  Recent years have seen an explosion of interest in organic radicals due to
their promise for highly efficient organic light-emitting diodes (OLEDs) and
molecular qubits. However, accurately and inexpensively computing their
electronic structure has been challenging, especially for excited states, due
to the spin-contamination problem. Furthermore, while alternacy or
`pseudoparity' rules have guided the interpretation and prediction of the
excited states of closed-shell hydrocarbons since the 1950s, similarly general
rules for hydrocarbon radicals have not to our knowledge yet been found. In
this article we present solutions to both of these challenges. We firstly
combine the extended configuration interaction singles (XCIS) method with
Pariser-Parr-Pople (PPP) theory to obtain a method which we call ExROPPP
(Extended Restricted Open-shell PPP theory). We find that ExROPPP computes
spin-pure excited states of hydrocarbon radicals with comparable accuracy to
experiment as high-level GMC-QDPT calculations but at a computational cost that
is at least two orders of magnitude lower. We then use ExROPPP to derive
widely-applicable rules for the spectra of alternant hydrocarbon radicals which
are completely consistent with our computed results. These findings pave the
way for the highly accurate and efficient computation and prediction of the
excited states of organic radicals.
