Vibrational Spectroscopy:
Vibrational spectroscopy can be thought of by starting with a simple harmonic
oscillator (SHO) model. In this model, we consider two atoms joined by a bond to be
equivalent to two masses joined by a spring. The spring can be compressed, forcing
the spheres close to each other - stretched, moving them apart - or allowed to freely
come to rest in the spheres' equilibrium positions. This can be shown in a potential
energy curve:
However, not all energies are possible. The possible vibrational states are given by the
vibration quantum number, v, and vibrational selection rule Δv = ±1.
The energy of each level, E
, is given by
h
E
2
1
(the second ν is the Greek character nu, the fundamental frequency). The
fundamental frequency is given by the equation ν = (1/2π)×(k/μ)1/2, where μ is the
reduced mass of the molecule, and k is thebond force constant. Notice that this k is
similar to the spring force constant from Hooke's Law F = -kx. This leads to a potential
energy surface that looks like this:
As you can see, the energy levels are equally spaced, and the equilibrium bond length
is constant for all energy levels. However, this model is imperfect - it does not account
for the posibility of bond dissociation (under this model, the bond would never break,
no matter the magnitude of the vibrational energy). It also does not account for extra
repulsive effects at very small bond lengths caused by the electroweak force. This is
the force which prevents atoms for being forced together as the distance between them
gets very small (the reason nuclear fusion only occurs at very high temperatures, for
example). A model which takes into account these factors, and which more accurately
models a vibration diatomic molecule, is the Anharmonic Oscillator (AHO), and the
corresponding potential energy surface called the Morse potential. The potential
energy curve now looks like this:
Notice how the expectation (or average) bond length, rexp is no longer the same for all
energy levels. One effect of the anharmonicity is this deformation as the energy of the
vibrations increases. Another is that now, the vibrational energy levels are no longer
equally spaced, but instead get closer together as the vibrational quantum number
increases. This model also illustrates that there aren't an infinite number of vibrational
energy levels - above some energy, the bond breaks and the molecule dissociates.
Vibrational energies occur roughly in the 100 - 4000 cm-1 (about 1 - 50 kJ mol-1),
or infra-red (IR) region of the electromagnetic spectrum. The fundamental principle
for obtaining a vibrational spectra is that the electric dipole moment of the molecule
must change during the vibration. If there is no change in dipole moment, then this
particular vibration will not give rise to any absorption in the IR region. An example
of this can be seen below for the symmetric stretch (Σg+) mode of carbon dioxide.
Because both C=O bond lengths change exactly in phase, there is never a net dipole
moment on the molecule. A homonuclear diatomic molecule such as dioxygen O2
has
zero dipole moment, so it has no IR spectrum.
The number of modes (types) of vibration can be predicted for a molecule, containing
N atoms, using the following general expressions.
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