Convergence of all-order many-body methods: Coupled-cluster study for Li

We present and analyze results of the relativistic coupled-cluster calculation of energies, hyperfine constants, and dipole matrix elements for the $2s$, $2p_{1∕2}$, and $2p_{3∕2}$ states of Li atom. The calculations are complete through the fourth order of many-body perturbation theory for energies and through the fifth order for matrix elements and subsume certain chains of diagrams in all orders. A nearly complete many-body calculation allows us to draw conclusions on the convergence pattern of the coupled-cluster method. Our analysis suggests that the high-order many-body contributions to energies and matrix elements scale proportionally and provide a quantitative ground for semiempirical fits of ab initio matrix elements to experimental energies.

Figure 1

Representative diagrams for various classes of contributions of the valence triples. The wavy lines represent the residual Coulomb interaction and those capped with the heavy square represent a one-body interaction (e.g., hyperfine interaction).

Figure 2

(Color online) Convergence pattern of the CCSDvT method as a function of MBPT complexity for the HFS constant of the ground state of Li. Breit, QED, recoil, and basis-set corrections are included in all theoretical values.

Figure 3

(Color online) Variations from the the final CCSDvT values of $A_{2s}$ and $E_{2s}$ in different approximations, e.g., $\Delta X_{\mathrm{SD}}=X_{\mathrm{SD}}-X_{\mathrm{CCSDvT}}$. The variations in matrix elements and energies are correlated and exhibit linear dependence.