Resolving all-order method convergence problems for atomic physics applications

The development of the relativistic all-order method where all single, double, and partial triple excitations of the Dirac-Hartree-Fock wave function are included to all orders of perturbation theory led to many important results for the study of fundamental symmetries, development of atomic clocks, ultracold atom physics, and others, as well as provided recommended values of many atomic properties critically evaluated for their accuracy for a large number of monovalent systems. This approach requires iterative solutions of the linearized coupled-cluster equations leading to convergence issues in some cases where correlation corrections are particularly large or lead to an oscillating pattern. Moreover, these issues also lead to similar problems in the configuration-interaction (CI)$+$all-order method for many-particle systems. In this work, we have resolved most of the known convergence problems by applying two different convergence stabilizer methods, namely, reduced linear equation and direct inversion of iterative subspace. Examples are presented for B, Al, Zn${}^{+}$, and Yb${}^{+}$. Solving these convergence problems greatly expands the number of atomic species that can be treated with the all-order methods and is anticipated to facilitate many interesting future applications.

Figure 1

The failure of the LCCSD straightforward iteration procedure for the $3s$ state in boron. The calculated correlation energy is plotted as a function of the iteration number. The dashed (red) line indicates the value of the experimental correlation energy.

Figure 2

Comparison of the RLE5, DIIS5, and DIIS7 schemes for the $3s$ state of boron. The correlation energy is given in atomic units.

Figure 3

Comparison of the DIIS5, RLE8, and DIIS8 schemes for the $3d_{3/2}$ state of boron. The correlation energy is given in atomic units.

Figure 4

Comparison of the CIS, RLE5, and DIIS5 schemes for the Yb${}^{+}$ core. The correlation energy is given in atomic units. RLE5 and DIIS5 data appear identical at this scale and are shown as a single curve.