x, y, z, x 2 , y 2 , z 2 , xy, yz, zx

4 Exercise - Introduction to Group Theory

4.1 (vibrational selection rules)

To nd vibrational selection rules, we haveto determine whether integrals of the types

R ψ ⁰ _ν f ψ ¹ _ν

^are

non-zero,withfunction

f

^being

x, y, z, x ² , y ² , z ² , xy, yz, zx

^oranycombinationthereof.Let

ψ ⁰ _ν

^betotally

symmetric and

ψ _ν ¹

^{may belong to any} irreducible representation. We are now meant to identify the irreducible representationsto which

ψ _ν ¹

^{may belong in order to give non-zerointegralsformolecules of}

symmetry

C 4 v

^and

D 3 d

^{.Westartoutrstdenotingthecharactertablesof}

C 4 v

^:

C 4 v E 2C ⁴ (z) C 2 2σ v 2σ d

^{linearfcts,rotations quadraticfcts cubicfunctions}

A 1 1 1 1 1 1 z x ² + y ² , z ² z ³ , z x ² + y ²

A 2 1 1 1 − 1 − 1 R z − −

B 1 1 − 1 1 1 − 1 − x ² − y ² z x ² − y ²

B 2 1 − 1 1 − 1 1 − xy xyz

E 2 0 − 2 0 0 (x, y) (R x , R y ) (xz, yz) xz ² , yz ²

xy ² , x ² y

x ³ , y ³

Werststartcalculatingallpossibledirectproducts:

A 1 ⊗ A 1 = A 1

A 1 ⊗ A 2 = A 2

A ¹ ⊗ B ¹ = B ¹ A 1 ⊗ B 2 = B 2

A 1 ⊗ E = E

further:

E ⊗ A 1 = E E ⊗ A 2 = E E ⊗ B 1 = E E ⊗ B ² = E

E ⊗ E = A 1 + A 2 + B 1 + B 2

goingonwith:

B 1 ⊗ A 1 = B 1

B 1 ⊗ A 2 = B 2

B ¹ ⊗ B ¹ = A ¹ B 1 ⊗ B 2 = A 2

B 1 ⊗ E = E

and

B 2 ⊗ A 1 = B 2

B 2 ⊗ A 2 = B 1

B ² ⊗ B ¹ = A ² B 2 ⊗ B 2 = A 1

B 2 ⊗ E = E

atlast

A 2 ⊗ A 1 = A 2

A 2 ⊗ A 2 = A 1

A 2 ⊗ B 1 = B 2

A 2 ⊗ B 2 = B 1

A 2 ⊗ E = E

Therefore

A 1

^and

E

^{symmetryvibrationswillbeIRactiveand}

A 1

^,

B 1

^,

B 2

^and

E

^{symmetryvibrations}

will beRamanactive.Weusedthat thedirect product should containtheirreduciblerepresentation of

theirr.rep.wehavelookedat forthetransition.

Nowwedenotethecharatertable of

D 3 d

^:

D 3d E 2C 3 3C 2 ⁰ i 2S 6 3σ d

^{linears,rots quadratics cubicfunctions}

A 1 g 1 1 1 1 1 1 − x ² + y ² , z ² −

A 2 g 1 1 − 1 1 1 − 1 R z − −

E g 2 − 1 0 2 − 1 0 (R x , R y ) x ² − y ² , xy

(xz, yz) −

A 1 u 1 1 1 − 1 − 1 − 1 − − x x ² − 3y ²

A 2 u 1 1 − 1 − 1 − 1 1 z − y x ² − 3y ²

, z ³ , z x ² + y ² E u 2 − 1 0 − 2 1 0 (x, y) − xz ² , yz ² xyz, z x ² − y ²

x x ² + y ²

, y x ² + y ²

Wecanagaincalculateallpossibledirectproducts:

A 1 g ⊗ A 1 g = A 1 g

A ¹ g ⊗ A ² g = A ² g

A 1 g ⊗ E g = E g

A ¹ g ⊗ A ¹ u = A ¹ u

A 1g ⊗ A 2u = A 2u

A 1 g ⊗ E u = E u

next

A 2 g ⊗ A 1 g = A 2 g

A 2 g ⊗ A 2 g = A 1 g

A 2g ⊗ E g = E g

A 2 g ⊗ A 1 u = A 2 u

A ² g ⊗ A ² u = A ¹ u

A 2 g ⊗ E u = E u

last

g

E g ⊗ A ¹ g = E g

E g ⊗ A 2 g = E g

E g ⊗ E g = A 1 g + A 2 g + E g

E g ⊗ A 1u = E u

E g ⊗ A 2 u = E u

E g ⊗ E u = A ¹ u + A ² u + E u

nowthe

u

^-terms:

A 1 u ⊗ A 1 g = A 1 u

A ¹ u ⊗ A ² g = A ² u

A 1 u ⊗ E g = E u

A 1 u ⊗ A 1 u = A 1 g

A 1 u ⊗ A 2 u = A 2 g

A 1 u ⊗ E u = E g

next

A 2u ⊗ A 1g = A 2u

A 2 u ⊗ A 2 g = A 1 u

A 2u ⊗ E g = E u

A 2 u ⊗ A 1 u = A 2 g

A ² u ⊗ A ² u = A ¹ g

A 2u ⊗ E u = E g

last

u

^-term:

E u ⊗ A ¹ g = E u

E u ⊗ A 2g = E u

E u ⊗ E g = A 1 u + A 2 u + E u

E u ⊗ A ¹ u = E g

E u ⊗ A 2 u = E g

E u ⊗ E u = A 1 g + A 2 g + E g

A 2u

^and

E u

^{symmetryvibrationswillbeIRactive.}

A 1g

^and

E g

^{symmetryvibrationswillbeRaman}

active.

4.2 (magnetic dipole allowed transitions for

T ^d

^-symmetry)

Foramoleculeof

T d

^{symmetrywecandeterminewhat pairsofstatescouldbeconnectedbyamagnetic}

dipole allowedtransition,whileforexampleMethanegot

T d

^{symmetry. Thecharactertableof}

T d

^:

T d E 8C 3 3C 2 6S 4 6σ d

^{linears,rots quadraticfcts cubicfunctions}

A 1 1 1 1 1 1 − x ² + y ² + z ² xyz

A 2 1 1 1 − 1 − 1 − − −

E 2 − 1 2 0 0 − 2z ² − x ² − y ² , x ² − y ²

− T 1 3 0 − 1 1 − 1 (R x , R y , R z ) −

x z ² − y ²

, y z ² − x ²

, z x ² − y ² T 2 3 0 − 1 − 1 1 (x, y, z) (xy, xz, yz) x ³ , y ³ , z ³

x z ² + y ²

, y z ² + x ²

, z x ² + y ²

Thedirect productsare:

T ¹ ⊗ A ¹ = T ¹ T 1 ⊗ A 2 = T 2

T 1 ⊗ E = T 1 + T 2

T 1 ⊗ T 1 = A 1 + E + T 1 + T 2

T 1 ⊗ T 2 = A 1 + E + T 1 + T 2

thereforethefollowingtransitionsaremagneticdipole allowed:

T 1 → A 1

T 1 → E T ¹ → T ¹ T 1 → T 2

this followsfrom thecondition, that we needto have at least one

T 1

^{in the direct} product-reduced form,fortheintegralnotto vanish.