Dirac Coulomb energies

This equation has the same contributions of order (Za)" as in (3.4), but formally this expression also contains nonrecoil and recoil corrections of order Zaf" and higher. The nonrecoil part of these contributions is definitely correct since the Dirac energy spectrum is the proper limit of the spectrum of a two-particle system in the nonrecoil limit m/M = 0. As we will discuss later the first-order mass ratio contributions in (3.5) correctly reproduce recoil corrections of higher orders in Za generated by the Coulomb and Breit exchange photons. Additional first order mass ratio recoil contributions of order (Za) ... 

Numerical Dirac-Fock-Coulomb (DFC) energies from [65] except for Eka-Rn. DFC calculation performed with Molfdir [66] using the basis given in the... 

Dirac-Hartree-Fock-Coulomb (DHFC) energy -528.68445... 

The consideration of the nonrelativistic limit of the Dirac energy eigenvalue for the hydrogen-like atom with a Coulombic potential for the electron-nucleus attraction, Eq. (6.3), demonstrates the effect of subtracting the rest energy mgC and leads us to a discussion of the reference energy in the following section 6.7. 

The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

A particularly interesting feature of the theory [9] is the incorporation of deviations from Coulomb scattering due to the nonvanishing size of the projectile nucleus. The very fact that the theory is based on the Dirac equation and that spin dependences enter nontrivially indicates that quantum mechanics is essential here. Moreover, at the highest energies considered, pair production becomes important, i.e., an effect that does not have a classical equivalent [57]. 

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.

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

Coulomb exchange effects are commonly introduced by means of the Dirac-Slater expression for the exchange energy of a electron gas ... 

Table 1 Xenon, comparison of orbital energies for numerical Dirac and ZORA and non relativistic calculations with basis set ZORA calculations in different Coulomb matrix approximations in the UGBS basis set...

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