Theoretical Transition Energies
In this section, we present a schematic description of the procedures that are
used to generate the theory portion of this database [88].
Uncorrelated Dirac-Fock Energies
The first step in the calculation minimizes the energy (with relativistic
corrections) in the independent electron approximation, for each hole state.
This provides a suitable starting point for adding many-body and QED
contributions. We follow the Dirac-Fock method, which allows treatment of
arbitrary atoms with arbitrary structure [89,90]. We
have included full exchange and relaxation (to account for inactive orbital
rearrangement due to the hole presence). The electron-electron interaction used
in this program contains all magnetic and retardation effects, which is very
important for large Z. The magnetic interaction is treated on an equal
footing with the Coulomb interaction, to account for higher order effects in
the wavefunction (that are also useful for evaluating radiative corrections to
the electron-electron interaction). All of these calculations must be done with
proper nuclear charge models, to account for finite-nuclear-size corrections to
all contributions. Nuclear deformations must be included for heavy nuclei
[91,92]. For all elements for which experiments have
been performed, we used experimental nuclear charge radii. For others we used a
formula from Johnson & Soff, corrected for nuclear deformations for
Z > 90 [93]. Contributions of
deformation to the rms radius (the only parameter of importance to the atomic
calculation) are roughly constant (0.11 fm) for Z > 90.
There is an unknown region, between Bi and Th
(83 < Z < 90) where deformation effects start
to be important, but for which they are not known. When experiments are
available for a particular isotope, we calculated separately the energies for
each isotope.
There are special difficulties involved when dealing with atoms with open outer
shells (obviously this is the most common case). Computing all energies
EJ for total angular momentum J would be both
impossible and useless. The Dirac-Fock method circumvents this difficulty. One
can evaluate directly an average energy that corresponds to the barycenter of
all EJ with weight (2J + 1).
There are still a few cases for which the average calculation cannot converge
(when the open shells have identical symmetry). In these cases the outer
electrons have been rearranged in an identical fashion for all hole states of
the atom, to minimize possible shifts due to this procedure.
Correlation and Auger Shifts
Once the Dirac-Fock energy is obtained, many-body effects beyond Dirac-Fock
relaxation must be taken into account. These include relaxation beyond the
spherical average, correlation (due to both Coulomb and magnetic interaction),
and corrections due to the autoionizing nature of hole states (Auger shift).
Since the many body generalization of the Dirac-Fock method, the so-called
Multiconfiguation Dirac-Fock (MCDF), is very inefficient for hole states, we
turned to relativistic many-body perturbation theory (RMBPT) to evaluate these
quantities. These many-body effects contribute very significantly to the final
value [94-97]. As these calculations are very time
consuming, they are performed only for selected Z and interpolated.
Since the Auger shifts do not always have a smooth Z-dependence, care
has been taken to evaluate them at as many different Zs as practical to
ensure a good reproduction of irregularities.
QED Corrections
The QED corrections originate in the quantum nature of both the electromagnetic
and electron fields. They can be divided into two categories, radiative and
non-radiative. The first one includes one-electron and electron-electron
radiative corrections. The one-electron radiative corrections (self-energy and
vacuum polarization) scale as Z4/n3 (n
being the principal quantum number) and are thus very important for inner
shells and high Z. For the self-energy and Z > 10
one must use all-order calculations [98-104]. Vacuum
polarization can be evaluated at the Uehling [105]
and Wichmann and Kroll level [106]. The
electron-electron radiative correction scales as
Z3/n3. Ab initio calculations
have been performed only for a few-electron ions [107,108]. Here we use the Welton approximation that
has been shown to reproduce very closely ab initio results in all
examples that have been calculated [91 and
109-112].
The second category (non-radiative) is composed of corrections to the
electron-electron interaction that cannot be accounted for by RMBPT and MCDF.
These corrections start at the two-photon interaction and include three-body
effects. The two-photon, non-radiative QED contribution has been calculated
recently only for the ground state of two-electron ions and cannot be evaluated
in practice for atoms with more than two or three electrons [113,114].
