Vibrations of polyatomic molecules

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Vibrationsofpolyatomic molecules
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Outline
Normal modes Selection rules Group theory
(Tjohooo!) Anharmonicity
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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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Total energy
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Total energy
We can now write the total vibrational energy as
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A look at CO2
Vibrations of the individual atoms
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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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Normal modes of CO2
3 x 3 - 5 4 vibrational modes
Symmetric stretch Anti-symmetric
stretch Orthogonal bending
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QM
Since the total energy is just a sum of terms, so
is the Hamiltonian of the vibrations. We write it
as
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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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Harmonic oscillator
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Harm. Osc.
We know the ground state
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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Selection rules
Molecular dipole moment depends on displacements
of the atoms in the molecule Taylor expand...
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Selection rules

Selection rules for IR absortion
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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Group theory and vibrations
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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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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Continued water example
The representation reduces to Gred3A1A22B13B2
Gtrans A1B1B2
GrotA2B1B2
Gvib2A1B2
Modes left for vibrations
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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
For a transition to be IR active, the normal mode
must be parallel to the polarisation of the
radiation.
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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.
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Water again...
A1
Gvib2A1B2
A1
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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Anharmonicity
Electric anharmonicity occurs when our expansion
of the dipole moment to first order is not valid.
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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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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.
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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
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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.