  We report the first investigation of the performance of EOM-CC4 -- an
approximate equation-of-motion coupled-cluster model which includes iterative
quadruple excitations -- for vertical excitation energies in molecular systems.
By considering a set of 28 excited states in 10 small molecules for which we
have computed CCSDTQP and FCI reference energies, we show that, in the case of
excited states with a dominant contribution from the single excitations, CC4
yields excitation energies with sub-kJ~mol$^{-1}$ accuracy (i.e., error below
$0.01$ eV), in very close agreement with its more expensive CCSDTQ parent.
Therefore, if one aims at high accuracy, CC4 stands as a highly competitive
approximate method to model molecular excited states, with a significant
improvement over both CC3 and CCSDT. Our results also evidence that, although
the same qualitative conclusions hold, one cannot reach the same level of
accuracy for transitions with a dominant contribution from the double
excitations.
