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Symmetry operations

Point group

Characters

+1 symmetric behavior

-1 antisymmetric

Mülliken symbols

Each row is an irreducible representation

Symmetry of translations (p orbitals)

Classes of operations

Rx, Ry, Rz: rotations

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

dx2- y2 behaves as x2 – y2

dz2 behaves as 2z2 - (x2 + y2)

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

s behaves as x2 + y2 + z2

Effect of the 4 operations in the point group C2v

on a translation in the x direction

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 designated A; antisymmétric ones are designated B
- Subscripts 1 or 2 (applied to A or B refer) to symmetric and antisymmetric representations with respect to C2 Cn or (if no C2) to svrespectively
- Superscripts ‘ and ‘’ indicate symmetric and antisymmetric operations with respect to sh, respectively
- In groups having a center of inversion, subscripts g (gerade) and u (ungerade) indicate symmetric and antisymmetric representations with respect to i

Character Tables

- Irreducible representations are the generalized analogues of s or p symmetry in diatomic molecules.
- Characters in rows designated A, B,..., and in columns other than E indicate the behavior of an orbital or group of orbitals under the corresponding operations (+1 = orbital does not change; -1 = orbital changes sign; anything else = more complex change)
- Characters in the column of operation E indicate the degeneracy of orbitals
- Symmetry classes are represented by CAPITAL LETTERS (A, B, E, T,...) whereas orbitals 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

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 is a member of the group, usually denoted by a*b, such that the following properties are satisfied :
- (Associativity) (a*b)*c = a*(b*c) for all a, b, c belonging to G.
- (Identity) There exists e belonging to G, such that e*g = g = g*e for all g belonging to G.
- (Inverse) For each g belonging to G, there exists the inverse of g,g-1, such that g-1*g = g*g-1 = e.
- If commutativity is satisfied, i.e. a*b = b*a for all a, b belonging to G, then G is called an abelian group.

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

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)

Classes

Two members, c1 and c2, of a group belong to the same class if there is a member, g, of the group such that

g*c1*g-1 = c2

Properties of Characters of Irreducible Representations in Point Groups

- Total number of symmetry 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.

Properties of Characters of Irreducible Representations in Point Groups - 2

The number of irreducible reps equals the number of classes. The character table issquare.

1 + 2 + 3 = 6

3 by 3

1

1

22

6

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

Properties of Characters of Irreducible Representations in Point Groups - 3

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

Properties of Characters of Irreducible Representations in Point Groups - 4

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

Properties of Characters of Irreducible Representations in Point Groups - 5

Each group has a totally symmetric irreducible rep having all characters equal to 1

Reduction of a Reducible Representation

Irreducible 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

representations

These are block-diagonalized matrices

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

Reducible Rep

C2v Character Table to be used for water

Symmetry operations

Point group

Characters

+1 symmetric behavior

-1 antisymmetric

Mülliken symbols

Each row is an irreducible representation

Symmetry and molecular vibrations

Symmetry and molecular vibrations

A molecular vibration is IR active

only 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 representation

with the same symmetry as an x, y, z coordinate (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

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 are either unchanged or have become the negatives of themselves by a symmetry operation contribute to the character.

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.

Apply each symmetry operation in that point group to the molecule

and determine how many atomsare not moved by 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

Finding the reducible representation

E = 3

s = 1

C2 = -1

i = -3

C3 = 0

3x3

9

1x-1

-1

3x1

3

1x1

1

(# atoms not moving x char. contrib.)

G

9

-1

3

1

Now separate the reducible representation into irreducible ones

to see how many there are of each type

S

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

A2 =

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

Symmetry of molecular movements of water

Vibrational modes

Often you analyze selected vibrational modes

Example: C-O stretch in C2v complex.

n(CO)

2 x 1

2

0 x 1

0

2 x 1

2

0 x 1

0

G

Find: # vectors remaining unchanged after operation.

G

2

0

2

0

A1 is IR active

= A1 + B1

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

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

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

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

Only one IR active band and no Raman active bands

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

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

Atomic orbitals interact to form molecular orbitals

Electrons are placed in molecular orbitals

following the same rules as for atomic orbitals

In terms of approximate solutions to the Scrödinger equation

Molecular Orbitals are linear combinations of atomic orbitals (LCAO)

Y = caya + cbyb (for diatomic molecules)

Interactions depend on the symmetry properties

and the relative energies of the atomic orbitals

As the distance between atoms decreases

Atomic orbitals overlap

Bonding takes place if:

the orbital symmetry must be such that regions of the same sign overlap

the energy of the orbitals must be similar

the interatomic distance must be short enough but not too short

If the total energy of the electrons in the molecular orbitals

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

Bonding

More generally:

Y = N[caY(1sa) ± cbY (1sb)]

n A.O.’s

n M.O.’s

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

Electrons in antibonding orbitals cause mutual repulsion between the atoms

(total energy is raised)

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

(total energy is lowered)

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

s (and s*) notation means no

change of sign upon rotation

p (and p*) notation means

change of sign upon C2 rotation

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 highest

Maximum spin multiplicity

Electrons have different quantum numbers including spin (+ ½, - ½)

1/2 (10 - 6) = 2

A double bond

Or counting only

valence electrons:

1/2 (8 - 4) = 2

Note subscripts

g and u

symmetric/antisymmetric

upon i

When two MO’s of the same symmetry mix

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

due to mixing

E (Z*)

DE s > DE p

C2pu2 pu2 (double bond)

C22- pu2 pu2sg2(triple bond)

O2pu2 pu2 p*g1p*g1 (double bond)

paramagnetic

O22- pu2 pu2 p*g2p*g2 (single bond)

diamagnetic

N2

sg (2p)

p*u (2p)

pu (2p)

sg (2p)

pu (2p)

Very involved in bonding

(vibrational fine structure)

s*u (2s)

s*u (2s)

(Energy required to remove electron, lower energy for higher orbitals)

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