Dirac-Coulomb -Breit Hamiltonian

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... 

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.

From a formal point of view, four-component correlation calculations [5, 6] based on the Dirac-Coulomb-Breit (DCB) Hamiltonian (see [7, 8, 9, 10, 11] and references therein) can provide with very high accuracy the physical and chemical properties of molecules containing heavy atoms. However, such calculations were not widely used for such systems during last decade because of the following theoretical and technical complications [12] ... 

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

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). 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

If a multiparticle system is considered and the election interaction is introduced, we may use the Dirac-Coulomb-Breit (DCB) Hamiltonian which is given by a sum of one-particle Dirac operators coupled by the Coulomb interaction 1 /r,7 and the Breit interaction Bij. Applying the Douglas-Kroll transformation to the DCB Hamiltonian, we arrive at the following operator (Hess 1997 Samzow and Hess 1991 Samzow et al. 1992), where an obvious shorthand notation for the indices pi has been used ... 

The Dirac-Coulomb-Breit (DCB) Hamiltonian for an N-electron system is... 

The spectrum of the single-electron Dirac operator Hd and its eigenspinors (/> for Coulombic potentials are known in analytical form since the early days of relativistic quantum mechanics. However, this is no longer true for a many-electron system like an atom or a molecule being described by a many-particle Hamiltonian H, which is the sum of one-electron Dirac Hamiltonians of the above kind and suitably chosen interaction terms. One of the simplest choices for the electron interaction yields the Dirac-Coulomb-Breit (DCB) Hamiltonian, where only the frequency-independent first-order correction to the instantaneous Coulomb interaction is included. 

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. 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

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. 

In the case of a relativistic system, as a first (and useful) approximation, the zero-order spectrum can be taken as the nonrelativistic one, with Hq defined explicifly as fhe Coulomb Hamiltonian. Then, the perturbation V is also written explicitly as the relativistic Breit-Pauli operators, and it is this perturbation that turns the initially discrete state into a resonance. For example, this type of advanced calculation, with multichannel coupling included, has been shown to explain quantitatively the positions and lifetimes of the relativistic levels of mefastable states in negative ions [90]. However, if the more accurate four-component relativistic Dirac treatment for each electron is invoked for cases of high effective nuclear charge, then the stability against autoionization implies not only the exclusion of components representing decay to... 

Abstract Variational methods can determine a wide range of atomic properties for bound states of simple as well as complex atomic systems. Even for relatively light atoms, relativistic effects may be important. In this chapter we review systematic, large-scale variational procedures that include relativistic effects through either the Breit-Pauli Hamiltonian or the Dirac-Coulomb-Breit Hamiltonian but where correlation is the main source of uncertainty. Correlation is included in a series of calculations of increasing size for which results can be monitored and accuracy estimated. Examples are presented and further developments mentioned. 

By substituting in Eq.4.18 the Coulomb interaction operator by its relativistic extension (4.19) one obtains the so-called Dirac-Breit many electron Hamiltonian... 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

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