SLIDE 1
Recall Character table
Binary basis emerging from tensor product
(x,y) component of electric dipole moment p vector belongs to irrep E z-component of p belongs to A1 All observables can be associated with irreps
E x E reducible repn P E v = ? What the two binary basis of irrep - - PowerPoint PPT Presentation
E x E reducible repn P E v = ? What the two binary basis of irrep - - PowerPoint PPT Presentation
E x E reducible repn P E v = ? What the two binary basis of irrep E? P E v Similarly, P A1 projects the vector v to and P A2 projects the vector v to Recall Character table Binary basis emerging from tensor product (x,y)
SLIDE 2
SLIDE 3
Recall Character table
Binary basis emerging from tensor product
What about quadrupole moment tensor- binary basis
SLIDE 4
For operator f, whether this is non-zero/zero. Gives allowed/forbidden transitions SELECTION RULES
SLIDE 5
Selection rules
SLIDE 6
Selection rules contd
For a system with group symmetry G, the transition
from an initial state to final state due to interaction is
Is this zero or non-zero? Note that initial state belongs to one irrep αof G Final state also to an irrep β of G The interaction operator belongs to an irrep γ The integrand belongs to The transition is allowed if the tensor product allows A1
SLIDE 7
Examples
Let us look at electric dipole moment transitions for
systems with group symmetry C4v
A1
A2
Is this allowed or forbidden?
SLIDE 8
Electric dipole moment belongs to irrep F1 of O
F1×A1=F1, F1×A2=F2, F1×E=F1 + F2 F1×F1=A1 +E+ F1+F2 , F1×F2=A2 +E+ F1+F2 , A1 to A2 electric dipole moment transition is forbidden.
Belongs to F2 of O
SLIDE 9
Polar vector Vs Axial vector
Polar vectors are same as axial vectors for molecules
with no inversion symmetry/mirror symmetry.
For group O, the selection rules is same for electric
dipole moment and magnetic dipole moment.
For group Td , the selection rule for electric dipole
moment transitions is different from the selection rule for magnetic moment transitions.
Write the selection rules for electric dipole and magnetic
dipole transitions for group D3d
SLIDE 10
Molecular vibrations
Classical problem of two masses connected by spring Frequency of oscillation/vibration is Where ϰ is stiffness constant and μ is reduced mass of
the system
As the number of masses in the system increase, the
number of degrees of freedom (dof) increase and
- scillatory motion becomes complicated
The number of vibrational degree of freedom is 3N-6-
Why? What will be the number of vibrational dof for linear molecule
SLIDE 11
Molecular vibrations
For a complex system with s dof Let (x1 , x2, … xs ) denote small excursions of mass
points whose Lagrangian is
We can diagonalise this so that where ηɭ are the normal coordinates
SLIDE 12
Vibrations of the non-linear molecule
For non-linear triatomic molecule, there will be 3
vibrational modes
Bond length changes or bond angle changes
SLIDE 13