(* K atom nach Kimball, Aussen-sp-Kugeln 19.12.2011, W. Heinzelmann, 1963 *)
Clear[k1,k2,k3,k4,sig1,sig2,sig3,c,z,R1,R2,R3,R4,a,b,k];
k = {k1 -> 1.0, k2 -> 1.0, k3->1.0, k4->1.0, sig1 -> 0.4, sig2 -> 0.4, sig3 -> 0.4};
z=19.0;
T = 2.25*k1/R1^2+9.0*k2/R2^2+9.0*k3/R3^2+1.125*k4/R4^2 /. k;
a = (R1+R2)/3+Sqrt[(R2+R3)^2-8*(R1+R2)^2/9];
b = a/3+Sqrt[(R3+R4)^2-8*a*a/9];
c = b+Sqrt[((R1+R2)/3)^2+8*(R1+R2)^2/9];
Vee = 3.0*sig1/R1+4.0*3.0*sig2/R2+4.0*3.0*sig3/R3+16/(R1+R2)+24/((R1+R2)*Sqrt[8/3])+
      16/a+48/(R2+R3)+16/(R1+R2+a)+24*Sqrt[3/8]/a+2/b+6/(R3+R4)+2/(a+b)+2/(b-R1-R2)+6/c /. k;
Vne = -3.0*z/R1-8*z/(R1+R2)-8*z/a-z/b;
Vir = (Vee+Vne)/T;
func = T+Vee+Vne;

t = FindMinimum[func, {R1,{0.072,0.0734}}, {R2,{0.245,0.255}}, {R3,{0.68,0.70}}, {R4,{1.93,1.94}}, {Method -> Automatic},{MaxIterations -> 500}]


N[-Vir /. t[[2]],10]

Show[Graphics[{Circle[{0,0},R1],Circle[{0,0},R2],Circle[{0,0},R3],Circle[{0,R1+R2},R2],
       Circle[{-0.866*(R1+R2),-(R1+R2)/2},R2],Circle[{0.866*(R1+R2),-(R1+R2)/2},R2],
        Circle[{-0.866*(R2+R3),(R2+R3)/2},R3],Circle[{0.866*(R2+R3),(R2+R3)/2},R3],
        Circle[{0,-(R2+R3)},R3],Circle[{0,0},R3],Circle[{-0.866*(R3+R4),-(R3+R4)/2},R4],
        Disk[{0,0},0.04],Disk[{-0.866*(R3+R4),-(R3+R4)/2},0.04],Disk[{0,R1+R2},0.04],
        Disk[{0,-(R2+R3)},0.04],Line[{{0,0},{-0.866*(R3+R4),-(R3+R4)/2}}],
        Line[{{0,R1+R2},{-0.866*(R3+R4),-(R3+R4)/2}}],Line[{{0,-(R2+R3)},{-0.866*(R3+R4),-(R3+R4)/2}}],
        Text[StyleForm["R3+R4", FontSize->16,FontWeight -> "Bold"],{-1.5,-1.3},{-1,0}],
        Text[StyleForm["b",FontSize->16,FontWeight -> "Bold"],{-1.3,-0.9},{-1,0}],
        Text[StyleForm["c",FontSize->16,FontWeight -> "Bold"],{-1.3,-0.4},{-1,0}]}] /. t[[2]],{AspectRatio -> Automatic,
    Axes -> True,GridLines -> Automatic, PlotRange -> {{-4.5,2.0},{-3.5,1.5}}, Frame -> True}]