BREAKDOWN OF THE BORN - OPPENHEIMER APPROXIMATION : THE INTERACTION ROTATIONS AND VIBRATIONS

According to Born-Oppenheimer approximation rotations and vibrations can process independently of each other. A molecule vibrates some 10³ times during a rotation. so the length and moment of inertia and B also change continually during the rotation of the vibration is simple harmonic the bond length is the same as the equilibrium bond length and will not vary with vibrational energy. However rotational constant depends on 1/r². The average value of 1/r² is not same as 1/r²ₑq is the equilibrium bond length. Further an increase in the vibrational amplitude and hence the value of B depends on the vibrational quantum number v.

In the anhormonic vibrations, the situation more complex increases in vibrational energy increases the average bond length (rₐᵥ). The rotational constant varies even more vibrational energy. Since rₐᵥ increases with vibrational energy, B is smaller in the upper vibrational state than in the lower vibrational state. As an approximation of the value Bᵥ the rotational constant in the vibration level v is given by,

Bᵥ = Bₑ - α(v+1/2)..............(1)

Bₑ - equilibrium rotational constant α - a small positive constant.

Consider the fundamental vibrational change ie the change v = 0 →v = 1, and the respective B values B₀ and B₁. For this transition.

∆∈ = ∈ⱼ', v = 1 − ∈ⱼ", v =0 = ω̄₀ + B₁j' (j'+1) −

B₀ j" (j"+1) cm⁻¹ where ω₀ = ωₑ (1−2xₑ)

When then have two cases

(i) ∆J = +1, J' = J" + 1

∆∈ = v̄R = ω̄₀ + ( B₁ + B₀) ( J" + 1)² cm ⁻¹      ( J" = 0, 1,2,............)......... (2)

(ii) ∆J = −1, J' = J" + 1

∆∈ = v̄P = ω̄₀ + (B₁+B) (J' +1) + (B₁ + B₀)

(J' + 1)² cm⁻¹ (J' = 0,1,2.....)..........(3)

Where we have written vP and vR to represent the wave number respectively two equation can be determined as

V̄ P, R = ω̄₀ + ( B₁ + B₀) m (B₁ − B₂) m² cm⁻¹

     ( m = ±1, ±2, ±3...........)...........(4)

Where positive m values refer to the R branch negative to P branch lines. Since B₁ < B₂. The last terms, above equation is negative. It's effect on the spectrum of diatomic molecule is to crowd the rotational lines in the R branch lines more widely spaced as negative in increases.