  Besides tensor contractions, one of the most pronounced computational
bottlenecks in the non-orthogonally spin-adapted forms of the quantum chemistry
methods CCSDT and CCSDTQ, and their approximate forms---including CCSD(T) and
CCSDT(Q)---are spin summations. At a first sight, spin summations are
operations similar to tensor transpositions; a closer look instead reveals
additional challenges to high-performance calculations, including temporal
locality as well as scattered memory accesses. This publication explores a
sequence of algorithmic solutions for spin summations, each exploiting
individual properties of either the underlying hardware (e.g. caches,
vectorization), or the problem itself (e.g. factorizability). The final
algorithm combines the advantages of all the solutions, while avoiding their
drawbacks; this algorithm, achieves high-performance through parallelization,
vectorization, and by exploiting the temporal locality inherent to spin
summations. Combined, these optimizations result in speedups between 2.4x and
5.5x over the NCC quantum chemistry software package. In addition to such a
performance boost, our algorithm can perform the spin summations in-place, thus
reducing the memory footprint by 2x over an out-of-place variant.
