Calculating departure coefficients, bn, as well as bnl, for non-LTE gases/plasmas, is a fundamental computational problem in radio astronomy, physics of the interstellar medium, and for diagnostics of plasmas of nuclear reactors. Most work in this area was done in the 1960s and 1970s. Recent advances in computing technology have rendered the technology used in these two decades obsolete. Hence we ask if the approximate techniques developed to compensate for the technological limitations of the 1960s and 1970s are still needed.

In this thesis we introduce modern computational techniques to solve, exactly, the computational problems relating to departure coefficients. Specifically, we have made use of arbitrary precision arithmetic as well as introducing GPU & parallelization techniques to already established solutions.

We investigated the problem of the hypergeometric function which arises as the solution of the wave equation for hydrogen, which is the key component in Einstein coefficients, radiative recombination rates and the Stark broadening theory. Furthermore, we implemented, optimized and compared two different techniques for calculating bn coefficients and developed a matrix approach for dealing with the bnl problem.

We hope that the solutions resulting from this thesis will pave the way for further development in the outlined area, allowing for exact solutions up to n = 1000 and greater.