In this report, we motivated and verified two changes of bases in order to express the Coulomb collisonal operator in terms of Hermite-Laguerre polynomials using two coefficients, $T_{lk(m)}^{pj}$. Implementation of these coefficients was performed using the Wolfram Mathematica and Fortran programming languages, valuable for its numerical accuracy and its computational performance respectively. These computations were shown numerically unstable in floating-point arithmetic, requiring the use of a multiple-precision library in Fortran. Finally, the performance of the code was optimized by implementing an alternate derivation of the $T_{lkm}^{pj}$ coefficients, as well as suggesting two postulates.
Further work could be conducted, especially to prove postulates \labelcref{eq:t4postulate,eq:t5postulate}. Investigating the existence of recursion relations between coefficients \labelcref{eq:T4lorenzo,eq:t5jorge,eq:t5stenger} could also valuable to reduce computational times. An increase of performance could be obtained easily by parallelising the code, eg. by using a \emph{Message Passing Interface} (MPI) implementation, since the computation of any given coefficient is independent from the others. This independence suggests an ideal speedup, corresponding to the number of allocated CPUs, could be obtained.