IX  Symmetry, point groups, and character tables. Part II: Classification of molecules into point groups
ABSTRACT Examines translational symmetry, introduces point groups, and presents a convenient procedure for assigning the point group of a molecule. Keywords (Audience): UpperDivision Undergraduate
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Resource PaperslX
Prepared under the sponsorship of
The Advisory Council on College Chemistry
Milton Orchin
Cincinnati, Ohio 45221 I
and H. H. Jaff5 I Symmetry, Point Groups, and Character Tables
University of Cincinnati
Part 111, Character tables and their significance
In preceding sections we have discussed
the symmetry properties of molecules in terms of their
geometry and have analyzed the dipole moments of
molecules. The geometry and the dipole moment are
static properties of the molecule. In order to analyze
the dynamic properties of molecules, we must consider
whether the particular property is symmetric (un
changed) or antisymmetric under the symmetry opera
tions appropriate to the molecule. All properties must
be (or must be decomposable into components which
are) either symmetric or antisymmetric with respect to
each symmetry operation that can be perf~rmed.~ In
order to determine whether a property is symmetric or
antisymmetric, we must have a clear understanding of
these terms. We will first examine motions, specifically
translations, of objects or molecules.
If one stood in front of a mirror and threw a ball up
in the air parallel with the mirror, one sees that, when
the hall moves upward, so does its reflection in the mir
ror, and with the same speed. When the ball reaches its
apogee and starts to descend, the image likewise changes
its direction and descends with the same speed as the
ball. The up and down motion of the hall (molecule)
is said to be symmetric with respect to reflection in the
mirror parallel to the motion because the actual motion
and its reflection always travel in the same direction
with the same speed (or magnitude). Now let us throw
the ball directly at the mirror, and analyze this motion
which is now perpendicular to the mirror. The re
flection or image travels at the same speed as the ball,
but the ball is moving toward the mirror and away from
the thrower, while the reflection is moving in t,he oppo
site direction, so that the ball and its image collide at
the mirror. The motion of the image has the same mag
nitude but is now opposite in direction or sign to the
real motion. Thus, motion (or translation) perpendic
This threcyart paper represents the last in n series of
"Resource Pi~pe1.s" prepared under the spo~isor~hip
Advisory Coonril on College Chemistry (AC3) which is
supporlcd by the National Science Foundation. Profes
sor L. Cnl.roll King of Northwestern Uuivelaity is tho
chairman.
Single copy reprints of this paper are being sent to chem
istry department chairme11 of every US. Institution of
feriq college chemistry courses and to others on the ACa
mailing list,.
This is Serial Pahlieation No. 49 of t,he Advisory Council.
of tho
ular to the mirror is said to be antisymmetric with
respect to reflection in the mirror. If we placed these
motions in a coordinate system with the z coordinate
vertical and in the plane of the paper, the x coordinate
perpendicular to the paper, and the y direction hori
zontal and in the plane of the paper, we see that, with
respect to reflection in a vertical plane, translational
motion in the z and y directions, which are parallel
with the mirror plane, are symmetric, while transla
tional motion in the x direction, perpendicular to the
mirror plane, is antisymmetric.
When a molecule has a center of symmetry and we
wish to discuss the symmetry behavior of some dy
namic property of the molecule, we use the special
terms gerade (German for even) and ungerade (German
for odd) for symmetric and antisymmetric behavior,
respectively. With respect to motion in the x, y, or z
direction, reflection through the center of symmetry
always reverses the direction of the motion, and hence
such motion is always ungerade. Since orbitals or
wave functions describe electronic motion, the molecu
lar orbitals of molecules with a center of symmetry
must be either gerade or ungerade. Thus, the a orbital
in ethylene, Figure 5A, is ungerade, a", while the a*
Figure 5. Gerade and ungerode orbital%
orbital in Figure 5B is gerade, a : ,
orbitals, Figure 5C. In Figure 5B, for example, if we
look at a point in the upper left positive lobe and draw
a line from it through the center of the molecule and

continue an equal distance in the same direction, we
encounter a point with the same sign; hence, this orbital
is gerade.
as are all d atomic
Parts I and I1 of this series appeared in the April and May
issues, respectively, and dealt with the topics of symmetry opera
tions and classification of molecules into point aroups. All foot
notes, figures, tables, and stmctmes are numbered Eonsecutively
throughout the series. The material in fine print indicated by
a. 5 adds rigor to some of the arguments hut is not considered
essential for the introductory course in organic chemistry 8s it
is now generally strootwed.
'An exception are properties of molecules with C , or S , with
p > 2 (i.e., molecules belonging to degenerate point groups), which
are treated in B later section.
1
1
510 / Journal of Chemicol Education
Page 2
8 Symmetric and Antisymmetric Character of Singlet
and Triplet States
In dealing with electronic absorption spectroscopy and photo
chemistry, we are generally cur~cerned with promoting one elec
tron from an orbital with a certain energy into a vacant (or vir
tual) orbital with higher energy. Although the two electrons
which share the original orbital must, of necessity, possess oppo
site spin, when the two electrons occupy different orbitals, this
restriction no longer applies. The spin of the two electrons in
the excited state can be eit,her the same (i.e., both
'/%), or the two elect,ronx can have opposed spin (i.e., if one is
+xil
the other is '/% or vice versa). The multiplicity of a
state, J, is equal to 2 8 ) + 1 where S is the sum of tho spin num
bers of either f'/2.
When both electrons have the same spin,
21S1 + 1 = 3 and we have the socalled t,riplet state. When
the elect,rons have opposed spin, 281 + 1 = 1 and we have the
socalled singlet state.
In a. common convention, the two possible spins of an electron,
and I/%,
are denoted as o and 8, respectively. Since two
electrons are involved. we mav call these electrons 1 and 2. thouzh
~~~
of course we cannot distinguish them. The possible spin combi
nations of the two elect,rons in different orbitals are
or both
~

Thus, in (a) the two electrons both have spin of '/z, in (h) both
 1 . In either (a) or (b), it doesn't make any difference which
of the two electrons (I) or (2) is being used, since in both cases
both electrons have the same spin. Thus, in (a), o(2)ar(l) is
identical with a(l)a(2). Both (a) and (b) are therefore sym
metric with respect to exchange of electrons!
and (d) we have two equivalent expressions and neither can he
considered done; s . combination of the two is necessary lo de
scribe the situation. Thus. if we exchsneed electrons in (c) we
However, in (c)
antisymmetric with respect to exchange of electrons. In such
cases, we must take linear combinations of the two functions,
i.e., we add them together to give one combination and then sub
tract them to give the second linear cambinstion
(c + dl: o(1)8(2) + a(2)8(1)
(c  dl: a(l)P(2)  o(2)8(1)
(symmet~ic, triplet)
(antisymmetric, singlet)
Kow let ur iwcitlg:ue the symm~tri~
of tlwr two spit) fuuvtioni with IC~XYI 10 es<l.a~~gc tlnr two
elwtroni. \\'c see tlmr if we e*ch:ilqe the rlturo~i 1 kl.J (2
in the first combination (c + d) we get no change in sign and
hence this function is also symmetric. But now let us exchange
the electrons (1) and (2) in (c  d), whereuponweget o(2)@(1) 
o(l)p(2), which is precisely the result obtained by multiplying
(C  d) by 1; hence (e  d) is antisymmetric with respect to
exchange of electrons. The antisymmetric spin function charac
terizes the singlet state (the total wave function must be anti
symmetric, and, in the singlet state, the orbit81 part of the wave
function is symmetric and the spin part of the wave function is
antisymmetric) and the symmetric spin function characterizes
the triplet state. As a point of fact, the energies of the total wave
function, of which the spin functions (s); (b), and (e + d) are a.
part, are all equal (degenerate) (in the absence of a magnetic
field) and together constitute the triplet state. Accordingly, we
may define the triplet state a3 the state whose spin function is
symmetric with respect to exchange of electrons.
ur ,~utir?mnertvic ~1111ravter

'The symmetry discussed here is of a somewhat different
nature than the symmetry discussed in the rest of this paper.
The operators used in the remaining parts are rotation, reflection,
inversion, and rotationreflection. In this section, the operat,or
is a more subtle one, one which exchanges the coordinates of elec
tron 1 and those of electron 2. Otherwire, the treatments are
equivalent.
When the p orbitals are shown as two circles of opposite sign
touching at a. point, the angular dependence of the wave funct,ion
is being illustrated.
Character Tables
In the study of the properties of molecules we are
frequently interested in the motion of the molecule it
self and in the motion of the atoms relative to one
another (vibrations, ir absorption) as well as the motion
of the electrons in the molecule (molecular orbital
theory and electronic spectra). When we are dealing
with a molecule like water which belongs to point group
C2" (no degeneracies) all motions of the molecnle must
be either symmetric or antisymmetric with respect to
each of the four symmetry operations of the group.
We may for our purposes here agree to characterize
symmetric behavior as +I and antisymmetric behavior
as 1 and to call the +1 and 1 the "character" of
the motion with respect to the symmetry operation.
For example, let us examine the behavior of the p,
orbital6 in water, Figure 6, under some of the symmetry
Figure 6. The p* orbit01 in woter.
operations of point group Cza. Rotation around the
zaxis changes the signs of the two lobes, hence under
C 2 ' the p, orbital is antisymmetric and has the char
acter 1.
Similarly, reflection on a%, gives a change in
'sign and hence is 1, while reflection in av, transforms
the orbital into itself and hence has the character +l.
The other symmetry operation in point group Czar
CSz,
as2, and aY~,
is the trivial identity operation I, which,
since it leaves the molecule unchanged, obviously has
the character +l. The total number of symmetry
operations in s. point group is called the order of the
point gronp; in Cz,
the order is four.
We stated earlier that one of the properties of a
group is that the product of any two elements in the
gronp is also an element of the group. In Cz,
three nontrivial elements, and since the third is the
product of any of the other two, we need only be con
cerned with the character of two elements or operations.
Since we have two possible independent operations and
each can have the character of + 1, we have a total of
four possible combinations. Let us arbitrarily consider
CzS
and a=, as the two independent operations; both can
be +1, and they can also be +1,  1; 1 1
 1, 1. We can put this information into a table
called a character table, and for point group Cz,,
is such a table. This tahle shows that under the iden
tity operation, I, every property is symmetric, as ex
pected, since the I operation does nothing. The four
possible combinations of C z ' and us, are shown, and it is
readily verified that in each case the character of u,, is
the product of C S ' X a,,.
heside
we have
or
Table 3
Symmetry Species of Irreducible Representations
We earlier examined the behavior of the p, orbital in
the water molecule, (Fig. G), and showed that under
the symmetry operations of point gronp C2",
the orbital is +I with respect to I, 1 with respect to
CE',  1 with respect to u , , , and + 1 with respect to u , , .
This +1, 1, 1, +l behavior is one of the four pos
sible ways t,hat every property of the molecule can be
Table 3,
Volume 47, Number 7, July 1970 / 51 1
Page 3
described. These four distinct behavior patterns are
called symmetry species or irreducible representations,
and their number is equal to the order of the group.
The symmetry species have, for convenience, been
given shorthand symbols which are shown in the case of
C2, in the first column of Table 3. All symmetry species
that are symmetric with respect to the highest rotational
axis are designated by A, and those antisymmetric are
designated by B. In the case of point groups like Dl,,
where there are three twofold axes, and therefore no
rotational axis that is of highest order, only the sym
metry species that is symmetric to all three Cp axes is
designated A. Where more than one species or repre
sentation is symmetric with respect to the highest
rotational axis, as in C2,, they are distinguished in the
subscripts (or sometimes by primes) and the totally
symm~tric species, i.e., the species as in C2" which is
plus with respect to every operation, is always the A,
species. Subscripting of the R species is more arbitrary.
In the case of molecules belonging to CZ,,
orientation given earlier is usually unambiguous and
after setting up the coordinate system, the B species
that is symmetric to a,, is called B1.
In all point groups with a center of symmetry, the
subscripts u and g are used for all species; the assign
ment depends upon the behavior with respect to the
center of inversion, i. Thus, consider the character
table of point group Cnn, Table 4. In this point group
there are four operations: I; CZe; u,, (or a,,); i; and
again four irreducible representations. The two A
species are here designated A, and A,;
totally symmetric, i.e., symmetric with respect to all
operations,
Let us examine some property of a molecule in point
group CZ*. We choose stransbutadiene and analyze
the lowest energy a bonding molecular orbital (Fig. 7).
(The molecule is shown in the plane of the paper and
the pa system is perpendicular to the paper.) Under
the symmetry operations I, C2', an, i, this orbital trans
forms as +1, +1, 1, 1, and hence belongs to sym
metry species A..
the rules for
the former is
Toble 3. Chorocter Table for Point Group C , ,
    
 
Cz,
I
C2(4

A,
+1
+ 1
Aa
+ 1 +1
HI
+ 1
 1
H,
1

~
, , J = d
Gv(~d
+1
 1
 1
+ 1


+ 1
 1
+ 1
 1
z
Rz
2, Rr
Y, Rz
 + 1
Table 4. Chorader Table for Point Group C2*
   
Cm
I
C2m
A,
+1
+l
A,
+1
+ 1
R,
+1
1
H"
+1

.

 
q(=~)
+ 1
 1
 1
+ 1
i
+ 1
1
+1
1
R,
B
RZ, Rv
2 , Y
 
1
Figure 7. The mrnt bonding molecular orbital in stransbutodiene, C,lu
In a point group like Dzh, Table 5, where there are six
B species, both numbers and the g and u letter designa
tion are required.
Now let us examine two of the possible ir vibrations of
ethylene, a molecule which belongs to point group Dz,.
An out of plane deformation mode shown in Figure SA
transforms as species BZc. The plus signs indicate
Figure 8. Symmetry species of two vibrational modes in ethylene, D z ~ .
motion out of the plane of the paper toward the ob
server, and the minus signs indicate motion behind the
plane away from the observer. Thus, C1 is moving
toward, C2 moving away, HI moving away, H, toward,
H a away, H, toward the observer. Reflection of these
motions through the xyplane gives motions at each of
the atoms which are exactly the reverse of the original
at that atom, and hence this out of plane bending mode
is antisymmetric with respect to reflection on uZ, and
has the character of  1. Testing this bending mode
under each of the operations listed in character table
D,, shows that it transforms as species Bz,. The vibra
tional mode shown in Figure SB can be similarly dem
onstrated to belong to species Bz,.
5 Degenerate Point Groups
The discussion of character tables up to thk point can he a g
plied to point groups CI, C,, C;, CZ, Csh, Ds, and Dlh. AU of these
point groups do not involve a symmetry axis C or S greater than
twofold (DZa has an & axis). As soon as an axis greater than
twofold arises, the problems of symmetry species become much
more difficult.
Let us use, as an example, the ion PtCLS and examine how the
three p orbitals of the platinum atom transform under the sym
Table 5. Character Table for Point Group Dxh 

c (XZ 1
~ Y Z ?
.

+1
+1
 1
 1
 1
1
+1
f l
+1
 1
 1
+l
 1 +I
+ 1
 1
Vh
DZL
A.
A.
Bu
HI"
B1.
B 2 "
B,,
B1"
E VL
I
+Y)
+l
 1
+1
1
 1
+1
 1
+ 1
i
c*(~,
+1
+1
+1
+ 1
 1
 1
 1
 1
C@
+ 1
1
 1
+1
+l
 1
1
c+'
+l
+ 1
 1
 1
1
1
+I
+l
+ 1
+1
+1
+1
+ 1
+1
+I
+1
+1
 1
+1
 1
+1
 1
+I
1
+1
. . .
...
R,
e
R.
Y
R,
2
5 12 / Journal of Chemical Education
Page 4
If the multiplication of the 2 X 2 metrix (I
O

")bytbe
(dl
i f 1
Figure 9. Thep orbitals of PI in[PfClrlz,
metry operations appropriate to the point group Dl*, Table 6,
to which this square planar ion, Figure 9A, belongs. The z and
yaxes (coordinate system Fig. 9B) are CS axes; the C9 axes which
bisect the angles between bonds are designated Cn' axes. The
two vertical planes of symmetry that include the z, y, axes a t r e o .
and those that include the C9' axes are called ad in the character
table. The p, orbital, Figure 9C, presents no particular prob
lem.' If we apply all the symmetry operations on the p, orbital
in turn as listed in the chracter table, we get, starting wit,b I,
+1, +l, +1, 1, 1, 1, +1, +1, 1, andfori, 1, which
tells us that p, belongs to species Al,.
Now let us examine the behavior of the p, orbital, Figure 9D.
(For convenience, the coordinate sysbem in B has been rotated
90' around the yaxis to give E, which places t,he zy plane in the
plane of the paper.) If we perform a 90" clockwise rotation, C , :
we see that p, is transformed to p, and hence p, is neither sym
metric nor antisymmetric under the operation. However, the
C4' operation simultaneously trsnsforms p, into p,;
the transformations are related and the two orbitals transform
together with t,he result shown in Figure SF.
Although the above transformations can he discussed in general
terms in vector notation, we shall develop the trsnsfarmations
ming the familiar orbitals. When we perform Z I clockwise rota
tion of 90' on the p, orbital, we get a new orbital which has none
of the old orbital in it and is exactly equal to the original pp,
orbital. If we call the new orbital p',, we may state the fact
mathematically
p'. = OP, + IP,
Similarly, t,he transformation of t,he old p, before the C, into
the new p', may be written
= Ips + OP,
A special met,hod of writing these equations is possible
obviously,
(1)
P ' ~
(2)
Table 6. Character Tab
D d n
I
+1
+1
+1
+ 1
+ 1
+1
+l
+l
+2
+2
ZC,(z)
C,¶ = C*" ZC2
+1
+1
 1
 1
+1
+1
1
 1
0
0
2C2'
+ 1
 1
 1
1
1
+1
+1
0
0
dl.
At,
As
A%"
%
RI"
&
Bs"
En
E"
+1
+1
+1
+1
1
 1
 1
 1
0
0
+1
+1
+ 1
+1
+1
+ 1
+1
+1
2
2
+l
>
2 X 1 matrix or column vector
to the rules of mstrix multiplicetian, the two equations (1) and
(2) would be obtained. The sum of the numbers appearing in
the diagonal from upper left to lower right of the transformation
matrix is called the trace (German spur) of the matrix and the
actual number found by the addition is called the character of
the transformation matrix. In the present example the charac
ter is 0. Accordingly, if we refer to the charscter table of Dl*,
we must look for a symmetry species which under the C4 opera
tion has a character of 0.
Here we can use a shortcut to determine the correct symmetry
species. When we are dealing with s . pair of orbit& of equal
energy that transform together we have a socalled degenerate
set. A set of two degenerate orbitals always belongs to an E
species (a set of three degenerate orbitals to n l'species). Hence,
our orbitals belong to one of the E species in Dlh, and it remains
only to specify the behavior under the operation i. A p orbital
is always antisymmetric to a center of symmetry, snd hence our
p,, p, arbitdls together belong to species E, in point group D,&,
the characters of which are found in the last row of the character
table. As an exercise let us determine the transformation matrix
of the p,, p, orbitals under the operation i
character is 2 as shown in the character table. Referring to
Figure SF we see that under i
P'= = IPS + OP,
P'. =
OP,  lp,
and the character of the transformat,ion mstrix
were carried out according
ti)
to confirm that the
Let us, instead of rotating the p,, p, orbitals 90' in the clock
wise direction as we did earlier, rotate in the cor~nterclackwise
direction, i.e., instead of C4 do C',. The transformation may be
written
P'. = OP,  IP, transformation matrix
P'" = lP, + OP,
The character of the t,ransformation matrix is again zero even
though the offdiagonal elements are now different, than those in
the transformation matrix obt,ained from the C, operat,ion. We
now see why C, and C', belong to the same "class"; they have
the sitme character. In the mathematics of group theory, a
"clad' includes all the group elements which are "conjugate"
to each other. Any two elements X and Y of a group are "conju
gate" to each other if there exists some element of the group,
say Z, such that
( 3
In our example C4 and C', are conjugate to each other and belong
'When the p orbit,sl is drawn dumbbell shaped, the square of
the angular dependence of the wave function is being illustrated.
This representation is appealing because the square of the wave
function, being a probability, has physical significance. How
ever, the two parts of the dumhbell should then both he positive.
In keeping with the common but incorrect practice of mixing the
representation of the angolar part of the wave function and its
square, we draw the dumbbell with different signs in the two
halves.
le for Point Group Dan
Volume 47, Number 7, July 1970 / 51 3
Page 5
to the same class; hence there are 2C4 because C4 and C', both
have the same character.
The Ca and C'. or 90' clockwise and counterclockwise rata
tions give, respectively, the transformat,ion matrices
These numbers correspond to the values of the sin and eos of
&90' and indeed that is their origin. We may generalize the
2 X 2 transformation metrix for any degree of rotat,ion by using
the matrices
eos 8 sin 8
(sin 8 cos 8) and (sin 8
and in either ease 2 cos 8 corresponds to the character. Thus,
in any doubly degenerate species, E, the character for the appro
priate rotation of 8' either clockwise or counterclockwise around
the C, axis is 2 cos 8. In these examples we have rotated the
orbitals (or more generally, the vectors), hut frequently one ro
tates the coordinate system instead. The choice affects the sign
of the offdiagonal elements but is immaterial, since the diagonal
elements which determine the character remain the same.
In the nondegenerate point groups discussed earlier, the trans
formation matrices are all 1 X 1 matrices and so we could im
mediately assign +1 to symmetric and 1 to antisymmetric
behavior.
cos 8 sin 8
cw 8
Symmetry Properties of Tmnslational and
Rotational Motion
It will be noted that character tables such as those
shown in Tabks 36 all include in a last column the
notations x, y, z and R,, R,, R,.
assigned to particular symmetry species in each point
group. They inform us to which symmetry species the
translations (of a molecule, for example) along the x,
y, zaxes belong, and to which symmetry species the
rotations R,, R,, R, around the x, y, and zaxes,
respectively, belong. Let us examine the motion of the
water molecule (point group CZa)
of the Cartesian axes. First we will ascertain how a
translation along the positive yaxis (Fig. 10A) trans
These symbols are
in the direction of each
( 0 )
( b l
( = )
Figure 10. Behovior of tranrloiional motion in the y direction on CP.
forms under the symmetry operations of Cz,. To sim
plify the analysis, we draw an arrow from the center of
the molecule along the yaxis. Under I the molecule is
unchanged, so the motion is +1 with respect to I.
Under C * ' we transform Figure 10B to Figure 10C and
we see that now the arrow is pointing in a direction
opposite to the original, i.e., the vector has changed
sign and the motion in the y direction is thus antisym
metric with respect to Czi
and hence has the character
1.
On reflection in the xzplane, the arrow would
again be pointing in the direction opposite to the orig
inal, as in CzZ,
and again the character would be 1.
Finally, reflection in the yzplane leaves the arrow un
changed .and hence under this operation, the character
is +l. The behavior under the four operations is thus
+1,  1,  1, f l , which belongs to symmetry species
R,. In the character table for CZu,
last column of the row of characters in the representa
tion B1 we have the symbol y, which tells us that motion
in the y direction of the water molecule transforms as
we see that in the
R1. Motion in the x and z directions can be similarly
analyzed, and we see that such motion belongs to the
representations BI and A1,
respectively. Since the p,,
p,, and p, orbitals behave like translations in these
directions, this notation also tells us how these orbitals
transform.
Similar transformations of rotational motion around
the three Cartesian coordinates can be assigned to
symmetry species. Let us analyze how rotation around
the zaxis transforms (Fig. 11). Again, to help us
analyze the situation, we employ arrows and show them
going in and out of the plane of the paper at each hydro
gen. The @ sign indicates motion upward out of the
plane; the Q sign indicates motion downward away from
the plane. The molecule is now rotating around the z
axis in a clockwise direction so that Ha is moving toward,
and H. moving away from the observer. In performing
the operations I, C2%,
havior of the arrows; if they change direction after the
operation the character is  1 and if the arrows remain
pointing in the same direction as the original, the rota
tion is +1 with respect to the operation. The be
havior of the rotational motions after each of the opera
tions is shown in Figure 11. On the CZz
exchange H. and Ha, but the atom on the left is still
going away, and the atom on the right is still coming
toward the observer; thus, although we exchanged
atoms, the atoms are indistinguishable, and the direc
tion of motions of the atoms on the left and right are
identical after the operation to that before the opera
tion. Hence, the Cz2
operation is +1 with respect to
rotational motion around the zaxis. The results of
reflections in the two planes of symmetry are shown in
Figure 11, and the characters belong to symmetry spe
cies AD If we look at the CZ, character table, we see
R, in the last column of the AZ species. The species
assignments for R, and R, are also indicated in the table.
UZZ, uyl, we can focus on the be
operation, we
$ Some Features of Character Tables and
Irreducible Representations
We have mentioned explicitly some properties of character
tables and symmetry species and implied others, and it is perhaps
Table 7. Character Table for Point Group Td
I
8C3
66
684
+l
+1 +1
Ts
3S2 = 3Ca
+ 1
+1
+2
 1
 1
A,
+1
E
T, +3
T z
+2
0
0
0
0
1
+1
+l
1
R*, R,, R,
5111.2
i3
c:
itll
 . *
e 11
111
Figure 11.
symmetry operations of C s . .
Behovior of rotational motion around the roxis under the
51 4 / Journal of Chemical Education