Group Theory & SpectroscopyMolecular Symmetry:

Symmetry elements (15.1)Point Groups (15.2)

Applications of Group Theory:Character Tables (15.4)Translational, Rotational and Vibrational Character

Introduction: Matrix Representation of Group TheoryDirect ProductsIrreducible Representations

Normal Mode Analysis

Molecular SymmetryRotational, vibrational, electronic and NMR spectroscopy,as well as molecular orbital theory, make extensive use ofmolecular symmetry to simplify calculations and aid ininterpretation of relatively complex spectra.

Common question: “Why is symmetry regarded as soimportant when most molecules considered in theirentirety have no symmetry at all?”

Many small molecules have very high symmetry, but manylarger molecules do not possess any overall symmetry. Nonetheless, the local symmetry or pseudo-symmetrywithin certain molecules is very important inunderstanding the spectroscopy of the molecule, which inturns yields information about the structure and function ofthe molecule.

NH2COCH2CH2

H H

H CH2OHOH

H

CH3

CH3

H3C CH

CH2

NH

CO

CH2

CH2

NC

H3C

H

HCH3NH2COCH2

H

H3C

NH2COCH2

H3CCH3

CH2CONH2

CH3

H

H

CH2CH2CONH2

CH3

CH3

CH2CH2CONH2

NN

NN

N

OO

OPO

O

N

Co+

NNC

NN

N

Co+

For example, in consideringelectronic or 59Co NMRspectroscopy of vitamin B12,the pyramidal symmetry atthe Co atom is paramount indetermining spectralcharacteristics.

Group TheoryIntuitively, it is easy to identify objects/entities that havesymmetry. Group theory is a mathematical treatment thatprovides us with a formal means of describing thesymmetry of objects (such as molecules).

C CH

H

H

HC C

F

H

H

HC C

F

F

H

H

C CH

F

H

FC C

F

H

H

F

(a) (b) (c)

(e)(d)

Group theory was formally developed as a mathematicaltheory in the 19th century, and was not applied tomolecules until the 1920's and 1930's.Looking at the molecules below, we see how ethylene (a)is like a rectangle and 1,2-trans-difluoroethylene (d) is likea parallelogram. The 1,1-difluoroethylene (b) and 1,2-cis-difluoroethylene (e) have symmetries similar to asymmetric trapezoid. Group theory allows us to formallyclassify the symmetry of objects and molecules.

Symmetry Operations & ElementsAn operator is symbol that tells you to do something towhatever follows it (e.g., d/dx says differentiate w.r.t. x)A symmetry operation is an operation which moves themolecule into a new configuration which is indistinguish-able from its original configuration.A symmetry element is a point, line or plane with respectto which a symmetry operation is performed.

BF2 F3

F1

BF1 F2

F3

For example, consider the planar BF3 molecule. There is athree-fold rotation axis (symmetry element) about whichwe can rotate the molecule by 120o or 240o (symmetryoperation) and generate the original molecule.

rotation by 2B/3“3-fold” rotation

Symmetry elements: Symmetry operations:E Identity Identity (do nothing)F Plane of symmetry Reflection through a plane

(mirror plane)Cn Proper axis of Rotation about an axis by

rotation 2B/n radiansi Centre of inversion Inversion of all atoms

(Centre of symmetry) through centreSn Improper rotation Rotation about an axis

followed by reflectionthrough plane z to this axis

n-Fold Rotational Axis, CnIf a molecule has an n-fold rotational axis, rotation of themolecule by 2B/n radians (n = 1, 2, 3, ..., 4) generates anorientation of the molecule which is indistinguishable fromthe original orientation.The Schönflies symbol for this rotation is Cn (Hermann-Mauguin notation is favoured for crystallography, we shalluse the Schönflies symbols for spectroscopy and MO’s)

HO

H

O

X eF F

FF

C

H

H HH

H C N

C2 C3 C4

C6

C4

Consider again the planar BF3 molecule:

BF2 F3

F1

BF1 F2

F3

rotation by 2B/3“3-fold” rotation

BF3 F1

F2

rotation by 4B/3“3-fold” rotation

The highest fold rotation axis is the principal axis (forexample, benzene has a bunch of C2 axes, but its principalaxis is the C6 axis).

Plane of Symmetry, FReflection of all of the atoms in a molecule through aplane of symmetry or mirror plane produces aconfiguration indistinguishable from the initial molecule.

BF F

F

Fv

Fv

Fv

Fh

Fv

Fv Fv

FvFv

C2

C2C2

F

F

F

P tC l C l

C l C l

C3, S3

C2

C2O, Fd

C2N, Fv

C2O, Fd

C2N, Fv C4, S4

Fv is a reflection plane vertical w.r.t. principal axis,Fh is a reflection plane horizontal w.r.t. principal axis,Fd is are reflection planes which bisect angles betweendihedral axes, which are C2 axes at equal angles w.r.t. oneanother and perpendicular to the principal axis.

Fv

Fv

Fv

C3, S3

C2C2

C2

Fh

Symmetry Elements of ObjectsReflection planes in arectangular box

FhFv

Rotational symmetryelements in a cube

Rotational symmetryelements in a box

Note that the convention is that the principal axis is the zaxis, and x and y axes lie along other symmetry elements.

Reflection planes w.r.t.the principal axis

Also, Fv is normally positioned along bonds, and Fd bisectsthe angle between the bonds

Identity and Inversion, E and iThe identity operator, E, does nothing to an object and isprimarily used for mathematical completion. Objectswhich can be only operated upon by the identity operator(i.e., none of the other operations can be done withoutchanging the configuration of the object) are said to beasymmetric.

The inversion operator takes the coordinates of an atomor nucleus through the centre of symmetry of themolecule and produces a configuration identical, butinverted, to the original one. In other words,

atom at x, y, z atom at -x, -y, -zinversion operation

C

CC

C

H

H

H

H

H

H

Ci

F

S

F

F F

FF

A molecule with an centre of inversion is known as acentrosymmetric molecule.

Ci

C4 Fh

Improper Rotation, SnThe so-called improper rotation, Sn, is better referred toas the rotation-reflection operator. The operator Snmeans to rotate by 2B/n radians about the principal axis,followed by reflection through a plane perpendicular to theaxis and through the centre of the molecule. The result is aconfiguration identical to the original one.

In the example below, a tetrahedral molecule is rotatedwith a C4 operation, followed by a Fh reflection:

The plane across which the reflection occurs may or maynot be a symmetry element.If there is a Fh plane perpendicular to the highest fold Cnaxis, then this axis must be an Sn axis also (e.g., BF3). It issaid that Cn and Fh generate Sn:

Fh × Cn = Sn:That is, if we carry out Cn followed by Fh, this is the sameas perform an Sn operation. It follows that F = S1 and i =4S2, so S1 and S2 are never used, and F and i are their ownsymmetry elements.

Generation of ElementsHere is an example of how C2 and Fv in difluoromethanegenerate FvN (note that the prime symbols can be used todifferentiate different mirror axes)

H1 H2

F2F1

H2 H1

F1F2

H2 H1

F2F1C2 Fv

H1 H2

F2F1

H2 H1

F2F1

FvN

Reflection through FvN (F1 and F2 are in the mirror plane):

is the same as Fv × C2 = FvN:

similarly: FvN × C2 = Fv and FvN × Fv = C2

Elements Cn and Sn can also be used to generate otherelements by raising them to powers 1, 2, 3, ..., (n-1). Forexample, a C3

2 operation, which is clockwise, is the sameas 2 consecutive clockwise C3 rotations by 2B/3

C32 = C3 × C3

or an anticlockwise rotation by -2B/3 can be written asC3

-1 / C32

or generally:Cn

n-1 / Cn-1

Point GroupsIn order to classify molecules according to their symmetry,the symmetry elements for each molecule are collected ingroups. The nature of the group to which a moleculebelongs is determined by the symmetry elements itpossesses.All of the elements of symmetry which any molecule mayhave constitute a point group, which is so called, becausewhen all of the operations of the group are carried out, atleast one point is unaffected.

A set of operations possessed by a molecule (and thereforea point group) must satisfy 4 requirements:

ClosureCombination of any 2 operations is same as anotheroperation in the group (i.e., they generate another operator)

IdentityThere must be an identity element. e.g. EA = A

AssociativityIf C(BA) = X then (CB)A = X. May perform multiplicationsin any grouping, but the order of the operators must remainunchanged (i.e., it is possible that AB … BA)

ReciprocalityEvery element has an inverse., i.e. AA-1 = E. Some are self-inverse, e.g., ii=E, FF=E

Ci 1̄

Cs mC1 1 C2 2 C3 3 C4 4 C6 6

C2v 2mm C3v 3m C4v 4mm C6v 6mmC2h 2/m C3h 6̄ C4h 4/m C6h 6/m

D2 222 D3 32 D4 422 D6 622

D2h mmm D3h 6̄2m D4h 4/mmm D6h 6/mmmD2d 4̄2m D3d 3̄m S4 4̄/m S6 3̄

T 23 Td 4̄3m Th m3O 432 Oh m3m

Names of Point GroupsPoint groups are denoted by Schönflies symbols(individual molecules) or by the Hermann-Mauguinsystem (crystal symmetry).

In the Hermann-Mauguin system, the number n denotes thepresence of an n-fold axis and m denotes a mirror plane. The diagonal line / indicates a mirror plane perpendicularto the symmetry axis. It is important to distinguishsymmetry elements of the same type but for differentclasses, such as for 4/mmm, in which there are three classesof mirror plane: Fv, Fh and Fd. A bar over a numberindicates that the element is combined with an inversion. The only groups listed above are the so-calledcrystallographic point groups.

The Non-Rotational GroupsPoint groups: C1, Cs and Ci

These point groups are quite common, and contain theidentity operator E or E plus one other element.

C1: E asymmetricCs: E, F single mirror plane onlyCi: E, i inversion centre only

C1 is a trivial point group indicating that the molecule hasno symmetry: no reflections, rotations, inversions orcombination thereof.

Cs has a plane of reflection, F, as its only non-trivialsymmetry element.

Ci has an inversion centre, i, as its only non-trivialsymmetry element.

NCl H

F

F

ClH F

Cl H

C O

H

ClS

O FF

Ci

Cs

C1

Cs

Single-Axis GroupsPoint Groups: Cn, Cnv, Cnh, C4v, S2n

Cn E, Cn C2, C3, etc., commonCnv E, Cn, nFv C2v, C3v very common Cnh E, Cn, Fh, Sn uncommonC4v E, C4, 4Fv Common in linear systemsS2n E, S2n S4: very rare

All of these groups have a single principal axis of rotation. The order of the rotation is given by n, and the Cn rotationis cyclic: to completely rotate full circle, perform Cn

n:

O O

H HB

O O

OH

OH

H C NC C

F

H

H

F

C2: H2O2 C3: Boric acid

C3C2

C2v: Water

C2, Fv, FvN

C2h: trans-CHCl=CHCl C4: HCN C4: 1,3,5,7-tetrafluoroCOT

C2, Fh, S2 C4, 4Fv

F

F

F

F

S4

Dihedral GroupsPoint groups: Dn, Dnd, Dnh, D4h

All of these possess a principal rotation axis of order nwith nC2 axes perpendicular to the principal axis.

Dn E, Cn, nC2Dnd E, Cn, nC2, S2nDnh E, Cn, nC2, Fh, nFv D2h, D3h, D4h, etc.D4h E, C4, 4Fv, Fh, i linear, symmetric

Co

C

H

HH

H

H

CH

C

C

H

H

BF F

F

C C C

H

H

HH

D4h: HCCHD3d: ethane, staggeredD3: Co(en)33+

D2d: alleneD6h: benzeneD3h: BF3

C3

S6, C3C4

Fh

C3

Fh Fh

C6

S4, C2

Special Symmetric GroupsPoint groups: Td, Oh, Ih, Kh

These highly symmetric groups are special cases thatcontain many different symmetry elements, and alwaysfeature multiple C2 axes.Td E, 4C3, 3C2, S4, 6Fd tetrahedralOh E, 3C4, 4C3, 6C2, octahedral

S6, S4, 3Fh, 6Fd, iIh E, 6C5, 10C3, 15C2, icosahedral

6S10, 10S6,15F, iKh All spherical

C

H

H HH

F

S

F

F F

FF

Oh: SF6Td: CH4

Ih: icosahedron (B12H122- cluster) Ih: buckminsterfullerene

The Kh point group is not applied to molecules - onlyatoms and spherical objects (sometimes called R3)

Determining the Point Group

C2v E C2 σv(xz) σNv(xz)A1 1 1 1 1 Tz αxx, αyy, αzz

A2 1 1 -1 -1 Rz αxy

B1 1 -1 1 -1 Tx, Ry αxz

B2 1 -1 -1 1 Ty, Rx αyz

Character TablesPoint groups are divided into two sets: non-degenerate anddegenerate. A degenerate point group has a Cn axis with n> 2, or an S4 axis. A molecule belonging to such a pointgroup may have degenerate properties: i.e., differentvibrational or electronic states with identical energies.

We know how to classify a molecule in terms of itssymmetry using group theory. However, molecules mayhave properties, such as electronic or vibrational wavefunctions, which do not preserve all of the symmetryelements attributed to the molecule.These properties can be easily classified using charactertables. The classic examples to be considered are the:C2v, C3v, and C4v character tables. (Classic case: H2O)point group label symmetry operations

symmetry species

character, P

symmetry species oftranslation, rotation andpolarizability

irreduciblerepresentation of B2

or generating elements

Pieces of the C2v Character TableLet’s say that H2O has a wavefunction that may or may notpreserve an element of symmetry.

Preserves the element: Does not preserve element:

RvFv6 (%1)Rv Rv

Fv6 (&1)Rv

Rv is symmetric to Fv Rv is asymmetric to Fv+1 is the character of -1 is the character ofRv w.r.t. Fv, P = +1 Rv w.r.t. Fv, P = -1

Any two of the E, C2, Fv or FNv elements can be regarded asgenerating elements, and the order of the point group is 4.

In each column, there are four possible combinations ofcharacters with respect to these generating elements.

Each of the four rows of the point group contain anirreducible representation of the group, each of which isrepresented by a symmetry species: A1, A2, B1, B2A1 is said to be totally symmetric (all characters are 1)A2, B1, B2 are non-totally symmetric

Conventions of labels:A indicates symmetry w.r.t. C2B indicates antisymmetry w.r.t. C21 indicates symmetry w.r.t. Fv2 indicates antisymmetry w.r.t. Fv