Recent significant advances in our understanding of the early Universe can be attributed to impressive developments in numerical hydrodynamics, radiative transfer, and nucleosynthetic modeling, as well as spectacular observational efforts.  However, while it is accepted that microphysical processes play an important, if not controlling, role in the formation and evolution of the first stars, improvements in the treatment of chemistry and molecular excitation have been lacking.  This program of new atomic and molecular calculations, in conjunction with astrophysical modeling of early star formation and evolution, will bring the treatment of primordial microphysics to state-of-the-art.  Explicit quantum-mechanical calculation of various hydrogen and deuterium reactions will be combined with current chemistry and cooling models and added into hydrodynamical calculations of a range of early star formation and evolution pictures.  Emphasis will be placed on non-LTE molecular level populations and the resultant molecular spectra.  The primordial chemical/cooling network will then be extended to include heavy elements such as carbon and oxygen.  Astrophysical studies including this state-of-the-art microphysics will investigate the photon escape fraction from Population III stars embedded in halos, the formation of low-mass Population III stars in fossil ionized hydrogen regions, and metal dispersal behavior from Population III supernovae.  This work will advance the understanding of primordial and low-metallicity chemistry by applying state-of-the-art atomic and molecular computations to new problems.Results will be added to a web-accessible database of reactions and code modules relevant to primordial astrochemistry.  The project includes the training of graduate students, and extending current activities for the participation of undergraduate students and under-represented groups, enhancing the infrastructure of astrochemical research and strengthening interdisciplinary collaborations.  In addition, the techniques and results are applicable to other areas such as fusion energy, combustion research, and atmospheric chemistry.