Four-current density functional theory

The convexity of E[A, Q] with respect to Q gives rise to a pair of mutual Legendre transforms 

Like —E, G is convex in A, hence another pair of mutual Legendre transforms is 

The general rule sup inf inf sup was used here, but a more subtle consideration proves equality to hold in the present case [2]. The last supremum now is over a linear function of C and hence is -foo unless the prefactor is zero. Hence, to obtain the infimum over J this prefactor must be zero, and the final result is 

In the non-relativistic theory, the kinetic energy is T = -EVf/2 , and T -t-oo implies n E L T ) for the density n via Sobolev s inequality [3]. The relativistic kinetic energy is (in the Schrodinger representation for ) Tr = ( I X) V j I ), and one is tempted to suppose G (T ). If for instance a bispinor orbital behaves like (f) r , then G implies s 1 for which indeed Tr +00. In the above exploited duality theory of Legendre transforms the variational space of admitted potentials is the dual of the variational space of admitted four-current densities, hence A G L T ) must be demanded. For a potential A this implies again s 1 and hence excludes Coulomb potentials although it permits to treat them as a limiting case. 

In a relativistic theory Coulomb potentials are a touchy case anyway. First of all, Ha would not be bounded below for A with s 1 an unlimited number of electrons (or positrons, depending on the sign of Aq) would plimge into the potential center as their mutual Coulomb repulsion could not compensate the attraction for small enough r. For s = 1, in the 

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

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. 

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. 

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,... 

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]... 

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