  The accurate description of biradical systems, and in particular the
resolution of their singlet-triplet gaps, has long posed a major challenge to
the development of electronic structure theories. Biradicaloid singlet ground
states are often marked by strong correlation and, hence, may not be accurately
treated by mainstream, single-reference methods such as density functional
theory or coupled cluster theory. The anti-Hermitian contracted Schr\"odinger
equation (ACSE), whose fundamental quantity is the two-electron reduced density
matrix rather than the N-electron wave function, has previously been shown to
account for both dynamic and strong correlations when seeded with a strongly
correlated guess from a complete active space (CAS) calculation. Here, we
develop a spin-averaged implementation of the ACSE, allowing it to treat higher
multiplicity states from the CAS input without additional state preparation. We
apply the spin-averaged ACSE to calculate the singlet-triplet gaps in a set of
small main group biradicaloids, as well as the organic four-electron biradicals
trimethylenemethane and cyclobutadiene, and naphthalene, benchmarking the
results against other state-of-the-art methods reported in the literature.
