Wavefunctions of the Morse Potential

Transcription

1 Wavefunctions of the Morse Potential The Schrödinger equation the Morse potential can be solved analytically. The derivation below is adapted from the original work of Philip Morse (Physical Review, 34, 57, 99). V( r) D e e β r r ( e) = Morse Potential V(r) qualitatively reproduces the potential energy surface (PES) diatomics: minima at r e, strong internuclear repulsion at r < r e, and dissociation at energy D e with increasing r. The parameter β determines the anharmonicity of the PES. Solution of the Schrödinger equation leads to an expression the radial wavefunction, Ψ, (angular momentum not considered) Ψ β, k, v, x and vibrational energy levels k k β e β x k v e ( k e β x := ) L v, k v, k e β x N( v) E v = ω e v + ωx e v + ω e is the harmonic constant, ωx e (ω e x e ) is the anharmonic constant, v is the vibrational quantum number, and x is the displacement from equilibrium (x = r r e ). The Morse potential is unique in that there exists a finite number of vibrational levels. The parameter k is twice v max. D e and β can be rewritten in terms of ω e and ωx e. µ is the reduced mass. ω e 8 π c µ ωx e ω e D e = β = v max = 4 ωx e h ωx e Nv is a normalization constant (which doesn't work right now...) N( v) Γ ( k v) v Γ ( k v + s ) = N( v) Γ ( s + ) s = 0 L(n,α,x) is the general Laguerre function, calculated via its recurrence relation, (Handbook of Mathematical Functions). L( n, α, x) Laguerre 0 Laguerre + α x m while m < n Laguerre m m + α + x m + α Laguerre m+ m + m m + Laguerre n Laguerre m Vibrational Wavefunctions of the Morse Potential.mcd Roy Jensen 09 0

2 Molecular and Spectroscopic Parameters mass := 4 gm mass := 4 gm weights of the atoms or effective masses in mol mol DIM approximation mass mass µ := µ =.000 gm reduced mass mass + mass mol Lower State Upper State r e.gs :=.04 Å r e.xs :=.067 Å equilibrium bond length ω e.gs := 86.3 cm ω e.xs := cm harmonic constant ωx e.gs := 35.3 cm ωx e.xs := 34.4 cm anharmonic constant ω e.gs ω e.xs vibs max.gs := vibs max.xs := maximum vibrational quanta in PES ωx e.gs ωx e.xs vibs max.gs = 6 vibs max.xs = 6 vibs max.gs := 4 vibs max.xs := 4 maximum vibrational quanta to consider Range & Graphing Parameters r min := 0.6 Å r max :=.0 Å points := 500 ω e.gs 4 ωx e.gs = cm.4 chart max := a 0 cm ω e.gs ( vibs max.gs + 0.5) ωx e.gs vibs max.gs t := 0.. chart max.4 a cm ω e.xs ( vibs max.xs + 0.5) ωx e.xs vibs max.xs max( a) factor := 300 expansion factor to display the wavefunction Vibrational Wavefunctions of the Morse Potential.mcd Roy Jensen 09 0

3 Electronic Potential, Wavefunctions, and Energies Ψ has the following column structure: x axis electronic PES v=0 wavefunction v= v=... /Å /cm - /Å β Ψ gen ω e, ωx e, r e, vibs 8 π c µ ωx e := constants h ω e k floor ωx e ω e D e 4 ωx e i 0.. points r max r min r i r min + points V i D e e β ( r i r e ) temp r Å temp augment temp, i 0.. vibs j 0.. points V i cm Å Ψ tempj Ψ β, k, i, r j r e temp augment( temp, Ψ temp ) x temp, 0 temp 0, 0 temp i 0.. vibs Norm x temp i+ temp i+ temp i+ Norm generates the x-range at regular intervals generates the PES generates the vibrational wavefunctions each vibrational level normalizes the wavefunction Ψ GS := Ψ gen ω e.gs, ωx e.gs, r e.gs, vibs max.gs generates the PESs the lower and upper states Ψ XS := Ψ gen ω e.xs, ωx e.xs, r e.xs, vibs max.xs Vibrational Wavefunctions of the Morse Potential.mcd 3 Roy Jensen 09 0

4 := temp submatrix Ψ 0 E gen ω e, ωx e, Ψ, vibs temp i 0.. vibs (,, points,, vibs + ) factor E i ω e ( i + 0.5) ωx e ( i + 0.5) temp i temp i E i + cm E GS := E gen ω e.gs, ωx e.gs, Ψ GS, vibs max.gs E XS := E gen ω e.xs, ωx e.xs, Ψ XS, vibs max.xs adds energy to the vibrational wavefunctions to separate when displaying Vibrational Wavefunctions of the Morse Potential.mcd 4 Roy Jensen 09 0

5 Morse Potential Wavefunctions. 0 4 Upper State Energy /cm- (& Wavefunction) + Te Lower State Energy /cm- (& Wavefunction) Internuclear Distance Vibrational Wavefunctions of the Morse Potential.mcd 5 Roy Jensen 09 0

6 Franck-Condon Factor (overlap of wavefunctions) x := Ψ GS 0 S := S i 0.. vibs max.gs j 0.. vibs max.xs points S i, j k = 0 Ψ GSk i + Ψ XSk x,, j+ Ψ GS0,, 0 v' v" S = Vibrational Wavefunctions of the Morse Potential.mcd 6 Roy Jensen 09 0