using SpecialFunctions, WignerSymbols
include("helper.jl")

# Gaussian potentials in HO space
inv_factorial(n) = Iterators.prod(inv.(1:n))
sqrt_factorial(n) = Iterators.prod(sqrt.(n:-1:1))
N_lnk(l, n, k) = 1/sqrt_factorial(n) * binomial(n, k) * sqrt(gamma(n + l + 3/2)) / gamma(k + l + 3/2)
Talmi(l, R, k1, k2; μω_gen=1.0) = (-1)^(k1 + k2) * (1 + 1/(μω_gen * R^2))^-(3/2 + l + k1 + k2) * gamma(3/2 + l + k1 + k2)
V_Gaussian(R, l, n1, n2; μω_gen=1.0) = (-1)^(n1 + n2) * better_sum([N_lnk(l, n1, k1) * N_lnk(l, n2, k2) * Talmi(l, R, k1, k2; μω_gen=μω_gen) for (k1, k2) in Iterators.product(0:n1, 0:n2)])

# for numerical evaluation of V matrix elements
sqrt_double_factorial(n) = Iterators.prod(sqrt.(n:-2:1))
sqrt_sqrt_pi = sqrt(sqrt(pi))
laguerre(l, n, x) = gamma(n + l + 3/2) * better_sum([(-x * x)^k / gamma(k + l + 3/2) * inv_factorial(n - k) * inv_factorial(k)  for k in 0:n])
ho_basis(l, n, x) = (-1)^n / sqrt_sqrt_pi * 2^((n + l + 2) / 2) * sqrt_factorial(n) / sqrt_double_factorial(2*n + 2*l + 1) * x^(l + 1) * exp(-x^2 / 2) * laguerre(l, n, x)

# for implementation of simple relative coordinates
double_factorial(n::Int) = Iterators.prod(big, n:-2:1)

"Gaussian integral for n ∈ Integers (Ref: Wolfram MathWorld + simplifications)"
gauss_int(a, n) = double_factorial(n - 1) / (2 * a)^((n + 1)/2) * (iseven(n) ? sqrt(π / 2) : 1)

"Gives ∫dp p u' u where u' and u' may have different l (Ref: worked out in Mathematica)"
function integral(np, lp, n, l, μω)
    s = [(-1)^(m + mp) * gauss_int(1, 2 * m + 2 * mp + l + lp + 3) * N_lnk(l, n, m) * N_lnk(lp, np, mp) for (m, mp) in Iterators.product(0:n, 0:np)]
    return 2 * sqrt(μω) * better_sum(s)
end

"Gives <n' l'|| p ||n l> for the HO basis"
reduced_matrix_element(np, lp, n, l, μω) = (-1)^lp * sqrt(2*lp + 1) * sqrt(2*l + 1) * wigner3j(Float64, lp, 1, l, 0, 0, 0) * integral(np, lp, n, l, μω)

"Matrix element <n1p l1p n2p l2p| p1⋅p2 |n1 l1 n2 l2> (Ref: de-Shalit & Talmi, Eq 15.5)"
function racahs_reduction_formula(n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, μ1ω1, μ2ω2)
    val = wigner6j(Float64, l1p, l2p, Λ, l2, l1, 1)
    if val != 0; val *= reduced_matrix_element(n1p, l1p, n1, l1, μ1ω1); end
    if val != 0; val *= reduced_matrix_element(n2p, l2p, n2, l2, μ2ω2); end
    return (-1)^(l1 + l2p + Λ) * val
end