Error with numerical integration for three parameters optimization

Question

I'm trying to optimize three parameters simultaneously using the following code:

(* Input  *)
NN = 4;
energy = 16;
α = 0;
β = 100;
z = 1;


(* Lambda values *)
λ1 = z;
λ2 = (2 energy + λ1^2)/6;
λ3 = (2 λ1*λ2 - α)/6;
λ4 = (4 λ2^2 + 6 λ1 *λ3 - 2 β)/
   20;


(* Find the optimized parameters *)
Clear[ψ, μ, f, Δ];
ψ[λ2_][λ3_][λ4_][r_] := 
  Exp[-r - λ2 r^2 - λ3  r^3 - λ4  r^4];

μ[λ2_][λ3_][λ4_][k_] := 
  NIntegrate[
   r^k ψ[λ2][λ3][λ4][r], {r, 
    0, ∞}];

f[λ2_][λ3_][λ4_][k_] := 
  k (k - 1) μ[λ2][λ3][λ4][k - 1] + 
   2 z μ[λ2][λ3][λ4][k] + 
   2 energy μ[λ2][λ3][λ4][k + 1] - 
   2 α μ[λ2][λ3][λ4][k + 2] - 
   2 β μ[λ2][λ3][λ4][k + 3];

Δ[λ2_][λ3_][λ4_] := 
  Sum[f[λ2][λ3][λ4][k]^2, {k, 1, NN}];

new = FindMinimum[Δ[λ2][λ3][λ4],\
 {λ2, λ3, λ4}] // Quiet;


λ2 = new[[2, 1]][[2]];
λ3 = new[[2, 2]][[2]];
λ4 = new[[2, 3]][[2]];

energy = λ2*3 - 1/2;
Print["New energy  ", energy];

but it returns errors like this

NIntegrate::inumr: The integrand E^(-r-(11 r^3)/6+(17 r^4)/5-r^2 λ2) has evaluated to non-numerical values for all sampling points in the region with boundaries {{∞,0.}}.

1- It is obvious that the integration procedure causes the errors, but I don't know how should I fix them?

2- Is my code to optimize three parameters true? (provided that I add a loop to my code) or does exist a more efficient way to do so?

Answer 1

Not completelly understood your computations, but this is my try. First, let me simplify a bit.

NN = 4;
energy = 16;
α = 0;
β = 100;
z = 1;

λ1 = z
λ2 = (2 energy + λ1^2)/6
λ3 = (2 λ1*λ2 - α)/6
λ4 = -(4 λ2^2 + 6 λ1*λ3 - 2 β)/20 (* changed sign here, otherwise the integral does not converge *)

Clear[ψ, μ, f, Δ];
ψ[λ2_?NumericQ, λ3_?NumericQ, λ4_?NumericQ, r_] := 
  Exp[-r - λ2 r^2 - λ3 r^3 - λ4 r^4];

μ[λ2_?NumericQ, λ3_?NumericQ, λ4_?NumericQ, k_] := 
  NIntegrate[r^k ψ[λ2, λ3, λ4, r], {r, 0, ∞}];

f[λ2_?NumericQ, λ3_?NumericQ, λ4_?NumericQ, k_] :=
 k (k - 1) μ[λ2, λ3, λ4, k - 1] + 2 z μ[λ2, λ3, λ4, k] + 
   2 energy μ[λ2, λ3, λ4, k + 1] - 
   2 α μ[λ2, λ3, λ4, k + 2] - 
   2 β μ[λ2, λ3, λ4, k + 3];

Δ[λ2_?NumericQ, λ3_?NumericQ, λ4_?NumericQ] := 
  Sum[f[λ2, λ3, λ4, k]^2, {k, 1, NN}];

min = NMinimize[{Δ[x2, x3, x4], x2 > 0 && x3 > 0 && x4 > 0}, {x2, x3, x4}]

{0.0000396903, {x2 -> 6.71696, x3 -> 0.000223156, x4 -> 0.000143381}}

And new energy:

newEnergy = (3 x2 - 1/2) /. min[[2]]

19.6509