## Atomic Reference Data |

One solves the Kohn-Sham orbital equations

(Eq. 25) |

with

(Eq. 26) |

The charge density *ρ* is
given by

(Eq. 27) |

where the 2 accounts for doubling the occupancy of each spatial orbital because
of spin degeneracy and *f _{i}* account for partial occupancy. The
potential, , is the
external potential; in the atomic case, this is

The various parts of the total energy are given by:

(Eq. 28) |

(Eq. 29) |

(Eq. 30) |

and

(Eq. 31) |

where *ε*_{xc}(*ρ*) is the exchange-correlation
energy per particle for the uniform electron gas of density *ρ*. This approximation for
*E*_{xc} is the principal approximation of the LDA.

For atoms, it is sufficient to pick an arbitrary spin-polarization direction, and to consider the local-spin-density, . Our choice of functional is that of Vosko, Wilk, and Nusair (1980) [4], as described above.

In general, the local-spin density approximation requires consideration
of the spin-density matrix, ,
where *σ* and
*σ*′ represent spin up or spin down. This leads to consideration of
a potential of the form, . This additional generality
is not required in the present work.

The relativistic local-density approximation [13] (RLDA) may be obtained from the (non-relativistic) local-density approximation (LDA) by substituting the relativistic kinetic-energy operator for its non-relativistic counterpart, and using relativistic corrections to the local-density functional. We use the relativistic corrections proposed by MacDonald and Vosko [7].

Here, we give the radial equations which are solved by our programs:

(Eq. 32) |

(Eq. 33) |

where *ε* is the eigenvalue in
Hartrees, and *c* is the speed of light; *ε* = 0 describes a free electron with zero kinetic
energy. The functions *G*(*r*) and *F*(*r*) are related to
the Dirac spinor by

(Eq. 34) |

where is a Pauli spinor [12].

Dirac's *κ* quantum number, along with
the azimuthal quantum number *m*, determines the angular dependence of the
state. For the central-field problem, the levels with various *m* are
degenerate and hence not solved for separately. The following table relates the
values of *κ* used in this project to the
more common spectroscopic notation.

κ |
state | κ |
state |
---|---|---|---|

-1 | s_{1/2} |
1 | p_{1/2} |

-2 | p_{3/2} |
2 | d_{3/2} |

-3 | d_{5/2} |
3 | f_{5/2} |

-4 | f_{7/2} |
||

The charge density is obtained from
where *µ* runs over the four components of the Dirac spinor.

The inclusion of relativistic effects doubles the number of degrees of freedom in atomic calculations. However, sometimes it is desirable to include some of the effects of relativity without increasing the number of degrees of freedom. Specifically, it is possible to neglect the spin-orbit splitting while including other relativistic effects, such as the mass-velocity term, the Darwin shift, and (approximately) the contribution of the minor component to the charge density.

Koelling and Harmon[14] have proposed a method to achieve this end, which we call the scalar-relativistic local-density approximation (ScRLDA). (Sc is used to avoid confusion with spin-polarization which is abbreviated S.) This is a simplified version of the RLDA. The equations to solve are:

(Eq. 35) |

where = -1 is the degeneracy-weighted average
value of the Dirac's *κ* for the two
spin-orbit-split levels, and *ε*
is the eigenvalue in Hartrees, with the same meaning as in the
RLDA.

The parameter *M* is given by

(Eq. 36) |

where *α* is the fine structure constant.
The charge density is related to *G* by the usual non-relativistic formula,

(Eq. 37) |

without an explicit contribution from the minor component *F*(*r*).