psi[n_,l_,m_,r_]=2/(n^2 Sqrt[Pochhammer[n-l,2l+1]]) (2r/n)^l LaguerreL[n-l-1,2l+1,2r/n] Exp[-r/n] SphericalHarmonicY[l, m, theta,phi]

psic[n_,l_,m_,r_]=2/(n^2 Sqrt[Pochhammer[n-l,2l+1]]) (2r/n)^l LaguerreL[n-l-1,2l+1,2r/n] Exp[-r/n] SphericalHarmonicY[l, m, theta,-phi]

lm={{0,0},{1,1},{1,0},{1,-1},{2,2},{2,1},{2,0},{2,-1},{2,-2},{3,3},{3,2},{3,1},{3,0},{3,-1},{3,-2},{3,-3}}

Table[Table[
Integrate[Integrate[Integrate[psic[4,First[lm[[i]]],Last[lm[[i]]],r]  psi[4,First[lm[[j]]],Last[lm[[j]]],r] r^2 Sin[theta],{phi,-Pi,Pi}],{theta,0,Pi}],{r,0,Infinity}],
{i,1,16}],{j,1,16}]
MatrixForm[%]

Table[Table[
Integrate[Integrate[Integrate[psic[4,First[lm[[i]]],Last[lm[[i]]],r] r Cos[theta] psi[4,First[lm[[j]]],Last[lm[[j]]],r] r^2 Sin[theta],{phi,-Pi,Pi}],{theta,0,Pi}],{r,0,Infinity}],
{j,1,16}],{i,1,16}]
Out[12]= {{0, 0, -6 Sqrt[5], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 

Out[5]//MatrixForm= 

                    0             0             -6 Sqrt[5]    0             0             0             0             0             0             0             0             0             0             0             0             0

                                                                                                   3
                                                                                          -12 Sqrt[-]
                    0             0             0             0             0                      5    0             0             0             0             0             0             0             0             0             0

                                                                                                          -24
                                                                                                        -------
                    -6 Sqrt[5]    0             0             0             0             0             Sqrt[5]       0             0             0             0             0             0             0             0             0

                                                                                                                               3
                                                                                                                      -12 Sqrt[-]
                    0             0             0             0             0             0             0                      5    0             0             0             0             0             0             0             0



                    0             0             0             0             0             0             0             0             0             0             -6            0             0             0             0             0

                                           3                                                                                                                                           2
                                  -12 Sqrt[-]                                                                                                                                 -12 Sqrt[-]
                    0                      5    0             0             0             0             0             0             0             0             0                      5    0             0             0             0

                                                  -24                                                                                                                                         -18
                                                -------                                                                                                                                     -------
                    0             0             Sqrt[5]       0             0             0             0             0             0             0             0             0             Sqrt[5]       0             0             0

                                                                       3                                                                                                                                           2
                                                              -12 Sqrt[-]                                                                                                                                 -12 Sqrt[-]
                    0             0             0                      5    0             0             0             0             0             0             0             0             0                      5    0             0



                    0             0             0             0             0             0             0             0             0             0             0             0             0             0             -6            0



                    0             0             0             0             0             0             0             0             0             0             0             0             0             0             0             0



                    0             0             0             0             -6            0             0             0             0             0             0             0             0             0             0             0

                                                                                                   2
                                                                                          -12 Sqrt[-]
                    0             0             0             0             0                      5    0             0             0             0             0             0             0             0             0             0

                                                                                                          -18
                                                                                                        -------
                    0             0             0             0             0             0             Sqrt[5]       0             0             0             0             0             0             0             0             0

                                                                                                                               2
                                                                                                                      -12 Sqrt[-]
                    0             0             0             0             0             0             0                      5    0             0             0             0             0             0             0             0



                    0             0             0             0             0             0             0             0             -6            0             0             0             0             0             0             0



                    0             0             0             0             0             0             0             0             0             0             0             0             0             0             0             0



Eigensystem[%]
Out[13]= {{-18, 18, -12, -12, 12, 12, -6, -6, -6, 6, 6, 6, 0, 0, 0, 0}, 
