Hydrogen molecule-ion

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

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

Δ = (E^(-D)) (1 + D + (1/3) (D^2))

Plot[-0.5 + (1/D) + ((J + K)/(1 + Δ)), {D, 0, 6}, 
      PlotRange -> {-0.6, -0.4}]

Plot[-0.5 + (1/D) + ((J - K)/(1 - Δ)), {D, 0, 6}, 
      PlotRange -> {-0.6, -0.4}]

Show[%, %%]

kinetic energy expectation values

Plot[(((1/2) + (((-1)/2) Δ - K))/(1 + Δ)), {D, 0, 6}]

Plot[(((1/2) - (((-1)/2) Δ - K))/(1 - Δ)), {D, 0, 6}]

Show[%, %%]

potential energies

Plot[((-1 + J + (1/D) + 2  K + (Δ/D))/(1 + Δ)), {D, 0, 6}, 
      PlotRange -> {-1.2, -0.5}]

Plot[((-1 + J + (1/D) - 2  K - (Δ/D))/(1 - Δ)), {D, 0, 6}, 
      PlotRange -> {-1.2, -0.5}]

Show[%, %%]

Hydrogen molecule

various integrals

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

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

Δ = (E^(-D)) (1 + D + (1/3) (D^2))

J2 = 
      (1/D) - (E^(-2 D)) ((1/D) + 
              (11/8) + (3/4) D + (1/6) (D^2))

Δ2 = (E^D) (1 - D + (1/3) (D^2))

K2 = 
      (1/5) (-(E^(-2 D)) (-(25/8) + ((23 D)/4) + 3 (D^2) + 
                  ((D^3)/3)) + (1/D) (6 ((Δ^2) (EulerGamma + Log[D]) + Δ(2^2) ExpIntegralEi[-4 D] - 
                      2 Δ Δ2 ExpIntegralEi[-2 D])))

symmetric spatial wave function

Plot[-1 + 
        (1/D) + ((2  J + J2 + 2  Δ K + K2)/(1 + (Δ^2))), {D, 0, 6}, 
      PlotRange -> {-1.2, -0.8}]

anti-symmetric spatial wave function

Plot[-1 + 
        (1/D) + ((2  J + J2 - 2  Δ K - K2)/(1 - (Δ^2))), {D, 0, 6}, 
      PlotRange -> {-1.2, -0.8}]

comparison

Show[%, %%]

Long-range behavior of the binding potential

long-range behavior of the binding potential, revealing that it does not correctly lead to the van der Waals potential.

Series[
        ExpIntegralE[1, z], {z, ∞, 2}] /. {z -> 
          2  D}

Normal[%]

Series[
        ExpIntegralE[1, z], {z, ∞, 2}] /. {z -> 
          4  D}

Normal[%]

Simplify[-1 + 
        (1/D) + ((2  J + J2 + 2  Δ K + K2)/(1 + (Δ^2))) + 1]

% /. {ExpIntegralEi[-4 
              D] -> -((E^(-4 D))/(16 (D^2))) + ((E^(-4 D))/(4 D)), 
        ExpIntegralEi[-2 
              D] -> -((E^(-2 D))/(4 (D^2))) + ((E^(-2 D))/(2 D))}

Simplify[Expand[%]]

it goes down exponentially because of (e^(2 D))factor in the denominator.

kinetic energy

symmetric

Plot[
      2 (( (1/2) + (((-1)/2) (Δ^2) - Δ K))/(1 + (Δ^2))), {D, 0, 6}, 
      PlotRange -> {0.8, 1.4}]

anti-symmetric

Plot[
      2 (( (1/2) - (((-1)/2) (Δ^2) - Δ K))/(1 - (Δ^2))), {D, 0, 6}, 
      PlotRange -> {0.8, 1.4}]

comparison

Show[%, %%]

potential energy

symmetric

Plot[((-2 + 2  J + J2 + (1/D) + 4  Δ K + K2 + (Δ^2) (1/D))/(1 + (Δ^2))), {D, 0, 6}, 
      PlotRange -> {-2.2, -1.6}]

anti-symmetric

Plot[((-2 + 2  J + J2 + (1/D) - 4  Δ K - K2 - (Δ^2) (1/D))/(1 - (Δ^2))), {D, 0, 6}, 
      PlotRange -> {-2.2, -1.6}]

comparison

Show[%, %%]