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)