The accuracy of electronic structure calculations are affected to some degree by numerical errors. Assessing whether these errors are at an acceptable level for chemical accuracy is difficult. This paper demonstrates how interval arithmetic can be used to address this issue in the context of a Hartree-Fock computation. Using the method proposed here, the effect of system size and basis set type on the overall numerical accuracy of the Hartree-Fock total energy is studied. Consideration is also given to the impact of various algorithmic design decisions. Examples include the use of integral screening, computing some integrals in single precision, and reducing the accuracy of the interpolation tables used to compute the reduced incomplete Gamma function required by some integral evaluation algorithms. All of these issues have relevance to the use of novel computing devices such as graphics processing units (GPU) and the Sony Toshiba IBM Cell Broadband, to exascale and green computing, and to the exploitation of the emerging generation of massively multicore processors.