## Atomic Reference Data |

We utilize the central-field approximation, with conventional labelling of
principal and angular momentum quantum numbers of electronic orbitals. We
limit our calculations to the ground-state electronic configurations of the first
92 neutral atoms and singly-charged cations of the periodic table; the
specific configurations used are described below. In cases of partially filled
electronic subshells, fractional occupancies are assigned to orbitals with
different azimuthal quantum number, *m*, to accomplish a spherical
averaging of the charge distribution. In the case of
RLDA, this extends to averaging over
subshells with the same orbital angular momentum *ℓ* but different values of total angular momentum *j*. Thus, for
example, if there were 2 electrons in a *p* shell, we assign an electron
population of 4/3 to the *p*_{3/2} states and 2/3 to the
*p*_{1/2} states.

The results presented here derive from four codes that were written independently. These were found to give results of good mutual consistency, provided that the numerical approximations within each code were varied until a very high degree of convergence was obtained within each code. The original authors of the four codes are, in alphabetical order

- Sverre Froyen
- Donald Hamann
- Eric Shirley
- Ilia Tupitsyn and Svetlana Kotochigova

These codes were used by us with the permission of their authors. However, all codes required modification to obtain numerical convergence to the target accuracy of 1 microHartree in total energy. These modifications were not subject to review by the original authors. Some of these codes circulate relatively freely within the electronic structure community, so we must caution readers that a given available version of any one of these codes need not yield results identical to those presented here. Our purpose in this study was to use robust tested tools to accomplish a specific task, not to provide a relative ranking of various codes. For this reason, we do not give details of individual code performance beyond what is necessary to describe the uncertainties in the results, and we refer to each code by a numerical label between 1 and 4, chosen arbitrarily.

The local-density approximation (LDA) requires that
the exchange-correlation potential be given as a function of the electron
density at a given point in space. For this part of our study, we use the form
of the exchange-correlation potential given by Vosko, Wilk, and Nusair (VWN)
[4]. The form is a fit to the
Ceperley-Alder electron gas study [6].
The VWN functional reproduces the random-phase-approximation (RPA) results
for a uniform electron gas in the high-density limit, it reproduces
the spin-stiffness constant calculated in the RPA in the paramagnetic
limit of a uniform electron gas, and it is uniformly differentiable as
a function of the electron gas parameter, *r _{s}*. It is also
in standard use, or available as an option, in many electronic structure codes, and
thereby provides a convenient reference potential for checking the accuracy of
numerical calculations.

We now summarize the form of the VWN functional. The exchange-correlation
energy per electron is separated into two parts, an exchange term and a
correlation term.

In the RLDA and
ScRLDA calculations, we use the
relativistic corrections to the energy-density
functional proposed by MacDonald and Vosko
[7].

(Eq. 22) |

with three parameters: the minimum radius, *r*_{min}, the maximum
radius, *r*_{max}, and the number of intervals, *N*. The
application of the exponential grid to the atomic Schrödinger equation has
been discussed by Desclaux [8]. For one code
we used *N* = 15788,
*r*_{min} = 1/(160 *Z*), and
*r*_{max} = 50. (All distances are in atomic units.)
Another code used *N* ≤ 8000,
*r*_{min} = 10^{-6}/*Z*, and
*r*_{max} = 800 *Z*^{-1/2}; in this
case, the energies were extrapolated to *n* → ∞ using an *N*^{-2} or
*N*^{-4} dependence, depending on the quantity in question.

Another code chooses a grid which is nearly linear near the origin,
and exponentially increasing at large *r*,

(Eq. 23) |

which again is determined by three parameters, *a*, *b*, and
*N*. This grid includes the origin explicitly as *r*_{0}.
In this case, we took *a* = 4.34 10^{-6}/*Z*,
*b* = 0.002 304, and *r*_{max} = 50,
leading to *N* = 7058 for H, increasing to *N* = 9021 for U,
and to *r*_{1} = 10^{-7} for H, decreasing to
1.1 10^{-9} for U.

A fourth code uses a change of variable technique:

. | (Eq. 24) |

A uniform grid is taken in the transformed variable from