  We present a basis set correction scheme for the coupled-cluster singles and
doubles (CCSD) method. The scheme is based on employing frozen natural orbitals
(FNOs) and diagrammatically decomposed contributions to the electronic
correlation energy that dominate the basis set incompleteness error (BSIE). As
recently discussed in [https://doi.org/10.1103/PhysRevLett.123.156401], the
BSIE of the CCSD correlation energy is dominated by the second-order
M{\o}ller-Plesset (MP2) perturbation energy and the particle-particle ladder
term. Here, we derive a simple approximation to the BSIE of the
particle-particle ladder term that effectively corresponds to a rescaled
pair-specific MP2 BSIE, where the scaling factor depends on the spatially
averaged correlation hole depth of the coupled-cluster and first-order pair
wavefunctions. The evaluation of the derived expressions is simple to implement
in any existing code. We demonstrate the effectiveness of the method for the
uniform electron gas. Furthermore, we apply the method to coupled-cluster
theory calculations of atoms and molecules using FNOs. Employing the proposed
correction and an increasing number of FNOs per occupied orbital, we
demonstrate for a test set that rapidly convergent closed and open-shell
reaction energies, atomization energies, electron affinities, and ionization
potentials can be obtained. Moreover, we show that a similarly excellent
trade-off between required virtual orbital basis set size and remaining BSIEs
can be achieved for the perturbative triples contribution to the CCSD(T) energy
employing FNOs and the (T*) approximation.
