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# The rigid rotator.pptx

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# The rigid rotator.pptx

Quantum chemistry

Quantum chemistry

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### The rigid rotator.pptx

1. 1. THE RIGID ROTATOR Name: D.Muthu pandi Roll no: 22APCH017 Class : I- M.sc Chemistry
2. 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. 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. 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. 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. 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. 7. THANK YOU!