Proving existence of negative Hydrogen Ions

Veryfying the integral ∫(d^3) x (d^3) 
          y Exp[-a | x | -b | y |  \(TraditionalForm\`1\/\(\(|\)\(x\&_\ \  - \ y\&_\)\(|\)\)\)

Assuming[{x > 0, y > 0}, 
      Integrate[
        Sin[θ]/(Sqrt[(x^2) + (y^2) - 2*x*y*Cos[θ]]), {θ, 
          0, π}]]

Assuming[{a > 0, b > 0}, 
      Integrate[
        x*y*(x + y - Abs[x - y])*Exp[-a*x]*Exp[-b*y], {x, 
          0, ∞}, {y, 
          0, ∞}]]

Everything  * (((4  π))^2)*1/2

The integral will be denoted by:

g[a_, 
        b_] := ((2 ((a^2) + 3 a b + (b^2)))/((a^2) (b^2) (((a + b))^3)))

Sprawdzenia

Exchange integral for the antisymmetric case

α^3*β^3*
      Factor[g[2  α, 2  β] - 
          g[α + β, α + β]]

Exchange integral for the symmetric case

α^3*β^3*
      Factor[g[2  α, 2  β] + 
          g[α + β, α + β]]

Antisymmetric Case

(x and y used instead of α and β)

Kin[x_, y_] := (x^2 + y^2 - 128*x^4*y^4/(x + y)^6)/2

Pot1[x_, y_] := x + y - 64*x^3*y^3/(x + y)^5

Pot2[x_, y_] := 
      x*y*(x - y)^2*(x^2 + 7  x*y + y^2)/(x + y)^5

SSNorma[x_, y_] := 1 - x^3*y^3*64/(x + y)^6

For the hydrogen ion:

Energy2[x_, 
        y_] := (Kin[x, y] - 
            e*(Pot1[x, y] - Pot2[x, y]))/(SSNorma[x, y] + 
            0.000001)

For the hydrogen atom:

Energy1[x_] := x^2/2 - e*x

e = 1/137.0

Energy1[e]*10^{5}

ContourPlot[-Energy2[x, y], {y, 0.0001, 0.009}, {x, 0.0001, 0.009}, 
      PlotPoints -> 30, Contours -> 90]

ContourPlot[-Energy2[x, y], {y, 0.0072, 0.00735}, {x, 0.0000001, 
        0.0002}, PlotPoints -> 30, Contours -> 90]

Energy2[0.0000001, e]*10^5

(minimum at the edge; Energy2=Energy1)

Symmetric Case

Clear[AKin, APot1, APot2, x, y, Energy2]

\(\(\(
\)\(AKin[x_, 
        y_] := \((x^2 + y^2 + 128*x^4*y^4/\((x + y)\)^6)\)/
        2


    APot1[x_, y_] := x + y + 64*x^3*y^3/\((x + y)\)^5


    APot2[x_, y_] := 
      x*y*\((x^4 + 5  x^3*y + 28*x^2*y^2 + 5  y^3*x + 
              y^4)\)/\((x + y)\)^5

    \)\)\)

ANorma[x_, y_] := 
      1 + x^3*y^3*64/(x + y)^6

AEnergy2[x_, 
        y_] := (AKin[x, y] - e*(APot1[x, y] - 1*APot2[x, y]))/
        ANorma[x, y]

ContourPlot[-AEnergy2[x, y], {x, 0.00002, 0.009}, {y, 0.00002, 
        0.009}, PlotPoints -> 30, Contours -> 190]

FindMinimum[AEnergy2[x, y], {{x, 0.00755}, {y, 0.00205}}]

Thus, the minimal AEnergy2= -2.73484*(10^(-5)) < Energy1 which proves, that the negative ion in the symmetric case has a stable bound state.

Chandrasekhar originally found the minima at x=e*1.039, y=e*0.283 (the same values as ours)