Relativistic excitation energies

The information obtainable from photoelectron polarization measurements is reviewed, for both atoms and molecules, by Heinzmann and Cherepkov (1996). Even at non-relativistic excitation energy, photoelectrons can be spin-polarized (Fano, 1969). For l / 0 atoms, due to the spin-orbit splitting of the initial atomic and/or the final ionic state, photoelectrons are in most cases highly spin-polarized (up to 100%) when photoexcited with circularly polarized light. Analogous effects occur in molecular photoionization, but systematic studies have only been made for hydrogen halide molecules, HX. The electronic ground state of HX+ is X2n. 

Abstract The equation of state (EOS) of nuclear matter at finite temperature and density with various proton fractions is considered, in particular the region of medium excitation energy given by the temperature range T < 30 MeV and the baryon density range ps < 1014 2 g/cm3. In this region, in addition to the mean-field effects the formation of few-body correlations, in particular light bound clusters up to the alpha-particle (1 < A < 4) has been taken into account. The calculation is based on the relativistic mean field theory with the parameter set TM1. We show results for different values for the asymmetry parameter, and (3 equilibrium is considered as a special case. 

Energy levels of heavy and super-heavy (Z>100) elements are calculated by the relativistic coupled cluster method. The method starts from the four-component solutions of the Dirac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. Simultaneous inclusion of relativistic terms in the Hamiltonian (to order o , where a is the fine-structure constant) and correlation effects (all products smd powers of single and double virtual excitations) is achieved. The Fock-space coupled-cluster method yields directly transition energies (ionization potentials, excitation energies, electron affinities). Results are in good agreement (usually better than 0.1 eV) with known experimental values. Properties of superheavy atoms which are not known experimentally can be predicted. Examples include the nature of the ground states of elements 104 md 111. Molecular applications are also presented. 

Equation (3) incorporates relativistic effects, effects of target density, and corrections to account for binding of inner-shell electrons, as well as the mean excitation energy C/Z is determined from the shell corrections, S/2 is the density correction, Ifj accounts for the maximum energy that can be transferred in a single collision with a free electron, m/M is the ratio of the electron mass to the projectile mass, and mc is the electron rest energy. If the value in the bracket in Eq. (4) is set to unity, the maximum energy transfer for protons... 

Figure 9 Scaled excitation energies (au) for different hydrogen-like ions against Debye shielding (au) showing explicitly relativistic effects. Reprinted with permission from [173] 2004, American Physical Society...

A relativistic Hartree-Fock-Wigner-Seitz band calculation has been performed for Bk metal in order to estimate the Coulomb term U (the energy required for a 5f electron to hop from one atomic site to an adjacent one) and the 5f-electron excitation energies (143). The results for berkelium in comparison to those for the lighter actinides show increasing localization of the 5f states, i.e., the magnitude of the Coulomb term U increases through the first half of the actinide series with a concomitant decrease in the width of the 5f level. 

In addition, there is interest in further extending the discussion to a variety of situations, that have recently gained much attention in the nonrelativistic case, as time-dependent systems [49], excited states [45] or finite temperature ensembles [110]. As an example of work along these lines we mention the gradient expansion of the noninteracting, relativistic free energy [110], leading to a temperature-dependent relativistic extended Thomas-Fermi model. 

Relativistic calculation have also been performed [8]. A simple Xa potential was used with the parameter a = 0.7 to obtain the first six excitation energies of the Cs atom. A comparison with experimental data shows that this simple approximation provides quite adequate results. 

Fig. 4. Relativistic shifts for s d"-s d" excitation energies of transition-metal atoms. Differences between excitation energies from Hartree-Fock and relativistic Hartree-Fock calculations are plotted. (Reproducedfrom Ref 174 by permission of the authors and the American Institute of Physics.)...

Nevertheless, the relativistic corrections are not negligible even for these 3d elements. In fact, in the case of FeO relativity reduces the excitation energy from the 5 A ground state to the first excited state (5 E) from the nonrelativistic value of 0.4 eV to 0.2 eV. On the other hand, the comparison of the LDA results with experiment clearly shows the need for nonlocal corrections. The GGA results are consistently closer to the experimental data, in particular for Re. The GGA values for >e are nonetheless not completely satisfying, which underlines the importance of the truly nonlocal contributions to Exc. 

The formulae for the level shifts derived from the relativistic calculations are given on p. 50. They are taken from Bethe, Brown and Stehn [8] where the computation of the average excitation energy for the 22S level in hydrogen is to be found. Calculations of the average excitation energy ir0, for the levels Is to 4p inclusive have been made by Harriman [56]. As anticipated (p. 47). these quantities are not very sensitive to n. 

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