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H2+ molecular ion overlap tunneling matrix element dependence on p+-p+ separation

  1. Apr 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Knowing that the wave function of the hydrogen atom is proportional to e-r/a0, estimate how t depends on the proton proton separation d. I need just to make a drawing of the integrand and make a reasonal approximation using my intuition (no numerical methods allowed to calculat the integral.)


    2. Relevant equations

    t is the overlap tunneling matrix element, with t= - <L|VL|R>

    with
    L= 1s- wavefunction centered on the left proton = psi1s(L-r)
    R= 1s- wavefunction centered on the right proton = psi1s(R-r)
    ( L or R is the position of the left or right proton, r is the position of the electron)

    VL= the attractive potential of the left proton on the electron= V(L-r)


    3. The attempt at a solution

    I just want to know whether my solution would be sufficient to this question, or whether I would need to do some more reasoning/ calculus/ plotting of functions.

    t= - <e-(L-r)/a0 | -e2/ (4*pi*e0*(L-r)) | e-(R-r)/a0>
    = e2/ (4*pi*e0) * (integral taken over dr) (1/(L-r)) * e-(L-r)/a0 -(R-r)/a0

    as d= | L-R |
    R= L-d

    so plugging that into the above equation for t gives:

    t= = e2/ (4*pi*e0) * (integral taken over dr) (1/(L-r)) * e(-L+r)/a0 (-L+d +r)/a0>

    When d decreases, the exponent of e gets larger , therefore t gets larger too. When d increases, the exponent of e gets smaller, therefore t gets smaller too.

    Is this it? I mean, the question said I needed to draw something and use my intuition, which I obviously did not do to solve this problem, this is just reasoning. What do I need to add?

    E.g. do I need to be more specific on the dependency other than 'getting bigger/ getting smaller', but say 'it behaves like an exponentially increasing function/ like a parabolic function etc'?

    Thanks in advance! I really appreciate your help!
     
  2. jcsd
  3. Apr 8, 2009 #2
    Please, help me, I keep getting stuck on this and the question is due tomorrow.

    Now I have the primitive function as the one in the attachment (with x for 'r', l for 'L' and r for 'R'). I don't know how to go further...
     

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