set9 - The solu)on for electrons and proton will be done in...

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The solu)on for electrons and proton will be done in steps H e ψ r e ; r p 1 , r p 1 ( ) = E e r p 1 , r p 2 ( ) r e ; r p 1 , r p 1 ( ) 1. 2. H p p r p 1 , r p 2 ( ) = 2 2 m p p 1 2 + p 2 2 ( ) + e 2 4 πε 0 R 12 + E e r p 1 , r p 2 ( ) p r p 1 , r p 2 ( ) = E p r p 1 , r p 2 ( ) An now to solve the wave func)on of the nuclei moving in an eFec)ve poten)al We have done that… U r p 1 , r p 2 ( ) = e 2 4 0 R 12 + E e r p 1 , r p 2 ( )
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Diatomic molecules A simple picture: two balls connected by a spring U(r) r D
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Harmonic oscillator U(r) r D Near the minimum the poten±al energy can be approximated by a parbola U ( r ) U r min ( ) + dU dr r = r min r r min ( ) + 1 2 d 2 U dr 2 r = r min r r min ( ) 2 The firste derivative is zero at the minimum U r ( ) U r min ( ) + 1 2 d 2 U dr 2 r = r min r r min ( ) 2 The value of the potential is a constant, that for the moment we do not care about U r ( ) k 2 r r min ( ) 2 r min
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Classical spring A: amplitude ; k: force constant ; m: mass ; ω: frequency Energy E = 1 2 mv 2 + 1 2 kx 2 Newton Eq.
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set9 - The solu)on for electrons and proton will be done in...

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