Reliable absolute intensities can be obtained in quite a number of ways, but today's successful methods are almost exclusively based on intensive computing, often containing massive Configuration Interaction (CI) calculations. Pure ab initio calculations, like Multi-Configuration Hartree-Fock (MCHF, Froese Fischer 1978) and the analogous relativistic Multi-Configuration Dirac-Fock (MCDF, Parpia et al.\ 1996; Desclaux 1975) are most suited to produce a few, but accurate results. MCHF is able to include a large amount of electrostatic correlation and is therefore, like the Configuration Interaction Version 3 (CIV 3) code (Hibbert 1975), successful in the field of "lighter", i.e. Z < 20, atoms. MCDF, on the other hand, is obviously better equipped for the heavier atoms. However, it can at the moment only handle a limited amount of electrostatic correlation, which means that fine structure is generally better described than absolute energies.
For the production of large quantities of reliable data, Cowan's RCN/RCG suite of programs (Cowan 1981) has gained widespread acceptance. Here, fitting to experimental energy levels with Slater-Condon parameters is vital to raise the accuracy, especially in complex energy structures with several d- or f-electrons outside closed shells where there are many levels of the same J-value in a relatively small energy interval.
The orthogonal operator method can be seen as a continuation of the Slater-Condon theory of atomic spectra. In the Slater-Condon theory, the Coulomb interaction between the electrons is described by means of the Slater F- and G-integrals, while the magnetic interaction due to the motion of the nucleus in the rest frame of an electron is translated into the well known spin-orbit interaction. These interactions, however, are not orthogonal which implies that adding another interaction to the system would influence the values of all previous interaction constants.
Roughly a decade ago (Judd et al. 1982; Hansen et al.\
1988a), this system was orthogonalized. As the parameters in the
new operator set are as independent as possible, the fits are highly
stabilized which offers the possibility to include new operators that
account for small interactions like higher order or pure relativistic
effects. As a result, the mean error 
of the fit is reduced by an
order of magnitude in a physically significant way.
Although the method is semi-empirical in character, ab initio calculations
do constitute
an important part of the procedure, especially to describe
relativistic, e.g. fine structure effects. More specifically in
connection with this work, two-
particle (like dd, dp or ds) magnetic effects are relevant.
Due to the small number of
experimental levels in the d
and dp configuration it is not possible
to fit these parameters. Therefore they were fixed at their MCDF ab initio\
calculated values. In spite of the sometimes large number of parameters,
the number of parameters actually varied is in many cases about the same as
in the traditional Slater-Condon method. Even without varying its
additional parameters, the orthogonal operator method still provides an
improved description of complex spectra (Raassen & Uylings
1996).
With respect to the classical Slater-Condon type calculations of transition probabilities, the differences are twofold: First, as the level positions and thereby the spacing between them are described in more detail, one may suppose that the corresponding eigenvectors are more accurate. These eigenvectors are used to transform the transition probabilities in pure SL-coupling to the real intermediate coupling scheme. Second, the transition integrals calculated by means of MCDF are corrected for the effect of core-polarization. We find that this correction, which is commonly a reduction of the transition integral, improves the agreement with absolute transition probabilities from experiment.
Of course there are difficulties as well, otherwise it would be hard to
explain that not everyone involved in this field is using orthogonal
operators. Compared to Cowan's programs, our procedure is less
automatic. Parameter values, especially if they are meant to be fixed,
have to be obtained either from extrapolation (which requires some experience
with the region under study) or from ab initio calculation. Programs to translate
Slater F- and G-integrals to their orthogonal counterparts are available.
We adapted the GRASP2 package (Parpia et al. 1996) to calculate
two-electron magnetic effects; a request to obtain this program should be
directed to Froese Fischer and the present authors.
However, effective (i.e. higher order) electrostatic parameters may pose
a problem if they can not be fitted directly, in which case one should apply
extrapolation or second order perturbation theory (Uylings et al.\
1993). Yet, we hope that this situation is temporary. Overview
articles containing numerous parameter values for d
(Hansen et al.\
1988b), d
s (van het Hof et al. 1991) and
d
p (Uylings & Raassen 1996) have been published and
can be used to find starting values for the parameters. Everybody interested
in working with orthogonal operators is invited to contact the authors or to
visit our Internet address ftp://nucleus.phys.uva.nl in the directory
pub/orth.