Collapse and fragmentation of molecular cloud cores. I - Moderately centrally condensed cores

3D calculations of the collapse of moderately centrally condensed molecular cloud cores with varied thermal and rotational energies are presented. The calculations are carried out using a newly developed and tested second-order accurate radiative hydrodynamics code. Because of the use of a second-order accurate numerical scheme and initial clouds that resemble both observed prolate molecular cloud cores and magnetically supported clouds at the initiation of the dynamic collapse phase, the new models provide a superior estimate of the likelihood of fragmentation as a mechanism for binary star formation.