Quintic-scaling rank-reduced coupled cluster theory with single and double excitations

We consider the rank-reduced coupled-cluster theory with single and double (RR-CCSD) excitations introduced recently [Parrish et al., J. Chem. Phys. 150, 164118 (2019)]. The main feature of this method is the decomposed form of doubly excited amplitudes, which are expanded in the basis of largest magnitude eigenvectors of MP2 or MP3 amplitudes. This approach enables a substantial compression of amplitudes with only minor loss of accuracy. However, the formal scaling of the computational costs with the system size (N) is unaffected in comparison with the conventional CCSD theory (∝N⁶) due to the presence of some terms quadratic in amplitudes, which do not naturally factorize to a simpler form even within the rank-reduced framework. We show how to solve this problem, exploiting the fact that their effective rank increases only linearly with the system size. We provide a systematic way to approximate the problematic terms using the singular value decomposition and reduce the scaling of the RR-CCSD iterations down to the level of N⁵. This is combined with an iterative method of finding dominant eigenpairs of MP2 or MP3 amplitudes, which eliminates the necessity to perform the complete diagonalization, making the cost of this step proportional to the fifth power of the system size, as well. Next, we consider the evaluation of perturbative corrections to CCSD energies resulting from triply excited configurations. The triply excited amplitudes present in the CCSD(T) method are decomposed to the Tucker-3 format using the higher-order orthogonal iteration procedure. This enables us to compute the energy correction due to triple excitations non-iteratively with N⁶ cost. The accuracy of the resulting rank-reduced CCSD(T) method is studied for both total and relative correlation energies of a diverse set of molecules. Accuracy levels better than 99.9% can be achieved with a substantial reduction of the computational costs. Concerning the computational timings, the break-even point between the rank-reduced and conventional CCSD implementations occurs for systems with about 30-40 active electrons.