Efficient singular-value decomposition of the coupled-cluster triple excitation amplitudes

We demonstrate a novel technique to obtain singular-value decomposition (SVD) of the coupled-cluster triple excitations amplitudes, $t_{ijk}^{abc}$. The presented method is based on the Golub-Kahan bidiagonalisation strategy and does not require $t_{ijk}^{abc}$ to be stored. The computational cost of the method is comparable to several CCSD iterations. Moreover, the number of singular vectors to be found can be predetermined by the user and only those singular vectors which correspond to the largest singular values are obtained at convergence. We show how the subspace of the most important singular vectors obtained from an approximate triple amplitudes tensor can be used to solve equations of the CC3 method. The new method is tested for a set of small and medium-sized molecular systems in basis sets ranging in quality from double- to quintuple-zeta. It is found that to reach the chemical accuracy ($\approx 1$ kJ/mol) in the total CC3 energies as little as $5-15\%$ of SVD vectors are required. This corresponds to the compression of the $t_{ijk}^{abc}$ amplitudes by a factor of ca. $0.0001-0.005$. Further benchmarks are performed to check the behaviour of the method in calculation of, e.g. interaction energies or rotational bariers, as well as in bond-breaking processes.