Current Density Relativistic Form

So it is also possible to use the form (139) for the vacuum charge current density, a form that eliminates any geometric unit such as Ar that is not fully relativistic. However, A is, strictly speaking, a potential energy difference and not a field. 

According to the Dirac [36] electron theory, the relativistic wavefunction has four components in spin-space. With the Hermitian adjoint wave function , the quantum mechanical forms of the charge and current densities become [31,40]... 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

It is directly possible to prove a HK-theorem for the form (3.55) using the density n and the gauge-dependent current jp — (c/e)V x m as basic DFT variables, but not for the form (3.54) which would suggest to use n and the full current j. One is thus led to the statement that the first set of variables can legitimately be used to set up nonrelativistic current density functional theory, indicating at first glance a conflict with the fully relativistic DFT approach. 

On the other hand, it should be pointed out that the relativistic current density can be recast in a form similar to that of the non-relativistic one through the Gordon decomposition [61,38]. Also, the non-relativistic current density depends explicitly on the external vector potential, whereas the relativistic one appears not to. However, also for the relativistic current density an explicit dependence can be brought out by considering the modification of the coupling of large and small components induced by the external vector potential. Finite basis set calculations rely on the kinetic balance relation [62,63]... 

The first term of Eq. (6) is just the non-relativistic current density for a spinless particle, while the second term arises from the electron spin and has the form of the curl of the spin density. In the relativistic case, this decomposition is at best an approximation since spin and linear motion of a particle are coupled. The kinetic energy (including the rest energy) of the Kohn-Sham reference svstem is given... 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

The functional dependence of Tg and xc on needs to be established. In Kohn-Sham applications, Tg is expressed directly in terms of spinor orbitals, so only Axe has to be considered. If one is aiming at setting up relativistic extensions of extended Thomas-Fermi models, one also has to consider dependence of Tg on the four-current. I shall present a few remarks on the density functional form of Tg, but first we look at the exchange and correlation energy. 

The second correction is a modification of the interaction energies. On the level of relativistic density functional theory for Coulomb systems this means, for instance, the replacement of the standard Hartree energy by its covariant form involving electron four-currents, j and the photon propagator,... 

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