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Near-exact CCSDT energetics from rank-reduced formalism supplemented by non-iterative corrections

We introduce a non-iterative energy correction, added on top of the rank-reduced coupled-cluster method with single, double, and triple substitutions, that accounts for excitations excluded from the parent triple excitation subspace. The formula for the correction is derived by employing the coupled-cluster Lagrangian formalism with an additional assumption that the parent excitation subspace is closed under the action of the Fock operator. Owning to the rank-reduced form of the triple excitation amplitudes tensor, the computational cost of evaluating the correction scales as $N^7$ with the system size, $N$. The accuracy and computational efficiency of the proposed method is assessed both for total and relative correlation energies. We show that the non-iterative correction can fulfill two separate roles. If an accuracy level of a fraction of kJ/mol is sufficient for a given system the correction significantly reduces the dimension of the parent triple excitation subspace needed in the iterative part of the calculations. Simultaneously, it enables to reproduce the exact CCSDT results to an accuracy level below 0.1 kJ/mol with a larger, yet still reasonable, dimension of the parent excitation subspace. This typically can be achieved at a computational cost only several times larger than required for the CCSD(T) method. The proposed method retains black-box features of the single-reference coupled-cluster theory; the dimension of the parent excitation subspace remains the only additional parameter that has to be specified.