We use a linearized coupled-cluster method to calculate the atomic properties of atoms and ions with one valence electron (monovalent systems). This method is also referred to as all-order in the literature as it involves summing series of dominant many-body perturbation terms to all orders. In the single-double (SD) all-order approach, single and double excitations of the Dirac-Fock orbitals are included. The SDpT all-order approach also includes classes of the triple excitations. Omitted higher excitations are estimated by the scaling procedure which can be started from either SD or SDpT approximations. We carry out four all-order computations for each of the electric-dipole matrix elements, ab initio SD and SDpT and scaled SD and SDpT. Either SD or SD scaled data are taken as final values based on the comparison of different contributions to the matrix elements. An algorithm is used to determine the uncertainties of the electric-dipole matrix elements based on the spread of the four results, size of the correlation correction, and comparison of different contributions to the matrix elements.

A detailed description of the SD and SDpT all-order approaches with all formulas is given in this pdf document, edited from edited from M. S. Safronova, High-precision calculations of atomic properties and parity nonconservation in systems with one valence electron, Ph. D. Thesis, University of Notre Dame (2001).

**References:**

[1] All-Order Methods for Relativistic Atomic Structure Calculations (review paper), M.S. Safronova and W.R.
Johnson,
*Advances in Atomic Molecular and Optical Physics series*, volume 55, 194 (2007).
Link

[2] Relativistic many-body calculation of energies, lifetimes, hyperfine constants, and polarizabilities in
^{7}Li
W. R.
Johnson, U. I. Safronova, A. Derevianko, and M. S. Safronova, *Phys. Rev. A* 77, 022510 (2008).
Link

[3] Relativistic many-body calculation of energies, lifetimes, polarizabilities, and hyperpolarizabilities in
Li-like
Be+, U. I. Safronova and M. S. Safronova, *Phys. Rev. A* 87, 032502 (2013).
Link

[4] High-accuracy calculation of energies, lifetimes, hyperfine constants, multipole polarizabilities, and
blackbody
radiation shift in ^{39}K, U.I. Safronova and M. S. Safronova, *Phys. Rev. A* 78, 052504 (2008).
Link

[5] Blackbody radiation shift, multipole polarizabilities, oscillator strengths, lifetimes, hyperfine constants,
and
excitation energies in Ca^{+}, M.S. Safronova and U.I. Safronova, *Phys. Rev. A* 83, 012503 (2011).
Link

[6] Critically evaluated theoretical energies, lifetimes, hyperfine constants, and multipole polarizabilities in
^{87}Rb,
M. S. Safronova and U. I. Safronova, *Phys. Rev. A* 83, 052508 (2011) .
Link

[7] Magic wavelengths, matrix elements, polarizabilities, and lifetimes of Cs, M. S. Safronova, U. I. Safronova,
and
Charles W. Clark *Phys. Rev. A* 94, 012505 (2016).
Link

[8] Calculation of parity-nonconserving amplitude and other properties of Ra^{+}, Rupsi Pal,
Dansha Jiang,
M.S.
Safronova, and U.I. Safronova, *Phys. Rev. A* 79, 062505 (2009).
Link