Dirac-Coulomb theory

Dirac-Coulomb theory within the mean field approximation (see Chapter 8) is routinely applied to molecules and allows us to estimate the relativistic effects even for large molecules. In the computer era. this means that there are computer programs available that allow anybody to perform relativistic calculations. 

Compared to typical chemical phenomena, the relativistic effects remain of marginal significance in almost all instances for the biomolecules or molecules typical in traditional organic chemistry. In inorganic chemistry, however, these effects could be much more important. Probably the Dirac-Coulomb theory combined with the mean field approach will remain a satisfactory standard for the vast majority of researchers, at least for the next few decades. At the same time, there... 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

A full account of the theory of relativistic molecular structure based on standard QED in the Furry picture will be found in a number of publications such as [7, Chapter 22], [8, Chapter 3]. These accounts use a relativistic second quantized formalism. For present purposes, it is sufficient to present the structure of BERTHA in terms of the unquantized effective Dirac-Coulomb-Breit (DCB) A-electron Hamiltonian ... 

A more precise value than in [63] of the nonlogarithmic correction of order a Za) for the IS -state was obtained in [66, 67], with the help of a specially developed perturbation theory for the Dirac-Coulomb Green function which expressed this function in terms of the nonrelativistic Schrodinger-Coulomb Green function [68, 69]. But the real breakthrough was achieved in [70, 71], where a new very effective method of calculation was suggested and very precise values of the nonlogarithmic corrections of order a Zo) for the IS -and 25-states were obtained. We will briefly discuss the approach of papers [70, 71] in the next subsection. 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

The Dirac-Coulomb-equation (i.e. the dimensionless equation with Coulomb potential) has exact solutions and can therefore be used as a good starting point for perturbation theory. 

We will start by reviewing some basic relativistic theory to introduce the notation and concepts used. The rest of the chapter is devoted to the three major post-DHF methods that are currently available for the Dirac-Coulomb-Breit Hamiltonian. All formulas will be given in atomic units. 

The simplest approximation is to combine the Dirac theory with the nrl of electrodynamics, which automatically leads to the Dirac Coulomb Hamiltonian... 

For further details the reader is referred to, e.g., a review article by Kutzel-nigg [67]. The Gaunt- and Breit-interaction is often not treated variationally but rather by first-order perturbation theory after a variational treatment of the Dirac-Coulomb-Hamiltonian. The contribution of higher-order corrections such as the vaccuum polarization or self-energy of the electron can be derived from quantum electrodynamics (QED), but are usually neglected due to their negligible impact on chemical properties. 

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