  In the present work we report an implementation of the rank-reduced
equation-of-motion coupled cluster method with approximate triple excitations
(RR-EOM-CC3). The proposed variant relies on tensor decomposition techniques in
order to alleviate the high cost of computing and manipulating the
triply-excited amplitudes. In the RR-EOM-CC3 method, both ground-state and
excited-state triple-excitation amplitudes are compressed according to the
Tucker-3 format. This enables to factorize the working equations such that the
formal scaling of the method is reduced to $N^6$, where $N$ is the system size.
An additional advantage of our method is the fact the accuracy can be strictly
controlled by proper choice of two parameters defining sizes of
triple-excitation subspaces in the Tucker decomposition for the ground and
excited states. Optimal strategies of selecting these parameters are discussed.
The developed method has been tested in a series of calculations of electronic
excitation energies and compared to its canonical EOM-CC3 counterpart. Errors
several times smaller than the inherent error of the canonical EOM-CC3 method
(in comparison to FCI) are straightforward to achieve. This conclusion holds
both for valence states dominated by single excitations and for states with
pronounced doubly-excited character. Taking advantage of the decreased scaling,
we demonstrate substantial computational costs reductions (in comparison with
the canonical EOM-CC3) in the case of two large molecules -- L-proline and
heptazine. This illustrates the usefulness of the RR-EOM-CC3 method for
accurate determination of excitation energies of large molecules.
