Single-particle equations

The next task is to derive an alternative form, more useful in practice, of the fundamental variational equations of Section 4.2.1. The basic idea is to represent the elementary density variables of RDFT in terms of auxiliary single-particle four spinors 

In the four-current version of RDFT the auxiliary spinors are chosen to reproduce the complete j 

Here Ts denotes the kinetic energy of the auxiliary particles , 

Eh is the covariant form of the Hartree energy, which can be split into the Coulomb contribution e j and a transverse part Ej, 

Finally, the xc-energy Exc, in which all many-body aspects beyond the Pauli principle are absorbed, is defined by (4.17) (the rest mass of the electrons has been subtracted from Etot). As the existence theorem (4.4) is equally valid for noninteracting particles, 

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The ROPM integral equation (3.6) has to be solved selfconsistently together with the single particle equations (2.6), i.e. (3.6) replaces the explicit evaluation of 5 ,. /8y/ of the conventional KS procedure. 

Quantum chemistry is most simply done with single-particle orbitals o,exchange-correlation energy Exc is then constructed from the orbitals, or from the spin densities raj and raj. The Hartree-Fock (HF) approximation neglects correlation but treats exchange exactly ... 

The generalized two-particle HF equations are seen to have a structure equivalent to their single-particle counterparts, exhibiting the presence of a direct term, written in terms of the density, and an exchange term. As the canonical HF equations, the present expressions do not contain spurious self-interaction terms. However, unlike the single-particle equations, they allow the determination of fully correlated two-particle states removing to this extent the most basic objection to the HF method. 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

Let us emphasize that the Kohn-Sham orbitals of DFT and the Hartree-Fock orbitals of ab initio theory, though normally quite similar numerically, are distinctly different from a theoretical point of view. Hartree-Fock orbitals satisfy a single-particle equation with a mathematically non-local (i.e. orbital-dependent)... 

The correct low concentration limits principle states that, in the limit as ak —> 0, the equations for the dispersed phases should approach the appropriate single particle equations, while the equations for the continuous phase should approach the correct equations for that single phase continuous fluid. 

This results in a set of single particle equations, known as the Kohn-Sham... 

Once the effective single particle equations have been derived, there are almost limitless possibilities in terms of the different schemes that have been invented to solve them. By way of contrast, the assertion is often made that if the calculations in these different realizations of the first-principles methodology are made correctly, then the predictions of these different implementations should be the same. 

Variants of the single-particle equations (4.20) are obtained for the other versions of RDFT. Starting from die zeroth component of (4.15),... 

In order to simplify the resulting single-particle equations, we next absorb E into Exc (MacDonald and Vosko 1979), relying on the fact that j is a unique functional of n, m. However, as for Equations (4.27), (4.28) this usually implies the neglect of E. With this redefinition/approximation, Equation (4.13) leads to (Eschrig et al. 1985 Ramana and Rajagopal 1981a)... 

We are thus led to consider the RDFT formalism for collinear m, Equation (4.14), which serves as a standard tool for the discussion of magnetic systems. The corresponding single-particle equations follow from Equations (4.29)- 4.33) by restriction to the z-component of m. A particularly useful form of the equations for collinear m is found in terms of the generalized spin-densities n ,... 

A sometimes overlooked fact is that the Kohn-Sham equation is exact at this stage. It is much easier to solve than the coupled Schrodinger equations that would have to be solved for the original system, since it decouples into single particle equations. The only problem is that we have introduced the exchange-correlation energy, which is an unknown quantity, and which must be approximated. Fortunately, it will turn out to be relatively easy to find reasonably good local approximations for it. 

These equations can be written more explicitly in terms of Kohn-Sham orbitals defined to be the eigenstates of the single particle equation... 

If one works out this expression one obtains equations that are identical to equations (316) and (317). These equations were first derived by Talman and Shadwick [45]. Since in our procedure we optimized the energy of a Slater determinant wavefunction under the constraint that the orbitals in the Slater determinant come from a local potential, the method is also known as the optimized potential method (OPM). We have therefore obtained the result that the OPM and the expansion to order e2 are equivalent procedures. The OPM has many similarities to the Hartree-Fock approach. Within the Hartree-Fock approximation one minimizes the energy of a Slater determinant wavefunction under the constraint that the orbitals are orthonormal. One then obtains one-particle equations for the orbitals that contain a nonlocal potential. Within the OPM, on the other hand, one adds the additional requirement that the orbitals must satisfy single-particle equations with a local potential. Due to this constraint the OPM total energy ) will in general be higher than the Hartree-Fock energy Fhf, i.e., Ex > E. We refer to Refs. [46,47] for an application of the OPM method for molecules. 

The basic idea in density functional theory is to replace the Schrodinger equation for the interacting electronic system with a set of single-particle equations whose density is the same as that of the original system. These equations are the Kohn-Sham equations[6], and may be written... 

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