Dirac-Hartree-Fock-Breit

Dirac-Hartree-Fock-Breit (DHFB) calculations of atoms and molecules across the Periodic Table. 

Tupitsyn, I. I. HFDB 2003. Program for atomic finite-difference four-component Dirac-Hartree-Fock-Breit calculations written on the base of the hfd code [110]. 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

In Table 2, the results of Hartree-Fock/Many-Body Perturbation Theory calculations for the argon atom are compared with two relativistic calculations the first using a Dirac-Hartree-Fock-Coulomb reference and the second using a Dirac-Hartree-Fock-Breit independent particle model. 

Figure 4.13 Excitation energies for the s-d and s-p gaps of the Group 11 elements. Experimental (Cu, Ag and Au) and coupled cluster data (Rg) are from Refs. [4, 91]. For the s-p gap of Rg we used Dirac-Hartree-Fock calculations including Breit and QED corrections.

Table 6 Matrix Dirac-Hartree-Fock (Edhf) and Hartree-Fock (Ehf) energies calculated using BERTHA. The Gaussian exponential parameters are those of the non-relativistic sets derived by van Duijenveldt and tabulated in Poirier et al [36]. Thejirst-order molecular Breit energy, Eb, v as calculated using methods described in [12] relativistic corrections to Ehf collected in the column labelled E energies are in atomic units.

Table 7 Estimates of total relativistic correction, E, and the first-order Breit energy correction,, obtained by combining the atomic or ionic contributions indicated by the second column. They may be compared with the values of the total relativistic correction, Ek. and thefirst-order Breit interaction, Eb, obtained directly from matrix Dirac-Hartree-Fock and Hartree-Fock calculations of the molecular structure using BERTHA [12], Only the results of the Iis7p2d atom-centred basis sets for Ek and Eb are quoted. All energies in atomic units.

Relativistic PPs to be used in four-component Dirac-Hartree-Fock and subsequent correlated calculations can also be successfully generated and used (Dolg 1996a) however, the advantage of obtaining accurate results at a low computational cost is certainly lost within this scheme. Nevertheless, such potentials might be quite useful for modelling a chemically inactive environment in otherwise fully relativistic allelectron calculations based on the Dirac-Coulomb-(Breit) Hamiltonian. 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

Dirac-Hartree-Fock-Coulomb energy plus first order Breit energy -528.552 09... 

Figure 6. The (Dirac) Hartree-Fock energy — hf and the first order Breit correction are given for helium-like ions. —grows approximately as and grows approximately as for large Z.

AE, all-electron results HF, Haitree-Fock DHF, Dirac-Hartree-Fock (spin-otbit averaged values) +QED, DHF including the Breit interaction and quantum electrodynamic corrections WB, Wood-Boring. 

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