Dressing lines and vertices in calculations of matrix elements with the coupled-cluster method and determination of Cs atomic properties

We consider evaluation of matrix elements with the coupled-cluster method. Such calculations formally involve infinite number of terms and we devise a method of partial summation (dressing) of the resulting series. Our formalism is built upon an expansion of the product (CC)-C-dagger of cluster amplitudes C into a sum of n-body insertions. We consider two types of insertions: particle (hole) line insertion and two-particle (two-hole) random-phase-approximation-like insertion. We demonstrate how to "dress" these insertions and formulate iterative equations. We illustrate the dressing equations in the case when the cluster operator is truncated at single and double excitations. Using univalent systems as an example, we upgrade coupled-cluster diagrams for matrix elements with the dressed insertions and highlight a relation to pertinent fourth-order diagrams. We illustrate our formalism with relativistic calculations of the hyperfine constant A(6s) and the 6s(1/2)-6p(1/2) electric-dipole transition amplitude for the Cs atom. Finally, we augment the truncated coupled-cluster calculations with otherwise omitted fourth order diagrams. The resulting analysis for Cs is complete through the fourth order of many-body perturbation theory and reveals an important role of triple and disconnected quadruple excitations.