Convergence of M{\o}ller-Plesset perturbation theory for excited reference states

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.