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- 1. THE RIGID ROTATOR Name: D.Muthu pandi Roll no: 22APCH017 Class : I- M.sc Chemistry
- 2. The rotation of diatomic molecule in Space where the bond length is assumed to remain unchanged during rotation The theory of Such rigid rotator Space is useful in dealing with rotational spectra of diatomic molecules. This the basic Principle of rotational spectroscopy. It occurs in Microwave region of the series. Rigid Rotator: Let us Consider homonuclear diatomic molecule There are 3 types of rotation. Along x, y and z axis, But the internuclear axis of rotation is negligible It can rotate about. an axis is perpendicular to the internuclear distance (i.e. y and z axis)
- 3. The Postulate of Quantum mechanics, ĤΨ = 𝐸Ψ …….(1) Ĥ = KÊ+𝑃E This case(rigid rotator motion) potential energy is zero, because no force of attraction. Why this called rigid rotator, The internuclear distance does not change so they called rigid rotator. K.E= 1 2 𝑚𝑣2 In linear momentum K.E= 1 2 𝐼ω2 Angular momentum (m=I) I = mr2 μ= 𝑚1 𝑚2 𝑚1 +𝑚2 I is moment of inertia In this case m is reduce mass KÊ = 𝐼(𝐼ω2 ) 2𝐼 = 𝐿2 2𝐼 = Ĥ (K.E= 𝑃2 2𝑚 )
- 4. The Schrodinger wave equation, can not be Solved readily Cartesian equation So we use for spherical Polar Coordinates, Instant of Cartesian Polar Coordinate (r,θ,Φ spherical polar coordinates) r= 0 to ∞ Θ= 0 to 2 Φ= 0 to 𝐿2= ħ2 [ 1 sin θ 𝜕 𝜕𝜃 (sin 𝜃 𝜕 𝜕𝑥 ) + 1 sin2 𝜃 𝜕2 𝜕2 ∅ ] ……(2) ħ2 2𝐼 [ 1 sin θ 𝜕 𝜕𝜃 (sin 𝜃 𝜕Ψ 𝜕𝑥 ) + 1 sin2 𝜃 𝜕2 Ψ 𝜕2 Φ ]= EΨ ……(3) The total wave function Ψ(𝜃, ∅)= θ 𝜃 Φ(∅) Total spherical harmonics Separation of variables: (L.H.S) sin 𝜃 θ 𝜕 𝜕𝜃 (sin 𝜃 𝜕θ 𝜕𝑥 ) + 8𝜋2 𝐼𝐸 ħ2 sin2 𝜃
- 5. = - 1 Φ 𝜕2 Φ 𝜕2 Φ (R.H.S) L.H.S= R.H.S = m2 m2= constant β= 8𝜋2 𝐼𝐸 ħ2 - 1 Φ 𝑑2 Φ 𝑑2 Φ = m2 𝑑2 Φ 𝑑2 Φ + m2Φ= 0 Φ(∅) =Ne ± im∅ ‘N’ is normalization constant The wave function is available of(Φ) is acceptable when m= 0,±1,±2,…. ∅= 0 to 2 by applying normalization condition the value of ‘N’ can be obtained Φ(∅) = 1 2𝜋 e ± im∅ (Normalized function) L.H.S. On solving LHS equation (depends on only θ) using assoiated legendre polynomial (PL 𝑚 cos 𝜃 )
- 6. θ(𝜃)= N PL 𝑚 cos 𝜃 N = ( 2𝑙+1 2 × (𝑙− 𝑚 ! (𝑙+ 𝑚 ! ) 1 2 β = l (l+1) ‘l’ is rotational quantum number The energy of eigen values are, β= 8𝜋2 𝐼𝐸 ħ2 = l (l+1) E = 𝑙 𝑙+1 ℎ2 8𝜋2 𝐼 (l=0,1,2,…..) In rotational(mw) spectroscopy, E = τ τ+1 h2 8π2 I (𝜏 =0,1,2,…….) I (𝜇r2) 𝜇 = 𝑚1 𝑚2 𝑚1 +𝑚2 From RS bond distance is obtained
- 7. THANK YOU!