Estimating t Dependency on Proton Separation d

[Lisa...]

Homework Statement

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.)

Homework Equations

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

with
L= 1s- wavefunction centered on the left proton = psi_1s(L-r)
R= 1s- wavefunction centered on the right proton = psi_1s(R-r)
( L or R is the position of the left or right proton, r is the position of the electron)

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

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!

 

[Lisa...]

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...