New alternatives for electronic structure calculations: Renormalized, extended, and generalized coupled-cluster theories

New single-reference coupled-cluster (CC) methods that can be applied to quasi-degenerate electronic states and molecular potential energy surfaces involving bond breaking and that can provide the virtually exact results for many-electron systems are reviewed. Three types of approaches are discussed: (i) the renormalized CC methods, which represent the approximate variants of the more general formalism of the method of moments of coupled-cluster equations (MMCC), (ii) the extended CC (ECC) methods, and (iii) the generalized CC (GCC) theory, in which many-electron wave functions are represented by the exponential cluster expansions involving general two-body operators. The main idea of the renormalized CC and other MMCC theories is that of the noniterative energy corrections which, when added to the energies obtained in the standard CC calculations, such as CCSD (CC singles and doubles), recover the exact or virtually exact energies. It is demonstrated that the completely renormalized CCSD(T) and CCSD(TQ) methods and their quadratic MMCC analogs represent "black-box" approaches, which are capable of removing the failing of the standard CCSD, CCSD(T), and similar methods at larger internuclear separations. The ECC methods, in which products involving low-order cluster components mimic the effects of higher-order clusters, are based on the idea of optimizing two cluster operators. It is shown that the ECC methods with singles and doubles (ECCSD), including the quadratic and bilinear ECCSD theories, provide great improvements in the poor description of multiple bond breaking by the standard CC approaches. Finally, we provide strong arguments that the GCC theory may represent the exact many-body formalism. This result may have a significant impact on future quantum calculations for many-electron and other pairwise interacting many-fermion systems.