Singlet fission offers an opportunity to improve solar cell efficiency, but its practical use is hindered by the limited number of known efficient materials. We look for chromophores that satisfy the desirable but rarely encountered adiabatic energy conditions, E(T-2) - E(S-0) > E(S-1) - E(S-0) approximate to 2[E(T-1) - E(S-0)], and are small enough to permit highly accurate calculations. We provide a rationale for the use of captodative biradicaloids, i.e., biradicals stabilized by direct interaction between their radical centers, which carry both an acceptor and a donor group. A computation of vertical excitation energies of 14 structures of this type by time-dependent density functional theory (TD-DFT) yielded 11 promising candidates. The vertical excitation energies from S0 and T1 were recalculated by complete-active-space second-order perturbation theory (CASPT2), and five of the compounds met the above energy criteria. Their adiabatic excitation energies from the S-0 into the S-1, S-2, T-1, and T-2 excited states were subsequently calculated, and three of them look promising. For 2,3-diamino-1,4-benzoquinone, adiabatic E(T-1) and E(S-1) energies were close to optimal (1.12 and 2.23 eV above the S0 ground state, respectively), and for its more practical N-peralkylated derivative they were even lower (0.63 and 1.06 eV above S-0, respectively). PCM/CASPT2 results suggested that the relative energies can be further tuned by varying the polarity of the environment.