1. Vibrations of polyatomic molecules
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2. Outline
* Normal modes * Selection rules * Group theory (Tjohooo!) * Anharmonicity
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3. Describing the vibrations
Molecule with N atoms has 3N-6 vibrational modes, 3N-5 if linear.
Find expression for potential energy.
Taylor expansion around equilibrium positions.
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4. Total energy ; where ; where Kinetic energy
Introduce mass weighted coordinates:
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5. Total energy We can now write the total vibrational energy as:
Nasty cross-terms when
What we want is to find set of coordinates where the cross-terms disappear.
Is this at all possible?
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6. A look at CO2 Vibrations of the individual atoms
Can be broken down as linear combinations of:
These modes do not change the centre of mass, and they are independent.
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7. Normal coordinates So, we can write the energy as:
where Q are the so called normal coordinates.
They can be a bit tricky to find, but at least we
know they are there.
Before we see how can use this, lets have a
look at the normal modes for our CO2.
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8. Normal modes of CO2 3 x 3 - 5 = 4 vibrational modes Symmetric stretch
Anti-symmetric stretch
Orthogonal bending
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9. QM
Since the total energy is just a sum of terms, so is the Hamiltonian of the vibrations. We write it as:
Also the vibrational wavefunction separates into a product of single mode wavefunctions:
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10. Schrödinger equation The Scrödinger equation then becomes:
… and this we recognise, right?
Harmonic oscillator with unit mass and force constant k.
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11. Harmonic oscillator Energy levels: Wavefunctions:
Total vibrational energy:
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12. Harm. Osc. … We know the ground state: Ground state energy:
Ground state wavefunction:
All normal modes appear symmetrically, and as squares
The ground state is symmetric with respect to all symmetry operations of the molecule.
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13. Selection rules
Molecular dipole moment depends on displacements of the atoms in the molecule: Taylor expand...
Dipole transition matrix element of a particular mode:
0 if , due to orthogonality if
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14. Selection rules ; Selection rules for IR absortion:
Similarly, by observing that
we get selection rules for Raman activity:
It can be hard to see which vibrations are IR/Raman active, but, as we have seen before, Group Theory can come to rescue.
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15. Group theory and vibrations
First find a basis for the molecule. Let’s take the cartesian coordinates for each atom. x1 x3 x2 y1 y2 y3 z1 z2 z3
Water belongs to the C2v group which
contains the operations E, C2, sv(xz) and sv’(yz).
The representation becomes
E C2 sv(xz) sv’(yz)
Gred 9 -1 1 3
The details of a normal mode depend on the strength of the chemical bonds and the mass of the atoms. However the symmetries are just a function of geometry.
Example: H2O (the following stolen from Hedén)
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16. Continued water example
Character table for C2v.
Now reduce Gred to a sum of irreducible representations. Use inspection or the formula.
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17. Continued water example
The representation reduces to Gred=3A1+A2+2B1+3B2
Gtrans= A1+B1+B2
Grot=A2+B1+B2
Gvib=2A1+B2
Modes left for vibrations
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18. What to use this for?
We know that that the ground state is totally symmetric: (A1)
First excited state of a normal mode belongs to the same irred. repr. as that mode because
So for , and must span the same irreducible representation for their product to be in A transform as translations, so:
For a transition to be IR active, the normal mode must be parallel to the polarisation of the radiation.
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19. What more to use this for?
By the same argument one can come the the conclution that
For a transition to be Raman active, the normal mode must belong to the same symmetry species as the components of the polarisability
These scale as the quadratic forms x2, y2, xy etc.
This also leads to the exclusion rule:
In a molecule with a centre of inversion, a mode cannot be both IR and Raman active.
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20. Water again... A1 A1 B2 Gvib=2A1+B2
All three modes are both IR and Raman active, no centre of inversion.
(a) and (b) are excited by z-polarised light, and (c) by y-polarised. B2
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21. Anharmonicity
Electric anharmonicity occurs when our expansion of the dipole moment to first order is not valid.
At this point overtones like can be allowed, since the matrix element containing the quadratic term Qi2 not necessarily vanishes. This can not be determined from group theory, but must be calculated for every molecule.
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22. Anharmonicity
We also see from the presence of QiQj cross-terms can cause a mixing of normal modes.
In a perfectly harmonic molecule, energy put into one normal mode stays there. Anharmonicity causes the molecule to thermalise.
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23. Anharmonicity
Also mechanical anharmonicity can lead to mixing of levels if one needs to add cubic and further terms in the expression for the potential. 0a 0b 1a 2a 1b
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24. Inversion doubling
Consider ammonia: pyramidal molecule with two sets of vibrational levels:
Coupling between the levels lead to mixing of up and down wavefunctions which lifts the degeneracy of the levels

25. Summary
Harmonic approximation of energy gives transition rules for IR and Raman activity.
Group theory can help us figure out which transition are active.
However, anharmonic terms can come in play and mess everything up.