“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 $C^{\mathrm{†}}C$ 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 $A\left(6s\right)$ 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.

Figure 1

Dominant LCCSD contributions for the matrix elements. The double arrows represent the valence state, crosses represent matrix elements $z_{\mathit{ij}}$, and heavy horizontal lines represent cluster amplitudes. In particular, the RPA diagram involves valence doubles and the BO diagram involves valence singles. Here and below we do not draw the exchange variants for the diagrams.

Figure 2

Schematic dressing of particle and hole lines in the CC diagrams for matrix elements.

Figure 3

Dressing of particle and hole lines in the singles-doubles approximation. The upper and lower panels represent dressing of particle and hole lines, respectively.

Figure 4

Dressing of particle-hole lines in the dominant LCCSD diagrams, Fig. 1. The circled crosses represent line-dressed matrix elements, Eq. (23), and the double-lined valence cluster amplitude is the dressed amplitude, Eq. (24).

Figure 5

Graphical equation for RPA-dressed insertion. The dashed horizontal line is the dressed insertion ${\mathcal{T}}_{\mathit{nbma}}$, while the bare object $c_{\mathit{nbma}}$ is represented by a wavy line.

Figure 6

Dressing particle-hole vertex of a diagram with the RPA insertion, Fig. 5.

Figure 7

Topological structure of fourth-order diagrams of class $Z_{1\times 2}\left(D_{\mathit{nl}}\right)$ derived in Ref. 10. In these diagrams horizonal lines denote Coulomb interaction and the topological objects coming from the lowest-order approximation to $C^{\mathrm{†}}C$ are highlighted. The upper three diagrams arise in the leading order of the line-dressing scheme and the two diagrams in the middle are related to the RPA-dressing scheme.