# 1.3 Summary of Symmetry Operations, Symmetry Elements, and

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1.3 Summary of Symmetry Operations, Symmetry Elements, and Point Groups.

Rotation axis. A rotation by 360˚/n that brings a three-dimensional body into an equivalent configuration comprises a C^ n symmetry operation. If this operation is performed a second time, the product C^ nC^ n equals a rotation by 2(360˚/n), which may be written as C^ n2. If n is even, n/2 is integral and the rotation reduces to C^ n/2. In general, a C^ nm operation is reduced by dividing m and n by their least common divisor (e.g., C^ 96 = C^ 32). Continued rotation by

360˚/n generates the set of operations: C^ n, C^ n2, C^ n3, C^ n4, ... C^ nn
where C^ nn = rotation by a full 360˚ = E^ the identity. Therefore C^ nn+m = C^ nm.

Operations

resulting from a Cn symmetry axis comprise a group that is isomorphic to the cyclic group of

order n. If a molecule contains no other symmetry elements than Cn, this set constitutes the

symmetry point group for that molecule and the group specified is denoted Cn. When additional

symmetry elements are present, Cn forms a proper subgroup of the complete symmetry point

group. Molecules that possess only a Cn symmetry element are rare, an example being Co(NH2CH2CH2NH2)2Cl2+, which possesses a sole C2 symmetry element.

Cl

N N

Co

Cl N N

One special type of C^ n operation exists only for linear molecules (e.g., HCl). Rotation by any angle around the internuclear axis defines a symmetry operation. This element is called C• axis and an infinite number of operations C^ •f are associated with the element where f denotes rotation in decimal degrees.
We already saw that molecules may contain more than one rotation axis. In BF3, depicted below, a three-fold axis emerges from the plane of the paper intersecting the center of the

F

F

B

F currentpoint 192837465 triangular projection. Three C2 axes containing each B-F bond lie in the plane of the molecule perpendicular to the three-fold axis. The rotation axis of highest order (i.e., C3) is called the principal axis of rotation. When the principal Cn axis has n even, then it contains a C2 operation associated with this axis. Perpendicular C2 axes and their associated operations must be denoted with prime and double prime superscripts.

Reflection planes. Mirror planes or planes of reflection are symmetry elements whose associated operation, reflection in the plane, inverts the projection of an object normal to the mirror plane. That is, reflection in the xy plane carries out the transformation (x,y,z) ∅ (x,y,-z). Mirror planes are denoted by the symbol s and given the subscripts v, d, and h according to the following prescription. Planes of reflection that are perpendicular to a principal rotation axis of even or odd order are named sh (e.g., the plane containing the B and 3F atoms in BF3). Mirror planes that contain a principal rotation axis are called vertical planes and designated sv. For example, in BF3 there are three sv planes, each of which contains the boron atom, fluorine atom, and is perpendicular to the molecular plane. Notice that for an odd n-fold axis there will be n sv mirror planes which are similar. That is if we have one sv plane, then by operation of Cn a total of n times, we generate n equivalent mirror planes. The n sv operations comprise a conjugate class. For even n the sv planes fall into two classes of n/2 sv and n/2 sd planes, as illustrated earlier in the chapter.

Rotoreflection axes or improper rotation axes. This new operation is best thought of as the stepwise product of a rotation, and reflection in a plane perpendicular to the rotation axis. We need to introduce it since a product such as C^ 6^sh is not equal to either a normal rotation or reflection. To satisfy group closure a new symmetry operation must be introduced. An
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inversion operation is the simplest rotoreflection operation and is given the name S^2 or ^i . That S^2 = ^i is immediately apparent because ^i .(x,y,z) = (-x,-y,-z) and C^ 2(z)^s(xy).(x,y,z) = C^ 2(z).(x,y,-z) = (-x,-y,-z). Because C^ n and ^sh commute we have S^n = C^ n^sh for the general rotoreflection operation. Since ^shn = E^ for even m and for m odd ^shm = ^sh, an Sn axis requires simultaneous independent existence of a Cn/2 rotation axis and an inversion center when n is even. For example, an S6 axis generates the operations for the point group S6. They are shown below.
S^6, S^62 = C^ 32^sh2 = C^ 32, S^63 = S^2 = C^ 2^sh = î, S^64 = C^ 32, S^65 = C^ 65^sh, S^66 = E^. An odd order rotoreflection axis requires independent existence of a Cn axis and a sh symmetry element. An S7 axis generates the following operations.
S^7, S^72 = C^ 72, S^73, S^74 = C^ 74, S^75, S^76 = C^ 76, S^77 = C^ 77^sh7 = ^sh, S^78 = C^ 78 = C^ 7, S^79 = C^ 72^sh, S^710 = C^ 73, S^711 = C^ 74^sh, S^712 = C^ 75, S^713 = C^ 76^sh, S^714 = E^ The S7 axis generates a cyclic group of order fourteen and requires the independent existence of C7 and sh. This group is denoted C7h and not S7.
Conjugate Symmetry Elements and Operations. In the symmetry group of BF3 the sequence of operations C^ 3´-1^svC^ 3 means to rotate by 120 ˚ perform ^sv, then rotate back 120˚. This sequence of operations is equivalent to reflection in a plane rotated 120˚ from the initial one, as shown below. The similarity transformation (C^ 32)-1^svC^ 32 generates an operation, which is the third vertical mirror plane. The three equivalent sv symmetry operations belong to the same conjugate class. Symmetry equivalent symmetry operations (i.e., operations belonging to symmetry elements that are interchanged by symmetry operations of the group) belong to the same conjugate class. In BF3 all the operations associated with the equivalent C2 elements belong to the same class. This may be verified by the similarity transformations C^ 3-1C^ 2C^ 3 and (C^ 32)-1 C^ 2C^ 32 operating on any of the three C2 axis.
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sn

1

^

C3

sn 2 ^sv

sn

2

^C -1

3

3

2

1

3

3

1

3

sv

1

generated

conjugate sv

from a

conjugate sv

1

2

3

2

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molecular plane and four perpendicular C2 axis as shown. Clearly the pair of axis labeled C2´

C2'

C2"

C2'

Cl

Cl

Pt

Cl

Cl

C2" y

x currentpoint 192837465 are equivalent, as are the pair of axis C2" since the four-fold rotation interchanges elements
among the pairs; however, the C2´ and C2" axis are not equivalent. There is no operation that
will interchange these elements. Primes designate C2 axes perpendicular to an even principal axis. The C^ 42 operation associated with the four-fold axis is called C^ 2. You might be surprised
to find that a C4 axis and one perpendicular C2´ axis generate the remaining three "primed" twofold axes. Action of C^ 4 on one C2´ axis generates the other C2´ axis. With the application of C^ 4 or C^ 2 no new axes are generated; however, consider the product of operators C^ 4C^ 2´ acting on
the point (x,y,z). Take one C2´ axis to lie along the x axis and choose C4 coincident with the z

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axis. The operation C^ 2´ on the general point (x,y,z) produces a new point at (x,-y,-z). The C^ 4 operation does not change z and rotates the molecule in the x,y plane. So C^ 4 (x,y,z) ∅
y (-y,x,z) • • C^ 4 x x
(-y,x,z) and the product C^ 4C^ 2´(x,y,z) = C^ 4(x,-y,-z) = (y,x,-z) is equivalent to rotation around the C2" axis bisec1[]gt1dir39ng64[1e0t0nh93dIe8691x107y2007q006u2a1079d72r24a600n04t030o69fD10tSh420te097[c61o1o28Ir0d3Ai]9n2ra003t0e7cs72hy1s0et8me6m0d0bDi8ce0Stctab3uDe9segb0iC^0n26"SP0(x8,y0,z6) ≠A(ry,3x2,-2z)0. 6660 2380 6660
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C2" (x,y,z) •x
This proves that the combination of a C^ 2´ operation with C^ 4 generates the C^ 2" operation, and that the two C2´ belong to the same conjugate class. You may prove that the two C^ 2" comprise a second class.
There are likewise two classes of vertical reflection planes. The two planes containing the C4 axis and one C2´ axis are called sv. Those mirror planes containing the C2" axis are denoted sd. This illustrates an interesting consequence of a vertical reflection plane. Consider the effect of ^sv or ^sd on rotation around the principal axis. The effect of ^sv is to change the sense of
sv

Cnm

Cn-mcurrentpoint 192837465

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rotation from counterclockwise to clockwise. This means that rotation by 360˚/n and -360˚/n are equivalent. In other words a rotation C^ nm and its inverse C^ n-m belong to the same conjugate class when a mirror plane contains the rotation axis. For example, the operations of a C7 axis fall into the classes (C7, C76); (C72, C75); (C73, C74) and for a C6 axis the classes are (C6, C65); (C3, C32); (C2) when a sv or sd mirror plane also exists.
Molecular symmetry groups. When you search for molecular symmetry elements, look for rotations, reflections and rotoreflection elements, which interchange equivalent atoms in a molecule and leave others untouched. The short hand notation for the various point groups uses abbreviations, which often specify the key symmetry elements present. The complete set of symmetry operations of a molecule defines a group, since they satisfy all the necessary mathematical conditions. When confronted with a molecule whose symmetry group must be determined, the following approach can be used.
Special symmetry groups usually are self-evident. The group D•h is the point group of homonuclear diatomic molecules and other linear molecules that possess a sh plane (e.g., CO2). For linear molecules an infinite number of sv planes always exist. The symbol D in a group name always means that there are n-two-fold axes perpendicular to the n-fold principal axis. The presence of sh and sv planes in a molecule always requires the presence of n^C´2 axes, since one can show that ^sh^sv = C^ ´2; however, the converse is not true. Therefore, there are an infinite number of C2 axes perpendicular to the C• axis of the molecule. The C• axis and sh generate an infinite number of S•f operations; one is S^2 = î.
If a linear molecule does not contain a sh plane, e.g., HCl or OCS, then it belongs to the C•v point group. This point group contains a C• axis and an infinite number of sv planes. Clearly C•v is a proper subgroup of D•h.
A free atom belongs to the symmetry group Kh, which is the group of symmetry operations of a sphere. We shall not be concerned with this point group; however, a
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consequence of spherical symmetry is the requirement that the angular wavefunctions of atoms behave like spherical harmonics.

There are seven special symmetry groups, which contain multiple high order (Cn, n > 3)

axes. These are the only possibilities for 3-dimensional objects. The groups may be derived

from

octahedron

tetrahedron

cube

Oh 3 C4 dodecahedron

Td 4 C3

Oh 3 C4 icosahedron

Ih 12 C 5 axes

Ih 12 C 5 axes

Figure 1.16. The Five Platonic Solids and Their Point Groups,

with The High Order C

n Axes Listed.

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the symmetry operations of the five Platonic solids (octahedron, tetrahedron, cube, icosahedron, dodecahedron). Platonic solids are solids, shown in figure 1.16, whose edges and verticies are all equivalent and whose faces are all some regular polygon. The symmetry operations of the tetrahedron comprise the group Td. The cube and octahedron possess equivalent elements and their operations define the Oh group.
The pentagonal dodecahedron and icosahedron likewise define Ih. We will restrict our discussion to the Oh group and its subgroups, O, Td, Th, and T, since Ih and its subgroup I are encountered less frequently. The collection of groups {O, Td, Th, and T} are often referred to as cubic groups. The symmetry elements belonging to an octahedron are: three C4 axes that pass through opposite verticies of the octahedron; four C3 axes that pass through opposite triangular faces of the octahedron; six C2 axes that bisect opposite edges of the octaedron; four S6 axes coincident with the C3 axes; three S4 axes coincident with the C4 axes; an inversion center or S2 axis coincident with C4; three sh which are perpendicular to one C4 axis and contain two others. Therefore, a single plane defined as sh, also can be thought of as sv with respect to the other C4 axis. Similarly, there are six planes called sd, which are also sv planes containing a C3 axis. Subgroups of Oh are generated by removing the following symmetry elements:

O= Td = Th = T=

lacks i, S4, S6, sh and sd and is called the pure rotation subgroup of Oh. this group lacks C4, i and sh and is the group of tetrahedral molecules, e.g., CH4. this uncommon group is derived from Td by removing S4 and sd elements. the pure rotation subgroup of Td contains only C3 and C2 axes.

If a molecule belongs to none of these special groups, then focus on the highest order rotation axis. Those rare molecules that possess only one n-fold rotation axis and no other symmetry elements belong to the Cn symmetry group, which is cyclic. If only a rotation axis exists and no mirror planes or other elements are obvious, a last check should be made for an Sn axis of order higher than the obvious rotation axis. When a higher order Sn axis can be found,
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then the Sn groups (n = even) are obtained. Naturally n must be even since S^nn = ^sh when n is odd and both Cn and s h exist independently. These groups (n = odd) are conventionally designated Cnh. Also, when n = 2, the S2 group (remember S2 = i) is conventionally called Ci. If we add to a Cn group nsv (n odd) or n/2sv and n/2sd (n even) mirror planes, but no other rotation axes, then the Cnv groups are generated. If to the elements of Cn we add a horizontal mirror plane, then we generate the Cnh groups. The product of C^ n and ^sh also generate an Sn symmetry element and its associated operations.
When additional n-two-fold axes perpendicular to Cn are present, but no mirror planes, then the Dn groups are generated. Addition of a sh plane to the rotation symmetry elements of a Dn group generates the Dnh groups. Similar to the Cnh groups an Sn symmetry axis is also produced (if n is even an inversion center will be present). The product C^ 2^sh (where C2 is associated with n ^ C2 axes in the Dnh group) generates vertical mirror planes so that the Dnh group also contains nsv planes (n odd) or n/2sv and n/2sd planes (n even). If in addition to the elements of a Dn group an S2n axis is present, then the Dnd groups are obtained. The product S^nC^ 2 (C2 belongs to a C2 axis perpendicular to the n-fold axis) yields the operation ^sd so that an additional n dihedral mirror planes are required.
If a molecule contains as the only symmetry element a mirror plane, then the group is called Cs. Finally, if no symmetry elements, other than E, exist, then the molecular symmetry group is the trivial one called C1. Table I.1 summarizes the groups we just discussed and their associated elements. It provides a useful reminder of all the necessary elements when assigning symmetry point groups.
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Table 1.1 Common Point Groups and Their Symmetry Elements

Point Group

Symmetry Elements Present

C1

Cs

Ci

Cn

Dn

n = odd

Dn

n = even

Cnv

n = odd

Cnv

n = even

Cnh

n = odd

Cnh

n = even

Dnh

n = odd

Dnh

n = even

Dnd

n = odd

Dnd

n = even

Sn

n = even only

T Th Td O Oh I Ih Kh

E E, sh E, i E, Cn E, Cn, n^C2 E, Cn, n/2^C2´, n/2^C2´´ E, Cn, nsv E, Cn, n/2sv, n/2sd E, Cn, sh, Sn E, Cn, sh, Sn, i E, Cn, sh, n^C2, Sn, nsv E, Cn, sh, n/2^C2´, n/2^C2´´, Sn, n/2sv, n/2sd, i E, Cn, n^C2, i, S2n, nsd E, Cn, nC2´, S2n, nsd E, Sn, Cn/2 and i if n/2 odd E, 4C3, 3C2 E, 4C3, 3C2, 4S2n, i, 3sh E, 4C3, 3C2, 3S4, 6sd E, 3C4, 4C3, 6C2 E, 3C4, 4C3, 6C2, 4S6, 3S4, i, 3sh, 6sd E, 6C5, 10C3, 15C2 E, 6C5, 10C3, 15C2, i, 6S10, 10S6, 15s E, infinite numbers of all symmetry elements

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RotationSymmetry ElementsMoleculeOperationsAxis