Dirac -Coulomb-Breit

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.

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

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, ... 

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 ... 

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. 

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). 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

Dependent Terms of the Dirac-Coulomb-Breit Hamiltonian. 

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. 

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