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Chem120A+Notes+-+Rigid+Rotor

Chem120A+Notes+-+Rigid+Rotor - Rotational Energy Levels 2D...

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Rotational Energy Levels
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2D Rotation r φ O x y The moment of inertia 2 2 2 2 d E I d φ Φ - = Φ h and the periodic boundary condition ( ) ( 2 ) φ φ π Φ = Φ + Solution: 2 I mr = Schrodinger Equation 2 2 ( ) e , 0,1,2, 2 in n n E n I φ φ - Φ = = = h L Wavefunction Energy Rotational excitation energy decreases as the moment of inertia increases.
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Estimation of Rotation Excitation Energy 2 2 2 2 2 ( 1) 2 2 1 ( ) 2 n n E I I n I + = - = + h h h Excitation Energy Given that the reduced mass of a diatomic molecule is typically around 10 -25 kg, and that the bond length is around 1 Å, the excitation energy for n = 0 to 1 can be calculated as 2 34 2 24 25 10 2 10 8 2 10 (1.05 10 J s) 5 10 J 2 2 10 kg (10 m) = 5 10 Hz 2 2 3.14 3 10 = 4 10 m 5 10 E I E c ϖ π λ ϖ - - - - - × = = × × × = × × × × = × × h h Rotational transitions of diatomic molecules occur in the microwave region, and Rotational excited states are thermally accessible at room temperature. (vibrations are in the time scale of ps)
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Rigid Rotor in 3D Space A particle rotates on a sphere surface around the origin point. Two degrees of freedom are the polar angle θ and the azimuthal angle φ . θ φ y x z O r Solution: 2 2 2 2 1 1 sin ( , ) ( , ) 2 sin sin Y EY I θ θ φ θ φ θ θ θ θ φ - + = h Schrodinger Equation Energy 2 ( 1) 2 l l l E I + = - h 1/ 2 | | (2 1)( | |)! ( , ) (cos ) 4 ( | |)! m m im l l l l m Y P e l m φ θ φ θ π + - = + Spherical harmonic wavefunction | | ( ) m l P x The associated Legendre ploynomials 0, 1, 2, , m l = ± ± ± L 0,1,2, l = L
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