  We embark on a quest to identify small molecules in the chemical space that
can potentially violate Hund's rule. Utilizing twelve TDDFT approximations and
the ADC(2) many-body method, we report the energies of S$_1$ and T$_1$ excited
states of 12,880 closed-shell organic molecules within the bigQM7$\omega$
dataset with up to 7 CONF atoms. In this comprehensive dataset, none of the
molecules, in their minimum energy geometry, exhibit a negative S$_1$-T$_1$
energy gap at the ADC($2$) level while several molecules display values $<0.1$
eV. The spin-component-scaled double-hybrid method, SCS-PBE-QIDH, demonstrates
the best agreement with ADC(2). Yet, at this level, a few molecules with a
strained $sp^3$-N center turn out as false-positives with the S$_1$ state lower
in energy than T$_1$. We investigate a prototypical cage molecule with an
energy gap $<-0.2$ eV, which a closer examination revealed as another false
positive. We conclude that in the chemical space of small closed-shell organic
molecules, it is possible to identify geometric and electronic structural
features giving rise to S$_1$-T$_1$ degeneracy; still, there is no evidence of
a negative gap. We share the dataset generated for this study as a module, to
facilitate seamless molecular discovery through data mining.
