Current nonrelativistic

No calculation has been made which includes all of the quantities listed in eqs. (3.100) and (3.102). Table 3 provides a summary of several current nonrelativistic optical potential models for proton-nucleus scattering and indicates which terms in eqs. (3.98)-(3.103) are included and notes the key approximations made. Further progress toward a calculation along the lines of eqs. (3.100) and (3.102) is expected in the near future. 

Solving this iterative process gives rise to a set of orbitals to construct the ground state four-current, J (x), including vacuum polarization corrections due to the external field as well as the field mediating the interaction between the electrons. As the charge density in the nonrelativistic case, the four-current has the form of a reference noninteracting A-electron system,... 

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

For the case of a purely electrostatic external potential, P = (F , 0), the complete proof of the relativistic HK-theorem can be repeated using just the zeroth component f (x) of the four current (in the following often denoted by the more familiar n x)), i.e. the structure of the external potential determines the minimum set of basic variables for a DFT approach. As a consequence the ground state and all observables, in this case, can be understood as unique functionals of the density n only. This does, however, not imply that the spatial components of the current vanish, but rather that j(jc) = < o[w]liWI oM) has to be interpreted as a functional of n(x). Thus for standard electronic structure problems one can choose between a four current DFT description and a formulation solely in terms of n x), although one might expect the former approach to be more useful in applications to systems with j x) 0 as soon as approximations are involved. This situation is similar to the nonrelativistic case where for a spin-polarised system not subject to an external magnetic field B both the 0 limit of spin-density functional theory as well as the original pure density functional theory can be used. While the former leads in practice to more accurate results for actual spin-polarised systems (as one additional symmetry of the system is take into account explicitly), both approaches coincide for unpolarized systems. 

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. 

Quite generally, it must be stated that some additional effort is required to develop the RDFT towards the same level of sophistication that has been achieved in the nonrelativistic regime. In particular, all exchange-correlation functionals, which are available so far, are functionals of the density alone. An appropriate extension of the nonrelativistic spin density functional formalism on the basis of either the time reversal invariance or the assembly of current density contributions (which are e.g. accessible within the gradient expansion) is one of the tasks still to be undertaken. 

Figure 10. The various contributions to the present universal mass/energy density, as a fraction of the critical density (Q), as a function of the Hubble parameter (Ho). The curve labelled Luminous Baryons is an estimate of the upper bound to those baryons seen at present (z ( 1) either in emission or absorption (see the text). The band labelled BBN represents the D-predicted SBBN baryon density. The band labelled by M (Om = 0.3 0.1) is an estimate of the current mass density in nonrelativistic particles ( Dark Matter ).

Using pseudopotentials has several major beneficial consequences (i) Only the valence electrons must be treated explicitly, thus the number of equations to be solved [Eqs. (13)] can be reduced drastically (ii) the pseudoorbitals are very smooth near the atomic core, and thus Tout can be reduced drastically and (iii) important relativistic effects of the core electrons of heavy elements such as the 5d elements can be included in nonrelativistic calculations. The major downsides are that the potential v(r) in Eq. (3) must be replaced with a more complicated and computationally expensive nonlocal pseudopotential and, more importantly, that the transferability of the pseudopotential, i.e., its accuracy in different bonding environments, may not be perfect. Developing highly transferable pseudopotentials that can be used at as low an cut as possible is a major current topic of research. 

Figure 1.1 Electron current density as a function of time. The laser pulse is characterized by a frequency co = 27.21 eV and an intensity I = 3.51 x 1021 W cm-2 = 105 a.u. Nonrelativistic (left part) and relativistic (right part) results are shown.

The basic CDFT Hamiltonian in Equation (5.6) does not rule out the existence of a finite orbital magnetic moment in the nonrelativistic limit. With the help of model calculations, it could be demonstrated that this is not the case for bulk Fe, Co and Ni (Ebert et al. 1997a). Starting an SCF calculation with a finite spin-orbit-induced orbital current density and switching off the spin-orbit coupling during the SCF-cycle the orbital magnetic moment vanished (see also next section). 

To study the magnetic properties of matter one would often like to be able to obtain information on the currents in the system and their coupling to possible external magnetic fields. Important classes of experiments for which this information is relevant are nuclear magnetic resonance and the quantum Hall effects. SDFT does not provide explicit information on the currents. RDFT in principle does, but standard implementations of it are formulated in a spin-only version, which prohibits extraction of information on the currents. Furthermore, the formalism of RDFT is considerably more complicated than that of SDFT. In this situation the formulation of nonrelativistic current-DFT (CDFT), accomplished by Vignale and Rasolt [140, 141], was a major step forward. CDFT is formulated explicitly in terms of the (spin) density and the nonrelativistic paramagnetic current density vector jp(r). Some recent applications of CDFT are Refs. [142, 143, 144, 145]. E. K. U. Gross and the author have shown that the existence of spin currents implies the existence of a link between the xc functionals of SDFT and those of CDFT [146], Conceptually, this link is similar to the one of Eq. (99) between functionals of DFT and SDFT, but the details are quite different. Some approximations for xc functionals of CDFT are discussed in Refs. [146, 147, 148]. 

The implementation follows current practices in nonrelativistic codes, but the internal structure, which has to carry relativistic spinor properties, is obviously different. However the computational overheads, which are much smaller than has usually been supposed, are offset by techniques to avoid calculating classes of integrals which make no detectable physical contribution and by improved representation of the physics. Some implementation issues are discussed in this section. 

In addition to the dominant PNC interaction given in Eq. (45), there are other smaller PNC interactions that must be considered. First, there is the interaction between the nuclear axial-vector current and the electron vector current from Z exchange. In the limit of nonrelativistic nucleon motion, this interaction is given by the spin-dependent Hamiltonian... 

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