Symmetry, Point Groups, and Character Tables 11, Classification of ...

Resource Pa pers-IX
Prepared under the sponsorship of
The Advisory Council on College Chemistry
Milton M. Orchin
ond H. H. Jaffe
University of Cincinnati
Cincinnati, Ohio 45221
Symmetry, Point Groups, and Character Tables
11, Classification of molecules into
point groups
After the introduction, in part I, to
point symmetry operations and their applications to
determining optical activity, classificat,ion of isomers,
nmr studies, and dipole moments, we are ready to take
a brief look at translational symmetry and then move
on to a general discussion of point groups and molecular
point group classification.
§Translational Symmetry
We have thus far concerned ourselves with point symmetry hecause
we have been discussing the symmetry properties of individual
molecules. If we wish to consider the fixed arrangement
of molecules with respect to one another (crystallography), then
we must consider space symmetry. The conventions used to
denote various symmetry elements and operations which we have
considered earlier are C,, S,, and i. In discussing space symmetry,
these point symmetry operations are indicated by other
symbols, the so-called ITermann-Mauguin (H-M) symbols instead
of the point symmetry symbols used so far. The symmetry
operations a.nd the symbols for them are shown in Table 1.
From this table it can be seen that what we have earlier designated
as C, and c in the paint symmetry notation are p and m in
H-M nomenclature. Also, it should he noted that in H-M
nomenclature there is no i. One of the big differences hetween
the H-M nomenclature aud the point symmetry notation described
ezrlier lies in the handling of improper axes. In t,he H-M
notat,ian. . the imnroner .~ .~ rotation is defined ss a rotation-inver-
~
nper%~iun in plwe irf the rl,lalit,lr-r~crtint~ S, trpcl.nrim
$MYI in the point s\.nisne

iCI
Figure 3. A, Translational symmetry. B, Behavior of a mirror plane
under translational symmetry. C, Molecular arrangement showing o glide
line.
equal intervals. In this case there is no other symmetry present
besides translational symmetry. If the molecule, e.g., H1O, subjected
to the trsnslation has alredv a mirror ulane normal to the
direction of trsnslation (shown as a solid k i n the first section
of Fig. 3B), this mirror plane is, of course, repeated by the translation,
as indicated by the dashed lines through the oxygens in
Figure 3B. However, the translational operation creates a. new
set of mirror planes, at half the translationel distance, which are
shown as long dashed lines. Translation cen also he combined
with a reflection across the translation, as in Figure 3C. Here,
pure translation generates the alternate, equally-oriented molecules,
while the reflection, coupled with trenslstion by onehalf
the repeat distance, generates the molecules of the other orientation.
The operation of translation plus reflection to give the copy
of the original is cdled the glide translation; it is necessarily half
the true translation. In onedimensional symmetry symbolism
this is called g for glide line.
We have thus far in this section discussed translational symmetry
in one direction. A second, non-parallel translation vector
may bedded to the fist to generate 8. pattern which fills a plane.
Finally, the translation can also be made in a direction perpendicular
to the plane, and one would then be concerned with many
real situations dealing with space symmetry and crystallography.
We will not take these subjects up again, but it is helpful to
realize the irn~r.rlsnr rrlrtirm.~hip bctwccr. pollll +)mrnc,ry,
~~wful in the cuuicler~titn. of iillgle molcr.llcs, xnd

whereupon the z-axis is now actually vertical, and oh is
actually horizontal.
(b) If the molecule has several rotational axes, the one of
highest order is taken as t.he vertical and z-axis. Thus,
[PtC1412- hes a 4-fold axis and four 2-fold axes, two of
them coincident with the coordinate axes directed at
corners, and the ot,her two bisecting opposite sides of the
square, XLI. Accordingly, the Cfold axis in the plane of
XLIV. Hence, in these molecules if the z-axis is placed
vertical and the molecule is drawn in the plane of the
paper, the x-axis is the axis perpendicular to the paper.
(h) If the molecule is plmar and the z-axis is perpendiculer to
lhis plane, the x-axis (which together with the p-axxis must
lie in the molecular plane) is chosen to pass through the
largest number of atoms. Thus, in trans-dichloroethylene,
XLb, the z-axis passes through the carbon atoms, and in
benzene, XLII, it passes through opposite carbon atoms.
With [PtCld2- the choices me equivalent and hence the
assignment is srbitrary.
"L8 XLlL
the paper is t,he z-axis, since the molecule is shown oriented
such that it,s molecular plane is perpendicular to the
plane of the paper. The molecular plane is perpendicular
to the vertical C, axis and hence is horisontal, irh. Benzene
hes a six-fold axis, s threefold and a two-fold axis coincident
with it, and six two-fold axes, three bisecting
opposite bonds and three passing through opposite atoms.
If the benzene molecule is placed in the plane of the paper,
the z-axis is perpendicular to it; if it is drawn perpendicular
to t,he plane of the paper, XLII, then the z-axis is in
the plane of the paper. The molecular plane is on in either
~ ~
case.
(e) If there are several rotational axes of the highest order,
the axis which pa3ses through the greatest number of
atoms is taken as the z-axis. Thus, in ethylene, XLIIIa,
@O/ ,j-,
XL,"
where there are three equal two-fold axes, the z-axis, the
vertical axis, is taken as the one which is coincident with
the C-C bond. If one wishes to have the z-axis actually
vertical, then the molecule is rotated 90' around the
z-axis, XLIIIb. Naphthalene, XLIV, has three CI axes;
the one passing t,hrough t,he bridging csrhon atoms is considered
to be the z-axis. The three two-fold axes in both
nmhthalene and ethvlene are coincident with the z-, y-
and z-axes.
Many molecules which have a Cn also have an SX,
(P > 1) coincident with the C,. In such cases, there is no
problem. Thus, chair cyclohexane, XLV, has 6 and Ss
It* X,",~ XW,b
coincident, and thin is tsken as the z-axis. However, in
tetrahedral molecules, for example, fhere are Ca axes, the
axes which include the C-H bonds, XLVIa, as well as S,
axes, the axes which bisect the A-C-I1 bond angles. In
such cases, it is convenient and cust,omary to take one of
the equivalent & axes as t,he z-axis, XLVIb; this is the
axis which pa3ses through opposite faces of the cube in
which the tet,rahedron may be inscribed.
3. The assignment of the x-axis is made as follows
(a) If the molecule is planar and the z-axis lies in this plane,
the z-axis is chosen t,o be normal to this plane. This is the
situation in water and phenanthrene, XXXVIII and
XXXIX, and ethylene and naphthalene, XLIII and
' In the point group DI~
(vide infra), where there are three twotold
rotational axes and two olanes of svmmetrv, ..
the axis in which
the two planes of symmetry intersect is chosen as tho z-axis
Finally, a word must he said about the assignment of
the positive direction of the axes. The convention
usually adopted is the so-called right-hand rule. The
thumh, index, and middle fingers of the right hand are
extended in three mutually perpendicular directions.
The directions in which the thumh, index, and middle
fingers - are uointina - then become, res~ectivelv, the uositive
x, y, and z directions. he ciordina& systems
shown with XXXVIII-XLIV obey the right-hand rule
convention; the head of the arrow indicates the positive
direction.
Despite the general acceptance of the above rules, it
is essential for an author to explicitly indicate his assignment
of the coordinate system for the molecule under
discussion.
Assignment of Point Groups
Every molecule can he characterized by the symmetry
operations that can be performed on it. If precisely
the same operations can be performed on two
molecules, the molecules, no matter how different
chemically, are symmetry-related and must be classified
together.
We have already indicated that H20, XXXVIII, and
phenanthrene, XXXIX, possess the same elements of
symmetry, and we can perform the following (and only
these) symmetry operations on them: Cz', a., and a,,.
In addition, the identity oueration, I, can be ~erformed
" -
Properties of o Group
. .
on every molecule, and hence there are four symmetry
operations that can be performed. The four operations
together constitute a group and because each of the
operations leaves the center of gravity unchanged, the
group is called a point group. The four operations or
elements constituting the group must satisfy the mathematical
properties of a group.
For our purposes, the most important requirement
for a set of symmetry operations to properly constitute
a group is the requirement that if two operations in the
group are multiplied together, the product must also he
an element (operation) in the group. Let us test this
requirement against the four symmetry operations constituting
the group to which water and phenanthrene
belong, using the coordinate system shown with
XXXIX. Let us first note that the operation C1'
changes a point x, y, z to -x, -y, z; a, changes a
point x, y, z to -x, y, z; UZ, changes a point x, y, z to
x, -y, z; and finally, the operation I changes a point
x, y, z to x, y, z, i.e., it leaves the point unchanged. Now
if I, C2', azl, uY. constitute the elements in the group,
multiplication of any two must give a third. In performing
multiplications of symmetry operations it
should he borne in mind that each operation is a mathe-
374 / lournol of Chemicol Education

matical operator, not a quantity. The operation is an
instruction to do something; C2' says "rotate 180"
round the z-axis"; u, says "reflect every point in the
molecule through the vertical planc xz," etc. The operation
requires us to operate on an operand, which is
written to the right of the operation. Thus, when forming
a product, the elements are taken from right to left;
Czz X UZ, means first to reflect on u,, then to rotate
lSOo around the z-axis. Let us now take any point, x,
y, z; u,, gives x, -y, z and the C2' converts x, -y, z to
-x, y, z. Now the conversion of x, y, z to -x, y, z by
CnE X UZ, is exactly equivalent to a third operation in
the group, uv,, since u,, also takes the original x, y, z
into -x, y, z; hence, C2V US, = o,. If we wish to
make use of our atoms in the water molecule, XXXVIII,
we can focus on a point on the front part of the left
hydrogen atom: a, takes the point to the front of the
right H atom; Czl takes the point to the rear of the left
H atom, where we expect to find the point if the original
point on the front of the left H atom is reflected
through 9,. The necessity for including the operation
I in a point group is now clear. In the present point
group the square of every element gives the original:
CsV C2' = I, etc.
$ We indicated above that there are other formal mathematical
requirements for s set of elements to constitute a. group. These
other requirements are
1. There is an element I such that 11.1 = X.1 = X. I is the
identity operation and Xis my operation in the set.
2. The associative law holds, i.e., abc = (ab)(c) = a(bc). (The
commutative law: ab = ba, does not always hold in groups.
In theease of water above, the four elements do commute and
in such cases the group is said to he Abelian.) All point. groups
that do not have an ares higher t,han two-fold are Ahelian.
3. For each element, X, there must he another element which is
its inverse, Y = X-1 such that YX = I. The inverse operation
was aescrihed earlier. In the present example of water,
each element is its own inverse, e.g., 180' rotation around the
z-axis in either direction followed by 180" rotation in the
opposite direction returns the molecule to the orientstion
identical with the original.
Classification of Point Groups
In our classification scheme we will start with point
groupsJ that have the least symmetry and finish with
those point groups which characterize molecules of the
highest symmetry. It is convenient for such classification
to use the rotational axis (axes) as the principal
criterion of classification.
Type I: No rotational axis greater than one-fold:
Poznt groups C,, C,, Ci (See XLVII-LI).
(a) CI. This point group has no elements of symmetry, and
hence all compounds which belong to this point group are
symmetric.
(h) C,. This group has only a single plane of symmetry, o (which
as we showed earlier is equivalent to St).
(c) C;. This group has only a center of inversion (which is
equivalent to A).
Type II: Only one axis of rotation greater than one-fold:
Point groups C,, S,, C,,, Cph.
(a) C,. Molecules in these point groups (LII-LIV) have only a
smgle rotational axis and all such molecules are necessarily
dissymmetric and hence optically acl.ive. If 1,2-dichloraethane
were fixed in the eonform.ztion show" in IXb, it would
belong to this point, group.
,,,- LIIb , 8 8 8 LC"
(h) 8;. There are only a few known molecules thst belong t,o
these point. groups. A molecule with S4 har also C2 and one
with &has Ca, hot if these are the only symmetry elements
prosent, the point groups are & and SI. Of special interest is
the molecule ~hown in Figure 2, which belongs to & and
which war synthesized explicitly to test for optical activity;
the molecule is inactive because the & axis assures that its
mirror imqe is superimposahle.
(c) CnO. M~lecules belonging to ihese point groups have the
symmetry elements C, and p vertical planes (s,) int,ersecting
in the axis. We have already described in detail the molecules
H.0 and phenanthrene which belong to C*,. Molecules
in the point groups C,. are very common and same are shown
in LV-LXIII. A suecia1 case is a = rn which is the uoint
group of all linear molecules which do not have a center of
symqetry, e.g., heteronuclear diatomic molecules.
(d) Cgh. molecule^ belonging to such point groups have the
symmetry element C, and at right angles to this vertical axis
a mirror or horizontal plane m. Since C, X ah = Ssr they
also have S,. When p is an even number, the presence of r~
implies i. Some examples are shown in LXIV-LXVI.
li
ci
Type 111: One p-fold axis and, p two-fold axes perpendicular
to the principal axis: Point groups D,, D,,, and
DM.
The symbols for the symmetry point groups thst are used
here are the so-called Schoenflies symbols, named after their inventor.
The alternate notation, which is used to describe space
symmetry groups is the Hermann-Msuguin notation.
Volume 47, Number 5, May 1970 / 375

(a) Dll. If, to the principal axis C,, are added p two-fold axes ns
the only symmehry elements, the molecule belongs to the
point gronp D,. This is not a common point group and Da is
encountered more frequently than other D, point groups.
hIoleoules belonging to Dn normally have two equivalent
halves twisted with respect to each other at an angle greater
than zero and less than 90'. Thus, twisted ethylene and
twisted biphenyl belong t,o D2. The Newman projection of
twisted ethylene is shown in LXVII. The principal or vertical
axis is the carbon-carbon bond axis and the two two-fold
axes are in a. plme perpendicular to this axis and are perpendicular
to each other. They are shown as broken Lines.
iu splane horizontal to the major axis and me perpendicular to
each other. These two two-fold axes bisect t,he angles between
the planes of symmetry; they are difficult to see without
a model bot are shown as broken Lines in the bottom diagram
beside LXXIX which is a top view of the diagram
above it. Now, since it is the Cz axes which are the frame of
reference, the planes are considered to bisect the angles belween
the axes, and the Cr axes are the coordinate axes.
Thus, the plenes of symmetry become diagonal with reference
to the C2 axes and hence the designation Dad, where d stands
for diagonal planes of symmetry. Completely staggered
ethane, LXXX, belongs to Dad. Here the principal axis is
C3 and includes the carbon-carbon bond. At right angles to
this axis are three C1 axes shown as broken lines in the Newman
project,ion formula. The three plenes of symmetry,
ud, bisect the angles between mecessive twofold axes. Chair
cyclohexane, LXXXI, has vertical C. axis, 3C9 axes at right
angles which bisect opposite carbon-carbon bonds and 3ad
which pass through opposite carbon atoms. Ferrocene, in
the staggered conformation, LXXXIIa, belongs to point
group Dad, which point group is eh~racterised by one five-
(h) D,n. If eh i~ added to D, the molecules having C,, p two-fold
axes, and s. mirror plane perpendicoler to the principal axk
(C,) belong to DPh; these symmetry elements imply, in addition,
p vertical planes, PC,. Many molecules belong to this
point group, especially to Dnr. In Dm we find one two-fold
axis and a mirror plane perpendicular to this axis which also
contains two bwo-fold axes. In Dlh all three axes %re CI so no
one axis stands out ar the principal axis. The principal or
vertical axis is arbitrarily chosen as the one (in the molecular
plane if one exists) which passes through the largest number
of atoms. Examples of Dph molecules are shown in LXVIII-
LXXVI. Thus, benzene has s Cs axis (thevertical axis) and
in the symmetry plane whieh is the molecular plane and
which is at right angles to the C. axis, there are six two-fold
axes: three which pass 1,hraagh opposit,e.osrbon atom and
three which bisect opposite bonds. If p is even, i is present.
A special case is p = m, giving Dm+,, which is the point group
to which all linear molecules with a center of symmetry helone:
- homonuclear diatomies; carbon dioxide; acetylene;
etc.
(c) Dpd. Molecnles in these point groups, in addition to having
the axes defining D,, have p diagonal planes, od, which bisect
the angles between successive two-fold axes. Also, because
S, X C2 = SzD, they have 2p-fold alternating axes. In Du
there are normally two equivalent halves of a. molecule
t,wisted exactly 90° with respect to one another. The spiran
shown in LXXVII; allene, LXXVIII; and biphenyl twisted
exactly 90°, LXXIX, are examples. Actually, in these mole-
fold principal and vertical %xis, five two-fold axes at right
angles to it and five diagonal planes of symmetry. The view
looking down on the moleoule is shown in LXXXIIb. The
heavy lines show the five od; the five two-fold axes bisect the
angles between these planes. This view is particularly favorable
for recognizing the presence of Sio which must be present
because in Dja there must be &, = Sm
Type IV: More than one axis higher than two-fold:
Poznt groups T, and 0, and also T, T,, 0, I, I, and K,.
(a) Ta Tetrahedral molecules with identical atoms or groups of
atoms around 8. central atom belong to this point group.
Some common molecules are methane, carbon tetrachloride,
and nickel tetracarbonyl. The symmetry properties of point
group Td are more readily recognized when the molecule is
inscribed in acube such as shown for methaneinXLVIh. The
symmetry elements present are 4C3, 3Ca (which coincide with
3&) and 60. The axis through a corner and opposite face is a
CJ axis, and since there are four corners, there are 4C3. These
4Ca symmetry axes permit 8Ca operations, since in the case
of s. Ca axis the clockwise and counterclockwise rotations give
equivalent but not ldenticd orientations. The 3C2 symmetry
axes and the 3& axes coincident wlth them give rise to 3Cs
and 6S4 operations. The axes through the midpoints of
opposite edges are the C* axes, and since there are six edges,
there are 3CI. Each edge lies in one mirror plme, and since
there are six edges, there are 6.r. Organic molecules of the
high symmetry required for Td %re not common (adamantsne
is one) hut there are many examples of inorganic molecules
(especially of silicon and tetracoordinsted aluminum) with
this symmetry.
(b) Oh. All symmetrical octahedral compounds belong to this
point group, as do the cube and the octahedron. Octahedral
molecules are fairly common in inorganic chemistry; some
examples are shown in LXXXIV and LXXXV. The sym-
cdes it is usually easier to recognize the planes of symmetry
than the C, axes perpendicular to the vertical axes. Each of
the molecules in LXXVII-LXXIX has two planes of symmetry,
the one in the plane of the paper and a plane perpendicular
to the plane of the paper. The principal CV axis coincides
with the intersection of these two planes. The two C1
axes, which together wi1.h the principal CZ axis define Dn, are

metry elements which characterize this point group and
which are perhaps most readily recognized in the octahedron
itself are 6C4 (operations 3C4 and 3C:); 8Cs (operations 4C8
and 4C'a); 9C2 (of which three are coincident with C4); i;
and 90. The C, axes psss through the opposite corners of
the octahedron (six faces, hence 3C4 axes). The Ca axes pass
through pairs of opposite faces and the center of the octshedron
(eight faces, hence 4Ca axes). The C2 bisects pairs of
opposite edges (twelve edges, hence 6C2; the three C, axes
are also CI axes). There are two sets of mirror planes-one
set goes through the centers of four faces and two edges (one
set of six ea) and the other set consists of three wt, each of
which is defined by four edges, for a total of 9 ~ . All 9v transform
in the same way and belong to the same class. The
other point groups belonging to typeIV are of less importance
because only a few molecules belong to them. A boron compound
with the symmetry of an icosahedron (In) har been
reported. Such a geometry involves 20 faces that are equilateral
triangles, snd the point group In to which it belongs
has 120 symmetry operations.
(e) Kn. This point group characterizes the sphere, the geometry
possessed by dl free atoms. All possible symmetry elements
belong to this point group.
A Convenienf Procedure for Assigning the Poinf Group
of o Molecule
We shall now describe a quick method for classification
of molecules into point groups based on their symmetry
properties. This method will probably take care
of most of the molecules that need to be classified.
(1) If a molecule is linear, it belongs to point group C, if it does
not have s. horizontal plane of symmetry (e.g., HC1) and to
point group Dmh if it does (acetylene).
(2) If the molecule is tetrahedral it belongs to Ts (methane), and
if octahedral, On (chromium hexecarbonyl).
(3) If the molecule has only one axis which is two-fold or higher
(C, p > 2), look for p two-fold axes at right angles to the C,
8x1s. If these exist, the molecule belongs to one of the D
groups. If there is a horizontal plane, the molecule is Dnh
(PtCL, Dl*). If there is no horizontal plane hut p vertical
planes, the molecule belongs to D,s (staggered ethane). If
there me no planes, it is D,. In the special but common case
where the molecule has three C2 axes ail at right angles to
each other, the molecule belong to Dnn if it also har three
Figure 4.
44
Special Groups*
A Row chart forslarrifying molecular symmetryinto point groups.
planes of symmetry (ethylene), and to DW if the molecule has
two equal halves twisted 90' with respect to each other
(ailene). Three (2% axes and no planes is D*.
(4) If the molecule has only a rotational axis, C,, p > 2, the
molecule helongs to point group C, (trans-1,2-dichlorocyc10-
propane, CI). If in addition to 8 single C. axis there is one
horizontal plane (often the moleculm plane), the molecule
belongs to C-A (trans-dichloroethylene, CSA). If there is no
horizontal plane hut p vertical planes, the molecule belongs
to C,. (ammonia, Ca.).
(5) If there is no rotational axis (other than the infinite number
of CI axes) but the molecule has a plane of symmetry, it belongs
to C.; if it has only a center of symmetry, Ci; and if it
is truly asymmetric, it belongs to Ci.
A systemat.ic flow chart arrangement for classification
is shown in Figure 4.
Volume 47, Number 5, May 1970 / 377