Download Vibrations of some flat molecules: Symmetry types and characters of

The visual derivation of characters for irreducible
representation of D6h group
Tadeusz Bancewicza)
Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
Adrian Kaminski a)
Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
Abstract
A visual description of finding characters of irreducible
representation (CIR) of the D6h point group is given. A similar
approach has been earlier discussed for the D3h group in CEJ.
The method applied is based on transformations of the plane
figure, a counterpart to a given molecule by the operations of
its symmetry, because to both (the figure and the molecule) with
the same operations are assigned. The present model visualizes
the approach applied by The Group Theory and gives the same
results.
I. INTRODUCTION
In spectroscopy1 the molecule and the symmetry of its vibrations play
very important role. Molecules of the same symmetry have similar spectra.
There is a field of mathematics called Group Theory which enables a
description of molecules and their vibrations according to their symmetry.
Vibrations performed by the molecules perform (we usually take into
account the so – called normal vibrations2) reflect their spatial shape and
the Group Theory offers a possibility to determine which of them are
active (observable) in infrared (IR), Raman and hyper-Raman spectra. The
spectra provide much information on the molecules, e.g. on their structure,
the type and the strength of bonds, the density of the electric charge
distribution and a possible dipole and/or higher multipole moments. This
information permits prediction of the chemical reactivity. The symmetry
allows us to answer the question of what types of vibrations the molecule
performs and how many of them are manifested in the spectrum. But
instead of exploring directly the molecule we focus on the geometrical
figure which has the same symmetry operations. Next, we will perform
the figure’s transformations through the symmetry operations. The results
will allow us to obtain CIR and determine the number of vibrations and
their types. The method has been earlier applied for the C2v, C4v and D3h
point groups.3,5 In this article we show how to expand the visual
1
approach of obtaining CIR for the D6h point group. The approach is
simple in its application and gives almost immediate results. Thereby, it
can be applied for introductory molecular chemistry (physics) classes. In
addition, there are available computer simulations (animations) and
software which refer to the D6h group.4 The simulations illustrate the
oscillations of benzene (it belongs to the D6h group) and transformations
of a chosen geometrical figure through all symmetry operations assigned
to it. We observe results of transformations which allow us to find CIR.
Symmetry operations and CIR serve also as data for the software which
enables us to get the number and the symmetry type of given
oscillations.
The paper is organized as follows:
Section II. Presents the types of figures in relation to the types of
vibrations and describes the way of finding CIR on the basis
of transformations of geometrical figures.
Section III. Gives a mathematical insight into the method applied.
II. CHARACTERS OF IRREDUCIBLE REPRESENTATIONS OF
THE D6h GROUP
Symmetry operations5 of a molecule create a point group and correspond
to a matrix. Such a matrix shows the way the coordinates of a point get
transformed in an operation considered. We can choose, for example, one
of the molecule’s atoms (except the central atom) as a point of reference
for transformations. Transformation matrices are one of possible
representations of a point group. The essential quantity in the Theory of
Representations is the character. It is the sum of diagonal elements of a
given matrix. The character, enables us to determine the number and the
type of vibrations which a given molecule performs. The method applied
in this work is based on transformations of a figure which has the same
symmetry operations as the molecule considered. Here we consider a
hexagon. We divide the figure into parts but its outline remains
unchangeable. To this parts we assign the signs (+) and (–). If, as a
result of transformation of the figure which represents non-degenerate
vibrations, we obtain unchanged figure, then to the result of this
transformation the value 1 is assigned. Whereas, when the signs change,
e.g. (+) is exchanged for (–) and vice versa, then the value –1 is assigned
to the result of the transformation.
In the case of figures which represent degenerate vibrations we use sets
of numbers given by Eq.(1-4). In addition, in this case as well, signs
plus and minus may appear in the front of a figure as well as on its
back (see Fig. 1).
2
A. Symbols of different vibration symmetry types and kinds of
figures
An exemplary molecule belonging to the D6h group is that of benzene. Its
shape corresponds to a hexagon and therefore we take for our
transformations this figure. Each vibration symmetry type refers to one
figure. The figure’s outline may be divided into parts and to each part
the sign (+) or (–) is assigned. The way of the division and the
arrangement of signs is related to the symbols and indexes which serve
to identify the type of symmetry. In order to define the symbols and
indexes, the designation “symmetric figure” and “antisymmetric figure” is
used. The symmetric figure means that the signs of the figure after
transformation are unchanged. In the antisymmetric figure signs are
exchanged i.e. (+) changes into (–) and vice versa. In general we use the
standard notation for molecular vibrations. We briefly describe their
meaning:
(i) The symbol (A, B) – A denotes symmetric and B antisymmetric
vibrations with respect to the highest multiplicity axis. The figures
of type A are symmetrical with respect to the rotation by 180°
(we make this rotation with respect to the highest multiplicity
axis), whereas the figures of type B are antisymmetric with
respect to the same axis of rotation. It means that the result of
the rotation of the type A figure gives the figure unchanged and
the result of the rotation of the type B figure gives the figure
with the opposite signs (the place marked by (+) changes into (–)
and vice versa).
(ii) The symbol (E) denotes double degenerate vibrations. It leads
to the necessity of considering two groups of figures.
(iii) Index (g, u) – index g denotes symmetricity and u antisymmetricity
of a vibration with respect to the molecule’s center of symmetry.
Index g, with reference to a figure, means that the figure must
have such a form that one gets the same figure after performing
the inversion with respect to the figure’s centre. Index u defines a
figure that yields the opposite figure when again the operation of
inversion is performed. For example, the inversion of the figure
within B1g yields the same figure because of g symmetry (see Fig.
1(h)), whereas the figure within B2u is antisymmetrical with
respect to inversion leading to exchange of signs because of u
symmetry (Fig. 1(g)). For figures that deal with the degenerate
vibrations we take two figures, the so – called Introductory Figures
3
(IF) (see section II(B)) and perform the transformation. Each of
the IF leads separately to the special way of attributing
numerical values to transformations considered.
(iv) Index (1, 2) – index 1 denotes symmetricity and 2 antisymmetricity
of a vibration with respect to the axis other than the highest
multiplicity one. First, we will consider figures which refer to
nondegenerate vibrations. For such figures the transformation
corresponding to the axis other than the highest multiplicity one is
the C2 rotation with respect to the axis passing through the
midpoints of the hexagon’s opposite sides or with respect to the
axis passing through the opposite vertices of the hexagon. The
choice of one out of the two axes is arbitrary. The way of
utilizing indexes (1, 2) is different for A and B types of figures.
For type A – index 1 means that the figure obtained is the same as
the initial one (after performing transformation of the initial
figure), whereas index 2 (for the same type) means that the figure
obtained is the opposite of the initial one (the signs are
exchanged, i.e. (+) changes into (–) and vice versa).
For type vibration of the type B – the effects of transformations
are exactly the opposite to the ones for type A.
For degenerate vibrations and figures which correspond to the
vibrations, the principle of indexing (1, 2) is as follows:
we consider rotations by the odd multiplicities of 60°, i.e. 60°,
180° and 300° with respect to the highest multiplicity axis. The
angles of rotations refer to the operations C6, S6, C2, C65, S65.
Index 1 means that if the Introductory Figure is rotated by 60°
(C6, S6) or by 300° (C65, S65), then neither of the operations change
the number’s sign assigned to IF, and change the sign into
opposite one if IF is rotated by 180° (C2). Index 2 informs that if
IF is rotated through the same operations as for the 1 case, then
the results are opposite to the ones obtained for 1, or there is no
change of the number’s sign assigned to IF for the operation C2
and change of the sign for C6, C65, S6, S65.
We emphasize that the number’s sign is discussed here but not the
number’s value.
B. Figures and the way of their transformations
Let’s consider the figures used to obtain CIR of non – degenerate
vibrations, Fig. 1. They can have the same signs on both the planes (the
front – plane sign is the same as the back – plane one, Fig. 1(a),(c),(e),(g))
or the opposite signs on the two planes, (Fig. 1(b),(d),(f),(h)). If we
4
transform a chosen figure and obtain unchanged one then the numerical
result equal to 1 is assigned to this transformation, however if after the
transformation the signs are exchanged, namely (+) changes into (–) and
vice versa, then we assign the value –1 to this transformation.
_+
_
_+
+
_
+_
_ +
+
(a)
_
(c)
_+
+
+_
FRONT
BACK
+
(b)
_+
_
+
_
+
_
+
_ +
+
_
+ _
FRONT
+_
_+
+
BACK
_
_
_
_
+
(d)
_
+
+
+
_
_
+
(e)
+
_
_
_
+
+
_
FRONT
+
(g)
_
+
_
_
_
+
+
_
+
BACK
(f)
_
+
FRONT
+
+
_
_
+
BACK
(h)
Fig.1. Figures used to obtain CIR of non – degenerate vibrations
(a) The figure leading the characters of A1g symmetry type
(b) The same figure as in (a) but with the opposite signs on both sides
of its plane. The figure permits getting the characters of A2u
(c) This figure is used to obtain the characters of A2g symmetry type
(d) The same figure as in (c) but with the opposite signs on both sides
of its plane, it permits getting A1u
(e) The figure for finding the characters of B1u symmetry type
(f) The same figure as in (e) but with the opposite signs on both sides
of its plane. It permits getting the characters corresponding to B2g
(g) The figure enables getting the characters of B2u symmetry type
5
(h) The same figure as in (g) but with the opposite signs on both sides
of its plane. It permits getting the characters of B1g
Now, let’s analyze the degenerate vibrations. In order to get CIR which
correspond to these vibrations we make use of the figures discussed
below. The figures that enable us to obtain the characters of E1u and E1g
symmetry types are presented in Fig. 2.
 

a

a


p


b


c

q


 
r


s
FRONT

 
c
d


s



q
d


b


p
 

r


e




e


f

t



u
(i)
 

a'
f

t



u


p'



b'


q'
c'

 
d'

r'


s'



f'
e'




t'

u'
(ii)
BACK
Fig.2. Figures used to obtain CIR of E1u type (i) and E1g type (ii)
There are two groups of figures (a, b, c, d, e, f) and (p, q, r, s, t, u) that can
be used to obtain CIR of E1u type (see Fig. 2(i)). Each figure (hexagon)
in a given group is rotated with respect to the previous one by 30° in
the counterclockwise direction. Each hexagon is divided into two parts and
to each part a sign is assigned. The two signs are the opposite. The front
– plane sign is the same as the back – plane one. The procedure to obtain
CIR is as follows:
we choose one hexagon from each group, for example a and p are the
so – called Introductory Figures (IF), and make transformation according
to D6h symmetry operations. If one obtains IF after the conversion then
the numerical value equal to 1 is assigned, whereas if one obtains the
opposite figure to IF then we assign value –1 to the result. For instance,
the opposite figure to a is d, and the opposite to p is s. When the
resultant figure (the final figure) is rotated with respect to IF by 60˚ and
300˚ then to the result of this transformation the numerical value 1/2 is
6
assigned and for the opposite case the numerical value equal to –1/2 is
assigned, so if IF are a and p then:
a
  1,
p
d
  1,
s
b
f
 1
 ,
q 2
u

c
e
1


r
2
t

(1)
The hexagons appear in pair composed of two figures namely the
hexagon from the first group is located on top of hexagon from the
second group (see Fig. 2). It means that it is possible to choose (a, p) or
(c, r) but not e.g. (a, t). It should be remembered that if we make the
transformations with respect to the diagonal axis of the hexagon which
belongs to the first group then the transformations of the second group
must be made with respect to the same axis.
There are two numbers which result from the transformation of a figure
from the first group and from the second group, respectively. We add
together these two numerical results and we get CIR. For 3C2’, 3C2”, 3σd,
3 σv one finds the characters in the following way. Three transformations
are performed for each axis defining operation C2’ and three for the
operation C2”, as well as three transformations for each plane of σd
symmetry and three for σv one. Let’s consider, for example, the operation
C2’, and then the first group chosen figure and the second group one are
rotated with respect to the three axes of symmetry – each figure
separately. The result of each of the three transformations (in one group)
is the number given by Eq. (1). So, there are three numbers for one
group. The three found numbers are added up to get the numerical result
of the transformations for the first group of hexagons. Analogous
procedure is carried out for the second group. Finally, we add up the two
numbers found (numerical results for the two groups) obtaining CIR for a
given symmetry operation.
In order to get CIR of E1g symmetry type we make use of figures shown
in Fig. 2(ii). They are the same figures as for E1u but with the opposite
signs on the front and back plane of a figure. The assignment of the
numbers is the same as in E1u. If we choose (a, p) as IF, then:
7
a  d '
  1,
p  s '
b  e' 
f  c'
 1
 ,
q  t'  2
u  r' 

d  a'
  1,
s  p '
c  f '
e  b' 
1


r  u' 
2
t  q' 

(2)
The procedure for obtaining CIR is the same as for E1u.
The last two symmetry types and their characters can be found on the
basis of the figures shown in Fig. 3.
+
+
_
_
+
_
+
+
a
b
c
_
+
+
p
+
_
+
+
+
_
+
_
_
+
q
+
+
_
+
(i)
r
_
_
_
+
_
+
+
+
+
_
_
a
b
c
a'
b'
c'
_
+
+
_
_
+
+
p
_
+
+
q
FRONT
_
+
r
_
_
p'
_
+
q'
_
+
r'
BACK
(ii)
Fig.3. The above figures permit getting CIR of (i) E2g and (ii) E2u types
Let’s consider figures corresponding to E2g. We have two groups of
figures (a, b, c) and (p, q, r) – see Fig. 3(i). In each group the subsequent
figure has the signs rotated with respect to the previous one by 120˚.
The group (p, q, r) can be obtained by defining the (a, b, c) group. For
8
example, we can take a from the first group and reflect it with respect
to the horizontal plane which is parallel to the bottom side of the triangle
inscribed inside the hexagon a. We get the figure p by this operation.
The other members of the figures of the second group are obtained by
120° and 240° rotation of the figure p, respectively. Another procedure is
to transform b with respect to the plane parallel to the right side of the
triangle to get q or transform c with respect to the plane parallel to the
left side of the triangle to get r. The remaining figures are obtained by
120° and 240° rotation of the first one. The assignment of numbers as
follows:
a
 1
p
b, c 
1

q, r 
2
(3)
The procedure of transformations is the same as for E1u and E1g.
Fig. 3(ii) presents the same figures as in E2g but with the opposite signs
on both sides of the plane which distinguish the front and the back of
the figures. The numbers assigned to the figures are the same as in E2g,
except for those which correspond to the so – called primed ( ' ) figures.
For these figures the opposite numbers are assigned (see (4)). The way of
the transformations of the figures corresponding to E2u is the same as for
E1u, E1g and E2g
a
 1
p
b, c 
1

q, r 
2
a' 
  1
p'
b' , c' 1

q' , r ' 2
(4)
III. THE MATHEMATICAL INSIGHT
The symmetry operations of a molecule belonging to the D6h group can
be described in terms of matrices. Let’s assume that the molecule lies on
XY plane, then the matrices are given by expressions (6).
The superscripts ( ' ) and ( " ) (see matrices (6)) refer to the rotation with
respect to the axis passing through the opposite sides of the hexagon and
the diagonal axis, respectively. The matrix elements are found according
to the literature known methods with slight modifications for C2’ and C2”,
for which the angles used to calculate the matrices are as follows:
For ( ' ) the angles satisfy the expression:
180˚+(2n+1)·60˚, “n” is an integer. E.g.
n = 0, then 180˚+(2·0+1)·60˚ = 240˚, which refers to M(C2 + 6),
n = 1, then 180˚+(2·1+1)·60˚ = 360˚, → M(C2 + 3•6),
n = 2, then 180˚+(2·2+1)·60˚ = 480˚, → M(C2 + 5•6).
9
For ( " ) the angles satisfy the expression:
180˚+2n·60˚,
n = 0, then 180˚+2·0·60˚ = 180˚, which refers to M(C2),
n = 1, then 180˚+2·1·60˚ = 300˚, → M(C2 + 2•6),
n = 2, then 180˚+2·2·60˚ = 420˚, → M(C2 + 4•6).
The usage of other “n” doesn’t change the value of the sine and the
cosine because of their periodicity (see the rotation matrix (5)).
cos
M Cz   
 sin 
 
 sin  
cos 
(5)
For 3σd and 3 σv the same numbers appear on the main diagonal of their
matrices as for 3C2’ and 3C2”.
The numbers on the main diagonal are the same as those found as a
result of the E1u figures transformation process (see (1)).
Whereas, for A1g the determinants of the matrices (5) are calculated to
get the characters.
10
1 0
M (E)  

0 1 
 1
3



2 
M (C6 )   2
1 
 3
 2
2 
 1
3


2 
M (C65 )   2
 3 1 
 2
2 
 1
3



2 
M (C3 )   2
 3 1 
 2
2 
 1
3



2 
M C32   2
 3  1 
 2
2 
 
 1 0 
M i   

 0  1
 1
3



2 
M S 3    2
 3 1 
 2
2 
1 0
M  h   

0 1 
 
 1 0 
M C 2''  M (C 2 )  

 0  1
 1
3




2 
M C 2'  M (C 26 )   2
 3 1 
 2
2 
 1
3


2 
M C 2''  M (C 2 26 )   2
 3 1 
 2
2 
 
 
 
1 0
M C 2'  M (C 236 )  

0 1 


M C 2''  M (C 2 46 )  


 
1
2
3
2

3

2 
1 
2 
(6)
 1
3




2 
M C 2'  M (C 256 )   2
 3 1 
 2
2 
 1
3



2 
M S 32   2
 3  1 
 2
2 
 1 0 
M (C 2 )  

 0  1
 
 
Detailed depiction of interrelation between matrices and numbers assigned
to the results of the transformations, for degenerate vibrations, can be
found in the appendix.
IV. SUMMARY
This paper presents a visual method of obtaining characters for all the
D6h group irreducible representations. Similar method has been previously
11
applied for the D3h group. The model is based on taking advantage of the
figure (figures) corresponding to a given type of symmetry and its (their)
transformations through all symmetry operations which are assigned to an
examined molecule. In order to make use of such a possibility one needs
to notice interrelation between symbols and indices which depict the
symmetry type and the form of the corresponding figure. By this means,
we have an opportunity to find CIR in visual and transparent way, what
makes the model clear and applicable even for introductory courses of
physics and chemistry. What’s more, the method has been supported by a
number of computer simulations and a software, giving the readers a
chance to explore the discussed topic by him – or herself. The
mathematical insight (see sec.III) – based on results of Group Theory6 –
proves the correctness of the model. The model isn’t associated with one
point group only but can applicable for other point groups, too.
a)
Electronic mail: T.Bancewicz: [email protected], A.Kaminski: [email protected]
Gordon M. Barrow, Introduction to Molecular Spectroscopy, (McGraw-Hill, New York, 1962).
2
.M. Hollas, Basic Atomic and Molecular Spectroscopy, (Royal Society of Chemistry, 2002).
3
T. Bancewicz, A. Kaminski, “The geometrical model for obtaining the D3h group’s characters
of irreducible representations and symmetry types”, Chem.Educ.J. (12) 1, (2009).
4
Computer simulations and software are available at < www.matphys.kki.pl>. You have to
write down the address in the field of addresses of www page. Don’t write down it in other
fields, for example “Google”.
5
P. Kowalczyk, Fizyka cząsteczek (Physics of Molecules), (PWN SA, Warsaw, 2000), Chap.1,
p.12. and pp.35 – 44.
6
J. F. Cornwell, Group Theory in Physics, (Academic Press, London, 1984).
1
Recommended literature:
□ H. Haken i H. C. Wolf, Molecular Physics with Elements of Quantum Chemistry, (PWN SA,
Warsaw, 1998).
□ R. L. Carter, Molecular Symmetry and Group Theory, (J. Wiley and Sons, N.York, 1997).
□ J. F. Cornwell, Group Theory in Physics, (Academic Press, London, 1984).
□ D.R. Lide, Handbook of Chemistry and Physics, (CRC Press, London, N.York, Washington
2004).
□ Z. Kęcki, Elementy Spektroskopii Molekularnej (Elements of Molecular Spectroscopy), (PWN
SA, Warsaw, 1998).
□ L. Landau i E. Lifszyc, Quantum Mechanics, (PWN, Warsaw, 1958).
□ I. W. Sawieliew, Physics Lectures, (PWN, Warsaw, 1998).
□ R. Liboff, Introduction to Quantum Mechanics, (PWN, Warsaw, 1987).
Appendix
The way of finding numbers in Eqs. (1) – (4) on the basis of the
transformation matrices, the symbols and indices which determine the type
of symmetry:
12
Type E1u
There are two numbers on the main diagonal of matrices (6); the first
one refers to the result of the operation in the first group of hexagons
and the other to that in the second. These numbers are the same as in
Eq. (1).
The numerical assignments for the remaining degenerate
symmetry (Eqs.(2 – 4)) are derived on the basis of E1u.
types
of
Type E1g
The way of looking for the numbers corresponding to this type of
symmetry (Eq. 2) is the same as for E1u except for those transformations
which cause the front – back alteration (i, 2S3, 2S6, σh), where the sign
change is needed.
Type E2g
This type of symmetry has also the same numbers (Eq. 3) as for E1u, or
the numerical results of the operations E, 2C3, 3C2', 3C2'', 2S3, 2S6, σh, 3σd,
3σv are the same for the two types (E1u, E2g), except for i, 2C6, C2, 2S6.
For inversion (i) we have index g which means symmetricity with respect
to the centre of symmetry, or the inversion matrix has the form:
1 0
M (i)  

0 1
In other words, there are the opposite numbers on the main diagonal of
the E2g inverse matrix to those of the E1u inverse matrix. Besides, the
type E2g has index 2 meaning antisymmetricity or the change of the
number’s sign at the rotations 2C6, C2, 2S6 (see II.A.(iv)) with regard to
those for E1u.
Type E2u
The way of looking for the numbers corresponding to this type of
symmetry (Eq. 4) is the same as for E1g except for (i, 2S3, 2S6, σh),
where the sign change is needed. The procedures of inversion of the
figures corresponding to E2u and E1g are different because indices “g”
and “u” have slightly different meaning for the two types (see II.A.(iii)).
Whereas, the transformations (2S3, 2S6, σh) cause the change of the
figure’s sign (the figure (figures) transformed through these operations
13
have the opposite signs on both sides of the plane in which the figure is
located), and thereby we observe the change of the number’s sign with
regard to the same operations for E1g.
14