Lecture 9/11

Intensities

A (A = X, Y, Z) is a space-fixed component of the

molecular dipole moment

Cre is the charge, Ar the A

coordinate of particle r 𝜇𝐴 = 𝐶𝑟𝑒𝐴𝑟 𝑟

Vanishing integral rule

The quantum mechanical integral

must vanish (i.e., be = 0) unless the integrand

contains a totally symmetric component in the

symmetry group(s) of the Hamiltonian

𝐼 = 𝜓′ 𝑂𝜓′′𝑑𝜏 *

𝜓′ 𝑂𝜓′′ *

Selection rules for transitions

So the intensity of a rotation-vibration transition is

proportional to the square of

Vanishing integral rule: For the integral to be non-

vanishing, the integrand must have a totally symmetric

component.

𝐼TM = Φrve 𝜇𝐴 Φrve d𝜏 ´* ´´

Symmetry of A generally

A has symmetry *

P A = A; P „pure“ permutation

P* A = A; P* permutation-inversion

* has character +1 under all „pure“ permutations P,

1 under all permutation-inversions P*

Symmetry of A for H2O

E* A = A

(12)* A = A

(12) A = A

* = A2

Selection rules for H2O

Apply vanishing integral rule to

For H2O we have S Φrve = cS Φrve

and so S ( rve rve ) = ( rve rve ) The condition for the integral to be non-vanishing = 1 for all S

𝐼TM = Φrve 𝜇𝐴 Φrve d𝜏 ´* ´´

Φ′∗ 𝜇𝐴Φ′′ Φ′∗ 𝜇𝐴Φ

′′ 𝒄𝑺′ 𝒄𝝁𝑨 𝒄𝑺′′

𝒄𝑺′ 𝒄𝝁𝑨 𝒄𝑺′′

All cS values are real for H2O

Selection rules for H2O

Condition: = 1 for S = E, (12), E*, (12)*

Selection rules: A1 A2

B1 B2

𝒄𝑺′ 𝒄𝝁𝑨 𝒄𝑺′′

𝒄𝝁𝑨

Selection rules for H2O and all other molecules

Further selection rule J = J´ – J´´ = -1, 0, +1

derives from rotational symmetry

Group K(spatial) of all rotations in space has irreducible

representations D(J)

Dipole moment has symmetry D(1)

Vanishing integral rule, now for K(spatial), requires that

D(J´) D(1) D(J´´) contain the totally symmetric

representation D(0) for the integral not to vanish

This requires J = J´ – J´´ = -1, 0, +1 and J´ + J´´ 1

Selection rules for H2O

Selection rules: A1 A2

B1 B2

J = J´ – J´´ = -1, 0, +1

and J´ + J´´ 1

Selection rules for NH3

Selection rules: A1´ A1 ´´

A2´ A2 ´´

E ´ E´´

𝒄𝝁𝑨

General selection rules for PH3

Selection rules: A1 A2

E E

𝒄𝝁𝑨

Calculation of intensities

Molecule-fixed axis system xyz follows the molecular

rotation

Molecule-fixed dipole

moment components

x,y,z are related

to space-fixed

components

X,Y,Z

𝜇𝐴 = 𝜆𝑥𝐴𝜇𝑥 + 𝜆𝑦𝐴𝜇𝑦 + 𝜆𝑧𝐴𝜇𝑧

= 𝜆𝛼𝐴𝜇𝛼 , 𝐴 = 𝑋, 𝑌, 𝑍𝛼=𝑥,𝑦,𝑧

x

y

z

X

Y

Z

N

Zero-order approximation

Molecular wavefunction

Lower state

Upper state

The line strength is the square of

𝜓′′ = Φelec,𝑛′′ Φvib,𝑛′′𝑣′′Φrot,𝑛′′𝑟′′ rve

𝜓′ = Φelec,𝑛′ Φvib,𝑛′𝑣′Φrot,𝑛′𝑟′ rve

𝐼TM = Φvib,𝑛′𝑣′ Φelec,𝑛′ 𝜇𝛼 Φelec,𝑛′′ Φvib,𝑛′′𝑣′′

𝛼=𝑥,𝑦,𝑧

Φrot,𝑛′𝑟′ | 𝜆𝛼𝜁| Φrot,𝑛′′𝑟′′

0

Zero-order approximation for transition within one electronic state

Analyze the line strength contribution

J = J´ – J´´ = -1, 0, +1 etc. Apply vanishing integral rule

where

Φvib,𝑛′𝑣′ 𝜇𝛼 Φvib,𝑛′′𝑣′′ Φrot,𝑛′𝑟′ | 𝜆𝛼𝜁| Φrot,𝑛′′𝑟′′

𝜇𝛼 = 𝜇𝛼 (𝑄1, 𝑄2, 𝑄3, … ) = Φelec,𝑛′|𝜇𝛼|Φelec,𝑛′′el, 𝛼 = 𝑥, 𝑦, 𝑧

Dipole moment components along molecule-fixed axes (for H2O)

(12) x = x

(12) y = -y

(12) z = -z

E* x = x

E* y = -y

E* z = z

(12) x = x

(12) y = -y

(12) z = -z

(12)* x = x

(12)* y = y

(12)* z = -z

A1

B1

B2

Dipole moment components along molecule-fixed axes (for H2O)

In general

For H2O

Γ(Φvib,𝑣′) Γ(Φvib,𝑣′′ ) ⊃ 𝐴1 for 𝜇𝑥

Γ(Φvib,𝑣′) Γ(Φvib,𝑣′′ ) ⊃ 𝐵1 for 𝜇𝑦 (but 𝜇𝑦 = 0)

Γ(Φvib,𝑣′) Γ(Φvib,𝑣′′ ) ⊃ 𝐵2 for 𝜇𝑧 *

*

*

For H2O

𝑥 = 𝑏 𝑦 = 𝑐

𝑧 = 𝑎

Thus far:

(Fairly) general considerations

Remaining topics:

Application to molecules

- Electronic wavefunctions

- Vibrational wavefunctions

- Rotational wavefunctions

Molecular wavefunction?

The total internal wavefunction

Ψrve is a solution of 𝐻rve Ψrve = 𝐸rve Ψrve

Ψnspin is a nuclear spin function

Better approximation

Hyperfine structure, ortho-para interaction

Matrix diagonalization, vanishing integral rule

Ψint = Ψrve Ψnspin

Ψint = 𝑐𝑝 Ψrve Ψnspin𝑝 (𝑝) (𝑝)

expansion cofficients

electronic coordinates

nuclear coordinates

Ψrve (𝑹e, 𝑹n) obtained in

the Born-Oppenheimer Approximation

Ψrve (𝑹e, 𝑹n) = Ψe (𝑹e, 𝑹n) Ψn (𝑹n)

Ab initio (electronic structure) calculation

nuclear positions fixed in space

r

r

VBO

𝐻elec = 𝑇elec(𝑹e, 𝑷e) + 𝑉Coulomb(𝑹e, 𝑹n ) (0)

(0) 𝐻elec Ψe (𝑹e, 𝑹n ) = 𝑉BO(𝑹n ) Ψe (𝑹e, 𝑹n ) (0) (0)

Nuclear-motion calculation

from ab initio calculation

„observable“ energy

𝐻n = 𝑇n(𝑹n, 𝑷n) + 𝑉BO(𝑹n)

𝐻n Ψn (𝑹n ) = 𝐸neΨn (𝑹n)

Nuclear-motion wavefunction

describes vibrational and rotational motion.

Simplest approximation

Better approximation

„Rotation-vibration“ interaction

Matrix diagonalization, vanishing integral rule

Ψn (𝑹vib, 𝑹rot) = Ψvib (𝑹vib) Ψrot (𝑹rot)

Ψn (𝑹vib, 𝑹rot) = 𝑐𝑝 Ψvib (𝑹vib) Ψrot (𝑹rot)𝑝 (𝑝) (𝑝)

expansion cofficients

Born-Oppenheimer Approximation

Beyond

the Born-Oppenheimer Approximation

expansion cofficients

„Born-Oppenheimer breakdown“

Renner effect, Jahn-Teller effect, ....

Matrix diagonalization, vanishing integral rule

Ψrve (𝑹e, 𝑹n) = Ψe (𝑹e, 𝑹n) Ψn (𝑹n)

Ψrve (𝑹e, 𝑹n) = 𝑐𝑝Ψe (𝑹e, 𝑹n) Ψn (𝑹n)𝑝 (𝑝)

(BO)

(NBO)