The one-electron equation

The no-pair DCB Hamiltonian (7) is used as a starting point for variational or many-body relativistic calculations [9]. The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10]. The radial four-spinor has the so-called large component Puk in the upper two places and the small component in the lower two. The quantum number k (with = j -H 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining 

Effect of nuclear model on DF calculated properties of Fm (ElOO). Values from ref. [18]. Absolute and percent differences are shown. Energies in hartree, distances in bohr units. 

The terms etc. in (10) represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (4) in addition to the electron repulsion l/rjj. The radial functions Pn ( ) and Qn/c( ) may be obtained by mmierical integration [20,21] or by expansion in a basis (for more details see recent reviews [22,23]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [24,25], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [26,27]. In the nonrelativistic limit (c oo), the small component is related to the large component by [24] 

Ishikawa and coworkers [15,28] have shown that G-spinors, with functions spanned in Gaussian-type functions (GTF) chosen according to (15), satisfy kinetic balance for finite c values if the nucleus is modeled as a uniformly-charged sphere. 

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Although the methods suggested above are not, by any means, completely satisfactory, they are sufficient to describe the main qualitative aspects of the problem. In the remainder of this section, therefore, we shall discuss methods of solving the one-electron equations (12), with h defined by (13). We shall assume that the Coulomb repulsion term can be neglected and, therefore, will not consider Hartree-Fock solutions. Details of the latter can be found in Refs. 25-27. 

Now suppose that we have solved the one-electron equations (26) and suppose that the subsidiary conditions ... 

The one-electron equations (60) offer a new possibility for finding the optimal NSOs by iterative diagonalization of the Fockian, Eq. (66). The main advantage of this method is that the resulting orbitals are automatically orthogonal. The first calculations based on this diagonalization technique has confirmed its practical value [81]. 

Each icrm is a function of only the coordinates r,- of a single electron, so each term must be individually zero, giving the one-electron equation for each electron, If these onc-eicctron solutions are substituted back into Eq. (A-1), we see that we have obtained a solution. 

In the present calculation, the parameter a was fixed at 0.7. The one-electron equation for (p/Cr), a wave function with orbital energy e , is... 

If we require the KS energy (Eq. (5)) to be a minimum and the one-electron equations (6) to hold, then self-consistency requires that... 

According to the LCAO methods used for theoretically investigating of B32 type Zintl phases, the wave functions are also expanded in spherical harmonics. However, the basis functions are properly chosen atomic orbitals with radial parts that are not exact solutions for the crystal potential used in the one-electron equation. In the first detailed charge analysis of LiAl by Zunger the discrete variational energy-band 1 s, 2 s, 2 p and Alls,2s,2p,3s,3p and 3d orbitals were used for these investigations. 

The radial solutions uA E,r) to the one-electron equation inside the spheres are of the form... 

The one-electron equation associated with the fully-relativistic nDKH+SNSO approximation described as, and implemmted in GTOFF, may be written as... 

Let us assume for a while that we have found such a wonder potential wo(f). We will worry later about how to find it in reality. Now we assume the problem has been solved. Can we find p Of course, we can. Since the Kohn-Sham electrons do not interact between themselves, we have only to solve the one-electron equation (with the wonder Uf))... 

In recent years, the DFT method involving the Kohn-Sham equation with LDA (or better gradient corrected) forms for the exchange-correlation functional as given by the one-electron equation... 

The Hohenberg-Kohn and Kohn-Sham theorems simply state that the total energy can be obtained by applying the variation principle to the total energy density functional and suggest that the one-electron equations obtained in this way also account for electronic correlation. HKS derive another theorem that the total energy is uniquely determined by the density. 

Substitution of the above expression in the one-electron equation yields... 

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