Four-current density functional

A BRIEF INTRODUCTION TO FOUR-CURRENT DENSITY FUNCTIONAL THEORY... 

The four-current density functional H[J] is split into an orbital variation part K and a density integral L ... 

In the relativistic version of DFT, the ground state energy of the system is a unique functional of the fermion four-current density, / (/-) ... 

The energy (3.32) is, via the (rather involved) dependence of the RKS-orbitals on the four current density, a functional of this quantity,... 

We see here that the real and imaginary parts of the complex invariant I correspond to the two invariants of the standard form of electromagnetic theory, namely, the scalar and the pseudoscalar terms. They appear together here in a single complex function because of the reflection-nonsymmetric feature of this theory. The invariant h corresponds to the real-valued modulus of the four-current density of the standard theory. 

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]... 

Fig. 10.4. Cathodic OSWV peak current densities for the reduction of Cu2+ at MPA-Gly-Gly-His modified electrodes as a function of Cu2+ concentration. Error bars are +1 standard deviation of the current densities of four individual electrodes. Inset shows the peak current density in the region between 0 and 400 nM Cu2+. OSWVs were measured in 50 mM ammonium acetate (pH 7.0) and 50 mM NaCl at a pulse amplitude of 0.025 V, a step of 0.004 V and frequency of 25 Hz.

Such polymers might be of substantial technological interest as electrically conductive molecular wires. Density functional (DFT) calculations indicated that the metal-based HOMO of 26 is delocalized across all four of the central copper(I) ions, as shown in Figure 1.6. Calculations also indicate that removal of one electron from this HOMO likewise results in a fully delocalized singly occupied orbital. This delocalization could provide a path for conductivity, allowing an electric current to flow from one end of a helicate polymer strand to the other. 

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. 

In the past four decades, we have witnessed the significant development of various methods to describe microporous solids because of their important contribution to improving of adsorption capacity and separation. Various models of different complexity have been developed [5]. Some models have been simple with simple geometry, such as slit or cylinder, while some are more structured such as the disk model of Segarra and Glandt [6]. Recently, there has been great interest in using the reverse Monte Carlo (MC) simulation to reconstruct the carbon structure, which produces the desired properties, such as the surfece area and pore volume [7, 8]. Much effort has been spent on studies of characterization of porous media [9-15]. In this chapter we will briefly review the classical approaches that still bear some impact on pore characterization, and concentrate on the advanced tools of density functional theory (DFT) and MC, which currently have wide applications in many systems. 

It must be emphasized that the restriction to an external potential of the type = (V°, 0) does not imply that the system cannot have some magnetic moment. Rather, the spatial components of the four-current must be viewed as functionals of the density, jln = [n] j 4>ln])jL0. 

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