This thesis describes several new theoretical developments that facilitate
the computation of the exact exchange energy, a vital component of accurate
molecular simulations. The primary technique on which these
developments are based is the resolution of the identity approximation,
particularly the pair atomic resolution of the identity approximation (PARI).
We prove that computation of exact exchange using the PARI approximation
is variationally stable, and provide benchmarks of the performance and accuracy
of our implementation. We then show that the most commonly used SCF convergence acceleration algorithm, DIIS,
enables the design of a new fast exchange algorithm that we designate as occ-RI-K.
Next, we combine the preceding occ-RI-K algorithm with the PARI approximation
to create a linear-scaling
exact exchange algorithm for the specific case of large weakly-interacting systems.
Finally, we discuss our development of a high-level, object-oriented software library for
block-sparse tensor operations. This library provides the underlying implementation
for all of the algorithms discussed in this thesis.