(* Na metal as {[Na(+1)(e)6/6}infG}, face centered cubic, ES 1963/2012 *)
Clear[k1,k2,k3,sig1,sig2,c,Z,R1,R2,R3];
c = {k1 -> 0.94009, k2 -> 1.3, k3 -> 1.017, sig1 -> 0.30, sig2 -> 0.30}; 
Z=11.0; M=1.74756; ENa=-4419.785; ad = Sqrt[3./8.];

(* He and Neon shell of Na+ *)
T = 2.25*k1/R1^2+9.0*k2/R2^2 /. c;
(* single electron in metal-orbital *)
T = T + 1.125*k3/R3^2 /. c;

Vee=3.0*sig1/R1+12.0*sig2/R2+16/(R1+R2)+24*ad/(R1+R2) /. c;
Vne=-3.0*Z/R1-8.0*Z/(R1+R2);
S2 = R2*4^(1/3);
(* Madelung lattice energy *)
MadE = - M*(Z-10.0)/(S2+R3) /. c;

(* Total energy, to be minimized *)
Etot = T + Vee + Vne + MadE ;

t = FindMinimum[Etot, {R1,0.125}, {R2,0.64}, {R3,2.56},{Method -> Automatic},{MaxIterations -> 500}]

(* Virial Ratio *)
N[-(Vee+Vne+MadE)/T /. c /. t[[2]],10]
(* Radius of Na+ in metal *)
N[S2 /. c /. t[[2]],5]
(* Lattice constant, angstrom *)
N[2*0.529177*(S2+R3) /. t[[2]],5]
(* Lattice energy at 0 K, eV *)
Del = t[[1]]*27.2114-ENa
(* Sublimation energy at 0 K, kJ/mol, kcal/mol *)
N[-Del*96.48532,5]
N[-Del*23.06055,5]