Transformation and Unique Labeling for Energy Levels

The JJ2LSJ program, which is important not only for the GRASP2K package but for the atom theory in general, is presented. The program performs the transformation of atomic state functions (ASFs) from a jj-coupled CSF basis into an LSJ-coupled CSF basis. In addition, the program implements a procedure that assigns a unique label to all energy levels. Examples of how to use the JJ2LSJ program are given. Several cases are presented where there is a unique labeling problem.


Introduction
In principle, any valid coupling scheme can be used to represent the wave function in atomic structure calculations.Levels of an energy spectrum are identified and labeled with the help of sets of quantum numbers describing the coupling scheme used for the wave function.However, these quantum numbers are exact only for the cases of pure coupling.In a calculations of energy spectra one has to start with the coupling scheme closest to reality [1].The most frequently used coupling schemes in atomic theory are the LSJ and jj.In atomic spectroscopy, the standard LSJ notation of the levels is frequently applied for classifying the low-lying level structures of atoms or ions.
Calculations may be performed in the relativistic (jj-coupling) scheme in order to get more accurate data that include relativistic effects.Thus, after a multiconfiguration Dirac-Hartree-Fock (MCDHF) or relativistic configuration interaction (RCI) [2] calculation the transformation to LSJ-coupling is needed.The JJ2LSJ code in GRASP2K [3] does this by applying a unitary transformation to the relativistic configuration state function (CSF) basis set which preserves orthonormality.The unitary transformation selected is the coupling transformation that changes the order of coupling from jj to LSJ, a transformation that does not involve the radial factor, only the spin-angular factor.
An energy level is normally assigned the label of the leading CSF in the wave function expansion.For many systems, two or more wave functions have the same leading CSFs giving rise to non-unique labels for the energy levels.We have such a situation for Si-like ions [4] and some other systems [5].The new JJ2LSJ program implements a procedure that resolves these problems, assigning a unique label to all energy levels.

Transformation from jj-to LSJ-Coupling
Each nonrelativistic nl-orbital (except for ns) is associated with two relativistic orbitals l ± ≡ j = l ± 1/2.In the transformation of the spin-angular factor |l w αLS into a jj-coupled angular basis, two subshell states, one with l − ≡ j = l − 1/2 and another one with l + ≡ j = l + 1/2, may occur in the expansion.This shell-splitting obviously conserves the number of electrons, provided (w = w 1 + w 2 ), with w 1 (max) = 2l and w 2 (max) = 2(l + 1).Making use of this notation, the transformation between the subshell states in LSJand jj-coupling can be written as |(l which, in both cases, includes a summation over all the quantum numbers (except of n, l − , and l + ).
Here, |(l J is a coupled angular state with well-defined total angular momentum J which is built from the corresponding jj-coupled subshell states with and the total subshell angular momenta J 1 and J 2 , respectively.An explicit expression for the coupling transformation coefficients in ( 2) and (3) can be obtained only if we take the construction of the subshell states of w equivalent electrons from their corresponding parent states with w − 1 electrons into account.In general, however, the recursive definition of the subshell states, out of their parent states, also leads to a recursive generation of the transformation matrices (4).These transformation coefficients can be chosen real: they occur very frequently as the building blocks in the transformation of all symmetry functions.The expressions and values of these coefficients are published in [6,7].These transformation matrices, which are applied internally by the program JJ2LSJ, are consistent with the definition of the coefficients of fractional parentage [8,9] and with the phase system used in the [10].So the program presented in the paper supports transformation from jjto LSJ-coupling if ASF (which needs transformation) was created using the approach [7][8][9][10].Otherwise the program may perform the transformation incorrectly.

Unique Labeling
An energy level is often given the label of the leading CSF in the wave function expansion.But it sometimes happens that two wave functions have the same largest CSF in LSJor jj-coupling, and then classification in energy spectra is not unique.The simplest way to have a unique identification of an energy level would be use a position number (POS) and symmetry J.But to get the energy spectra with unique labels in LSJ-coupling we should re-classify levels.For that purpose JJ2LSJ transformation with the unique labeling option can be used.To obtain unique labels the algorithm proposed in [11,12] is used: for a given set of wave functions with the same J and parity, the CSF with largest expansion coefficient is used as the label for the function containing this largest component.Once a label is assigned, the corresponding CSF is removed from consideration in the determination of the next label.In such a way we will get energy levels with unique labels.In this process, cases where one CSF is dominant (defines more than 50 % of the wave function composition) that CSF will give the label for the corresponding energy level, but when the composition is spread over a number of CSFs, and none particularly large, the label is defined by the algorithm.Thus labeling is done by blocks of levels, each of the same J and parity.The first step is to order the levels by energy and assign the POS (position) identifier with the lowest having POS = 1, the second POS = 2, etc. and then proceed with determining the label.
In the Section 4 we will present a few examples where wave functions have the same dominant term and where the unique labeling algorithm is needed.

The JJ2LSJ Program
JJ2LSJ program is intended to perform the transformation of ASFs from a jj-coupled CSF basis into an LSJ-coupled CSF basis.This program is written in FORTRAN90 and is included in the GRASP2K package [3].It uses the same libraries as other programs in GRASP2K.The program is based on the earlier published LSJ program [13], but modified for speed up.The new program transforms only the most important components of large expansions.In addition, the new program provides an option to choose unique labeling versus labeling by the leading CSF in the wave function expansion.

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The program can also be used in non-default mode.The typical run proceeds as follows: Default settings?(y/n) >>n All levels (Y/N) >>y Maximum % of omitted composition >>0. 5 What is the value below which an eigenvector component is to be neglected in the determination of the LSJ expansion: should be smaller than: 0.00500 >>0.003What is the value below which an eigenvector composition is to be neglected for printing?>>0.0005Do you need the output file *.lsj.c? (y/n) >>y Do you need the output file *.lsj.j? (y/n) >>y The non-default mode is useful in several cases: (1) The present code allows the user, through the first parameter (0.5), to select the maximum percentage of the ASF composition that can be omitted.Given this information and with the help of the second parameter (0.003), it is easy to derive the largest small coefficient in the CSF expansion that may be included.However, with many components of about the same size, smaller values may be needed to meet the original objective.In this implementation, the user specifies the CSFs that can be omitted.The remaining CSFs define the basis that is to be transformed.By transforming this basis in decreasing order of importance, the desired percentage of the wave function can be transformed.A third parameter (0.0005) controls the printing of expansion coefficients in the LSJ basis and their contribution to the composition of the wave function.The default is to transform at least 99% of the wave function composition and print components in LSJ that contribute more than 0.1% to the composition.The cut-off for the jj-expansion has the value of 0.005, whereas the cut-off for printing is 0.001.
(2) In particular, the user may request a complete transformation, with a resulting list of CSFs in LSJ-coupling in name.lsj.c and their expansion coefficients in name.lsj.j.The two files have the same format as in ATSP2K [14].Complete expansions are feasible only for small expansions.In this case the first and second parameter should be 0.
(3) The non-default option should be used if we choose a unique labeling option, but the program will not give the unique identification for all levels.In this case we need to transform a larger amount of ASF with larger number of expansion coefficients.It can be done with help of the first and second parameter.

Results
In the recent calculations of energy spectra for Sr XXV [4] two pairs of odd levels with J = 2 had the same label and were separated by adding subscripts 'a' and 'b'.In the NIST database [15] for two (3s 3p 3 ( 2 1 P) 3 P o 2 and the 3s 2 3p 3d 3 F o 2 ) of these levels there is no data and the 3s 2 3p 3d 1 D o 2 level is not identified.
Running the JJ2LSJ program for Sr XXV levels 13 and 25 are relabeled in the Table 1.Table 1 gives also the labels from [4].As we see level 13 had the same label as level 14, and 25 was labeled as 17.In Table 1 also the compositions in LSJcoupling are given.In the Table 2 transition data of E1, M1, M2 transitions for relabeled levels are presented.
Table 1.Energy levels in cm −1 and LSJ-composition for Si-like Sr.In the original data levels 13 and 14 had the same label and subscripts 'a' and 'b' were introduced to separate the levels.Using the JJ2LSJ program levels 13 and 14 are now assigned unique labels.

No. Level [4]
Level (Relabeled) LSJ-Composition n = 7 Another example for which problems with unique labels occur is P-like W. Calculations using the MCDHF and RCI methods show that there are many levels with the same labels [16].
Table 3 presents the part of energy spectra with unique labels and LSJ-composition.The levels which were relabeled are marked with grey color.

Conclusions
In this paper, a new version of the JJ2LSJ program, consistent with the approach described in [7][8][9][10], is presented.The program performs the transformation of ASFs from a jj-to LSJ-coupling and provides the option to assign all level unique labels.Examples of the program use and explanations of possible options are given.In the paper, a few cases (Si-like Sr and P-like W) where the problem with unique labeling in energy spectra occur, are discussed and new labels are assigned.
The program is freely distributed.It may be obtained from the corresponding author.

>>jj2lsjjj2lsj:
Transformation of ASFs from a jj-coupled CSF basis into an LSJ-coupled CSF basis (Fortran 95 version) (C) Copyright by G. Gaigalas and Ch.F.

Table 2 .
Transition data for Si-like Sr where each level has been assigned a unique label.