Dirac Breit Hamiltonian

In contrast to the one-electron terms, the reduction of the 4x4 Dirac-Breit Hamiltonian to the 2x2 Breit-Pauli Hamiltonian is very tedious for the two-electron terms as each interaction term has to be transformed according to the Foldy-Wouthuysen protocol. As the derivation can be found for example in Refs. (56-58) and in detail in Ref. (21), we only present here the transformed terms and discuss their dimension. The two-electron Breit-Pauli operator gBP (i, j) reads... 

Relativistic corrections of order v2/c2 to the non-relativistic transition operators may be found either by expanding the relativistic expression of the electron multipole radiation probability in powers of v/c, or semiclas-sically, by replacing p in the Dirac-Breit Hamiltonian by p — (l/c)A (here A is the vector-potential of the radiation field) and retaining the terms linear in A. Calculations show that in the general case the corresponding corrections have very complicated expressions, therefore we shall restrict ourselves to the particular case of electric dipole radiation and to the main corrections to the length and velocity forms of this operator. 

Naturally, the field-dependent Breit-Pauli Hamiltonian automatically results as the low-order limit of the Foldy-Wouthuysen-transformed field-dependent Dirac-Breit Hamiltonian. Once this has been carried out along... 

For the calculations of relativistic density functions we used a multi-configuration Dirac-Fock approach (MCDF), which can be thought of as a relativistic version of the MCHF method. The MCDF approach implemented in the MDF/GME program [4, 27] calculates approximate solutions to the Dirac equation with the effective Dirac-Breit Hamiltonian [27]... 

The Dirac-Breit-Coulomb (DBC) Hamiltonian (Hdbc) is a good starting point for a discussion of many-fermion relativistic Hamiltonians, not only because of its historical importance and widespread use in molecular relativistic calculations (in spite of its known failures) but also as an excellent example of a disastrous... 

The Dirac-Coulomb-Breit Hamiltonian rewritten in second-quantized... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

From Dirac-Breit to Breit-Pauli Hamiltonians 189... 

By inserting the equations defining the kinetic energy operators and the pairwise interaction operators into Eq. (8) we obtain the Dirac-Coulomb-Breit Hamiltonian, which is in chemistry usually considered the fully relativistic reference Hamiltonian. 

Up to this point, we have presented the fully relativistic Hamiltonian. Of course, we could set out to calculate energies of molecules employing this Hamiltonian. However, the various spin-spin interactions are easier described in terms of a perturbation picture rather than as excited states of the full-fledged Hamiltonian. Especially for the fully relativistic Dirac-Coulomb-Breit Hamiltonian, the latter calculations would be computationally very demanding. 

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). 

Dependent Terms of the Dirac-Coulomb-Breit Hamiltonian. 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

The Breit Hamiltonian for two electrons consists essentially of a Dirac Hamiltonian for each electron, with interaction terms. It may be written... 

It is possible to obtain the nuclear spin magnetic interaction terms by starting from the Breit equation. We recall that the Breit Hamiltonian describes the interaction of two electrons of spin 1 /2, each of which may be separately represented by a Dirac Hamiltonian ... 

In chapter 3 we showed how the relativistic Breit Hamiltonian can be reduced to non-relativistic form by means of a Foldy Wouthysen transformation. We obtained equations (3.244) and (3.245) which represent the non-relativistic Hamiltonian for two particles of masses m, and nij and electrostatic charges and —ej and from this Hamiltonian we were able to derive the interelectronic interactions. We could, however, consider using (3.244) and (3.245) as the Hamiltonian for an electron of charge —e, = — e and mass m, = m, and a nucleus of mass nij = Ma and charge —ej = + 7. e. As before, we make the assumption that the nucleus has spin 1 /2, behaves like a Dirac particle and has an anomalous magnetic moment compared with that given by the Dirac theory. Consequently we may rewrite (3.245) by making the replacements... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

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. 

Nieuwpoort, W. C., Aerts, P. J. C. and Visscher, L. (1994) Molecular electronic structure calculations based on the Dirac-Coulomb-(Breit) Hamiltonian. In Malli (1994), pp. 59-70. 

Since this only affects the one-electron portion of the Hamiltonian, its implementation in DFT is straightforward for atomic calculations. However the eigenvalues of this relativistic Hamiltonian also correspond to a negative continuum [24]. A more sophisticated Hamiltonian is the non-virtual pair approximation or the projected Dirac-Coulomb-Breit Hamiltonian [24] ... 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

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. 

This is the leading relativistic correction of 0[c ) to the energy, based on the Dirac-Coulomb-Hamiltonian. We shall later see that there is another term of 0(c ) due to the Breit interaction. 

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. 

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,... 

The formalism described here to derive energy-consistent pseudopotentials can be used for one-, two- and also four-component pseudopotentials at any desired level of relativity (nonrelativistic Schrbdinger, or relativistic Wood-Boring, Douglas-Kroll-Hess, Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian implicit or explicit treatment of relativity in the valence shell) and electron correlation (single- or multi-configurational wavefunctions. The freedom... 

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