Chapter 10 Relativistic density functional theory: Foundations and basic formalism

Abstract

Abstract 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. In practice, on the other hand, radiative corrections are usually neglected in RDFT calculations. This neglect is formally introduced into RDFT via the no-pair approximation. Within this framework the main task is to find an appropriate approximation for the relativistic exchange-correlation energy functional. As explicit density functionals the relativistic local density approximation (RLDA) and the relativistic generalized gradient approximation (RGGA) are reviewed. Both their derivation from the properties of the relativistic homogeneous electron gas and a number of illustrative results are presented. In particular, it is shown that the RLDA does not provide an adequate description of the relativistic corrections in the case of atomic systems, while the RGGA performs as well for heavy atoms as the nonrelativistic GGA does for light atoms. Finally, a new generation of relativistic density functionals is discussed in which, in addition to the four current, the effective single-particle spinors are used for the representation of the exchange-correlation functional. The most prominent example for such an implicit density functional is the exact exchange. The actual application of implicit functionals requires the solution of an integral equation for the exchange-correlation potential (Optimized Potential Method), which is also introduced. On this basis a self-consistent treatment of the transverse exchange is possible, which allows a detailed investigation of the importance of transverse corrections.