Coulomb-Breit potential

Coulomb-Breit potential gives the following set of operators, where the QED correction to the electronic spin has been introduced by means of the ge pa factor. 

A judicious choice of the zero-order kernel (sum of the Coulomb and Breit potentials, for more detail see, e.g, [6, 7, 10]) generates a solvable unperturbed... 

The recoil correction in (4.19) is the leading order (Za) relativistic contribution to the energy levels generated by the Braun formula. All other contributions to the energy levels produced by the remaining terms in the Braun formula start at least with the term of order (Za) [17]. The expression in (4.19) exactly reproduces all contributions linear in the mass ratio in (3.5). This is just what should be expected since it is exactly Coulomb and Breit potentials which were taken in account in the construction of the effective Dirac equation which produced (3.5). The exact mass dependence of the terms of order Za) m/M)m and Za) m/M)m is contained in (3.5), and, hence,... 

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. 

The curve G corresponds to the first order Coulomb interelectron interaction, the curve C++ corresponds to the no pair approximation for the second-order Coulomb box interelectron interaction (Fig.7a). The curve B corresponds to the first order Breit interaction, the curve BC corresponds to the second-order Coulomb-Breit box interaction (exchange of the one Coulomb and one transverse photons) Fig.7c,d. The curve denoted by ( ) includes the contributions ( )= GC, BB, X, where GG is the negative-energy contribution to the Coulomb - Coulomb box interaction Fig.7a, BB is the Breit-Breit box interaction Fig.7g, X denotes all cross interactions Fig.7b,e,f,h. The order of magnitude of all ( ) corrections is defined by the high-energy intermediate electron state contributions. This means that the corresponding effective interelectron interaction potential does not depend on the ex-... 

Figure 1. ROPM exchange-only potentials for neutral Hg Selfconsistent Coulomb (C), Coulomb-Breit (C-l-B) and fully transverse (C-t-T) results in comparison with nonrelativistic limit (NR). At r = 10 Bohr one finds P = (3jr /j) /(ffic) =2.9 (for a finite nucleus).

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. 

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,... 

Ionization potentials and excitation energies of element 121 and its cations by the Dirac-Coulomb and Dirac-Coulomb-Breit CCSD method (in eV). 

The most advanced relativistic approach in relativistic calculations of X-ray spectra, is most likely that based on the Dirac-Coulomb-Breit Hamiltonian and quantum electrodynamic contributions accounted for. In addition, one should also carry out the corresponding correlated-level calculation within these relativistic formalism. To illustrate the role and size of relativistic and QED corrections the core and valence ionisation potentials and excitation energies of noble gases are shown. The relativistic fOTC CASSCE/CASPT2 method together with the restricted active space... 

The many-electron Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian is neither gauge invariant — it does not even contain vector potentials — nor Lorentz covariant as these symmetries have explicitly been broken in section 8.1. Moreover, the first-quantized relativistic many-particle Hamiltonians suffer from serious conceptual problems [217], which are solely related to the unbounded spectrum of the one-electron Dirac Hamiltonian h. ... 

These SCF equations contain the r variable only and are therefore onedimensional, which makes them particularly accessible by numerical solution methods. The corresponding Dirac-Coulomb-Brrif SCF equations [201] can be obtained in an analogous way from the energy expression that includes the Breit term. It is, however, interesting to note that some of the Breit potential energy terms enter the qff-diagonal and are therefore added to (r). This is a... 

The two relativistic four-component methods most widely used in calculations of superheavy elements are the no-(virtual)pair DF (Coulomb-Breit) coupled cluster technique (RCC) of Eliav, Kaldor, and Ishikawa for atoms (equation 3), and the Dirac-Slater discrete variational method (DS/DVM) by Fricke for atoms and molecules. " Fricke s DS/DVM code uses the Dirac equation (3) approximated by a Slater exchange potential (DFS), numerical relativistic atomic DS wavefunctions, and finite extension of the nuclei. DFS calculations for the superheavy elements from Z = 100 to Z = 173 have been tabulated by Fricke and Soff. A review on various local density functional methods applied in superheavy chemistry has been given by Pershina. ... 

Table 2.1 Ionization potentials (IP) and excitation energies (EE) of La and its cations (eV) by the nonrelativistic (NR), Dirac-Coulomb (DC), and Dirac-Coulomb-Breit (DCB) FSCCSD method...

The ionization potentials and low excitation energies calculated for El 22 are shown in Table 2.7. More values may be found in [60]. Intermediate Hamiltonian values for E122 and its monocation were calculated by the Dirac-Coulomb and Dirac-Coulomb-Breit schemes, to obtain the effect of the Breit interaction (2.2). The Breit term contribution is small (0.01-0.04 eV) for transitions not involving/ electrons but increases to 0.07-0.1 eV when/ orbital occupancies are affected, as observed above (Section 2.3.1). The ground state is predicted to be 8s 8p7d, in agreement with early Dirac-Fock(-Slater) calculations [55-57], and not the 8s 8p configuration obtained by density functional theory [58]. The separation of the... 

Modification of the potential operator due to the finite speed of light. In the lowest order approximation this corresponds to addition of the Breit operator to the Coulomb interaction. 

---