  This Article presents two optimized multi-GPU algorithms for Fock matrix
construction, building on the work of Ufimtsev et al. and Barca et al. The
novel algorithms, opt-UM and opt-Brc, introduce significant enhancements,
including improved integral screening, exploitation of sparsity and symmetry, a
linear scaling exchange matrix assembly algorithm, and extended capabilities
for Hartree-Fock caculations up to $f$-type angular momentum functions. Opt-Brc
excels for smaller systems and for highly contracted triple-$\zeta$ basis sets,
while opt-UM is advantageous for large molecular systems. Performance
benchmarks on NVIDIA A100 GPUs show that our algorithms in the EXtreme-scale
Electronic Structure System (EXESS), when combined, outperform all current GPU
and CPU Fock build implementations in TeraChem, QUICK, GPU4PySCF, LibIntX,
ORCA, and Q-Chem. The implementations were benchmarked on linear and globular
systems and average speed ups across three double-$\zeta$ basis sets of
1.5$\times$, 5.2$\times$, and 8.5$\times$ were observed compared to TeraChem,
GPU4PySCF, and QUICK respectively. Strong scaling analysis reveals over 91%
parallel efficiency on four GPUs for opt-Brc, making it typically faster for
multi-GPU execution. Single-compute-node comparisons with CPU-based software
like ORCA and Q-Chem show speedups of up to 42$\times$ and 31$\times$,
respectively, enhancing power efficiency by up to 18$\times$.
