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This paper suggests a method of evaluation of uncertainties in calculated transition probabilities by randomly varying parameters of an atomic code and comparing the results. A control code has been written to randomly vary the input parameters with a normal statistical distribution around initial values with a certain standard deviation. For this particular implementation, Cowan’s suite of atomic codes (R.D. Cowan,
With the rapid improvement in computer power and quality of atomic structure codes, calculations of atomic structure and transition properties now become widely used in largescale simulations of physical conditions in complex plasma environments, such as found in fusion devices. Such simulations have wide range of applications, from plasma diagnostics to prediction of technological characteristics of industrial devices. To assess the accuracy of these simulations, it is important to have welldefined uncertainties for all calculated atomic parameters. Thus, critical evaluation of atomic data, implying estimation of uncertainties, has recently become one of the top priorities in fusion research [
Currently existing methods of evaluation of uncertainties of calculated transition probabilities were summarized by Wiese [
In this paper, I suggest a new method of evaluation of uncertainties. It also relies on comparisons; however, these comparisons do not use any external data, but only the data produced by the same computational procedure. The base for comparisons is built from data generated with varied parameters of the atomic code. Application of this method is illustrated for the case of magneticdipole (M1) and electricquadrupole (E2) transitions within the ground configuration of titaniumlike iron (Fe V). The calculations were made with the suite of atomic structure codes by Cowan [
Calculation of transition probabilities involves several stages. These stages differ in different methods. In the nonrelativistic or quasirelativistic Hartree–Fock method, at first, the radial parts of wavefunctions are computed for each configuration in the singleconfiguration approximation. Then the Slater parameters and configuration–interaction (CI) parameters, as well as multipole transition integrals, are calculated from these radial functions. The parameters mentioned above are called hereafter the
The idea of the method is to vary the LSF parameters in a random fashion (using a normal statistical distribution centered at the LSF values with a width equal to the
It must be noted that this initial assumption of normal statistical distributions of all parameters is arbitrary to a large degree. One could assume, for example, that these distributions are uniform, meaning that all values of parameters within certain limits are equally probable. However, a normal statistical distribution seems to better reflect the common observation that the results of the LSF provide the best (most probable) values for the parameters insofar as the eigenvalues obtained with the fitted parameters exhibit symmetrical distributions around experimental values with the smallest standard deviations. The standard distribution has wellknown statistical properties such as its symmetry around the central value (mean), its width at halfmaximum (equal to 1.35 times the standard deviation around the mean) and the probabilities of occurrence of values deviating by more than one, two, or three standard deviations (32%, 5%, and 0.3%, respectively). Although other forms of distributions are possible, the normal distribution seems to be a good test case.
Cowan’s codes [
A few preliminary remarks should be made. The first one concerns the statistical distributions of the calculated quantities. In several recent papers, including [
The second note concerns the identification of transitions. Generally, when the Slater parameters are changed, diagonalization of the Hamiltonian results in changes in both eigenvalues and eigenvectors, as well as in the predicted transition wavelengths. Identification of transitions produced by such different calculations poses a serious technical problem, since the level ordering is not necessarily preserved. The method that works well for this purpose, is the eigenvector recognition used in my version of Cowan’s LSF procedure [
The third note concerns the calculated radiative rates,
Before describing the implementation of the method, a brief description of Cowan’s suite of codes is necessary. The suite consists of four separate programs that are run consecutively. The programs are named with Cowan’s initials, “RC”, followed by a letter symbol defining the code category (“N” for the program calculating singleconfiguration wavefunctions, “G” for the program diagonalizing the matrix of the Hamiltonian and computing the spectrum, “E” for the LSF program). An exception is made for the RCN2 program, which will be explained below. Each program has one or more input files and produces output files, some of which are intended to be input files for the next program in the chain. Thus, RCN has an input file that defines the configurations to be included in the calculation, as well as several parameters having technical purposes, such as the tolerance for iterations in the selfconsistent field calculations. As output, RCN produces a binary file defining the computed wavefunctions, which is then used by the next program, RCN2, as an input file. RCN2 also has an additional input (text) file that can be modified by the user. This file contains definitions of some adjustable parameters, such as scaling factors for Slater parameters. Input files that can be modified by the user have in their names letters “in” followed by the code symbol and sometimes by numerals defining the code version. Thus, the input file for RCG version 11, produced by RCN2, is named “ing11.” This file contains all parameters needed for the construction of the Hamiltonian matrix, such as Slater and CI parameters denoted below as
This organization of the codes allows the user to make either ab initio or semiempirically adjusted calculations. In particular, the leastsquaresfitted parameters
The procedure implemented here is as follows.
The main directory of the calculation is set up with the input and output files for the Cowancode calculation with LSF. The LSF for the even parity of Fe V was made earlier as described in [
The control code creates a separate subdirectory for each random trial and sets up all files necessary for calculations in each of these subdirectories. The maximum number of random trials used in this test implementation was 10,000, so there were 10,000 subdirectories.
The control code reads the input file for the matrixdiagonalization code RCG (ing11) from the main directory and prepares sets of randomly varied E2 matrix elements for each trial. The random variations are implemented using a standard random number generation routine (producing uniformly distributed random floatingpoint numbers in the interval between 0 and 1), converted to normally distributed numbers using the Box–Miller transformation [
The control code reads the output file of RCG (outg11) and other auxiliary files from the main directory (including ing11), reads the output file of the LSF code RCE (oute), finds the data block corresponding to the last LSF iteration, and reads the fitted parameter values (Slater and CI) and their
The control code identifies the eigenvectors produced by RCG with those produced by RCE (which are nearly identical, since the preliminary RCG calculation was made in the main directory using the parameter values from LSF), and thus establishes correspondence between the initial calculated eigenvalues and experimental energies.
The control code continues reading the part of the outg11 file from the main directory containing the transition data (initial and final energy levels, wavelengths,
In each trial subdirectory, the control code randomly varies the Slater and CI parameters using the same procedure as for the E2 matrix elements, except that the widths of normal statistical distributions are set to be equal to the corresponding
In each trial subdirectory, the control code runs RCG, reads the resulting outg11 file containing new eigenvectors and sets of transition data, identifies the new eigenvectors with the old ones, kept in memory from step 5, and, for each transition, rescales the new
The accumulated statistics data are processed and results are printed to an output file.
The control code was written in Perl programming language. It has about 1,000 lines of code and uses several external Perl utility codes developed earlier and included with the Cowan code package [
The 3d^{4} configuration of Fe V consists of 34 energy levels spanning the range from zero to 94,000 cm^{−1}, all of which are experimentally known with uncertainties ranging from 0.3 cm^{−1} to 1.5 cm^{−1} [
As described in [
Calculations of M1 transitions are intrinsically more accurate, because they involve only linear combinations of amplitudes of the eigenvector components of the lower and upper levels with coefficients that are functions of quantum numbers of the basis states. Thus, the calculated M1
The LSF parameters used as initial input for Monte Carlo trials are listed in
Parameters of the leastsquares fitting (LSF) used as input for Monte Carlo trials (in units of cm^{−1}).
Configurations  Parameter  LSF 

Group^{ a}  HFR  LSF/HFR  

3d^{4} 

36,510.2  43  0.0  
90,868.6  118  105,204.6  0.8637  
55,549.7  191  66,193.4  0.8392  

36.8  3  0.0  

599.6  61  0.0  

−7.7  0  14  0.0  

531.9  23  533.1  0.9977  
3d^{3}4s 

215,050.4  31  177,245.6  1.2133  
95,607.2  135  111,187.6  0.8599  
58,863.0  209  70,214.5  0.8383  

45.2  3  0.0  

634.1  60  0.0  

−7.7  0  14  0.0  

586.8  24  585.2  1.0027  
10,704.7  75  6  12,235.1  0.8749  
3d^{3}5s 

421,078.0  58  5  382,121.1  1.1019  
97,235.2  815  112,221.2  0.8665  
59,610.7  1157  70,920.1  0.8405  

52.9  7  0.0  

362.6  362  0.0  

−7.7  0  14  0.0  

568.4  31  592.4  0.9595  
3327.7  101  3315.5  1.0037  
3d^{3}4d 

387,478.0  30  348,713.5  1.1112  
96,771.8  95  1  112,024.1  0.8638  
59,294.2  174  2  70,789.8  0.8376  

45.5  3  12  0.0  

495.8  51  13  0.0  

−7.7  0  14  0.0  

589.0  16  3  590.3  0.9978  

75.8  fixed  76.5  0.9908  
16,966.1  126  7  19,123.3  0.8872  
8,158.7  177  8  8,715.8  0.9361  
5,223.2  28  9  8,263.5  0.6321  
6,488.4  103  10  7,994.5  0.8116  
5,643.9  132  11  5,877.2  0.9603  
3d^{3}5d 

486,707.5  67  5  450,809.1  1.0796  
97,062.4  96  1  112,360.5  0.8638  
59,483.7  174  2  71,016.1  0.8376  

45.5  3  12  0.0  

495.8  51  13  0.0  

−7.7  0  14  0.0  

592.0  17  3  593.3  0.9978  

33.9  fixed  34.2  0.9912  
6,810.0  51  7  7,675.8  0.8872  
3,290.4  71  8  3,515.1  0.9361  
1,960.2  10  9  3,101.2  0.6321  
2,624.3  42  10  3,233.5  0.8116  
2,337.2  55  11  2,433.8  0.9603  
3d^{2}4s^{2} 

447,291.0  62  5  411,400.0  1.0872  
101,108.2  100  1  117,043.9  0.8638  
62,117.5  182  2  74,160.5  0.8376  

45.5  3  12  0.0  

495.8  51  13  0.0  

−7.7  0  14  0.0  

639.4  18  3  640.8  0.9978  
3d^{2}4s4d 

623,244.0  86  5  587,320.0  1.0612  
101,728.1  100  1  117,761.5  0.8638  
62,532.2  183  2  74,655.6  0.8376  

45.5  3  12  0.0  

495.8  51  13  0.0  

−7.7  0  14  0.0  

644.1  18  3  645.5  0.9978  

85.4  fixed  86.2  0.9907  
18,159.0  135  7  20,467.8  0.8872  
8,755.3  189  8  9,353.2  0.9361  
10,739.7  75  6  12,275.2  0.8749  
5,514.6  29  9  8,724.6  0.6321  
6,889.7  110  10  8,488.9  0.8116  
6,021.5  141  11  6,270.5  0.9603  
30,469.1  fixed  33,996.9  0.8962  
3d^{2}4d^{2} 

811,506.1  112  5  775,546.8  1.0464  
102,346.9  101  1  118,477.8  0.8638  
62,946.2  185  2  75,149.9  0.8376  

45.5  3  12  0.0  

495.8  51  13  0.0  

−7.7  0  14  0.0  
33,810.3  fixed  39,233.4  0.8618  
23,909.6  fixed  26,600.9  0.8988  

649.0  18  3  650.4  0.9978  

90.6  fixed  91.4  0.9912  
18,870.5  141  7  21,269.8  0.8872  
9,140.4  198  8  9,764.6  0.9361  
5,687.7  30  9  8,998.4  0.6321  
7,150.7  114  10  8,810.5  0.8116  
6,265.5  147  11  6,524.5  0.9603  
Configuration interaction  
3d^{4}  –3d^{3}4s  2,217.4  95  15  3,336.6  0.6646  
–3d^{3}5s  1,219.4  52  15  1,834.8  0.6646  
–3d^{3}4d  1,821.3  78  15  2,740.5  0.6646  
12,948.9  557  15  19,484.5  0.6646  
8,883.0  382  15  13,366.4  0.6646  
–3d^{3}5d  1,047.9  45  15  1,576.8  0.6646  
7,329.0  315  15  11,028.1  0.6646  
5,046.3  217  15  7,593.2  0.6646  
–3d^{2}4s^{2}  10,004.3  430  15  15,053.6  0.6646  
–3d^{2}4s4d  6,419.5  276  15  9,659.5  0.6646  
–3d^{2}4d^{2}  4,952.9  213  15  7,452.7  0.6646  
6,174.4  266  15  9,290.7  0.6646  
4,676.0  201  15  7,036.0  0.6646  
3d^{3}4s  –3d^{3}5s  409.1  18  15  615.6  0.6645  
4,080.3  175  15  6,139.6  0.6646  
–3d^{3}4d  14,076.1  605  15  21,180.5  0.6646  
5,091.7  219  15  7,661.5  0.6646  
–3d^{3}5d  8,682.3  373  15  13,064.4  0.6646  
3,471.1  149  15  5,223.1  0.6646  
–3d^{2}4s^{2}  3,873.0  167  15  5,827.8  0.6646  
–3d^{2}4s4d  1,926.3  83  15  2,898.5  0.6646  
13,783.0  593  15  20,739.5  0.6646  
9,476.2  408  15  14,258.9  0.6646  
−8,252.2  355  15  −12,417.2  0.6646  
−796.0  34  15  −1,197.8  0.6646  
–3d^{2}4d^{2}  −3,905.8  168  15  −5,877.1  0.6646  
3d^{3}5s  –3d^{3}4d  3,396.2  146  15  5,110.4  0.6646  
2,475.6  106  15  3,725.1  0.6646  
–3d^{3}5d  4,378.2  188  15  6,588.0  0.6646  
1,743.1  75  15  2,622.8  0.6646  
–3d^{2}4s4d  185.6  8  15  279.3  0.6645  
2,099.6  90  15  3,159.3  0.6646  
–3d^{2}4d^{2}  954.7  41  15  1,436.6  0.6646  
3d^{3}4d  –3d^{3}5d  501.5  22  15  754.6  0.6646  
6,530.5  281  15  9,826.6  0.6646  
3,481.2  150  15  5,238.2  0.6646  
3,347.1  144  15  5,036.5  0.6646  
3,346.7  144  15  5,035.8  0.6646  
2,486.7  107  15  3,741.8  0.6646  
–3d^{2}4s^{2}  −7,061.3  304  15  −10,625.3  0.6646  
–3d^{2}4s4d  4,209.3  181  15  6,333.8  0.6646  
−6,388.1  275  15  −9,612.2  0.6646  
−2,959.1  127  15  −4,452.6  0.6646  
–3d^{2}4d^{2}  1,955.1  84  15  2,941.9  0.6646  
13,992.6  602  15  21,054.9  0.6646  
9,625.4  414  15  14,483.5  0.6646  
80.7  3  15  121.4  0.6647  
−2,931.1  126  15  −4,410.5  0.6646  
−2,127.2  91  15  −3,200.8  0.6646  
3d^{3}5d  –3d^{2}4s^{2}  −3,819.9  164  15  −5,747.9  0.6646  
–3d^{2}4s4d  −2,351.0  101  15  −3,537.6  0.6646  
−1,316.3  57  15  −1,980.6  0.6646  
–3d^{2}4d^{2}  1,550.6  67  15  2,333.2  0.6646  
−657.1  28  15  −988.7  0.6646  
−769.7  33  15  −1,158.1  0.6646  
3d^{2}4s^{2}  –3d^{2}4s4d  15,121.7  650  15  22,753.9  0.6646  
5,433.2  234  15  8,175.4  0.6646  
–3d^{2}4d^{2}  23,912.3  1,028  15  35,981.2  0.6646  
3d^{2}4s4d  –3d^{2}4d^{2}  15,680.1  674  15  23,594.1  0.6646  
5,650.7  243  15  8,502.7  0.6646  
23,674.3  1,018  15  35,623.1  0.6646  
E2 transition reduced matrix elements  
3d^{4}  –3d^{4}  (3d, 3d)  −1.22129  
3d^{4}  –3d^{3}4s  (3d, 4s)  −36.36873  
3d^{4}  –3d^{3}5s  (3d, 5s)  −9.56604  
3d^{4}  –3d^{3}4d  (3d, 4d)  1.04638  
3d^{4}  –3d^{3}5d  (3d, 5d)  0.40754  
3d^{3}4s  –3d^{4}  (4s, 3d)  −1.22729  
3d^{3}4s  –3d^{3}4s  (3d, 3d)  −1.10293  
3d^{3}4s  –3d^{3}4d  (4s, 4d)  5.48000  
3d^{3}4s  –3d^{3}5d  (4s, 5d)  0.09788  
3d^{3}4s  –3d^{2}4s^{2}  (3d, 4s)  −1.05636  
3d^{3}4s  –3d^{2}4s4d  (3d, 4d)  0.90280  
3d^{3}5s  –3d^{4}  (5s, 3d)  −0.09184  
3d^{3}5s  –3d^{3}5s  (3d, 3d)  −1.08225  
3d^{3}5s  –3d^{3}4d  (5s, 4d)  −9.51515  
3d^{3}5s  –3d^{3}5d  (5s, 5d)  17.67485  
3d^{3}4d  –3d^{4}  (4d, 3d)  1.04638  
3d^{3}4d  –3d^{3}4s  (4d, 4s)  5.48000  
3d^{3}4d  –3d^{3}5s  (4d, 5s)  −9.51515  
3d^{3}4d  –3d^{3}4d  (4d, 4d)  −10.89202  
3d^{3}4d  –3d^{3}5d  (4d, 5d)  6.90270  
3d^{3}4d  –3d^{2}4s4d  (3d, 4s)  −1.02367  
3d^{3}4d  –3d^{2}4d^{2}  (3d, 4d)  0.87600  
3d^{3}5d  –3d^{4}  (5d, 3d)  0.40754  
3d^{3}5d  –3d^{3}4s  (5d, 4s)  0.09788  
3d^{3}5d  –3d^{3}5s  (5d, 5s)  17.67485  
3d^{3}5d  –3d^{3}4d  (5d, 4d)  6.90270  
3d^{3}5d  –3d^{3}5d  (5d, 5d)  −36.43838  
3d^{2}4s^{2}  –3d^{3}4s  (4s, 3d)  −1.05636  
3d^{2}4s^{2}  –3d^{2}4s^{2}  (3d, 3d)  −1.00354  
3d^{2}4s^{2}  –3d^{2}4s4d  (4s, 4d)  5.14550  
3d^{2}4s4d  –3d^{3}4s  (4d, 3d)  0.90280  
3d^{2}4s4d  –3d^{3}4d  (4s, 3d)  −1.02367  
3d^{2}4s4d  –3d^{2}4s^{2}  (4d, 4s)  5.14550  
3d^{2}4s4d  –3d^{2}4s4d  (4d, 4d)  −9.84869  
3d^{2}4s4d  –3d^{2}4d^{2}  (4s, 4d)  4.98748  
3d^{2}4d^{2}  –3d^{3}4d  (4d, 3d)  0.87600  
3d^{2}4d^{2}  –3d^{2}4s4d  (4d, 4s)  4.98748  
3d^{2}4d^{2}  –3d^{2}4d^{2}  (4d, 4d)  −9.45614 
^{a }Parameters in each numbered group were linked together with their ratio fixed at the Hartree–Fock level.
Let us start with something familiar to atomic physicists, at least to some extent. Namely, let us consider how the standard deviations
Relative
For strong M1 transitions with
If we look closer at the layout of the data points in the upper part of
Large deviations of calculated
I tried to find a correlation between the strong cancellation and the high volatility of
I attempted to investigate the cancellation effects in more detail. For each transition, each trial calculation produces a somewhat different CF. The CF values appear to be distributed normally around the initial values for most of transitions. Only for about 10% of all transitions distributions of CF are far from normal. The widths (
Now let us take a look at statistical distributions of results for individual transitions, something that was never considered before in the literature. These distributions are shown in
Relative deviations
One can see from
To investigate how the different functions
Fractional number of outliers deviating from the mean by more than



−5  0.000000286652 
−4  0.000031671242 
−3  0.001349898032 
−2  0.022750131948 
−1  0.158655253932 
−0  0.500000000000 
+0  0.500000000000 
+1  0.158655253932 
+2  0.022750131948 
+3  0.001349898032 
+4  0.000031671242 
+5  0.000000286652 
One can see from
The ratios of
I attempted to divide all considered transitions in three groups based on the “best” function
Ratios of fractional number of counts
Furthermore, even when the best of the three functions is used for each transition, the number of far outliers is still much greater than for the normal distribution. No apparent connection was found between the choice of the best function and physical characteristics of a transition, such as the type (M1 or E2), line strength, wavelength, energy levels involved, and the degree of cancellation.
Thus, the logarithmic function, which I earlier assumed (in [
In retrospective, the failure to find one or a few simple functions of
However, it turned out possible to find the best function of a general type described by the Box–Cox transformation function [
Although the Box–Cox transformation is defined piecewise, it is a smooth function of
The optimal value of
An alternate, somewhat simpler method of finding the optimal Box–Cox transformation is to find the value of
For a normal distribution, the
Examples of normal probability plots for one transition (see text). (
For a normal distribution, thus defined
For transitions considered in this work, it turned out that, for the optimal values of
Two methods of finding the optimal parameter
As noted above, the two methods of optimizing the Box–Cox transformation work almost equally well for the considered transitions. However, to avoid ambiguity, hereafter, when mentioning the optimal parameter
The optimal values of
Distribution of optimal values of the Box–Cox transformation parameter
The distribution of optimal powers
After making several runs with 1,000 trials each, I noticed that for many transitions the optimal
Distribution of optimal values of the Box–Cox transformation parameter
Larger statistics revealed that for a few transitions the optimal Box–Cox transformation is far from normal. Examples of normal probability plots for some of such “abnormal” transitions are given in
Examples of normal probability plots for abnormal transitions (see text). (
When optimal Box–Cox transformations are applied to all transitions, the overall comparison of statistical distributions with normal (similar to the plot given in
Similar to
One can see from
Strictly speaking, the standard deviations of optimally transformed
None of these uncertainty contributions are considered in the present work. Nevertheless, the obtained optimal transformation parameters
A rough estimate of relative
The closeness of
As noted above, the standard deviations for straight
Calculated transition probabilities of M1 and E2 transitions of Fe V with estimated uncertainties and Box–Cox transformation parameters optimized for each transition, based on overall statistics for five sets of Monte Carlo trials, four with 1,000 trials each and one with 10,000 trials.
Transition  CF^{ c}  BF^{ d}  Type^{ e}  Fraction(E2)^{f} 



^{3}P2_{1}–^{1}S1_{0}  24,972.8  121,130.1  1,039.963  6.3 × 10^{−1}  9  0.0440(3)  0.0073(6)  M1  0  0.484(21)  9 
^{3}F2_{2}–^{1}S1_{0}  26,760.7  121,130.1  1,059.666  2.54 × 10^{−1}  9  0.0625(7)  0.00294(22)  E2  1  0.74(6)  9 
^{3}D_{2}–^{1}S1_{0}  36,758.2  121,130.1  1,185.229  8.1 × 10^{−1}  9  0.1150(13)  0.0094(8)  E2  1  0.40(4)  9 
^{1}D2_{2}–^{1}S1_{0}  46,291.1  121,130.1  1,336.20  75.7  3  −0.1904(19)  0.878(5)  E2  1  0.88(21)  3 
^{3}P2_{0}–^{1}D1_{2}  24,055.5  93,832.5  1,433.137  3.3 × 10^{−2}  8  0.0229(5)  0.00083(6)  E2  1  0.55(4)  8 
^{3}H_{4}–^{1}D1_{2}  24,932.4  93,832.5  1,451.377  2.10 × 10^{−1}  9  0.1449(8)  0.0052(4)  E2  1  0.42(3)  9 
^{3}P2_{2}–^{1}D1_{2}  26,468.2  93,832.5  1,484.466  6.6 × 10^{−2}  9  0.00398(5)  0.00165(14)  M1 + E2  0.092(3)  0.12(3)  9 
^{3}F2_{4}–^{1}D1_{2}  26,973.7  93,832.5  1,495.689  2.89 × 10^{−1}  6  −0.1997(12)  0.0072(4)  E2  1  0.91(5)  6 
^{3}G_{4}–^{1}D1_{2}  30,147.2  93,832.5  1,570.221  3.1 × 10^{−1}  12  0.2507(8)  0.0076(9)  E2  1  0.347(19)  12 
^{5}D_{1}–^{3}P1_{0}  142.4  63,419.8  1,580.343  8.6 × 10^{−1}  8  0.01802(15)  0.193(13)  M1  0  0.527(23)  8 
^{5}D_{2}–^{3}P1_{0}  417.5  63,419.8  1,587.244  4.5 × 10^{−2}  8  0.1840(6)  0.0101(7)  E2  1  0.48(3)  8 
^{5}D_{0}–^{3}P1_{1}  0.0  62,914.1  1,589.47  9.4 × 10^{−2}  8  0.00491(5)  0.0212(13)  M1  0  0.539(23)  8 
^{5}D_{1}–^{3}P1_{1}  142.4  62,914.1  1,593.075  9.4 × 10^{−3}  9  0.139(3)  0.00213(14)  M1 + E2  0.9781(18)  0.45(3)  9 
^{5}D_{2}–^{3}P1_{1}  417.5  62,914.1  1,600.087  6.9 × 10^{−1}  8  0.0634(4)  0.156(10)  M1 + E2  0.0191(4)  0.505(22)  8 
^{5}D_{1}–^{3}F1_{2}  142.4  62,321.1  1,608.268  2.15 × 10^{−2}  8  0.70(3)  0.00290(19)  M1 + E2  0.0398(13)  0.61(3)  8 
^{5}D_{3}–^{3}P1_{1}  803.1  62,914.1  1,610.021  7.6 × 10^{−3}  9  0.0853(4)  0.00171(11)  E2  1  0.43(3)  9 
^{5}D_{2}–^{3}F1_{3}  417.5  62,364.3  1,614.288  3.9 × 10^{−2}  8  0.0372(4)  0.0057(4)  M1  0.000396(21)  0.58(3)  8 
^{5}D_{2}–^{3}F1_{2}  417.5  62,321.1  1,615.415  4.6 × 10^{−2}  8  0.0164(4)  0.0062(4)  M1 + E2  0.0128(4)  0.526(24)  8 
^{5}D_{0}–^{3}P1_{2}  0.0  61,854.1  1,616.71  4.0 × 10^{−3}  10  0.112(3)  0.00112(8)  E2  1  0.39(3)  10 
^{5}D_{1}–^{3}P1_{2}  142.4  61,854.1  1,620.438  3.8 × 10^{−2}  9  0.00943(20)  0.0108(7)  M1 + E2  0.101(3)  0.492(21)  9 
^{5}D_{3}–^{3}F1_{3}  803.1  62,364.3  1,624.400  1.21 × 10^{−1}  9  0.00286(3)  0.0176(14)  M1 + E2  0.0132(4)  0.496(23)  9 
^{5}D_{3}–^{3}F1_{2}  803.1  62,321.1  1,625.540  1.62 × 10^{−2}  9  0.0152(4)  0.00219(18)  M1 + E2  0.1097(23)  0.407(17)  9 
^{5}D_{3}–^{3}F1_{4}  803.1  62,238.0  1,627.739  4.0 × 10^{−2}  8  0.01356(19)  0.0059(4)  M1 + E2  0.0379(8)  0.548(25)  8 
^{5}D_{4}–^{3}F1_{3}  1,282.7  62,364.3  1,637.154  2.35 × 10^{−2}  9  0.0330(5)  0.0034(3)  M1 + E2  0.194(4)  0.478(23)  9 
^{5}D_{3}–^{3}P1_{2}  803.1  61,854.1  1,637.975  4.9 × 10^{−1}  9  0.919(6)  0.138(9)  M1  0.00893(19)  0.470(22)  9 
^{5}D_{4}–^{3}F1_{4}  1,282.7  62,238.0  1,640.546  2.39 × 10^{−1}  9  0.00509(10)  0.035(3)  M1 + E2  0.0405(9)  0.450(22)  9 
^{3}P1_{1}–^{1}S1_{0}  62,914.1  121,130.1  1,717.74  3.6  9  −0.9999951(4)  0.042(4)  M1  0  0.448(18)  9 
^{1}G2_{4}–^{1}D1_{2}  36,585.6  93,832.5  1,746.819  26.5  2.0  0.3386(15)  0.660(4)  E2  1  0.29(13)  2.0 
^{3}D_{3}–^{1}D1_{2}  36,630.0  93,832.5  1,748.175  5.5 × 10^{−1}  9  −0.883(5)  0.0138(11)  M1  0.00534(17)  0.484(20)  9 
^{3}D_{2}–^{1}D1_{2}  36,758.2  93,832.5  1,752.102  1.85 × 10^{−1}  9  0.0826(14)  0.0046(4)  M1 + E2  0.407(6)  0.470(22)  9 
^{3}D_{1}–^{1}D1_{2}  36,925.2  93,832.5  1,757.244  5.6 × 10^{−1}  9  −0.9550(18)  0.0139(12)  M1  0.000143(12)  0.451(19)  9 
^{1}S2_{0}–^{1}D1_{2}  39,633.0  93,832.5  1,845.04  1.94  2.5  0.0641(10)  0.0482(7)  E2  1  0.09(10)  2.5 
^{5}D_{4}–^{1}F_{3}  1,282.7  52,732.6  1,943.64  9.0 × 10^{−4}  18  0.0320(7)  0.00138(12)  M1 + E2  0.0138(4)  0.237(10)  18 
^{1}D2_{2}–^{1}D1_{2}  46,291.1  93,832.5  2,102.76  6.76  1.9  0.3210(14)  0.1682(10)  E2  0.99953(9)  0.66(16)  1.9 
^{3}H_{4}–^{1}G1_{4}  24,932.4  71,280.3  2,156.92  1.08 × 10^{−1}  6  0.00095(7)  0.0228(9)  M1 + E2  0.0205(19)  0.98(3)  6 
^{3}H_{5}–^{1}G1_{4}  25,225.5  71,280.3  2,170.65  2.9 × 10^{−1}  10  1.0000000(0)  0.062(4)  M1  0.00286(11)  0.398(19)  10 
^{5}D_{2}–^{1}D2_{2}  417.5  46,291.1  2,179.22  2.0 × 10^{−3}  18  −0.003989(14)  0.00156(14)  M1  0.00082(4)  0.242(10)  17 
^{3}F2_{3}–^{1}G1_{4}  26,842.3  71,280.3  2,249.63  2.28 × 10^{−1}  10  −0.0598(12)  0.048(4)  M1  0.00560(16)  0.380(19)  10 
^{3}F2_{4}–^{1}G1_{4}  26,973.7  71,280.3  2,256.30  2.8 × 10^{−1}  9  −0.000310(9)  0.058(4)  M1  0.00086(16)  0.431(22)  9 
^{3}G_{3}–^{1}G1_{4}  29,817.1  71,280.3  2,411.04  2.30 × 10^{−1}  7  −0.274(14)  0.0485(21)  M1  0.00355(8)  0.76(3)  7 
^{1}F_{3}–^{1}D1_{2}  52,732.6  93,832.5  2,432.36  1.79  3  0.833(6)  0.0445(8)  E2  0.9980(4)  1.00(21)  3 
^{3}G_{5}–^{1}G1_{4}  30,429.9  71,280.3  2,447.22  2.81 × 10^{−1}  8  0.716(14)  0.059(3)  M1 + E2  0.0242(6)  0.537(19)  8 
^{3}P2_{0}–^{3}P1_{1}  24,055.5  62,914.1  2,572.66  1.72 × 10^{−2}  11  −0.000017(2)  0.0039(4)  M1  0  0.343(22)  11 
^{3}P2_{1}–^{3}P1_{0}  24,972.8  63,419.8  2,600.21  6.3 × 10^{−3}  6  0.0000020(1)  0.00142(9)  M1  0  0.32(6)  6 
^{3}P2_{0}–^{3}F1_{2}  24,055.5  62,321.1  2,612.53  1.98 × 10^{−1}  2.2  0.1249(15)  0.0267(6)  E2  1  0.75(13)  2.2 
^{3}P2_{1}–^{3}P1_{1}  24,972.8  62,914.1  2,634.86  3.23 × 10^{−1}  2.3  0.221(7)  0.073(3)  E2  0.9976(4)  0.03(15)  2.3 
^{3}P2_{0}–^{3}P1_{2}  24,055.5  61,854.1  2,644.81  3.07 × 10^{−1}  2.1  0.295(5)  0.087(3)  E2  1  −0.06(17)  2.1 
^{3}H_{4}–^{3}F1_{3}  24,932.4  62,364.3  2,670.72  5.34 × 10^{−1}  3  0.749(16)  0.0778(10)  E2  0.99917(18)  0.00(15)  3 
^{3}P2_{1}–^{3}F1_{3}  24,972.8  62,364.3  2,673.61  2.59 × 10^{−1}  2.1  0.1276(11)  0.0377(6)  E2  1  0.75(12)  2.1 
^{3}H_{4}–^{3}F1_{2}  24,932.4  62,321.1  2,673.81  2.69  2.1  0.763(10)  0.363(8)  E2  1  0.62(22)  2.1 
^{3}P2_{1}–^{3}F1_{2}  24,972.8  62,321.1  2,676.70  2.15 × 10^{−1}  2.2  0.1443(11)  0.0290(4)  E2  0.99926(12)  0.49(19)  2.2 
^{3}H_{4}–^{3}F1_{4}  24,932.4  62,238.0  2,679.77  8.5 × 10^{−3}  4  0.053(17)  0.00125(6)  M1 + E2  0.83(4)  −0.04(3)  4 
^{3}H_{5}–^{3}F1_{3}  25,225.5  62,364.3  2,691.80  2.40  2.0  0.804(6)  0.350(6)  E2  1  0.56(18)  2.0 
^{3}H_{5}–^{3}F1_{4}  25,225.5  62,238.0  2,700.99  4.05 × 10^{−1}  3  0.831(7)  0.0596(4)  E2  0.9975(3)  0.15(15)  3 
^{3}P2_{2}–^{3}P1_{0}  26,468.2  63,419.8  2,705.44  1.90  2.2  0.349(7)  0.425(9)  E2  1  0.20(4)  2.2 * 
^{5}D_{0}–^{3}D_{1}  0.0  36,925.2  2,707.37  2.54 × 10^{−1}  9  −0.647(11)  0.3478(13)  M1  0  0.508(20)  9 
^{3}P2_{1}–^{3}P1_{2}  24,972.8  61,854.1  2,710.60  6.72 × 10^{−1}  2.1  0.301(6)  0.190(4)  M1 + E2  0.963(4)  0.04(17)  2.1 
^{5}D_{1}–^{3}D_{1}  142.4  36,925.2  2,717.86  2.24 × 10^{−1}  8  −0.153(3)  0.3072(17)  M1  0.00242(7)  0.561(21)  8 
^{3}H_{6}–^{3}F1_{4}  25,528.4  62,238.0  2,723.28  2.58  1.9  0.9692(14)  0.380(6)  E2  1  0.41(16)  1.9 
^{3}F2_{2}–^{3}P1_{0}  26,760.7  63,419.8  2,727.03  4.04 × 10^{−1}  4  0.067(3)  0.090(4)  E2  1  7.05(9)  3 
^{5}D_{1}–^{3}D_{2}  142.4  36,758.2  2,730.25  2.12 × 10^{−1}  9  −0.1428(8)  0.3656(9)  M1  8.2(6) × 10^{−6}  0.479(20)  9 
^{5}D_{2}–^{3}D_{1}  417.5  36,925.2  2,738.34  3.3 × 10^{−3}  17  0.00369(17)  0.0045(4)  M1 + E2  0.045(4)  0.252(11)  16 
^{3}P2_{2}–^{3}P1_{1}  26,468.2  62,914.1  2,742.98  1.49  2.2  0.413(11)  0.336(10)  M1 + E2  0.9865(12)  2.35(22)  2.2 * 
^{5}D_{2}–^{3}D_{2}  417.5  36,758.2  2,750.92  1.70 × 10^{−1}  8  −0.01167(15)  0.294(3)  M1  0.00299(7)  0.572(21)  8 
^{5}D_{2}–^{3}D_{3}  417.5  36,630.0  2,760.66  1.06 × 10^{−1}  9  −0.0764(3)  0.1526(4)  M1  7(22) × 10^{−8}  0.444(19)  9 
^{3}F2_{2}–^{3}P1_{1}  26,760.7  62,914.1  2,765.17  1.38 × 10^{−1}  6  0.0713(22)  0.0312(13)  E2  0.9953(9)  −6.08(5)  5 * 
^{3}F2_{3}–^{3}P1_{1}  26,842.3  62,914.1  2,771.43  2.55 × 10^{−1}  2.5  0.0737(9)  0.0577(17)  E2  1  0.82(8)  2.5 
^{5}D_{3}–^{3}D_{2}  803.1  36,758.2  2,780.43  1.14 × 10^{−1}  11  −0.0543(7)  0.198(3)  M1  0.000132(6)  0.332(17)  11 
^{3}P2_{2}–^{3}F1_{3}  26,468.2  62,364.3  2,785.00  1.27 × 10^{−1}  3  0.132(6)  0.0185(5)  E2  0.9997(9)  8.19(18)  2.4 
^{3}P2_{2}–^{3}F1_{2}  26,468.2  62,321.1  2,788.35  3.3 × 10^{−2}  10  0.122(3)  0.0045(4)  M1 + E2  0.9793(25)  −4.53(4)  7 
^{5}D_{3}–^{3}D_{3}  803.1  36,630.0  2,790.38  9.8 × 10^{−2}  8  −0.003092(15)  0.1408(6)  M1  0.00552(11)  0.513(20)  8 
^{5}D_{3}–^{1}G2_{4}  803.1  36,585.6  2,793.84  1.08 × 10^{−3}  18  −0.03628(17)  0.00110(11)  M1  0.00299(6)  0.238(10)  18 
^{3}P2_{2}–^{3}F1_{4}  26,468.2  62,238.0  2,794.83  3.32 × 10^{−1}  2.1  0.1395(10)  0.0488(9)  E2  1  0.52(11)  2.1 
^{3}F2_{2}–^{3}F1_{3}  26,760.7  62,364.3  2,807.88  3.44 × 10^{−1}  3  0.196(9)  0.0501(9)  M1 + E2  0.744(15)  −3.21(20)  3 * 
^{3}F2_{2}–^{3}F1_{2}  26,760.7  62,321.1  2,811.29  8.01 × 10^{−1}  2.1  0.271(5)  0.1081(22)  E2  0.99969(5)  0.49(13)  2.1 
^{3}F2_{3}–^{3}F1_{3}  26,842.3  62,364.3  2,814.33  4.77 × 10^{−1}  3  0.170(8)  0.0695(20)  M1 + E2  0.979(4)  −0.02(15)  3 
^{3}F2_{3}–^{3}F1_{2}  26,842.3  62,321.1  2,817.76  9.7 × 10^{−1}  3  0.3846(24)  0.1305(14)  M1 + E2  0.929(3)  0.41(14)  3 
^{3}F2_{2}–^{3}F1_{4}  26,760.7  62,238.0  2,817.87  7.2 × 10^{−3}  10  0.14(3)  0.00106(12)  E2  1  5.04(3)  8 
^{3}F2_{3}–^{3}F1_{4}  26,842.3  62,238.0  2,824.37  3.09 × 10^{−1}  4  0.127(11)  0.0456(9)  M1 + E2  0.555(25)  −0.22(3)  4 
^{3}F2_{4}–^{3}F1_{3}  26,973.7  62,364.3  2,824.78  7.14 × 10^{−1}  3  0.267(11)  0.1039(12)  M1 + E2  0.801(12)  0.929(10)  3 
^{3}P2_{2}–^{3}P1_{2}  26,468.2  61,854.1  2,825.15  5.00 × 10^{−1}  2.4  0.338(5)  0.1411(24)  E2  0.99984(3)  −0.52(18)  2.4 
^{5}D_{4}–^{3}D_{3}  1,282.7  36,630.0  2,828.24  4.1 × 10^{−1}  9  −0.932(3)  0.5910(9)  M1  0.00234(5)  0.457(19)  9 
^{3}F2_{4}–^{3}F1_{2}  26,973.7  62,321.1  2,828.23  1.63 × 10^{−1}  7  0.503(8)  0.0220(12)  E2  1  0.29(4)  7 
^{5}D_{4}–^{1}G2_{4}  1,282.7  36,585.6  2,831.80  4.8 × 10^{−3}  18  −0.001868(12)  0.0049(5)  M1  0.000342(17)  0.250(11)  17 
^{3}F2_{4}–^{3}F1_{4}  26,973.7  62,238.0  2,834.90  7.71 × 10^{−1}  2.1  0.210(6)  0.114(3)  M1 + E2  0.971(5)  −0.02(16)  2.1 
^{3}F2_{2}–^{3}P1_{2}  26,760.7  61,854.1  2,848.70  1.06 × 10^{−2}  10  0.050(9)  0.0030(4)  M1 + E2  0.995(5)  4.45(4)  8 
^{3}F2_{3}–^{3}P1_{2}  26,842.3  61,854.1  2,855.34  5.12 × 10^{−2}  3  0.0505(25)  0.0145(6)  M1 + E2  0.982(4)  0.61(13)  3 
^{3}F2_{4}–^{3}P1_{2}  26,973.7  61,854.1  2,866.10  2.44 × 10^{−1}  2.5  0.0750(9)  0.0688(24)  E2  1  0.81(9)  2.5 
^{1}G2_{4}–^{1}G1_{4}  36,585.6  71,280.3  2,881.44  1.14 × 10^{−1}  7  0.081(7)  0.0241(16)  E2  0.9948(10)  0.38(6)  7 
^{1}I_{6}–^{1}G1_{4}  37,511.6  71,280.3  2,960.46  2.99  1.9  −0.9848(9)  0.629(18)  E2  1  0.21(15)  1.9 
^{3}G_{3}–^{3}P1_{1}  29,817.1  62,914.1  3,020.54  8.3 × 10^{−3}  10  −0.0745(9)  0.00187(13)  E2  1  0.415(14)  10 
^{3}G_{3}–^{3}F1_{3}  29,817.1  62,364.3  3,071.57  5.60 × 10^{−1}  4  0.385(17)  0.0815(23)  M1 + E2  0.654(14)  0.55(3)  4 
^{3}G_{3}–^{3}F1_{2}  29,817.1  62,321.1  3,075.65  1.130  2.0  0.33(3)  0.1525(7)  M1 + E2  0.724(23)  −0.67(16)  2.0 
^{3}G_{3}–^{3}F1_{4}  29,817.1  62,238.0  3,083.53  5.5 × 10^{−2}  8  0.277(19)  0.0081(5)  M1 + E2  0.564(20)  0.154(4)  8 
^{3}G_{4}–^{3}F1_{3}  30,147.2  62,364.3  3,103.04  6.29 × 10^{−1}  2.4  0.405(20)  0.0916(22)  M1 + E2  0.981(3)  −0.10(20)  2.4 
^{3}G_{4}–^{3}F1_{2}  30,147.2  62,321.1  3,107.21  3.16 × 10^{−1}  2.1  0.693(7)  0.0427(3)  E2  1  0.16(20)  2.1 
^{3}G_{4}–^{3}F1_{4}  30,147.2  62,238.0  3,115.25  4.73 × 10^{−1}  4  0.382(15)  0.0696(16)  M1 + E2  0.671(14)  0.651(20)  4 
^{3}P1_{2}–^{1}D1_{2}  61,854.1  93,832.5  3,126.20  9.4 × 10^{−2}  9  −0.00802(7)  0.00234(20)  M1  0.00695(21)  0.475(18)  9 
^{3}G_{5}–^{3}F1_{3}  30,429.9  62,364.3  3,130.51  3.00 × 10^{−1}  3  0.881(5)  0.0436(5)  E2  1  0.22(14)  3 
^{3}G_{5}–^{3}F1_{4}  30,429.9  62,238.0  3,142.94  1.163  1.9  0.783(5)  0.1713(8)  M1 + E2  0.831(13)  0.38(17)  1.9 
^{3}F1_{2}–^{1}D1_{2}  62,321.1  93,832.5  3,172.54  7.7 × 10^{−2}  9  0.0451(10)  0.00191(16)  M1 + E2  0.0316(6)  0.451(19)  9 
^{3}F1_{3}–^{1}D1_{2}  62,364.3  93,832.5  3,176.89  1.55 × 10^{−1}  9  0.98985(9)  0.0038(3)  M1 + E2  0.0162(6)  0.447(20)  9 
^{5}D_{2}–^{3}G_{4}  417.5  30,147.2  3,362.67  4.9 × 10^{−5}  9  0.115(4)  0.000453(13)  E2  1  0.47(3)  9 
^{5}D_{1}–^{3}G_{3}  142.4  29,817.1  3,368.91  3.3 × 10^{−5}  10  0.153(8)  0.000225(6)  E2  1  0.41(3)  10 
^{5}D_{3}–^{3}G_{5}  803.1  30,429.9  3,374.35  4.8 × 10^{−5}  10  0.136(4)  0.000349(12)  E2  1  0.40(3)  10 
^{5}D_{2}–^{3}G_{3}  417.5  29,817.1  3,400.43  8.4 × 10^{−3}  18  0.2067(5)  0.057(4)  M1 + E2  0.0101(9)  0.252(10)  18 
^{5}D_{3}–^{3}G_{4}  803.1  30,147.2  3,406.86  8.8 × 10^{−3}  16  0.1012(4)  0.082(5)  M1 + E2  0.0207(15)  0.310(11)  16 
^{5}D_{4}–^{3}G_{5}  1,282.7  30,429.9  3,429.88  8.2 × 10^{−4}  15  0.782(12)  0.0060(4)  M1 + E2  0.326(21)  0.216(14)  15 
^{5}D_{3}–^{3}G_{3}  803.1  29,817.1  3,445.62  2.0 × 10^{−2}  18  0.01776(8)  0.136(11)  M1  0.00104(13)  0.254(10)  18 
^{5}D_{4}–^{3}G_{4}  1,282.7  30,147.2  3,463.47  3.2 × 10^{−2}  16  0.00535(3)  0.295(19)  M1  0.00051(8)  0.315(12)  16 
^{5}D_{4}–^{3}G_{3}  1,282.7  29,817.1  3,503.54  2.9 × 10^{−3}  18  −0.02342(14)  0.0197(16)  M1  0.000037(23)  0.251(10)  18 
^{3}H_{4}–^{1}F_{3}  24,932.4  52,732.6  3,596.07  8.4 × 10^{−3}  15  0.179(7)  0.0129(8)  M1 + E2  0.348(19)  0.214(10)  15 
^{3}H_{5}–^{1}F_{3}  25,225.5  52,732.6  3,634.39  3.4 × 10^{−3}  8  0.676(10)  0.00516(17)  E2  1  0.469(18)  8 
^{1}D1_{2}–^{1}S1_{0}  93,832.5  121,130.1  3,662.28  5.07  1.9  0.9519(13)  0.0588(8)  E2  1  0.49(16)  1.9 
^{5}D_{0}–^{3}F2_{2}  0.0  26,760.7  3,735.76  2.0 × 10^{−5}  13  −0.0236(9)  0.000052(2)  E2  1  −0.62(3)  12 * 
^{5}D_{1}–^{3}F2_{3}  142.4  26,842.3  3,744.27  4.7 × 10^{−6}  7  −0.0040(3)  6.2(4) × 10^{−6}  E2  1  0.33(5)  7 
^{3}D_{2}–^{3}P1_{0}  36,758.2  63,419.8  3,749.65  1.058  1.9  0.761(7)  0.237(5)  E2  1  0.20(15)  1.9 
^{5}D_{1}–^{3}F2_{2}  142.4  26,760.7  3,755.75  1.10 × 10^{−1}  8  −0.955(3)  0.289(4)  M1  0.000223(7)  0.97(4)  8 
^{5}D_{2}–^{3}F2_{4}  417.5  26,973.7  3,764.53  6 × 10^{−8}  50  −0.00005(3)  6(4) × 10^{−8}  E2  1  0.606(3)  50 
^{3}D_{1}–^{3}P1_{0}  36,925.2  63,419.8  3,773.28  1.87 × 10^{−1}  9  −0.1172(19)  0.042(3)  M1  0  0.462(18)  9 
^{5}D_{0}–^{3}P2_{2}  0.0  26,468.2  3,777.05  7.6 × 10^{−5}  8  0.0780(19)  0.000087(2)  E2  1  0.83(6)  8 
^{5}D_{2}–^{3}F2_{3}  417.5  26,842.3  3,783.25  1.90 × 10^{−1}  9  −0.24931(7)  0.251367(17)  M1  0.000088(4)  0.505(20)  9 
^{5}D_{2}–^{3}F2_{2}  417.5  26,760.7  3,794.97  2.13 × 10^{−1}  9  −0.1413(5)  0.561(4)  M1  1(3) × 10^{−8}  0.463(20)  9 
^{5}D_{1}–^{3}P2_{2}  142.4  26,468.2  3,797.48  4.1 × 10^{−2}  8  −0.0174(3)  0.0466(10)  M1  0.00165(5)  −1.30(3)  8 * 
^{3}D_{3}–^{3}P1_{1}  36,630.0  62,914.1  3,803.50  4.31 × 10^{−1}  2.0  0.759(7)  0.097(3)  E2  1  0.27(15)  2.0 
^{3}P2_{2}–^{1}F_{3}  26,468.2  52,732.6  3,806.35  1.21 × 10^{−3}  19  0.34(10)  0.00185(18)  M1 + E2  0.0081(21)  0.156(10)  19 
^{5}D_{3}–^{3}F2_{4}  803.1  26,973.7  3,820.00  1.80 × 10^{−1}  9  −0.11327(12)  0.17361(9)  M1  0.000080(4)  0.510(20)  9 
^{3}D_{2}–^{3}P1_{1}  36,758.2  62,914.1  3,822.14  1.044 × 10^{−1}  2.3  0.778(9)  0.0236(3)  M1 + E2  0.985(3)  −0.10(20)  2.3 
^{5}D_{2}–^{3}P2_{2}  417.5  26,468.2  3,837.58  4 × 10^{−5}  200  0.00052(19)  0.00005(9)  M1 + E2  0.11(21)  0.096(2)  61 * 
^{5}D_{3}–^{3}F2_{3}  803.1  26,842.3  3,839.27  4.9 × 10^{−1}  9  −0.025879(5)  0.65408(6)  M1  4.55(15) × 10^{−5}  0.504(20)  9 
^{3}D_{1}–^{3}P1_{1}  36,925.2  62,914.1  3,846.71  5.41 × 10^{−1}  3  0.457(18)  0.1224(12)  M1 + E2  0.676(19)  −0.04(5)  3 
^{3}F2_{2}–^{1}F_{3}  26,760.7  52,732.6  3,849.22  1.87 × 10^{−3}  9  0.297(6)  0.00287(6)  M1 + E2  0.940(8)  0.51(3)  9 
^{5}D_{3}–^{3}F2_{2}  803.1  26,760.7  3,851.34  5.7 × 10^{−2}  15  0.070(6)  0.150(8)  M1  0.000149(17)  −1.192(17)  13 
^{3}F2_{3}–^{1}F_{3}  26,842.3  52,732.6  3,861.36  8.1 × 10^{−3}  17  0.023(4)  0.0123(10)  M1 + E2  0.126(15)  0.218(12)  17 
^{1}G2_{4}–^{3}F1_{3}  36,585.6  62,364.3  3,878.07  3.4 × 10^{−2}  11  0.0160(4)  0.0050(5)  M1 + E2  0.059(3)  0.303(22)  11 
^{3}F2_{4}–^{1}F_{3}  26,973.7  52,732.6  3,881.05  1.71 × 10^{−2}  14  0.0254(14)  0.0263(13)  M1 + E2  0.138(9)  0.340(10)  14 
^{3}D_{3}–^{3}F1_{3}  36,630.0  62,364.3  3,884.76  1.95 × 10^{−1}  7  0.0180(11)  0.0284(18)  M1 + E2  0.072(5)  0.602(21)  7 
^{5}D_{4}–^{3}F2_{4}  1,282.7  26,973.7  3,891.31  8.5 × 10^{−1}  9  −0.008406(8)  0.8191(4)  M1  0.000170(4)  0.509(20)  9 
^{3}D_{3}–^{3}F1_{2}  36,630.0  62,321.1  3,891.30  4.8 × 10^{−2}  9  0.0497(6)  0.0065(5)  M1 + E2  0.079(5)  0.386(15)  9 
^{5}D_{3}–^{3}P2_{2}  803.1  26,468.2  3,895.24  7.9 × 10^{−1}  8  −0.957(18)  0.901(4)  M1  0.000177(5)  0.85(3)  8 
^{1}G2_{4}–^{3}F1_{4}  36,585.6  62,238.0  3,897.17  3.8 × 10^{−2}  10  0.01470(24)  0.0056(5)  M1 + E2  0.0778(22)  0.327(25)  10 
^{3}D_{3}–^{3}F1_{4}  36,630.0  62,238.0  3,903.92  2.61 × 10^{−1}  8  0.910(12)  0.038(3)  M1 + E2  0.102(9)  0.391(18)  8 
^{3}D_{2}–^{3}F1_{3}  36,758.2  62,364.3  3,904.21  1.35 × 10^{−2}  3  0.083(4)  0.00197(6)  E2  0.99990(8)  0.79(15)  3 
^{3}D_{2}–^{3}F1_{2}  36,758.2  62,321.1  3,910.81  3.08 × 10^{−1}  8  0.0663(12)  0.042(3)  M1 + E2  0.052(4)  0.479(18)  8 
^{5}D_{4}–^{3}F2_{3}  1,282.7  26,842.3  3,911.32  7.0 × 10^{−2}  8  0.03872(4)  0.09226(13)  M1  0.000442(10)  0.519(20)  8 
^{5}D_{4}–^{3}F2_{2}  1,282.7  26,760.7  3,923.84  1.53 × 10^{−6}  11  0.0242(11)  4.03(16) × 10^{−6}  E2  1  −1.38(4)  10 * 
^{3}D_{1}–^{3}F1_{2}  36,925.2  62,321.1  3,936.53  2.07 × 10^{−1}  8  0.895(8)  0.0279(19)  M1 + E2  0.094(8)  0.402(18)  8 
^{3}D_{3}–^{3}P1_{2}  36,630.0  61,854.1  3,963.34  5.91 × 10^{−1}  2.3  0.870(5)  0.1668(16)  M1 + E2  0.793(13)  0.20(13)  2.3 
^{5}D_{4}–^{3}P2_{2}  1,282.7  26,468.2  3,969.42  9.7 × 10^{−6}  9  −0.0060(3)  1.11(5) × 10^{−5}  E2  1  0.56(3)  9 
^{3}D_{2}–^{3}P1_{2}  36,758.2  61,854.1  3,983.59  3.56 × 10^{−1}  4  0.376(23)  0.1006(8)  M1 + E2  0.641(23)  −0.39(5)  4 
^{1}D2_{2}–^{1}G1_{4}  46,291.1  71,280.3  4,000.60  1.36 × 10^{−2}  5  0.0258(14)  0.00286(16)  E2  1  0.34(5)  5 
^{5}D_{0}–^{3}P2_{1}  0.0  24,972.8  4,003.22  1.37 × 10^{−1}  8  −0.0323(3)  0.1033(10)  M1  0  0.612(22)  8 
^{3}D_{1}–^{3}P1_{2}  36,925.2  61,854.1  4,010.27  7.56 × 10^{−2}  2.5  0.379(20)  0.02134(18)  M1 + E2  0.709(18)  0.13(12)  2.5 
^{5}D_{1}–^{3}P2_{1}  142.4  24,972.8  4,026.18  2.7 × 10^{−4}  12  0.073(3)  0.000201(7)  M1 + E2  0.625(19)  0.275(13)  12 
^{5}D_{2}–^{3}P2_{1}  417.5  24,972.8  4,071.29  1.18  9  −0.2726(10)  0.8860(19)  M1  0.000131(3)  0.454(19)  9 
^{5}D_{2}–^{3}H_{4}  417.5  24,932.4  4,078.00  8.0 × 10^{−8}  7  0.0014(3)  0.000014(5)  E2  1  2.59(4)  7 
^{5}D_{3}–^{3}H_{5}  803.1  25,225.5  4,093.45  1.52 × 10^{−6}  6  0.0106(5)  0.00225(12)  E2  1  1.00(7)  6 
^{5}D_{4}–^{3}H_{6}  1,282.7  25,528.4  4,123.28  1.16 × 10^{−5}  10  0.0308(4)  0.0185(18)  E2  1  0.40(3)  10 
^{5}D_{3}–^{3}P2_{1}  803.1  24,972.8  4,136.25  5.5 × 10^{−5}  9  0.03538(23)  4.11(8) × 10^{−5}  E2  1  0.44(3)  9 
^{5}D_{3}–^{3}H_{4}  803.1  24,932.4  4,143.17  9.2 × 10^{−4}  25  0.1288(6)  0.1629(3)  M1  0.000017(4)  0.170(9)  24 
^{5}D_{4}–^{3}H_{5}  1,282.7  25,225.5  4,175.44  1.2 × 10^{−5}  27  0.9716(20)  0.018(5)  M1 + E2  0.0285(19)  0.167(7)  26 
^{5}D_{1}–^{3}P2_{0}  142.4  24,055.5  4,180.63  1.52  9  −0.1224(6)  0.999728(5)  M1  0  0.407(18)  9 
^{5}D_{4}–^{3}H_{4}  1,282.7  24,932.4  4,227.19  4.7 × 10^{−3}  25  0.01030(7)  0.8368(3)  M1  0.000280(7)  0.170(9)  24 
^{5}D_{2}–^{3}P2_{0}  417.5  24,055.5  4,229.29  4.1 × 10^{−4}  9  0.1029(5)  0.000272(5)  E2  1  0.41(3)  9 
^{1}S2_{0}–^{3}P1_{1}  39,633.0  62,914.1  4,294.12  1.57 × 10^{−1}  8  −0.0827(14)  0.0355(22)  M1  0  0.50(3)  8 
^{3}G_{3}–^{1}F_{3}  29,817.1  52,732.6  4,362.63  1.40 × 10^{−1}  9  −0.01243(24)  0.2141(7)  M1  0.00700(18)  0.484(20)  9 
^{3}G_{4}–^{1}F_{3}  30,147.2  52,732.6  4,426.40  1.93 × 10^{−1}  8  0.436(11)  0.2955(23)  M1  0.00030(6)  0.540(21)  8 
^{1}G1_{4}–^{1}D1_{2}  71,280.3  93,832.5  4,432.91  5.54 × 10^{−1}  2.0  0.867(5)  0.01377(11)  E2  1  0.46(16)  2.0 
^{3}G_{5}–^{1}F_{3}  30,429.9  52,732.6  4,482.50  9.1 × 10^{−4}  10  0.813(6)  0.00139(3)  E2  1  0.343(17)  10 
^{3}P2_{1}–^{1}D2_{2}  24,972.8  46,291.1  4,689.49  7.6 × 10^{−2}  8  0.313(5)  0.05932(17)  M1  0.00191(5)  0.494(19)  9 
^{3}P2_{2}–^{1}D2_{2}  26,468.2  46,291.1  5,043.26  2.14 × 10^{−1}  8  0.0215(6)  0.1662(22)  M1  0.000072(3)  1.33(5)  8 
^{3}F2_{2}–^{1}D2_{2}  26,760.7  46,291.1  5,118.80  2.48 × 10^{−1}  9  −0.03653(17)  0.193(3)  M1  0.000119(4)  −0.316(18)  9 * 
^{3}F2_{3}–^{1}D2_{2}  26,842.3  46,291.1  5,140.27  4.9 × 10^{−1}  8  0.999725(23)  0.3795(19)  M1  0.00106(3)  0.578(22)  8 
^{1}F_{3}–^{1}G1_{4}  52,732.6  71,280.3  5,390.01  1.108 × 10^{−1}  1.9  0.920(3)  0.0233(6)  M1 + E2  0.981(3)  0.25(14)  1.9 
^{1}D2_{2}–^{3}P1_{1}  46,291.1  62,914.1  6,014.10  1.33 × 10^{−1}  9  0.99741(21)  0.0301(20)  M1  0.00256(6)  0.408(25)  9 
^{3}G_{3}–^{1}D2_{2}  29,817.1  46,291.1  6,068.49  1.26 × 10^{−2}  17  −0.99957(4)  0.0098(9)  M1  0.0031(4)  0.269(11)  17 
^{1}G2_{4}–^{1}F_{3}  36,585.6  52,732.6  6,191.4  1.176 × 10^{−2}  2.1  0.2411(13)  0.0180(17)  E2  0.9942(13)  0.31(17)  2.1 
^{3}D_{3}–^{1}F_{3}  36,630.0  52,732.6  6,208.46  1.82 × 10^{−1}  9  0.019892(22)  0.2794(12)  M1  0.000269(7)  0.431(18)  9 
^{1}D2_{2}–^{3}F1_{3}  46,291.1  62,364.3  6,219.82  7.4 × 10^{−2}  9  −0.253(3)  0.0107(9)  M1  0.00210(5)  0.371(15)  9 
^{1}D2_{2}–^{3}F1_{2}  46,291.1  62,321.1  6,236.58  5.6 × 10^{−2}  10  0.02148(14)  0.0076(6)  M1  0.000023(3)  0.356(16)  10 
^{3}D_{2}–^{1}F_{3}  36,758.2  52,732.6  6,258.28  8.2 × 10^{−2}  9  0.815(6)  0.1251(4)  M1  0.00223(5)  0.462(19)  9 
^{1}D2_{2}–^{3}P1_{2}  46,291.1  61,854.1  6,423.72  1.88 × 10^{−1}  9  0.03917(5)  0.053(3)  M1  0.000694(21)  0.421(25)  9 
^{3}P2_{1}–^{1}S2_{0}  24,972.8  39,633.0  6,819.3  1.66  8  −0.999980(2)  0.99912(6)  M1  0  0.615(22)  8 
^{3}P2_{0}–^{3}D_{1}  24,055.5  36,925.2  7,768.1  5.7 × 10^{−2}  9  0.9999895(11)  0.0778(6)  M1  0  0.50(3)  9 
^{3}H_{5}–^{1}I_{6}  25,225.5  37,511.6  8,137.0  1.24 × 10^{−1}  9  1.0000000(0)  0.4314(3)  M1  1.4(5) × 10^{−7}  0.492(20)  9 
^{3}H_{6}–^{1}I_{6}  25,528.4  37,511.6  8,342.7  1.63 × 10^{−1}  9  0.0059430(0)  0.56728(17)  M1  1.14(4) × 10^{−5}  0.479(20)  9 
^{3}P2_{1}–^{3}D_{1}  24,972.8  36,925.2  8,364.2  1.34 × 10^{−1}  8  0.2511591(3)  0.1833(14)  M1  0.0041(4)  0.50(3)  8 
^{3}P2_{1}–^{3}D_{2}  24,972.8  36,758.2  8,482.7  7.2 × 10^{−4}  16  0.0076(9)  0.00125(9)  M1 + E2  0.098(16)  0.155(16)  16 
^{3}H_{4}–^{1}G2_{4}  24,932.4  36,585.6  8,579.0  1.74 × 10^{−1}  7  0.0082(3)  0.177(3)  M1  4.87(13) × 10^{−5}  1.02(3)  7 
^{3}H_{5}–^{1}G2_{4}  25,225.5  36,585.6  8,800.3  2.54 × 10^{−1}  8  −0.741(12)  0.2584(22)  M1  0.000105(3)  0.68(3)  8 
^{3}P2_{2}–^{3}D_{1}  26,468.2  36,925.2  9,560.3  3.7 × 10^{−2}  9  −0.1111(10)  0.0513(8)  M1  0.0031(5)  0.88(3)  9 
^{3}P2_{2}–^{3}D_{2}  26,468.2  36,758.2  9,715.5  5.9 × 10^{−2}  10  0.01476(11)  0.1024(17)  M1  0.0033(3)  −0.058(23)  10 
^{3}F2_{2}–^{3}D_{1}  26,760.7  36,925.2  9,835.5  1.80 × 10^{−2}  11  0.88(3)  0.0246(6)  M1 + E2  0.111(12)  −0.63(3)  10 * 
^{3}P2_{2}–^{3}D_{3}  26,468.2  36,630.0  9,838.1  6.1 × 10^{−2}  9  −0.945(6)  0.0869(12)  M1  0.0037(4)  0.42(3)  9 
^{3}F2_{2}–^{3}D_{2}  26,760.7  36,758.2  9,999.8  1.75 × 10^{−2}  8  0.067(6)  0.0303(7)  M1 + E2  0.041(4)  1.23(6)  8 
^{3}F2_{3}–^{3}D_{2}  26,842.3  36,758.2  10,082.0  2.96 × 10^{−3}  8  0.36(3)  0.00511(9)  M1+E2  0.55(4)  −0.404(23)  8 
^{3}F2_{3}–^{3}D_{3}  26,842.3  36,630.0  10,214.1  7.5 × 10^{−3}  8  0.048(6)  0.01083(10)  M1 + E2  0.069(6)  0.370(18)  8 
^{3}F2_{3}–^{1}G2_{4}  26,842.3  36,585.6  10,260.7  1.46 × 10^{−1}  7  0.599(18)  0.1487(21)  M1  1.50(8) × 10^{−6}  0.91(3)  7 
^{3}D_{3}–^{1}D2_{2}  36,630.0  46,291.1  10,348.0  1.16×10^{−1}  9  0.9537(17)  0.0902(8)  M1  0.000194(5)  0.473(21)  9 
^{3}F2_{4}–^{3}D_{3}  26,973.7  36,630.0  10,353.1  9.7 × 10^{−3}  6  0.399(11)  0.0139(4)  M1 + E2  0.214(14)  0.451(20)  6 
^{1}F_{3}–^{3}F1_{3}  52,732.6  62,364.3  10,379.5  1.24 × 10^{−2}  9  0.001661(3)  0.00181(14)  M1  0.000142(7)  0.430(18)  9 
^{3}F2_{4}–^{1}G2_{4}  26,973.7  36,585.6  10,400.9  3.3 × 10^{−1}  9  0.01357(12)  0.336(3)  M1  3.6(7) × 10^{−7}  0.453(20)  9 
^{1}F_{3}–^{3}F1_{2}  52,732.6  62,321.1  10,426.3  1.88 × 10^{−1}  8  0.9999919(9)  0.0254(18)  M1  1.35(4) × 10^{−5}  0.441(19)  8 
^{3}D_{2}–^{1}D2_{2}  36,758.2  46,291.1  10,487.1  2.17 × 10^{−2}  9  0.006205(20)  0.01686(16)  M1  0.000181(6)  0.432(20)  9 
^{1}F_{3}–^{3}F1_{4}  52,732.6  62,238.0  10,517.5  1.02 × 10^{−1}  8  0.999963(4)  0.0151(11)  M1  0.000285(8)  0.445(19)  8 
^{3}D_{1}–^{1}D2_{2}  36,925.2  46,291.1  10,674.1  1.04 × 10^{−1}  9  0.9869(6)  0.0804(7)  M1  2.24(8) × 10^{−5}  0.471(21)  9 
^{3}F1_{4}–^{1}G1_{4}  62,238.0  71,280.3  11,056.1  5.7 × 10^{−2}  9  −0.012325(3)  0.0120(7)  M1  3.74(8) × 10^{−5}  0.491(22)  9 
^{3}F1_{3}–^{1}G1_{4}  62,364.3  71,280.3  11,212.7  3.3 × 10^{−2}  9  −1.0000000(0)  0.0069(4)  M1  0.00102(3)  0.492(22)  9 
^{3}G_{3}–^{3}D_{1}  29,817.1  36,925.2  14,064.6  6.68 × 10^{−4}  2.1  0.753(9)  0.00091(8)  E2  1  −0.05(19)  2.1 
^{3}G_{5}–^{1}I_{6}  30,429.9  37,511.6  14,117.0  3.6 × 10^{−4}  17  1.0000000(0)  0.00126(10)  M1  0.0032(3)  0.264(11)  17 
^{3}G_{3}–^{1}G2_{4}  29,817.1  36,585.6  14,770.3  4.2 × 10^{−2}  12 *  −1.0000000(0)  0.0424(18)  M1  2.26(10) × 10^{−5}  0.305(14)  12 
^{3}G_{4}–^{1}G2_{4}  30,147.2  36,585.6  15,527.6  5.9 × 10^{−3}  14  −0.00145(4)  0.0060(4)  M1  0.000037(4)  0.322(13)  14 
^{3}G_{5}–^{1}G2_{4}  30,429.9  36,585.6  16,240.7  2.5 × 10^{−2}  12 *  −1.0000000(0)  0.0255(9)  M1  1.21(4) × 10^{−5}  0.333(17)  12 
^{3}H_{4}–^{3}G_{5}  24,932.4  30,429.9  18,185.1  1.00 × 10^{−3}  2.5  0.0044(3)  0.0073(5)  M1  0.000031(11)  1.32(18)  2.5 
^{3}H_{4}–^{3}G_{4}  24,932.4  30,147.2  19,171.0  3.44 × 10^{−2}  4  −0.0093(5)  0.321(19)  M1  0.000297(17)  3.40(8)  4 
^{3}H_{5}–^{3}G_{5}  25,225.5  30,429.9  19,209.3  4.7 × 10^{−2}  8  −0.0055920(0)  0.344(3)  M1  0.000144(16)  0.57(3)  8 
^{3}H_{5}–^{3}G_{4}  25,225.5  30,147.2  20,318  6.6 × 10^{−4}  19 *  0.080(22)  0.0061(6)  M1 + E2  0.102(21)  0.156(11)  19 
^{3}H_{6}–^{3}G_{5}  25,528.4  30,429.9  20,402  4.7 × 10^{−2}  8  1.0000000(0)  0.344(3)  M1  0.00156(13)  0.55(3)  8 
^{3}H_{4}–^{3}G_{3}  24,932.4  29,817.1  20,472  4.2 × 10^{−2}  6  −0.642(22)  0.283(12)  M1  0.00167(12)  0.97(4)  6 
^{3}F2_{4}–^{3}G_{5}  26,973.7  30,429.9  28,934  4.0 × 10^{−2}  8  1.0000000(0)  0.295(5)  M1  0.000053(5)  0.72(3)  8 
^{3}F2_{3}–^{3}G_{4}  26,842.3  30,147.2  30,258  9.7 × 10^{−4}  19 *  0.0071(8)  0.0090(9)  M1  0.0013(3)  0.267(10)  19 
^{3}F2_{4}–^{3}G_{4}  26,973.7  30,147.2  31,511  2.99 × 10^{−2}  7  −0.00800(5)  0.278(6)  M1  3.7(6) × 10^{−6}  0.84(3)  7 
^{3}F2_{2}–^{3}G_{3}  26,760.7  29,817.1  32,718  3.3 × 10^{−2}  9  −1.0000000(0)  0.225(3)  M1  0.000029(3)  0.572(24)  9 
^{3}F2_{3}–^{3}G_{3}  26,842.3  29,817.1  33,616  4.1 × 10^{−2}  9  −0.0332325(9)  0.278(4)  M1  4.4(5) × 10^{−6}  0.591(24)  9 
^{3}F2_{4}–^{3}G_{3}  26,973.7  29,817.1  35,169  1.6 × 10^{−4}  22 *  0.0015(4)  0.0011(4)  M1  0.0008(4)  1.015(7)  22 
^{3}H_{4}–^{3}F2_{4}  24,932.4  26,973.7  48,988  6.3 × 10^{−3}  15 *  −0.03761(23)  0.0061(4)  M1  1.7(10) × 10^{−8}  0.286(19)  15 
^{3}H_{4}–^{3}F2_{3}  24,932.4  26,842.3  52,359  1.6 × 10^{−3}  15 *  −1.0000000(0)  0.00218(17)  M1  9.1(6) × 10^{−6}  0.286(17)  15 
^{3}P2_{1}–^{3}P2_{2}  24,972.8  26,468.2  66,872  4.5406 × 10^{−2}  0.06  1.0000000(0)  0.052(4)  M1  9.9(6) × 10^{−8}  2621(13)  0.013 
^{3}P1_{2}–^{3}P1_{1}  61,854.1  62,914.1  94,340  2.6842 × 10^{−2}  0.012  1.0000000(0)  0.00607(17)  M1  6.95(15) × 10^{−7}  444(16)  0.012 
^{3}P2_{0}–^{3}P2_{1}  24,055.5  24,972.8  109,020  1.3837 × 10^{−2}  0.06  1.0000000(0)  0.0104(9)  M1  0  35(3)  0.06 
^{3}P1_{1}–^{3}P1_{0}  62,914.1  63,419.8  197,700  7.0044 × 10^{−3}  0.010  1.0000000(0)  0.00157(4)  M1  0  453(15)  0.010 
^{5}D_{3}–^{5}D_{4}  803.1  1,282.7  208,500  2.9885 × 10^{−3}  0.007  1.0000000(0)  0.9999994364(2)  M1  1.048(20) × 10^{−7}  594(20)  0.007 
^{5}D_{2}–^{5}D_{3}  417.5  803.1  259,300  2.6639 × 10^{−3}  0.010  1.0000000(0)  0.99999968(8)  M1  3.61(7) × 10^{−8}  325(13)  0.011 
^{3}G_{3}–^{3}G_{4}  29,817.1  30,147.2  302,900  9.212 × 10^{−4}  0.23  1.0000000(0)  0.0086(8)  M1  3.1(6) × 10^{−10}  17.5(6)  0.23 
^{3}H_{5}–^{3}H_{6}  25,225.5  25,528.4  330,100  6.144 × 10^{−4}  0.15  1.0000000(0)  0.9815(18)  M1  8(3) × 10^{−11}  24.6(6)  0.15 
^{3}H_{4}–^{3}H_{5}  24,932.4  25,225.5  341,200  6.625 × 10^{−4}  0.15  1.0000000(0)  0.979(5)  M1  2.8(4) × 10^{−10}  49(3)  0.15 
^{3}G_{4}–^{3}G_{5}  30,147.2  30,429.9  353,700  4.685 × 10^{−4}  0.4  1.0000000(0)  0.0034(3)  M1  7.4(11) × 10^{−10}  7.7(4)  0.4 
^{5}D_{1}–^{5}D_{2}  142.4  417.5  363,500  1.1839 × 10^{−3}  0.015  1.0000000(0)  0.99999992(8)  M1  6.89(13) × 10^{−9}  294(11)  0.015 
^{5}D_{0}–^{5}D_{1}  0.0  142.4  702,000  1.5517 × 10^{−4}  0.018  1.0000000(0)  1.000000000(0)  M1  0  264(7)  0.018 
^{a }The energy levels and Ritz wavelengths calculated from them are taken from [
As noted in
The number of comparisons required for a reliable estimation of uncertainty of calculated
Of course, the above considerations apply only to the test case considered here, the M1 and E2 transitions of Fe V calculated with Cowan’s codes. Other codes and other atomic systems may have different properties, and it would be interesting to obtain similar estimates for them. This would lead to better understanding of dependability of traditional methods of uncertainty estimation based on comparison of results of different atomic codes.
Statistical distributions of
The method suggested here was relatively easy to implement with Cowan’s suite of codes, which include the LSF procedure with welldefined uncertainties of the fitted parameters. Other codes also have adjustable parameters. For example, SUPERSTRUCTURE [
Adjustable parameters are also used by Hibbert in the CIV3 code [
Other sophisticated
In principle, a clever statistician should be able to predict the expected transformations of distribution functions of calculated
Solving these problems calls for development of a relatively new field of atomic physics,
The present work suggests a method for evaluation of sensitivity of calculated transition probabilities (
In order to determine the total uncertainties of the calculation, these internal uncertainty estimates should be combined with studies of effects of approximations made in the atomic model, for example, by comparing
One conclusion following from the present results is that uncertainties should be estimated not for the straight
When results of two different theoretical models are compared to each other, one can expect that the shapes of statistical distributions of the
There is an important implication for Monte Carlo simulations of plasma kinetics, namely, the distributions of the input atomic parameters (such as the
The author declares no conflict of interest.