In the local-density approximation (LDA), the many-electron problem is approximated by a set of single-particle equations which are solved with the self-consistent field method. The total energy is minimized. The total energy is taken to be the sum of a kinetic energy, T, the classical Hartree term for the electron density, E_{coul}, the electron-nucleus energy, E_{enuc}, and the exchange-correlation energy, E_{xc}, which takes into account approximately the fact that an electron does not interact with itself, and that electron correlation effects occur.
One solves the Kohn-Sham orbital equations
$$[-{1\over2}\nabla^2 + v_{\rm eff}(\vec r)] \psi_i(\vec r) = \varepsilon_i \psi_i(\vec r) ~ ,$$
(Eq. 25)
with
$$v_{\rm eff}(\vec r) = v(\vec r) + \int {\rm d}\vec r^\prime {{\rho(\vec r)}\over {|\vec r - \vec r' |}} + v_{\rm xc} (\vec r) ~ .$$
(Eq. 26)
The charge density ρ is given by
$$\rho (\vec r) = 2 \sum_i f_i \mid \psi_i (\vec r)\mid^2 ~.$$
(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, $$v(\vec{r})$$, is the external potential; in the atomic case, this is
The various parts of the total energy are given by:
$$ T = - 2 \sum_i f_i \int {\rm d}\vec{r} \, \psi_i^\ast (\vec{r}) \, {\textstyle{1\over2}} \, \nabla^2 \psi_i (\vec{r}) ~,$$
(Eq. 28)
$$ E_{\rm enuc} = \int {\rm d}{\vec{r}} \rho(\vec{r}) \, v(\vec{r}) ~ ,$$
(Eq. 29)
$$ E_{\rm coul} = \, {\textstyle{1\over2}} \, \int {\rm d}\vec{r} {\rm d}\vec{r}^\prime ~ {{\rho(\vec{r}) \rho(\vec{r}^\prime) }\over{\mid \vec{r} - \vec{r}^\prime\mid }} ~ ,$$
(Eq. 30)
and
$$E_{\rm xc} = \int d \vec r \rho(\vec r) \varepsilon_{\rm xc}(\rho) ~ ,$$
(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.
The local-spin-density approximation (LSD) is a generalization of the LDA in which the spin-degree of freedom is treated in a nontrivial way [10,11].
For atoms, it is sufficient to pick an arbitrary spin-polarization direction, and to consider the local-spin-density, $$\rho (\vec r, \sigma) ~ = ~ | \psi (\vec r, \sigma)|^2~.$$
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, $$| \psi (\vec r, \sigma)^* ~ \psi (\vec r, \sigma^\prime)|~,$$
where σ and σ′ represent spin up or spin down. This leads to consideration of a potential of the form, $$v_{xc}(\vec{r}, \sigma, \sigma^\prime) ~.$$
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:
$${{{\rm d}F}\over{{\rm d}r}} - {\kappa\over r} F= -c^{-1} (\epsilon-v(r) ) G , $$
(Eq. 32)
$${{{\rm d}G}\over{{\rm d}r}} + {\kappa\over r} G= c^{-1} (\epsilon-v(r)+2 c^2) F ,$$
(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
$$\psi = \pmatrix {G(r) r^{-1} {\cal Y}_{\kappa m} (\hat r) \cr
i F(r) r^{-1} {\cal Y}_{-\kappa m} (\hat r) \cr }$$
$$\rho(\vec{r})= 2\sum_i\,f_i \sum_\mu |\psi_\mu(\vec{r})|^2$$
(Eq. 34)
where $${\cal Y}_{\kappa m}(\hat r)$$
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 $$\rho (\vec r) = 2 \Sigma_i f_i \Sigma_{\mu} | \psi_{\mu}(\vec r)|^2$$
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:
$${{{\rm d}^2 G}\over{{\rm d}r^2}} - {{\ell(\ell+1)}\over{r^2}}~ G = 2 M (V-\epsilon) G + {{1}\over{M}} {{{\rm d} M}\over{{\rm d}r}} \left( {{{\rm d}G}\over{{\rm d}r}} + {{\langle \kappa \rangle}\over{r}}G \right) ~ ,$$
(Eq. 35)
where $${\langle \kappa \rangle} = -1$$
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
$$M = 1 + {{\alpha^2}\over{2}} (\epsilon-V) ~ ,$$
(Eq. 36)
where α is the fine structure constant. The charge density is related to G by the usual non-relativistic formula,
$$r^2 \rho(r) = G(r)^2$$
(Eq. 37)
without an explicit contribution from the minor component F(r).