  The accurate prediction of singlet and triplet excitation energies is of
significant fundamental interest and is critical for many applications. An area
of intense research, most calculations of singlet and triplet energies use
time-dependent density functional theory (TDDFT) in conjunction with an
approximate exchange-correlation functional. In this work, we examine and
critically assess an alternative method for predicting low-lying neutral
excitations with similar computational cost, the ab initio Bethe-Salpeter
equation (BSE) approach, and compare results against high-accuracy
wavefunction-based methods. We consider singlet and triplet excitations of 27
prototypical organic molecules, including members of Thiel's set, the acene
series, and several aromatic hydrocarbons exhibiting charge-transfer-like
excitations. Analogous to its impact in TDDFT, we find that the Tamm-Dancoff
approximation (TDA) overcomes triplet instabilities in the BSE approach,
improving both triplet and singlet energetics relatively to higher level
theories. Finally, we find that BSE-TDA calculations built on good DFT starting
points, such as those utilizing optimally-tuned range-separated hybrid
functionals, can yield accurate singlet and triplet excitation energies for
gas-phase organic molecules.
