The recently developed new approach to the many-electron correlation problem in atoms and molecules, termed the method of moments of coupled-cluster (CC) equations (MMCC), is reviewed. The ground-state MMCC formalism and its extension to excited electronic states via the equation-of-motion coupled-cluster (EOMCC) approach are discussed. The main principle of all MMCC methods is that of the non-iterative energy corrections which, when added to the ground- and excited-state energies obtained in the standard CC calculations, such as CCSD or EOMCCSD, recover the exact, full configuration interaction (CI) energies. Three types of the MMCC approximations are reviewed in detail: (i) the CI-corrected MMCC methods, which can be applied to ground and excited states; (ii) the renormalized and completely renormalized CC methods for ground states; and (iii) the quasi-variational MMCC approaches for the ground-state problem, including the quadratic MMCC models. It is demonstrated that the MMCC formalism provides a new theoretical framework for designing 'black-box' CC approaches that lead to an excellent description of entire potential energy surfaces of ground- and excited-state molecular systems with an ease of use of the standard single-reference methods. The completely renormalized (CR) CCSD(T) and CCSD(TQ) methods and their quadratic and excited-state MMCC analogues remove the failing of the standard CCSD, CCSD(T), EOMCCSD and similar methods at larger internuclear separations and for states that normally require a genuine multireference description. All theoretical ideas are illustrated by numerical examples involving bond breaking, excited vibrational states, reactive potential energy surfaces and difficult cases of excited electronic states. The description of the existing and well-established variants of the MMCC theory, such as CR-CCSD(T), is augmented by the discussion of future prospects and potentially useful recent developments, including the extension of the black-box CR-CCSD(T) method to excited states.

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