Four-component ab initio method

Various approaches can be pursued to compute spin-orbit effects. Four-component ab initio methods automatically include scalar and magnetic relativistic corrections, but they put high demands on computer resources. (For reviews on this subject, see, e.g., Refs. 18,19,81,82.) The following discussion focuses on two-component methods treating SOC either perturbationally or variationally. Most of these procedures start off with orbitals optimized for a spin-free Hamiltonian. Spin-orbit coupling is added then at a later stage. The latter approaches can be divided again into so-called one-step or two-step procedures as explained below. 

Four-Component Ab Initio Methods for Atoms, Molecules and Solids... 

M. Reiher, J. Hinze. Four-component ab initio methods for electronic structure calculations of atoms, molecules, and solids. In B. A. Hess, Ed., Relativistic Effects in Heavy-Element Chemistry and Physics, p. 61-88, Chichester, 2003. Wiley. 

In a rigorous treatment, one replaces the one-electron operator h by the four-component Dirac-operator hjj and perhaps supplement the two-electron operator by the Breit interaction term [15]. Great progress has been made in such four-component ab initio and DPT methods over the past decade. However, they are not yet used (or are not yet usable) in a routine way for larger molecules. 

Relativistic Methods 204 8.1 Connection Between the Dirac and Schrodinger Equations 207 8.2 Many-particle Systems 210 8.3 Four-component Calculations 213 11.4.1 Ab Initio Methods 272 11.4.2 DFT Methods 273 11.5 Bond Dissociation Curve 274 11.5.1 Basis Set Effect at the HF Level 274 11.5.2 Performance of Different Types of Wave Function 276... 

In addition to the ab initio approach to relativistic electronic structure of molecules, four-component Kohn-Sham programs, which approximate the electron-electron interaction by approximate exchange-correlation functionals from density functional theory, have also been developed (Liu et al. 1997 Sepp et al. 1986). However, we concentrate on the ab initio methods and refer the reader to Chapter 4, which treats relativistic density functional theory (RDFT). 

So far, we have only discussed the four-component basis-set approach in connection with the simplest ab initio wave-function model, namely for a single Slater determinant provided by Dirac-Hartree-Fock theory. We know, however, from chapter 8 how to improve on this model and shall now discuss some papers with a specific focus on correlated four-component basis-set methods. 

It was soon obvious that ab initio methods would be impractical for the study of large polyatomic systems. Attempts were made to use empirically determined data to approximate the complicated integrals used in ab initio theory. All of the difficult three- and four-center integrals were ignored, and the one- and two-center terms were approximated using a mixture of functions based on atomic spectra and on formal theory. Procedures of this type, which have both experimental and theoretical components, are called semiempirical methods. 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

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