Fundamentals of Infrared and Raman Spectroscopy, SERS, and Theoretical Simulations
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Infrared and Raman spectroscopy are two important vibrational spectroscopy methods. These investigation techniques provide complementary images of molecular vibrations, because the mechanisms of the interaction of light with molecules in those two spectroscopic techniques are quite different.
2.1 Vibrations of Molecules Molecules consist of atoms with a certain mass that can be considered as being connected by elastic bonds. As a result, they can perform periodic motions, and have vibrational degrees of freedom. All motions of the atoms in a molecule relative to each other are a superposition of so-called normal vibrations, in which all atoms are vibrating with the same phase and normal frequency. Their amplitudes are described by a normal coordinate. Polyatomic molecules with N atoms possess 3N-6 normal vibrations (linear ones have 3N-5 normal vibrations), which define their vibrational spectra (Demtröder 1981, Chalmers and Griffiths 2002, Hollas 1992, Bunker 1979, Niquist 2001). These spectra depend on the masses of the atoms, their geometrical arrangement, and the strength of their chemical bonds. Depending on whether the bond length or angle is changing, there are two types of molecular vibrations, stretching and bending. The stretching vibrations can be symmetric and asymmetric, whereas the bending vibrations can be subdivided in scissoring, rocking, wagging and twisting vibrations. For molecules with certain elements of symmetry, some vibrational modes may be degenerate, so that more than one mode has a given vibrational frequency, while others may be completely forbidden. Thus, because of degeneracy, the number of the observed fundamental absorption bands is often less than 3N-6. In “super molecules,” such as crystals or complexes, the vibrations of the individual components are coupled. The simplest model of a vibrating molecule describes an atom bound to a very large mass by a weightless spring (Schrader 1995). The force F, which is neces-
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2 Fundamentals of Infrared and Raman Spectroscopy, SERS, and Theoretical Simulations
sary to move the atom by a certain distance x from an equilibrium position, is given by Hooke’s law, and is proportional to the force constant f, which represents a measure of the bond strength: F = − fx .
(2.1)
By using Newton’s law, from where the force is also proportional to the mass M of the atom and its acceleration (the second derivate of the elongation with respect to the time, d2x/dt2), and solving the second order differential equation, which possesses the solution: x = x0 cos(2πνt + δ ) ,
(2.2)
where δ represents the phase angle, the frequency of the vibration of the mass connected to a very large mass by an elastic spring can be expressed (Schrader 1995) as follows:
ν=
1 2π
f . M
(2.3)
If one considers a diatomic molecule the mass M is the reduced mass of the diatomic molecule with the masses M1 and M2 and is given by the following formula (Schrader 1995): 1 1 1 . = + M M1 M 2
(2.4)
Equation 2.3 gives the frequency ν (in Hz, s–1) of the vibration. In vibrat
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