J. E. Hansen, 
B.R. Judd, 
A. J. J. Raassen, and P. H. M. Uylings
Van der Waals-Zeeman Instituut, Universiteit van Amsterdam,
1018 XE Amsterdam, The Netherlands
Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, Maryland 21218, USA
Received: August 6, 1997
Small discrepancies in the fitted energy levels of the configurations
of transition-metal ions are ascribed to effective three-electron
magnetic operators 
. Surprisingly it has been found that, of the
16 possible operators with ranks 1 in both spin and orbital spaces, four
operators labelled by the irreducible representation (irrep) (11) of SO(5) are sufficient to obtain results
which appear to be limited by the errors in the experimental
energy levels.
An interpretation is given involving products of operators labelled
by the irreps of SO(5) and the symplectic group Sp(10).
31.15.Hz, 31.15.Md,31.25.Eb
The use of group theory for the understanding of physical
phenomena has a long and distinguished tradition in quantum mechanics
[1]. Group theory has repeatedly been used to elucidate
patterns in experimental observations in all areas of modern physics.
In particular, atomic
spectra have from the beginning been a fertile hunting ground
for the effects of symmetry. As simple a configuration as
requires the introduction of a new quantum number, seniority,
to distinguish the two D terms. However, in contrast to the lowest
symmetries represented by the S and L quantum numbers, the higher
symmetries are not expected to be connected with constants of the motion
and are not represented by ``good'' quantum numbers. Nevertheless, the advantages
of using group theory for understanding
the structures of the configurations 
and 
are substantial, and
the simplifications afforded by the use of the Wigner-Eckart theorem,
when applied to the higher groups, are extremely large. Even so, it
is clear that it would be very interesting if group theory exposed
not only the beauty of the underlying structure, but also the
relative importance of the various interactions between the electrons. A
demonstration of the connection between these two aspects of atomic
physics forms the subject of the present Letter.
The power of group theory is particularly striking when the accuracy of the observations cannot be matched by the available ab initio theories. In this respect the unrivaled accuracy of spectroscopic observations, which is far beyond what can be reached in ab initio calculations for atoms with more than 2 electrons, is important. To take advantage of this precision we have recently introduced the ``orthogonal operator'' method [2, 3, 4, 5] for the description of complex configurations. In this method, which is based on parametric fitting to the observed energy level structure, the physical interactions are augmented with ``effective interactions'' which represent the effects of configuration interaction (CI). In the case of the d shell, all operators are classified with respect to SO(5) and Sp(10). This choice has the advantage that the operators in general are automatically orthogonal to each other in the sense of the theory i.e. the matrix elements corresponding to different operators are represented by orthogonal matrices. A practical advantage of orthogonality is that bringing new operators into play entails minimal adjustments to the values of the parameters associated with the operators already in use. More importantly in the present context, the addition of effective operators makes it possible to obtain a much more accurate description of the data and thereby, as we will show, allow the ``observation'' of very small relativistic and correlation effects that normally would disappear in the ``noise'' caused by the inability of more conventional methods to take into account even large higher-order effects. While any extension of the conventional parameter sets can be expected to lead to a decrease in the remaining discrepancies in the fit, the use of group theory to classify the operators makes it easy to ensure that the additional operators show a minimal correlation to, i.e. are orthogonal to, the old ones; but, as we will show, it also allows us to ascribe physical significance to the remaining discrepancies in the fit. Thus the orthogonal operator technique is crucial for making possible the observations reported here. Although of less significance in the present context it should be mentioned also that the wavefunctions obtained in fits including the new operators have been shown to be superior to the old ones if used to predict properties such as decay rates [6].
A fruitful area for the use of
the orthogonal operator technique has turned out to be
the transition-metal atoms and ions with an open 3d shell, for which complete
sets of energy levels are known in many cases.
For example, the configurations have been studied for several degrees
of ionization using a complete set of 21 electrostatic operators [7].
``Complete'' means in this context, that CI effects can be included to all
orders, if due to the Coulomb interaction. To include magnetic effects
to second order, it is necessary to add a set
of 9 operators [4].
In the transition metals this is sufficient to obtain
fits with deviations between observed and calculated energies, 
, of the order
of a few cm 
[7], an improvement with factors between 20 and 80
compared to previous parametric
fittings and fairly close to the uncertainties in the experimental energies.
However, the discrepancies
between theory and experiment, though small, have been found
to display patterns that suggest that further analysis might be useful.
For example, the 's for the individual J levels of similar terms
belonging to
mutually conjugate configurations (lying on either side of
the half-filled shell) exhibit sign reversals for the spectra that have been studied [7].
In addition, there is an obvious pattern in the values
within each multiplet with the 's decreasing in a regular
manner from
positive to negative values. Two examples are given in table I (the columns
labelled ``no three-body'') concerning the
F term in 
(Cr IV [8]) compared to the conjugate term in 
(Ni IV [9]) and the 
D
terms in 
(Fe V [10]) and 
(Ni V [11]). We repeat that without
a (nearly) complete set of orthogonal operators it would not be possible to
expose such minute physical effects from the configuration energies which
in Cr IV for example cover an energy range of more than 53000 cm and in Ni V
probably more than 130000 cm (the latter is an estimate since the highest
level, S 
, is unknown).
The J dependence evident in table I points to the importance of
spin-orbit effects; yet the fitting already includes the normal single-electron
spin-orbit interaction as well as the two-electron terms of the spin-other-orbit
and ELSO (electrostatically-correlated spin-orbit) type. However, operators
representing three-electron
magnetic effects of spin-orbit type have been neglected so far. Such operators would be expected to
come from a variety of interactions between and excited configurations,
and their diagonal matrix elements would possess the property of reversing
signs when going from to the conjugate configuration 
.
Using group theory to construct operators labelled by
irreducible representations
(irreps) of SO(5), the matrix elements of a complete set of 16 orthogonal three-electron
magnetic operators with rank 1 in both spin and orbital spaces have
been calculated for by Leavitt [12].
We will call the associated
parameter values for the normalized operators 
.
With this set of operators, where the
magnetic ones now are complete to third order, new fits have been made
for the configurations and we report here a remarkable simplification
which allows us to obtain fits with deviations which are very close to the
experimental energy uncertainties using only four of the 16 operators.
This simplification is unexpected since all 16 operators appear in the same order of
perturbation theory for a given selection of the perturbing operators [12].
Given the many operators available it is possible to obtain fits
with essentially zero for each level. However, in this case
the experimental errors in the energy level values will influence the
parameter values. It is therefore important to use other criteria to
determine the most appropriate fit. We use mainly the isoionic and
isoelectronic trends in the parameter values and we do not allow all parameters
to vary freely but the y(11) set is basically determined after the
appropriate values of the larger parameters have been obtained.
Table I shows also the results of adding subsets of the three-body operators
to the fitting for the four ions, as well as the residue 
that
gives an indication of the
remaining errors in the fit. is
defined as 
where the sum is over all levels of the
particular configuration. The 16 operators have, as mentioned, been
classified according to the irreps W of SO(5). There are four operators with
the label W=(11), which we label y(11), five with (21), three with (31), two with (32) and one of
each with the labels (33) and (41). These labels are assumed to have primarily
mathematical significance but table I shows that the effect
of introducing the y(11) set is to reduce the residue with factors
varying between 4 and 10. In contrast, introducing instead the five (21)
operators or the five operators with either (31) or (32) symmetry gives
at best an improvement by a factor of 1.8.
The most spectacular result is obtained for
the configuration in Ni IV where the y(11) set reduces
from 52.7 to 5.4 cm while the other sets do not lead to a
reduction below 41 cm . The larger errors for the and configurations compared
to the situation for the conjugate configurations,
and , can be expected because the lowest order electrostatic and
magnetic interactions are larger thus pointing to larger higher-order effects.
In Cr IV, Ni IV and Ni V, the 's in the
fit including the y(11) set are smaller than the estimated errors
in the experimental energies (roughly 0.4 cm ) signifying that the residues in these cases might be determined
by the experimental errors and not by neglected physical effects. However,
a closer look at the regularities in the 's indicates that
the energy level values perhaps are more accurate than the authors'
conservative estimates would suggest.
Given that there are 1820 possible combinations of 4 operators out of the 16
possible, it is clear that
it is very difficult to pick the most suitable set of operators by trial
and error. Even trying to determine whether it would be useful to add
a few of the remaining operators to the y(11) set would be a tedious task.
However, the results obtained with the y(11)
set indicate that it is unlikely that such an extension would
be useful.
Table II shows complete results for the configuration in Cr IV and
its conjugate configuration in Ni IV. The table shows the 's
with and without inclusion of the y(11) set. The results
obtained without the y(11) set show, particularly for the high J
values, clearly the sign reversal between and which led
to the present investigation in the first place. However, when we consider
the residuals obtained with y(11) included it can be seen that, although
most 's are smaller than the expected experimental accuracy, there
is a remarkable similarity between the two columns but this time
the deviations fairly consistently have the same sign, allowing us to conclude
that the remaining deviations in so far as they are real probably have a different origin.
As mentioned, the finding that a small subset of the operators is
sufficient to include the higher-order magnetic effects comes as a
surprise and it clearly would be interesting to understand why this
is so. It is highly suggestive that the
irrep (11) of SO(5) is exactly the same as that labelling the ordinary
single-electron spin-orbit interaction 
in . Thus a
perturbation in which is combined with SO(5) scalars would
preserve the label (11), since (00) 
(11) = (11). The Coulomb
interaction plays the main role in mixing configurations, but,
although scalar with respect to S and L, it is not a pure SO(5)
scalar. Within , there is a component belonging to the irrep (22) in
addition to (00). When configuration interaction is considered, the
situation is more complex. The mixing of 
into calls for
products of creation and annihilation operators of the types 
and 
, and the SO(5) label is the part of (10) 
(00) that
contains spin and orbital ranks of zero, namely the irrep (30). A
third-order mechanism involving 
s in which two Coulomb operators
and appear would be expected to lead to contributions to all
the , since (30) (11) contains all the irreps in the list of
operator labels [12].
Another type of single-electron excitation that preserves
parity is the mixing of 
into . It is here that a new
feature appears. The group SO(5) can be enlarged to include
the d' electron by adding a second set of group generators to those
referring to the d electron. The condition that the Coulomb
interaction be an SO(5) scalar can now be stated in the form of
conditions on the Slater integrals: namely X(2) = (5/9)X(4),
where 
, or 
. As is well
known, the condition on 
is equivalent to a delta-function
interaction [13]. However, this short range interaction may be closer to the Coulomb force than
appears at first sight because of screening effects.
In fact, the Coulomb energies of the terms of 
can be represented
by an operator whose largest component 
belongs to (00) [14, 15].
The dominance of effective operators belonging to (00) is apparent as
we proceed along the d shell: for example, of the effective
four-electron operators 
, the largest in the fits are 
,
and 
, all belonging to (00) [7].
If, then, we make the decision to restrict our attention to just those parts of the
Coulomb interaction belonging to (00) of SO(5), i.e. restricting the
contributions to just the y(11) set, we can calculate
the ratios of the parameters that
measure the strengths of the normalized operators under various assumptions. A few results
are compared to experiment in Table III. Since 
is the most
reliably determined experimentally, we arbitrarily set 
= 1. Case A is a third-order
calculation in which only is included (and not 
or the
cross-term 
). However, the null rank of
leads to large matrix elements when products of the type
are considered, and this gives the ratios
of case B. In both cases the absolute signs of agree with
experiment, as do the signs of all four for case B. It should
be mentioned that such third-order contributions are distinct from
second-order effects that come from 
excitations but have already been included by
means of two-electron magnetic operators [4].
Included in Table III is also the result of a more general
type of calculation (case C). This involves the symplectic group
Sp(10). Within , both and the delta-function interaction are
(different) mixtures of 2 two-electron operators, 
and 
, whose
respective symplectic labels are 
and
[5]. If we
attempt to represent the effects of third-order perturbation theory
by effective operators acting solely within , we are led to study
products of the types 
and 
. It turns out
that the first contributes only to 
. The second, however,
contributes to all members of the y(11) set. The results for the
symmetrized form, 
, are listed as case C in Table III.
The correspondence to the experimental ratios is remarkably good. We
should also note that the similarity of several ratios for cases A, B
and C is a direct consequence of the association of each with a
given irrep of Sp(10). In particular, the stability of 
(which corresponds quite well to the rather uncertain experimental
values) is due to the shared label 
for and 
,
and the
fact that is formed from symplectic components of similar
types for the three cases.
Until a complete third-order calculation is carried out, we
cannot properly assess the general applicability of the model based
on the irreps (00) of SO(5) and of Sp(10). Nevertheless, the
evidence clearly points to the excitations of the type as
making the most significant contributions to the parameters .
The extraction of physical information from detailed fits of theory
to experiment is a prime justification for the fitting procedure. In
fact, it could be argued that any discrepancies that remain in a fit
indicate that the experimental information is not being put to full
use. The magnitudes of the parameters constitute a separate issue,
one that has to be faced with the help of ab initio methods.
However, at present such methods cannot meet the level of precision
demanded by the parametric approach. The two methods are
complementary, not competitive.
Our success in reproducing the general trends of the four parameters of Table III provides a rationale for extending the use of effective operators with group labels elsewhere. One rather obvious extension is to the f shell, where group-theoretical methods are well established.
One of us (BRJ) acknowledges partial support from the United States National Science Foundation.