Relativistic Hohenberg-Kohn Theorem

Moreover, all corresponding counterterms (being expectation values) are independent of the representation, so that the renormalisation scheme remains unchanged and it is just a matter of convenience which representation is used. While the Heisenberg representation (2.20) is more suitable for the derivation of explicit functionals, the Hamiltonian (2.43) (together with Eq. (2.40)) can be utilised for the proof of a relativistic Hohenberg-Kohn theorem. 

Foundation containing some comments on the relativistic Hohenberg-Kohn theorem and indicating how the exact (but not easily solvable) relativistic Kohn-Sham equations (containing radiative corrections and all that) can be reduced to the standard approximate variant. 

The relativistic Hohenberg-Kohn theorem was first formulated by Rajagopal and Callaway [5,6] and by McDonald and Vosko [7]. As expected for a Lorentz covariant situation it states that the ground-state energy is a rniique functional of the ground-state four-current... 

This outlines the main steps in developing a relativistic Hohenberg-Kohn theorem. The requirements for a stringent derivation go considerably beyond this. A complete derivation would involve ... 

The Hohenberg-Kohn theorems were extended by Rajagopal and Callaway (25) to the more general relativistic case. Instead of the electron density they treated the 4-current in the same manner as Hohenberg and Kohn and obtained the energy expression,... 

An overview of relativistic density functional theory (RDFT) is presented with special emphasis on its field theoretical foundations and the construction of relativistic density functionals. A summary of quantum electrodynamics (QED) for bound states provides the background for the discussion of the relativistic generalization of the Hohenberg-Kohn theorem and the effective single-particle equations of RDFT. In particular, the renormalization procedure of bound state QED is reviewed in some detail. Knowledge of this renormalization scheme is pertinent for a careful derivation of the RDFT concept which necessarily has to reflect all the features of QED, such as transverse and vacuum corrections. This aspect not only shows up in the existence proof of RDFT, but also leads to an extended form of the single-particle equations which includes radiative corrections. The need for renormalization is also evident in the construction of explicit functionals. 

The relativistic generalisation of the Hohenberg-Kohn theorem states that the external four-potential is - except for a gauge transformation - determined by the four-current of the system. The first component e7° is the charge density while the spatial components, J are associated both with orbital currents and the spin density. In the non-relativistic limit, the coupling of electron spin to an external magnetic field is automatically retrieved. 

Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem, estabUshes the time-dependent case of the latter (see Sect. 6.4). 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

From what has been said already with respect to the variational collapse and the minimax principle, it is clear from the beginning that the standard derivation of the Hohenberg-Kohn theorems [386], which are the fundamental theorems of nonrelativistic DFT and establish a variational principle, must be modified compared to nonrelativistic theory [383-385]. Also, we already know that the electron density is only the zeroth component of the 4-current, and we anticipate that the relativistic, i.e., the fundamental, version of DFT should rest on the 4-current and that different variants may be derived afterwards. The main issue of nonrelativistic DFT for practical applications is the choice of the exchange-correlation energy functional [387], an issue of equal importance in relativistic DFT [388,389] but beyond the scope of this book. 

Modern DFT is founded on the Hohenberg-Kohn theorems and the Kohn-Sham equations. These are presented in detail in textbooks as well as in the review literature on the subject (Parr and Yang 1989, Koch and Holthausen 2001, Eschrig 1996, Gross and Kurth 1994, Salahub et al. 1994). However, to set the stage for a discussion of the relativistic case, a brief summary of the nonrelativistic foundations serves as a convenient starting point. 

The Hohenberg-Kohn Theorem for Relativistic N-Particle Systems... 

Starting from these considerations, we can repeat the reasoning of the nonrelativistic Hohenberg-Kohn theorem in a relativistic setting. We thus need to show the 1 1... 

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