. 3)
y
  120

axis 3



x
axis 2
axis 1
Fig.3. Location of axes due to NO3¯
The matrices simultaneously represent the described transformation of an
arbitrary point on the plane as follows:
7
1 0
M E   

0 1 
1 0 
M  h   

0 1 
 1
3



2 
M C3    2
 3 1 
 2
2 
 1
3



2 
M C32   2
 3  1 
 2
2 
 1
3



2 
M S 3    2
 3 1 
 2
2 
 1
3



2 
M S 32   2
 3  1 
 2
2 
 
For rotations
(2)
 
we use the known
cos
formula : M C z   
 sin 
 
 sin  
cos 
For rotations C2 and reflections σv, we have 3 transformations – with
respect to each axis (plane) presented in Fig. 3. Each axis is rotated with
respect to the previous one by about 120˚. Therefore by rotating C2 with
respect to axis 1, by about 180˚, with respect to axis 2 about
180˚+120˚=300˚, and with respect to axis 3 about 180˚+240˚=420˚ we
find the following result:
cos180  sin 180   1 0 
axis 1 : M (C 2 ' )  M C 2   
   0  1
sin
180
cos
180

 

 
axis 2 : M C2"
 1
cos300  sin 300   2
 M C 2  3   

sin
300
cos
300

  3
 2
 
axis 3 : M C 2"'

cos 420  sin 420  
 M C2  2  3   

 sin 420 cos 420  

1
2
3
2
3

2 
1 
2 

(3)
3

2 
1 
2 
The numbers within matrices correspond to the ones which have been
assigned to the result of transformation of geometrical figure. Let us take
into account e.g. E’ symmetry type. Each number on the main diagonal
is assigned to the result of the transformation of IF. These numbers
should be added together so as to obtain the character. For example,
revolution C3 of the first IF e.g. a (Fig. 2(i)) about 120˚ brings it to b.
This result is assigned as -1/2, according to Eq. (1), which corresponds to
8
the first number on the main diagonal of matrix M(C 3). The revolution
about 120˚ of the second IF i.e. r brings it to s, which is assigned as 1/2 Eq. (1). This corresponds to the second number on the main diagonal
of M(C3). When we add up these numbers we receive the result: CIR: (1/2) + (-1/2) = -1. A similar approach applies to other matrices except C2
and σv where one needs to perform three operations with respect to each
axis (plane) separately. The results cover the numbers on the main
diagonals of M(C2’), M(C2”), M(C2”’). We add up the obtained numbers
of the first group (Fig. 2(i)) and then the second group. Finally, we add
together the results of both groups.
See appendix, in order to get correlations between numbers in matrices
(2), (3) and the numbers which are assigned to the results of geometrical
figurers’ transformations.
B. Active vibrations and their frequencies
There exists a well known formula in Molecular Spectroscopy:
1
(I)
(R)
n(K)   hq  q  q ,
(4)
h q
which allows one to calculate the number of translations, rotations and
oscillations of a determined symmetry type. The meaning of the
consecutive symbols is as follows:
h – number of all symmetry operations in the point group,
hq – number of symmetry operations in qth class, where class is a set of
symmetry operations which have the same characters. E.g. if we take
into account NO3¯, then the class 3C2 means two proper rotations,
χq(I) – character of irreducible representation of the qth class (see Table 1),
χq(R) – character of reducible representation of the qth class. More
informations on the character of the reducible representation at
< www.matphys.kki.pl > ,
K – symmetry type.
One may find more details expanding on how to find these quantities in
Kęcki’s book on elements of molecular spectroscopy. 10 But, because the
considerations in this paper are oriented on oscillations only, and in
addition, active in IR, the Eq. (4) has been modified in such as way to
eliminate translations and rotations and receive only oscillations. Therefore
one obtains:
9
1

N (K)    hq  q (I)  q (R)   c ,
 h q

(5)
where c denotes the number of translations and rotations of a given
symmetry type. It turns out that c for vibrations active in IR and for Dnh
point groups – which the author deals with in this paper – remains constant
and equal 1. This is easy to verify by applying known methods.
Therefore, for NO3¯ (SO3) one receives: N (A2”) = 1, N (E’) = 2, or
equivalently:
N IR osc = A2” + 2E’
(6)
where, N IR osc denotes the total number of oscillations in IR. The
modification Eq. (4) into (5) allows one to get direct results, which also
could be found by examination, step by step, translations and rotations.12
Both ways lead to compatible results.
One may achieve the description of vibrations applying inner coordinates
(further denoted η). These coordinates refer to the change of the bond
lengths and angles between them. The number of inner coordinates
overlaps the number of degrees of freedom which is 3n – 6 for
polyatomic molecules. In case of D3h molecules there are 6 inner
coordinates. The way of vibrations calculation is based on Lagrange
equation (7)
d  dT  dU


0
dt  di  di
(7)
where i = 1, 2, …6 for molecules which belong to D 3h group. The
potential energy expression in harmonic approximation is given by Eq. (8)
U
1 6
 ki j  i  j
2 ij
(8)
In order to determine kinetic energy (T) one needs to know kinetic
coefficients ai j which are made on the base of known atomic masses,
bond lengths and angles between them. Then kinetic energy is given by
the Eq. (9)
T
d i d j
1 6
a
 i j dt dt
2 ij
(9)
10
The forms of potential (8) and kinetic (9) energies give us the possibility
to resolve Eq. 7 and obtain a set of equations (10)
Tab. 2 Vibrational frequencies in 1/cm
NO3¯
SO3
symmetric stretching
1065 symmetric stretching
degenerate deformation 530
degenerate stretching
antisymmetric degenerate (I)
1050
720
1391 antisymmetric degenerate (II) 1360
symmetric deformation 498
6
 (ki j  ai j  2 ) j  0
(10)
j
which enables one to find 6 unknowns ω2 in D3h molecule and
simultaneously 6 frequencies (ν) of vibrations because ω = 2πν. The
condition to resolve (10) is that the determinant (11) must be zero.
det ki j  ai j  2  0
(11)
The conducted calculations for molecule SO3 and ion NO3¯ give the
results which are gathered in Table 2. The results are compatible with
experiments, presented by Cabannes11 and also recommended by Lide12.
Fig. 4. presents sample pictorial spectrum of SO3, where 1/λ1 refers to
symmetric deformation vibration, 1/λ2 – degenerate deformation, 1/λ3 –
symmetric stretch, 1/λ4 – degenerate stretch vibration, respectively
1/λ1
1/λ2
1/λ3
1/λ4
Fig.4. Pictorial spectrum of SO3
11
IV. SUMMARY
This paper has discussed the method of finding the character of
irreducible representations – which are widely applied by Molecular
Spectroscopy – based on symmetry correlation between a molecule and it’s
corresponding figure. Associated with the topic Group Theory is a
powerful instrument for facilitating and enabling calculations. Some parts
of the Theory – which require certain mathematical apparatuses – have been
simplified in such a way that transformations of figures can be
considered, and thereby, to find CIR necessary to get the types and
number of vibrations. The modified way of calculating active vibrations in
IR connected with previously found CIR has been described. In addition,
there have been also calculated frequencies of vibrations of NO3¯ and
SO3. Such an approach gives the student the opportunity to become more
familiar with the theme. The presented method could be applicable to
theoretical physics teaching integrating problem simulation. That there is a
possibility of using simulations and software means that one could
consider introducing the enriched material at university level. For greater
clarity on Molecular Spectroscopy, I suggest Ref. 13. and for Group
Theory I suggest Refs. 9 and 14. The author concentrated on point
groups D3h but the considerations could easily be expanded to other
groups and molecules.
ACKOWLEDGMENTS
The authors wish to thank prof. W. Nawrocik and prof. H. Szydłowski from
the Institute of Physics ( Adam Mickiewicz University ) for some
significant suggestions on symmetry and computer simulations.
APPENDIX
1. E” symmetry type
This type of symmetry corresponds to the same figures as those for E’
but with a different sign on each surface of the same triangle (Fig.2(ii)).
The change of the sign occurs only at the reflection with respect to the
plane which contains a molecule, or σh, S3, S32. Therefore, the matrix
representations for E” symmetry are the same as those for E’, except for
the aforementioned σh, S3, S32, where we have opposite numbers on the
main diagonal. The numbers which are on the main diagonal correspond
to those at Eq. (1). In other words, they are the numbers assigned to the
result of each transformation.
12
 1 0 
M  h   

 0  1


M S 3   


 
M S 32
1
2
3
2
 1

 2
 3
 2

3

2 
1 
2 
(4)
3

2 
1 
2 
2. A1’ symmetry type
In order to get the characters for this type of symmetry we calculate the
determinants of matrices presented for E’ and E”. They simultaneously
constitute the numbers assigned to the result of each symmetry operation
for Fig.1(a).
3. A2” symmetry type
This type corresponds to the same figure as for A 1’ but with the opposite
signs on each surface of the same triangle Fig.1(b). This means that the
characters should be determined with the opposite number for operations
which cause the change of the sign at the triangle’s transformations. These
are: σh, S3, S32 and 3C2. The resulting numbers are those which serve to
determine CIR for this type.
4. A1” symmetry type
We establish the characters corresponding to a change of sign at the
deflection point along Z axis. In other words, we observe an alteration at
coordinate z. In addition, we use the condition of vectors orthogonality
along vertical columns in the table of characters. The established numbers
correspond to those ones assigned to the results of consecutive operations
for Fig.1.(d).
5. A2’ symmetry type
We take the same figure (Fig.1(c)) as for A1” type but with the same
sign on both planes/sides of the same triangle. Therefore the operations
which change a sign into an opposing one receive opposite numbers.
13
Those operations are: σh, S3, S32 and 3C2. This is in accordance with the
process described in the previous text (Sec. III (A)).
a)
Electronic mail: T.Bancewicz: tbancewi@zon12.physd.amu.edu.pl , A.Kaminski: matphys@go2.pl
H. Weyl, Symmetry, (Princeton University Press, New Jersey, 1983), Chap.1, pp.4 – 5. and
Chap.2, pp.41 – 42.
2
R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, (Addison –
Wesley, Reading, MA, 1963), Vol.1, Chap.11, p.168.
3
P. Kowalczyk, Physics of Molecules, (PWN SA, Warsaw, 2000), Chap.1, p.12. and pp.35 – 44.
4
Computer simulations and software are available at < www.matphys.kki.pl>. You can write
down the address in the line of addresses: http// of Internet Explorer or in other searching
areas, for example “Google”.
5
J. Hanc, S. Tuleja and M. Hancova, “Symmetries and conservation laws: Consequences of
Noether’s theorem,” Am. J. Phys. (72) 4, 428 – 433 (2004).
6
D. Siegel, “Cinematographic metaphors for the relativistic revolution,” Am. J. Phys. (74) 3, 173174 (2006).
7
H. Haken and H. C. Wolf, Molecular Physics and Elements of Quantum, (PWN SA, Warsaw,
1998), pp. 112 – 138.
8
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14
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