Diatomic molecule as non rigid rotator

In molecular physics and chemistry, the model of a diatomic molecule as a non-rigid rotator provides insight into how the molecule's rotation affects its energy levels. This model is an extension of the rigid rotor model and accounts for the fact that real molecules are not perfectly rigid due to vibrational motion and other factors.

Rigid vs. Non-Rigid Rotor Models

1. Rigid Rotor Model:

   • Assumes the bond length in the diatomic molecule is fixed and does not change.

   • Non-Rigid Rotor Model:

   • Accounts for the fact that the bond length can change due to rotational and vibrational coupling. In reality, as a diatomic molecule rotates, the bond length can vary slightly, leading to deviations from the rigid rotor model.

   • This model introduces the concept of the rotation-vibration coupling. As the molecule rotates, the bond length changes, which in turn affects the rotational energy levels.

Key Concepts in the Non-Rigid Rotor Model

1. Centrifugal Distortion:

   • As the rotational quantum number JJJ increases, the centrifugal force causes the bond length to increase. This effect is known as centrifugal distortion.

2. Rotational-Vibrational Coupling:

   • The non-rigid rotor model can be combined with vibrational models, leading to more complex energy expressions that include both rotational and vibrational contributions. The interaction between rotational and vibrational motions can be modelled by including terms that couple these two types of motion.

3. Energy Level Splitting:

   • The energy levels of a non-rigid rotator are split compared to a rigid rotor, and these splitting's can be observed experimentally in spectra. This allows for more accurate modelling of real molecular spectra.

     Accurate spectroscopic measurement reveal many deviation from rigid rotator approximation. A better model called non rigid rotator is represented in rotational system by a mass point. It is not connected with massless spring. As rotational quantum number increases action of centrifugal force increases internuclear distance there by decreasing the value of rotational constant. This change in energy level of diatomic is known as centrifugal distortion.

     The energy for non rigid rotator can be derived with the help of Schrodinger wave equation which is given as

     E_J=ℏ² IJ(J+1)/ 8𝜋²I - ℏ^4 / 32𝜋^4 I²r²k J²(J+1)² where E_J​ is the rotational energy, ℏ is the reduced Planck constant, I is the moment of inertia of the molecule, and J is the rotational quantum number.

     In term of term value

where B = h/ 8𝜋² I C is rotational constant and D = h³ / 32 𝜋^4 I²r²kc is centrifugal distortion constant which value is always positive. A massless spring like a chemical bond on the rotation if it will be stretched and compressed periodically with a fundamental vibrational frequency overline w then the force constant is given as

The variation between B, D and w bar is given as

D = 16B³𝜋²𝜇c² / k = 4B³ / overline w ....3

It is clear that if vibrational frequency increases then the centrifugal distortion constant decreases. Generally, the value of rotational constant is less than the vibrational frequency. let J and J+1 be the rotational constant for lower and upper level. Then the rotational term value for these level are given as

T_J = BJ(J+1) - DJ² (J+1)² ...4

and T_J+1 = BJ(J+1)(J+2) - D(J+1)²(J+2)²....5

The term shift for non rigid rotator is

ΔT = T_J+1 - T_J

= B(J+1)(J+2)- D(J+1)²(J+2)² - BJ(J+1)+ DJ²(J+1)²

= B(J+1)(J+2-J) - D(J+1)²[ (J+2)² - J²]

= 2 B(J+1) - 4D(J+1)³ ....6

Application

The non-rigid rotor model is especially useful in understanding the spectra of diatomic molecules and analysing rotational-vibrational spectra. It is more accurate for molecules with significant rotational effects on the bond length or those with high rotational quantum numbers. This model helps in predicting and interpreting the spectral lines observed in molecular spectroscopy.

In summary, the non-rigid rotator model extends the rigid rotor approximation by incorporating the effects of bond length changes due to rotational motion, providing a more accurate description of a diatomic molecule's rotational behaviour.

This note is a part of the Physics Repository