psi[1, 0] = 
        2 (a^(-3/2)) Exp[-r/a];
psi[2, 0] = (1/(Sqrt[2])) (a^(-3/2)) (1 - (r/(2  a))) 
          Exp[((-r)/(2  a))];
psi[2, 1] = (1/(Sqrt[24])) (a^(-3/2)) (r/a) 
          Exp[((-r)/(2  a))];
psi[3, 
          2] = (4/(81 (Sqrt[ 30]))) (a^(-3/2)) ((((r/a)))^2) 
          Exp[((-r)/(3  a))];

      
inList = {{1, 0}, {2, 0}, {2, 1}, {3, 2}};

ψ[n_, l_, m_] = 
        psi[n, l] SphericalHarmonicY[l, m, θ, ϕ];

fullScalarProduct[f_, g_] := 
        Integrate[
          Conjugate[f] g Sin[θ] (r^2), {ϕ, 0, 
            2  π}, {θ, 0, π}, {r, 0, ∞}, 
          Assumptions -> {a ∈  Reals, 
              a > 0}];
reducedScalarProduct[f_, g_] := 
Integrate[
          f g (r^2), {r, 0, ∞}, 
          Assumptions -> {a ∈  Reals,  
              a > 0}];
reducedNorm[
          f_] := 
Integrate[(f^2) (r^2), {r, 
            0, ∞}, 
          Assumptions -> {a ∈  Reals,  a > 0}];

Table[
      reducedNorm[psi[inList[[i, 1]], inList[[i, 2]]]], {i, 
        4}]

Off[ClebschGordan::tri];

      
R[initial_, final_] := 
        Module[{}, 
e (Sqrt[ 2  initial[[2]] + 
                  1]) (Sqrt[2  final[[2]] + 1])
ThreeJSymbol[{final[[2]], 0}, {initial[[2]], 
                0}, {1, 0}]
reducedScalarProduct[
              psi[initial[[1]], initial[[2]]],  
              r psi[final[[1]], 
                  final[[2]]]]
];

\(rate[j_, initial_, jp_, final_] := \ 
      Module[{temp1, temp2}, 
temp1 = 
          SixJSymbol[{j, jp, 1}, {final[\([2]\)], initial[\([2]\)], 1/2}]\^2; 
temp2 = 
          R[initial, final]\^2; 
 (*\ \(Uncomment\ if\ \(curious : \ Print[temp1, "and ", 
                  temp2]\);\)\ *) 
{Join[{j}, 
              initial] -> 
            Join[{jp}, 
              final], 
\((2  jp + 
                1)\) \(2\/ℏ\^4\) \(q\_\(initial -> final\)\^3\) \(2\/3\) temp1\ temp2}
]\)

rateTables = {rate[5/2, {3, 2}, 3/2, {2, 1}], 
        rate[3/2, {3, 2}, 1/2, {2, 1}], 
rate[3/2, {3, 2}, 
          3/2, {2, 1}]}

rateTables[[2, 2]] + rateTables[[3, 2]]

\(qPhoton[initial_, 
        final_] := \ \(\(\(α\^2\) 
              me\ c\)\/2\) \((1\/final[\([1]\)]\^2 - 
            1\/initial[\([1]\)]\^2)\); 
    rate2[j_, initial_, jp_, final_] := \ 
      Module[{temp1, temp2}, 
temp1 = 
          SixJSymbol[{j, jp, 1}, {final[\([\)\(2\)\(]\)], initial[\([\)\(2\)\(]\)], 1/2}]\^2; 
temp2 = 
          R[initial, final]\^2; 
 (*\ \(Uncomment\ if\ \(curious : \ Print[temp1, "and ", 
                  temp2]\);\)\ *) 
{Join[{j}, 
              initial] -> 
            Join[{jp}, 
              final], 
N[\((2  jp + 
                    1)\) \(2\/ℏ\^4\) \(qPhoton[initial, 
                      final]\^3\) \(2\/3\) 
                temp1\ temp2 //. {a -> ℏ\^2\/\(me\ e\^2\), 
                e -> \@\(α\ ℏ\ c\)}]}
];\)
finalConversion = {α -> 7.29735308 (10^(-3)),  
          c -> 2.99792458 (10^10), ℏ -> 
            1.05457266 (10^(-27)),  
          me ->  9.1093897 (10^(-28))};

rateTables = {rate2[5/2, {3, 2}, 3/2, {2, 1}], 
        rate2[3/2, {3, 2}, 1/2, {2, 1}], 
rate2[
          3/2, {3, 2}, 3/2, {2, 1}]}

rateTables /. finalConversion // N

(( (c^2) me )/ℏ) /. 
      finalConversion