Download 6. Molecular vibrations

6. Molecular vibrations
A molecular potential energy curve
can be approximated by a parabola
near the bottom of the well. The
parabolic potential leads to
harmonic oscillations. At high
excitation energies the parabolic
approximation is poor (the true
potential is less confining) and is
totally wrong near the dissociation
limit.
6.1 Harmonic oscillator
A particle undergoes harmonic
motion if F= - k x, k force constant.
The force F is related to the
potential energy by
F 
dV
dx
The parabolic potential energy of a
harmonic oscillator is
V = ½ k x2, where x is the
displacement from equilibrium. The
narrowness of the curve depends
on the force constant k: the larger
the value of k, the narrower the well.
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6.1.1 Schrödinger equation and eigenvalues
The SE for the oscillator is

k
m2
m1
 d  1 2
 kx   E
2 dx 2 2
2
2
m is the effective or reduced mass: m1m2/(m1+m2)
Quantization arises from the boundary conditions imposed by
the potential: =0 at x  
The energy levels are:
1
E n  ( n  ) 
2

k

n =0, 1, 2, …
6.1.2 The energy levels
The energy levels of a harmonic
oscillator are evenly spaced with a
separation ω , with w=(k/m)1/2.
Even in its lowest state, an oscillator
has an energy greater than zero!
The separation between adjacent levels is:
En 1  En  
The harmonic oscillator has a zero-point energy:
E0 
1

2
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6.1.3 The form of the wavefunctions
The normalized wavefunction
and probability distribution
(shown also by shading) for
the lowest energy state of a
harmonic oscillator.
The normalized wavefunction
and probability distribution
(shown also by shading) for
the first excited state of a
harmonic oscillator.
6.1.4 The probability distributions
The probability distributions for the first five states of a harmonic
oscillator and the state with n=20. Note how the regions of
highest probability move towards the turning points of the
classical motion as n increases.
‘Mechanical oscillator’ with ‘low’ quantum number??
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6.3 Molecular degrees of freedom
Translations: The molecule as a whole (strictly its centre of
mass) moves from place to place (temperature, heat).
Rotations: The molecule rotates (about its centre of mass).
Vibrations: The molecule's bonds stretch, contract and bend so
that the structure oscillates around the most stable configuration.
Vibration is sometimes called internal motion since it is possible
to define it in such a way as to separate it from translation and
rotation.
6.3.1 Number of molecular vibrational modes
What is the number of vibrational modes for a molecule?
Each atom has 3 degrees of freedom. A molecule of N atoms
has 3N degrees of freedom. There are 3 translations and-if the
molecule is not linear-also 3 rotational modes.
Number of vibrations for a non-linear molecule of N atoms:
3N-6
Number of vibrations for a linear molecule of N atoms: 3N-5
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6.3.2 Translations and rotations of 3-atomic non-linear
molecules
http://www-biol.paisley.ac.uk/chemistry/home/staff/bstewart/teaching/inorg3/VIBSPEC/VIBSPEC3.HTM
6.3.4 Vibrational modes of 3-atomic non-linear
molecules (water)
symmetric stretch
symmetric bend
anti-symmetric stretch
http://www-biol.paisley.ac.uk/chemistry/home/staff/bstewart/teaching/inorg3/VIBSPEC/VIBSPEC3.HTM
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6.4 Selection rules
The gross selection rule for the absorption of radiation by a molecular
vibration is that the electric dipole moment of the molecule must
change when the atoms are displaced relative to one another.
Such vibrations are infrared active.
Note that the molecule need not have a permanent dipole!
The oscillation of a molecule,
even if it is nonpolar, may
result in an oscillating dipole
that can interact with the
electromagnetic field.
6.4.1 Example - Vibrational modes of carbon dioxide
How many vibrational modes are expected for CO2 and which
modes are IR active??
IR active?
3N-5 = 3x3-5=4
vibrational modes are
expected for this
linear molecule:
no
yes
yes
yes
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6.4.2 Selection rules
The specific vibrational selection rule obtained from an analysis of the
expression for the transition moment and the properties of integrals
over harmonic oscillator wavefunctions for IR activity is:
n  1
Transitions with Dn=+1 correspond to absorption and those with Dn=-1
correspond to emission.
The corresponding frequencies are
v
E
h
and in the harmonic approximation there is only a single line.
6.5 Example ‘X‘-H bond
The force constant of a typical X-H chemical bond is around 500
N/m. Mass of the proton is about 1.7x10-27 kg w=5.4 x 1014 1/s
  5.7  10 20 J (around 360 meV)
kBT at room temperature is around 25 meV, hence almost all
molecules will be in their vibrational ground state and the dominant
transition will be the fundamental transition, 1 0.
the frequency of the transition is n=DE/h =86 THz
and l=c/n = 3.5mm.
Transitions between adjacent vibrational energy levels of molecules
are stimulated by or emit infrared radiation.
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6.6 Anharmonicity
Real potentials are anharmonic which is taken into account in other
potential models.
V  (1  e  a ( R  Re ) ) 2
The Morse potential energy curve
reproduces the general shape of a
molecular potential energy curve.
The corresponding SE can be
solved, and the values of energies
obtained. The number of bound
levels is finite.
6.7 Vibration-rotation spectra
6.7.1 Spectral branches
Analysis of the quantum mechanics of simultaneous vibrationalrotational changes shows that DJ= ±1during the vibrational
transition of a diatomic molecule.
If the molecule also possesses angular momentum about its axis,
then the selection rule also allows DJ=0 (as in the case of the
electronic orbital angular momentum of the paramagnetic NO)
When the vibrational transition n+1  n occurs, J changes by ±1
(and in some cases by 0). The absorptions then fall into three
groups called branches of the spectrum:
The P branch consist of all transitions with DJ=-1.
The Q branch consist of all transitions with DJ=0.
The R branch consist of all transitions with DJ=+1.
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6.7.2 Example HCl
A high resolution vibrationrotation spectrum of HCl. The
lines appear in pairs because
H35Cl and H37Cl both
contribute (their abundance
ratio is 3:1). There is no Q
branch because DJ=0 is
forbidden for this molecule.
The separation between the
lines in both branches gives
the value of B (and hence the
bond length can be deduced)
6.7.3 Vibration-rotation spectra
The transitions of P,
Q, and R branches in
a vibration-rotation
spectrum. The
intensities reflect the
populations of the
initial rotational levels
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6.8 Vibrational Raman spectra of diatomic molecules
The gross selection rule for vibrational Raman transitions is that the
polarizability should change as the molecule vibrates.
Homonuclear and heteronuclear diatomic molecules are vibrationally
Raman active!
The specific selection rule for vibrational Raman transitions is
n  1
6.8.1 Example - Vibrational modes of carbon dioxide
Which of the four vibrational modes of CO2 are Raman
active??
Raman active?
yes
no
no
no
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6.8.2 Vibration-rotation Raman spectra
In gas-phase spectra, the Stokes and anti-Stokes lines have a
branch structure arising from the simultaneous rotational transitions
that accompany the vibrational excitations.
The selection rules are DJ=0,±2 (as in pure rotational Raman
spectroscopy), and give rise to the O-branch (DJ=-2), the Q
branch (DJ=0), and the S branch (DJ=+2).
6.8.3 Vibration-rotation Raman spectra
The transitions of O, Q, and S
branches in a vibration-rotation
Raman spectrum of a linear
rotor. Note that the frequency
scale runs in the opposite
direction to that of a vibrationrotation IR spectrum, because
the higher energy transitions (on
the right) extract more energy
from the incident beam and
leave it at lower frequency.
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