  Excited states in molecules can be difficult to investigate and generally
require methods that are either computationally expensive or are not
universally accurate. Recent research has focused on using higher-energy Slater
determinants as mean-field representations of excited states, which can then be
used to define reference states for electron correlation techniques such as
M{\o}ller-Plesset theory. However, the convergence behaviour of these
excited-state perturbation series has not yet been explored, limiting our
understanding into the systematic improvability and reliability of such
methods. Here, we present a systematic analysis of the M{\o}ller-Plesset
perturbation series for closed-shell excited states using higher-energy
reference determinants identified with, and without, state-specific orbital
optimisation. We combine numerical calculations with complex analysis to show
that the excited-state M{\o}ller-Plesset theory diverges almost universally for
the stereotypical $\mathrm{H}_2$, $\mathrm{H}_2\mathrm{O}$, and $\mathrm{CH}_2$
molecules. Our results cast doubt on the suitability of higher-energy Slater
determinants as reference states for perturbative treatments of electron
correlation.
