1) when the bond is elastic, a molecule may have vibrational energy and vibrate with certain vibrational fundamental frequency dependent on the masses of the atoms and elasticity of the bond force constant. If the vibrational motion is simple harmonic, the force constant is given by
k= 4π²c²ω̄²μ
Where ω̄ is the vibrational frequency in cm⁻¹. The Variation of B with J is determining by the force constant. The weaker the bond will be more readily will it distort under centrifugal forces.
2) Due to elasticity, r and B vary during vibration. When these quantities are measured from the microwave techniques, many hundreds of vibrations occur during a rotational and hence the measured value is an average from the definition of B, we have
B= h/8π²lc
= h∕8π²cμr² (or) B α 1/r²
Since all other qualities are independent of vibration. But simple harmonic motion, a molecular bond is compressed and extended to an equal amount on each side of the internuclear distance and the average value of the distance is therefore unchanged, the average value of 1/r² is not equal to 1/re² where rₑ is the equilibrium internuclear distance.
Hence there will be three different B values. At equilibrium distance rₑ the rotational constant is B₀. When the molecule is in the ground vibration energy level, the average internuclear distance is r₀ with the rotational constant B₀. When the molecule has excess vibrational energy the quantities are rᵥ and Bᵥ where v is the vibrational quantum number.
Solving the Schrodinger wave equation for a non-rigid molecule, the rotational energy is given by
Eⱼ= h² J (J+1)− h² J²(J+1)²
8π²l 32π⁴ l²r²k
joules.............. (1)
Or ∈ ⱼ = Ej = BJ(J+1) - DJ² (J+1)² cm⁻¹
hc
..............(2)
Where B is the rotational constant and D is the centrifugal distortion constant, which is a positive quantity.
B= h/8π² lc and D= h²/32π²l²r²k/ cm⁻¹
The equation (1) is applicable to simple harmonic oscillator. If the force field is harmonic, the expression becomes,
∈ ⱼ = BJ(J+1) − DJ²+ (J+1)² +HJ³ (J+1)³+
kJ⁴ (J+1)⁴.............cm⁻¹
Where H, k etc are small constant depending upon the geometry and negligible compared to D. From the defining equation of B and D we have
D= 16B³π²μc²/k = 4B/ω̄²
Where ω̄ is the vibrational frequency. It's of the order of 10³cm⁻¹ while B is found to be order of 10cm⁻¹ and D will be of the order of 10⁻³cm⁻¹ and small compared with B. For small J values the correction term DJ²(J+1)² is negligible while for J values greater than 10, it may become appreciable,
The selection rule is ∆J = ±1
∆∈ ⱼ→ⱼ₊₁ =∈ⱼ₊₁−∈ⱼ=B [(J+1) (J+2)−J (J+1)]−D[(J+1)² (J+2)² − J² (J+1)²] =2B(J+1) − 4D(J+1)² = 2B (J+1) −4D(J+1)³ cm⁻¹
Thus the spectra of the non-rigid rotor is similar to that the rigid rotor except that each line is displaced slightly to lower frequency, the displacement increasing with (J+1)³. The rotational energy levels and the spectra of the rigid rotor and non-rigid rotor are shown below the figure.
The change in rotational energy levels and rotational spectrum and when passing from rigid to a non-rigid diatomic molecule. Levels on the right calculated using
D = 10⁻³B.
A knowledge of D gives two useful information. It shows us to determine the J values of lines from which they arise. Secondly a knowledge of D helps us to determine the vibrational frequency of a diatomic molecule though appropriately.
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