MIT OpenCourseWarehttp://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.δ g() Absorption Lineshape for the Displaced Harmonic Oscillator Model Finite temperature calculations Lets work in units of k := 1 hbar := 1 Define the frequency of the electronic transition:Ω0 := 10 ωeg := Ω0 and the vibrational frequency: ω0 := 1 The unitless displacement of the two harmonic wells is: D := 0.5 − 1 ⋅Define thermal occupation factor nT := ⎜⎛ exp ⎛ hbar ω0 ⎟⎞ ⎞( ) ⎜− 1⎟ ⋅ := if g = 0 , 1 , 0 ⎝ ⎝ kT⎠ ⎠ ( ) Set up frequency grid: i := 0 .. 100 ωi := −5 + Ω0 + 0.1 i⋅ Absorption lineshape: JK− D⋅(2n T⋅ ()+1)10 10 ⎡D+ J K ⎤ σω, T ) := π⋅e ⋅⋅( () + 1) ⋅ ()⋅ δω− ω − (J − K( ∑ ∑ ⎣⎢ J!⋅K! nT nT⎣⎡ ⎣⎡ eg )⋅ω0⎦⎤⎦⎤⎦⎥ J = 0 K = 0 Envelope of vibronic progression: π ⎡⎢ −(ω− ωeg − D⋅ω0)2 ⎤⎥Env(ω, T) := ⎡ 2 ⎤⋅exp⎢ 2 ⎥ ⎣D⋅ω0 ⋅( ⋅ () + 1)⎦ ⎣ ⋅⋅ω0 ⋅( ⋅⎦2nT2D () + 1)2nTPlot lineshapes for low, mid and high temperatures. Temperature is defined relative to nuclear frequency (T/ω) ω0 a T = ∞() := 0.01 T 3 σω( i , T) 2 Env(ωi , T)1 0− 6 − 4 − 2 0 2 4 6 ωi−ωeg ω0b T := 1 = 1() T 2 1.5 σωi , T( ) 1 Env(ωi , T) 0.5 0− 6 − 4 − 2 0 2 4 6 ωi−ωeg ω0 c T := 3 = 0.333() T 1.5 σω( i , T) 1 Env(ωi , T) 0.5 0− 6 − 4 − 2 0 2 4 6 ωi−ωeg2− 1− 0 1 σωi 2,( ) σωi 1,( )1−σωi 0.01,( )2−ω0 T 0.333= ω0 T 1= ω0 T ∞= D = 0.5 ω0 = 1 − 4 − 2 0 2 4 6
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