Vibrational Spectroscopy

Vibrational Spectroscopy
The vibrational spectroscopy involves transition between different vibrational energy levels. The vibrational energy is calculated from the Schrodinger equation for a simple harmonic oscillator.
1. Simple Harmonic Oscillator
The potential energy of a simple harmonic oscillator exhibits a quadratic dependence on the coordinate, e.g.:

………..(1)
where q, denotes the vibrational coordinate, i.e. inter-nuclear distance. Thus the Schrodinger equation for a simple harmonic oscillator is:

……..(2)
From the solution of this equation, the vibrational energy of a linear harmonic oscillator is obtained as:

……(3)
where n is the vibrational quantum number and v0 is the classical frequency of vibration.

…..(4)
In the last equation, k, denotes force constant and ? denotes reduced mass of the harmonic oscillator. The energy of the lowest state (n=0) is 1/2h?0 and is called zero point energy. The vibrational energy gap for the transition from the level n to n is given by:
….(5)
The selection rule for vibrational transition is ?n=±1. Thus the energy gap between two successive vibrational levels is h?0. Thus a pure vibrational transition should be observed at this energy. The vibrational energy h?0 is much greater than the thermal energy (kBT) at room temperature and hence, at room temperature only the zeroth vibrational (n=0) level is populated with very small.
2. Anharmonic Vibration
In many cases, the potential energy of an oscillator deviates from that given in equation (1) and consists of terms beyond the quadratic term. Such an oscillator is called an anharmonic oscillator. In general, the potential energy of an anharmonic oscillator may be written as:

………(6)
If one uses this potential in the Schrodinger equation one may show that the energy of an anharmonic oscillator is given by:

…(7)

In the last equation x, y, .. are known as the anharmonicity constants. Considering only the first two terms in equation (7) the energy change for transition from a state of vibrational quantum number n to the next one (n+1) is:

…..(8)
For a simple harmonic oscillator the vibrational energy gap between any two adjacent levels is the same (h?0). From equation (8) it is evident that with increase in n the vibrational gap between two adjacent levels decreases because of anharmonicity.
Another consequence of anharmonicity is follows. Addition of the higher terms in the potential, makes the transition ?n=±2, ±3, ±4,… allowed. If ?n=1, the transition is called fundamental transition and the corresponding frequency (En-E0)/h is denoted as ?1. For ?n=2, the transition is called first overtone with a frequency ?2. Similarly, for second overtone, ?n=±3 and the frequency is ?3 Evidently.

….(9)


Since x is small obviously ?1: ?2: ?3?1:2:3.

3. Vibrational Spectrum and force Constants
It is evident from the position of the vibrational line one gets ?0 and from this using equation (4) one obtains the force constant (k) of the vibration. In a molecule the force constants for various bonds and various functional groups are different. Thus the main application of vibrational spectroscopy is to detect different functional groups of a molecule. The most common vibrations are O-H stretching, C-H stretching and C=O stretching frequencies which appear at a frequency around 3500 cm-1, 3000 cm-1 and 1800 cm-1.
4. Isotope Effect in Vibrational Spectrum
For different isotopically substituted molecules, the force constant remains almost same because the latter depends on the electronic configuration. However the reduced mass of the system changes appreciably and this causes significant change in the vibrational frequency. If for two isotopes the reduced masses are ?1 and ?2, then the ratio of the vibrational frequencies ?01 and ?02 are

…..(10)