Solution to Midterm Problem 3 part a)

Plotting The Heitler-London Vs. Morse Potentials.

Defining all the special functions as in the notes:

J = -(1/D) + (E^(-2 D)) (1 + (1/D));

Jp = 
        (1/D) - (E^(-2 D)) ((1/D) + 
                (11/8) + (3/4) D + (1/6) (D^2));

K = -(E^(-D)) (1 + D);

Δ = (E^(-D)) (1 + 
              D + (1/3) (D^2));

Δp = (E^D) (1 - D + (1/3) (D^2));

Kp = (1/5) (-(E^(-2 D)) (((-25)/8) + (23/4) D + 
                    3  (D^2) + (1/3) 
                      (D^3)) + (6/D) ((Δ^2) (EulerGamma + Log[D]) + Δ(p^2) ExpIntegralEi[-4 D] - 
                    2  Δ Δp ExpIntegralEi[-2 D]));

Plugging into the definition of the symmetric potential, we get the following beast:

Vs = 
      (1/D) + ((2  J + Jp + 2  Δ K + Kp)/(1 + (Δ^2)))

And the anti-symmetric beast is:

Va = 
      (1/D) + ((2  J + Jp - 2  Δ K - Kp)/(1 - (Δ^2)))

Vsplot = 
      Plot[Vs - 1, {D, 0, 6}, 
        PlotRange -> {-1.2, -0.8}]

Vaplot = 
      Plot[Va - 1, {D, 0, 6}, 
        PlotRange -> {-1.2, -0.8}]

Show[Vsplot, Vaplot]

VMplot = 
      Plot[-1 + 
            0.11597 ((E^(-2 (x - 1.64254)/b)) - 
                  2  (E^(-(x - 1.64254)/b))) /. {b -> 
              0.917193}, {x, 0, 6}, PlotRange -> {-1.2, -0.8},  
        PlotStyle -> {Dashing[{0.02, 0.01}]}]

Comparing (Morse Potential is Dashed):

Show[Vsplot, Vaplot, VMplot]