Summary of previous lesson

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• Summary of previous lesson
• We introduced the basic formalism of quantum
  mechanics
• We showed that for a time-independent Hamiltonian
  
• We introduced the Born-Oppenheimer approximation
• We discussed the interaction of light and matter
  and introduced dipole moment, Einsteins
  coefficients and connections between them.

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Line Width Since every excited state has a
finite lifetime, ?, we can expect a finite
bandwidth for any transition involving this
state, due to the Heisenberg uncertainty
principle

• This broadening effect is called natural line
  broadening and leads to a Lorenzian shaped
  spectrum . However, there are additional, larger
  broadening effects, which include
• Doppler broadening can be evaluated from the
  Maxwell distribution of velocities of molecules
  in the gas phase.
• Pressure broadening collisions shorten the
  lifetime of excited states
• Inhomogeneous broadening effect of the
  environment in condensed phases on the spectrum

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The line width depends on the time of
observation! If the environment of the molecule
is static a broad, Gaussian line width. If the
environment of the molecule changes faster than
the observation time motional narrowing a
narrow, Lorenzian line width.
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Vibrational spectroscopy Diatomic molecule
(harmonic oscillator)
The wavefunctions are even/odd alternately.
Transitions between vibrational levels are
dictated by the following type of integral, which
involves the dipole moment
Thus transitions are driven by the gradient of
the dipole moment.
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The transition probability depends on integrals
of the type
This integral is non-zero only if the integrand
is even. A selection rule can be derived by using
the even/odd character of the wavefunctions, and
it is found that
This selection rule is compromised in an
anharmonic oscillator and we get overtone
transitions
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Polyatomic molecules and normal modes The above
description is fine for a diatomic molecule. In a
polyatomic molecule we need to introduce the
concept of normal modes, which are obtained from
the full potential of the molecule, which now
includes motions of all atoms

The Qks are the normal coordinates. There are
3N-6 normal modes of internal motion. The
Hamiltonian can be written as a sum of terms in
the separate Qs, so the overall wavefunction is
a product of the wavefunctions of separate normal
modes.
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The For example, for water we have 3 normal
modes
Symmetric stretch Asymmetric stretch Bend (3650
cm-1) (3760 cm-1) (1600 cm-1)
The dipole moment changes during all three
motions, so they can all be IR active, but that
also depends on the selection rule discussed
above.
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• Symmetry considerations
• Nondegenerate normal modes are either symmetrical
  or antisymmetrical with respect to a given
  symmetry.
• Degenerate normal modes are transformed by a
  symmetry operation into a combination of the
  members of the degenerate set.

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Thus for each symmetry group there is a set of
irreducible representations, or symmetry species.
It is found that the table of characters (traces
of the transformation matrices) of each group
allows to perform most of the required
operations. As an example lets calculate the
effect of a symmetry operation on some quantum
mechanical integrals. First the overlap integral
Assume that the functions are bases for the
irreducible one-dimensional representations ?a
and ?b , respectively. This means that
and
Therefore
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This can only be true if
for all Rs.
Now the characters are either 1 or -1 for a one
dimensional representation, so the above equation
can be true only if ab. Thus the overlap
integral must be zero if the two functions do not
belong to the same species. We can do the same
with the matrix elements of the dipole moment
operator to obtain selection rules for
vibrational transitions. Suppose the electric
field is oriented along the z coordinate of the
molecule. Then the relevant integral is
which can be transformed as before to get
so
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This is stated in a general form like this From
the point of view of symmetry, in order to
satisfy the selection rule integral for a
vibrational transition, the direct product of the
symmetry species of the two wave functions should
contain one of the symmetry species of the dipole
moment (in simpler language the product of the
wave functions should transform like the dipole
moment). This is simplified when one of the wave
functions is the ground state (i.e. for a
fundamental transition), because this state is
always totally symmetric. In this case we have
which implies that the excited state wave
function should belong to the same symmetry
species of the dipole moment element in order to
have an IR active transition. This is the
celebrated selection rule for vibrational
transitions.
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As an example, water belongs to the point group
with the following symmetry elements
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Looking at the character table for this group we
can see that the symmetric stretch and the bend
belong to A1, while the asymmetric stretch
belongs to B2. Since z and y also belong to these
representations, fundamental transitions of all
modes may be IR active.
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Or in the case of CO3- , it can be shown that Q1
is of symmetry A1, Q2 is of symmetry A1, and
the pairs Q3 ,Q4 and Q5 ,Q6 are of symmetry E .
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And looking at the character table of D3h we find
that transitions involving Q1 or Q2 are not
allowed in the IR, while the others are.