Dirac-Hartree-Fock approximation

An important problem in the application of QED methods to many-electron atoms is the choice of the zero-order approximation (actually the choice of the basis set of the one-electron relativistic wave functions in Eq(86). One natural choice is the approximation of noninteracting electrons when the potential V in Eq(2) is the Coulomb potential of the nucleus (28). This approximation is convenient for highly charged, few-electron ions. For a many-electron neutral atom a better choice is the Dirac-Hartree-Fock (DHF) approximation. 

The potential Vdhf can be defined similarly to the nonrelativistic case  

apart from the interelectron interaction we have to take into account also the additional interaction with the external field —eVoHF- In first order of the perturbation theory (in the interaction constant e) for the two-electron atom we have to consider the additional Feynman graph depicted in Fig.Sa, in second order the graph Fig.Sb and in third order the graphs Fig.8c,d. 

However we have to remember that the order of magnitude of the Feynman graph with 2n vertices and m external field hnes is The extra powers 

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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. 

The sum on the right hand side of this expression is precisely the expectation value of the Hamiltonian in the independent-particle approximation. Since the sum on the right-hand side of Eq. (83) is the starting point for a variational treatment of the (Dirac) Hartree-Fock approximation, it follows that choosing U = Vhf leads to - - = Ehf-... 

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

Apparently, a large number of successful relativistic configuration-interaction (RCI) and multi-reference Dirac-Hartree-Fock (MRDHF) calculations [27] reported over the last two decades are supposedly based on the DBC Hamiltonian. This apparent success seems to contradict the earlier claims of the CD. As shown by Sucher [18,28], in fact the RCI and MRDHF calculations are not based on the DBC Hamiltonian, but on an approximation to a more fundamental Hamiltonian based on QED which does not suffer from the CD. At this point, let us defer further discussion until we review the many-fermion Hamiltonians derived from QED. 

The relativistic or non-relativistic random-phase approximation (RRPA or RPA)t is a generalized self-consistent field procedure which may be derived making the Dirac/Hartree-Fock equations time-dependent. Therefore, the approach is often called time-dependent Dirac/Hartree-Fock. The name random phase comes from the original application of this method to very large systems where it was argued that terms due to interactions between many alternative pairs of excited particles, so-called two-particle-two-hole interactions ((2p-2h) see below) tend to... 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

In the case of a general polyatomic molecule, the I of the integral above may be located on different atoms, in the worst case giving rise to a four-center integral. For calculations within mean-field or independent particle approximations, such as Hartree-Fock or Dirac-Hartree-Fock (DHF), the bulk of the computational effort lies in the evaluation and handling of these two-electron integrals. This has consequences for our choice of expansion functions for the analytic approximation. 

Dirac s relativistic theory for the motion of electrons in molecules was introduced in the preceding chapters. The appearance of positron solutions and the four-component form of the wave function looks problematic at first sight but in practice it turns out that the real challenge is, like in non-relativistic electronic structure theory, the description of the correlation between the motion of electrons. The mean-field approximation that is made in the Dirac-Hartree-Fock (DHF) approach provides a good first step, but gives bond energies and structures that are often too inaccurate for chemical purposes. 

Dirac-Hartree-Fock and Dirac-Kohn-Sham methods By an application of an independent-particle approximation with the DC or DCB Hamiltonian, the similar derivation of the non-relativistic Hartree-Fock (HF) method and Kohn-Sham (KS) DFT yields the four-component Dirac-Hartree-Fock (DHF) and Dirac-Kohn-Sham (DKS) methods with large- and small-component spinors. 

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. 

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.

Equilibrium distances r, vibrational constants (Og and dissociation energies Dg for the ground state of molecular iodine, fium Ref. [102], All-electron (AE) calculations are obtained from either the scalar-relativistic Douglas-Kroll-HeB (DKH) approximation or 4-component Dirac-Hartree-Fock (DHF) correlated calculation taken from Ref [103]. 

Figure 1.1 Radial expectation values for the valence s- and p-orbitals in periods 2 and 3 of the periodic table (approximate numerical Dirac-Hartree-Fock values from Ref. [14]). Figure adapted from Ref. [13].

For molecules, Hartree-Fock approximation is the central starting point for most ab initio quantum chemistry methods. It was then shown by Fock that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, has the same antisymmetric property as the exact solution and hence is a suitable ansatz for applying the variational principle. 

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