710 Lecture 04

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Lecture 4

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Point group Symmetry operations

Characters

+1 symmetric behavior

-1 antisymmetricMlliken symbols

Each row is an irreducible representation

Character Table

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x, y, zSymmetry of translations (p orbitals)

Rx, Ry, Rz: rotations

Classes of operations

dxy, dxz, dyz, as xy, xz, yz

dx2- y2 behaves as x2y2

dz2 behaves as 2z2 - (x2 + y2)

px,py,pz behave as x, y, z

s behaves as x2 + y2 + z2

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Symmetry of Atomic Orbitals

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Effect of the 4 operations in the point group C2von a translation in the x direction

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Naming of Irreducible representations

One dimensional (non degenerate) representations are designated A or B.

Two-dimensional (doubly degenerate) are designated E.

Three-dimensional (triply degenerate) are designated T.

Any 1-D representation symmetric with respect to Cn is designatedA;antisymmtric ones are designated B

Subscripts 1 or 2 (applied to A or B refer) to symmetric and antisymmetricrepresentations with respect to C2Cn or (if no C2) to svrespectively

Superscripts and indicate symmetric and antisymmetric operations withrespect to sh, respectively

In groups having a center of inversion, subscripts g(gerade) and u(ungerade) indicate symmetric and antisymmetric representations withrespect to i

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Character Tables

Irreducible representations are the generalized analogues of s or p symmetry indiatomic molecules.

Characters in rows designated A, B,..., and in columns other than E indicate thebehavior of an orbital or group of orbitals under the corresponding operations (+1 =orbital does not change; -1 = orbital changes sign; anything else = more complexchange)

Characters in the column of operation Eindicate the degeneracy of orbitals

Symmetry classes are represented by CAPITAL LETTERS (A, B, E, T,...) whereasorbitals are represented in lowercase (a, b, e, t,...)

The identity of orbitals which a row represents is found at the extreme right of the row

Pairs in brackets refer to groups of degenerate orbitals and, in those cases, the

characters refer to the properties of the set

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Definition of a Group

A group is a set, G, together with a binary operation :such that the product of any two members of the group isa member of the group, usually denoted by a*b, suchthat the following properties are satisfied :

(Associativity) (a*b)*c= a*(b*c) for all a, b, cbelonging to G.

(Identity) There exists e belonging to G, such that e*g= g= g*efor all gbelonging to G.

(Inverse) For each gbelonging to G, there exists the inverse of

g,g

-1

, such that g

-1

*g= g*g

-1

= e. Ifcommutativityis satisfied, i.e. a*b = b*a for all a, b

belonging to G, then G is called an abelian group.

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Examples

The set of integers Z, is an abelian group

under addition.

What is the element e, identity, such that

a*e = a?

What is the inverse of the a element?

0

-a

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As applied to our symmetry

operators.

For the C3v point group

What is the inverse of each operator? A * A-1 = E

E C3(120) C3(240) sv (1) sv (2) sv (3)

E C3(240) C3(120) sv (1) sv (2) sv (3)

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Examine the matrix represetation of the C2v point group

1 0 0

0 1 00 0 1

xyz

=

xyz

-1 0 0

0 -1 00 0 1

xyz

=

-x-y

z

C2

1 0 0

0 -1 0

0 0 1

xy

z

=

x-y

z

sv(xz)

E

- xy

z

sv(yz)

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Multiplying two matrices (a reminder)

a11 a12a21 a22a31 a32

b11 b12 b13b21 b22 b23

=

c11 c12 c13c21 c22 c23c31 c32 c33

c11 = a11b11 + a12b21c12 = a11b12 + a12b22c13 = a11b13 + a12b23



1 0 0

0 1 00 0 1

Most of the transformation

matrices we use have the

form

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-1 0 0

0 -1 00 0 1

1 0 0

0 1 00 0 1

1 0 0

0 -1 00 0 1

C2E sv(xz) sv(yz)

What is the inverse of C2? C2

What is the inverse ofsv? sv

-1 0 0

0 -1 00 0 1

-1 0 0

0 -1 00 0 1

=

1 0 0

0 1 00 0 1

1 0 0

0 -1 00 0 1

1 0 0

0 -1 00 0 1

=

1 0 0

0 1 00 0 1

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-1 0 0

0 -1 00 0 1

1 0 0

0 1 00 0 1

1 0 0

0 -1 00 0 1

C2E sv(xz) sv(yz)

What of the products of operations?

E * C2 = ?C2

1 0 0

0 1 00 0 1

-1 0 0

0 -1 00 0 1

=

-1 0 0

0 -1 00 0 1

sv * C2 = ? sv

1 0 0

0 -1 00 0 1

-1 0 0

0 -1 00 0 1

=

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ClassesTwo members, c

1and c

2, of a group belong to the same class if there is a member,

g, of the group such that

g*c1*g-1 = c2

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Properties of Characters of Irreducible Representations in

Point Groups

Total number ofsymmetry operations in the group is called the order of

the group (h). For C3v, for example, it is 6.

1 + 2 + 3 = 6

Symmetry operations are arranged in classes. Operations in a class are

grouped together as they have identical characters. Elements in a class are

related.

This column represents three

symmetry operations having identical

characters.

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Properties of Characters of Irreducible

Representations in Point Groups - 2

The number of irreducible reps equals the number of

classes. The character table issquare.

3 by 3

The sum of the squares of the dimensions of the each irreducible rep equals theorder of the group, h.

1 + 2 + 3 = 6

1

1

22

6

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For any irreducible rep the squares of the characters summed over the symmetry

operations equals the order of the group, h.

A1: 12 + (12 + 12 ) + = 6

A2: 12 + (12 + 12 ) + ((-1)2 + (-1)2 + (-1)2 ) = 6

E: 22 + (-1)2 + (-1)2 = 6

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Irreducible reps are orthogonal. The sum of the products of the characters for

each symmetry operation is zero.

For A1 and E:

1 * 2 + (1 *(-1) + 1 *(-1)) + (1 * 0 + 1 * 0 + 1 * 0) = 0

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Each group has a totally symmetric irreducible rep having all characters equal

to 1

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Reduction of a Reducible

RepresentationIrreducible reps may be regarded as orthogonal vectors. The magnitude of the

vector is h-1/2

Any representation may be regarded as a vector which is a linear combination

of the irreducible representations.

Reducible Rep = S (ai * IrreducibleRepi)

The Irreducible reps are orthogonal. Hence

S(character of Reducible Rep)(character of Irreducible Repi) = ai * h

Or

ai =S(character of Reducible Rep)(character of Irreducible Repi) / h

Sym ops

Sym ops

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These are block-diagonalized matrices

(x, y, z coordinates are independent of each other)

1 0 0

0 -1 00 0 1

1 0 0

0 1 00 0 1

-1 0 0

0 -1 00 0 1

C2 sv(xz)E

-1 0 0

0 1 00 0 1

sv'(yz)

C2 sv(xz) sv'(yz)

x

y

z

E

1

1

1

3

-1

-1

1

-1

1

-1

1

1

-1

1

1

1

Irreducible

representations

Reducible Rep

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Point group Symmetry operations

Characters+1 symmetric behavior

-1 antisymmetricMlliken symbols

Each row is an irreducible representation

C2vCharacter Table to be used for water

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Lets use character tables!

Symmetry and molecular vibrations

# of atoms degrees offreedom

Translationalmodes

Rotationalmodes

Vibrationalmodes

N (linear) 3 x 2 3 2 3N-5 = 1

Example

3 (HCN)

9 3 2 4

N (non-

linear)

3N 3 3 3N-6

Example

3 (H2O)

9 3 3 3

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Symmetry and molecular vibrations

A molecular vibration is IR activeonly if it results in a change in the dipole moment of the molecule

A molecular vibration is Raman active

only if it results in a change in the polarizability of the molecule

In group theory terms:

A vibrational motion is IR active if it corresponds to an irreducible representationwith the same symmetry as anx, y, zcoordinate (or function)

and it is Raman active if the symmetry is the same as

x2, y2, z2, or one of the rotational functions Rx, Ry, Rz

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How many vibrational modes belong to each irreducible representation?

You need the molecular geometry (point group) and the character table

Use the translation vectors of the atoms as the basis of a reducible

representation.

Since you only need the trace recognize that only the vectors that areeither unchanged or have become the negatives of themselves by a

symmetry operation contribute to the character.

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Apply each symmetry operation in that point group to the molecule

and determine how many atomsare not movedby the symmetry operation.

Multiply that number by the character contribution of that operation:

E= 3

s = 1

C2 = -1

i = -3

C3 = 0

That will give you the reducible representation

A shorter method can be devised. Recognize that a vector is unchanged or

becomes the negative of itself if the atom does not move.

A reflection will leave two vectors unchanged and multiply the other by -1

contributing +1.

For a rotation leaving the position of an atom unchanged will invert the direction of

two vectors, leaving the third unchanged.

Etc.

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3x3

9

1x-1

-1

3x1

3

1x1

1

Finding the reducible representation

(# atoms not moving x char. contrib.)

E= 3

s = 1

C2 = -1

i = -3

C3 = 0

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Now separate the reducible representation into irreducible ones

to see how many there are of each type

A1 = 1/4 (1x9x1 + 1x(-1)x1 + 1x3x1 + 1x1x1) = 3

A2 = 1/4 (1x9x1 + 1x(-1)x1 + 1x3x(-1) + 1x1x(-1)) = 1

9 -1 3 1

S

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Symmetry of molecular movements of water

Vibrational modes

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IR activeRaman active

Which of these vibrations having A1 and B1 symmetry are IR or Raman active?

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ML C

CLO

O

C2

E

1

2

ML C

CLO

O

1

2

ML C

CLO

O1

2

ML C

CLO

O

1

2

ML C

CLO

O1

2

C2 sv(xz) sv(yz)

Often you analyze selected vibrational modes

n(CO)

Find: # vectors remaining unchanged after operation.

2 x 1

2

0 x 1

0

2 x 1

2

0 x 1

0

Example: C-O stretch in C2v complex.

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A1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1

A2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x-1) = 0

2 0 2 0

B1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1

B2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x1) = 0

ML C

CLO

O

A1 is IR active

B1 is IRandRamanactive

= A1 + B1

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MC L

CLO

1

2O

What about the trans isomer?

C2(z) sv(xz) sv'(yz)

x

E

1

1

2

1

-1

0

1

-1

0

D2h

Ag

B3u

C2(y) C2(x)

1

1

2

i

1

-1

0

sv(xy)

1

1

2

1

1

2

1

-1

0

Only one IR active band and no Raman active bands

Remember cis isomer had two IR active bands and one Raman active

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Symmetry and NMR spectroscopy

The # of signals in the spectrum

corresponds to the # of types of nuclei not related by symmetry

The symmetry of a molecule may be determined

From the # of signals, or vice-versa

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Molecular Orbitals

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Atomic orbitals interact to form molecular orbitals

Electrons are placed in molecular orbitalsfollowing the same rules as for atomic orbitals

In terms of approximate solutions to the Scrdinger equation

Molecular Orbitals are linear combinations of atomic orbitals (LCAO)

Y = caya + cbyb (for diatomic molecules)

Interactions depend on thesymmetry properties

and the relative energies of the atomic orbitals

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As the distance between atoms decreases

Atomic orbitals overlap

Bonding takes place if:

the orbitalsymmetry must be such that regions of the same sign overlap

the energy of the orbitals must besimilar

the interatomic distance must be short enough but not too short

If the total energy of the electrons in the molecularorbitals

is less than in the atomic orbitals, the molecule is stable compared with the atoms

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Combinations of two s orbitals (e.g. H2)

Antibonding

Bonding

More generally:Y = N[caY(1sa) cbY (1sb)]

nA.O.s nM.O.s

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Electrons in bonding orbitals concentrate between the nuclei and hold the nuclei together

(total energy is lowered)

Electrons in antibonding orbitals cause mutual repulsion between the atoms

(total energy is raised)

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Both s (and s*) notation means symmetric/antisymmetric with respect to rotation

s s* s*zC2 zC2 zC2

zC2Not s

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Combinations of twop orbitals (e.g. H2)

p (and p*) notation means

change of sign upon C2 rotation

s (and s*) notation means no

change of sign upon rotation

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Combinations of twop orbitals

zC2

zC2

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Combinations of two sets ofp orbitals

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Combinations ofs andp orbitals

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Combinations ofdorbitals

No interactiondifferent symmetry

d means change of sign upon C4

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NO NOYES

Is there a net interaction?

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Relative energies of interacting orbitals must be similar

Strong interaction Weak interaction

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Molecular orbitals

for diatomic molecules

From H2 to Ne2

Electrons are placed

in molecular orbitals

following the same rules

as for atomic orbitals:

Fill from lowest to highestMaximum spin multiplicity

Electrons have different quantum

numbers including spin (+ , -

)

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Bond order = # of electronsin bonding MO's# of electrons inantibonding MO's12

-

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O2 (2 x 8e)

1/2 (10 - 6) = 2A double bond

Or counting only

valence electrons:1/2 (8 - 4) = 2

Note subscripts

gand usymmetric/antisymmetric

upon i

Place labels or in this diagram

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Place labelsgoru in this diagram

sg

p*g

s*u

pu

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dg

d*u

p*g

pu

s*u

sg

goru?

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Orbital mixing

Same symmetry and similar energies !

shouldnt they interact?

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sorbital mixing

When two MOs of the same symmetry mix

the one with higher energy moves higher and the one with lower energy moves lower

M l l bi l

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H2sg2 (single bond)

He2

sg

2 s*u

2 (no bond)

Molecular orbitals

for diatomic molecules

From H2 to Ne2

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E (Z*)

DE s > DE pParamagneticdue to mixing

C2pu2 pu

2 (double bond)

C22- pu

2 pu2sg

2(triple bond)

O2pu2 pu

2 p*g1p*g

1 (double bond)

paramagnetic

O22- pu

2 pu2 p*g

2p*g2 (single bond)

diamagnetic

B d l th i di t i l l

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Bond lengths in diatomic molecules

Filling bonding orbitals

Filling antibonding orbitals

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Photoelectron Spectroscopy

hn

(UV o X rays) e-

Ionization

energy

hn

photons

kinetic energy of

expelled electron=-

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N2O2

s*u (2s)

pu (2p)

sg (2p)

s*u (2s)

sg (2p)pu (2p)

p*u (2p)

Very involved in bonding

(vibrational fine structure)

710 Lecture 04

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