  An accurate determination of singlet-triplet gaps in biradicals, including
cyclobutadiene in the automerization barrier region where one has to balance
the substantial nondynamical many-electron correlation effects characterizing
the singlet ground state with the predominantly dynamical correlations of the
lowest-energy triplet, remains a challenge for many quantum chemistry methods.
High-level coupled-cluster (CC) approaches, such as the CC method with a full
treatment of singly, doubly, and triply excited clusters (CCSDT), are often
capable of providing reliable results, but the routine application of such
methods is hindered by their high computational costs. We have recently
proposed a practical alternative to converging the CCSDT energetics at small
fractions of the computational effort, even when electron correlations become
stronger and connected triply excited clusters are larger and nonperturbative,
by merging the CC($P$;$Q$) moment expansions with the selected configuration
interaction methodology abbreviated as CIPSI. We demonstrate that one can
accurately approximate the highly accurate CCSDT potential surfaces
characterizing the lowest singlet and triplet states of cyclobutadiene along
the automerization coordinate and the gap between them using tiny fractions of
triply excited cluster amplitudes identified with the help of relatively
inexpensive CIPSI Hamiltonian diagonalizations.
