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Electronic spectrum of the A5Σ - X5Π transition of CrO

Simulation of figure 3
This example is based on A. S-C. Cheung, W. Zyrnicki and A. J. Merer, J. Molec. Spectrosc. 104 315 (1984), and shows an example of a  spectrum involving states with high multiplicity, and also demonstrates two alternative approaches to dealing with molecular parameters that are not presented in quite the same way as PGOPHER requires them. Five slightly different data files are given below. The difference between them is not really significant on the scale of the precision of the original data, but CrOAdjusted.pgo is the recommended one to use.

CrOoriginal.pgo. This gives the data in a form that is closest to the form presented in the paper, which requires special consideration as follows:

As an example of how this can be done, consider the diagonal Ω = 0 matrix elements given in the paper for the X5Π state (Table I):
+ (B-AD-2LambdaD+/-3*oD)*(J*(J+1)+5)
- D*(J*J*(J+1)*(J+1)+20*J*(J+1)+25)

PGOPHER gives the following for the same matrix elements (right click on X and select matrix elements):

<J,Lambda=1,Omega=0 + |H| + J,Lambda=1,Omega=0> = 
+ Origin
+ B*(J*(J+1)+5)
+ -A
+ + 3*o
+ -6*gamma
+ D*(-J*(J*(J*(J+2)+21)+20)-25)
+ (AD+2*LambdaD)*(-J*(J+1)-5)
+ +3*oD*(J*(J+1)+5)

A little algebra will show that these two expressions are identical apart from o and oD (discussed below) and the J independent terms. Equating these shows that  TO = Origin-A-6*gamma. Repeating the process with Ω = 1 shows that T1 = Origin-6*gamma. The difference between these, T1 - T0 =  A, so given that the paper fixes T0 = 0 it follows that A can be set to T1. The paper has chosen to take the energy zero for the state at Ω = 0, while PGOPHER will take it at Σ = 0 component (Ω = 1 here). To align the two scales, set the PGOPHER Origin = A+6*gamma. Repeating the process for the other Ω components gives the following set of equations:

T(-1) = Origin-2*A-4*gamma+<v=0|Om=-1|v=0>
T2 = Origin+A-4*gamma+<v=0|Om=2|v=0>
T3 = Origin+2*A+<v=0|Om=3|v=0>

Note that the homogeneous perturbations such as <v=0|Om=3|v=0> have been added here, and each of these is essentially indicating how far the component has been shifted from the equal spacing predicted from the simple spin orbit energy formula AΛΣ. The required value for each component can be worked out by rearranging the above giving, for example, <v=0|Om=2|v=0> =  T2-T0 - (2A+2*gamma).

CrOoriginalAlternate.pgo. Thus file uses an alternate method of positioning the Ω components by setting A to zero, and using OmegaOrder set to Regular to force the correct state ordering. This was essentially generated from the file above by setting A and the origin to zero, and adding A(Ω-1) to all the J independent perturbations, though for exact agreement the origin needs to be set to .0912 = 6γ. CrOoriginalAdjusted.pgo. This is the original data file, with some minor adjustments to give slightly better agreement with the energy levels and transitions listed in the paper. To do this, the information in table VIII and the appendix was converted to a form PGOPHER could use. Using the simplified format the energy level list can be input as follows:
FrequencyOffset -54.43912
- - 1 F1f 0.0002
- - 2 F1f 2.025
- - 3 F1f 5.0621
- - 4 F1f 9.1117
- - 14 F1f 105.3185
- - 15 F1f 120.5138

- - 0 F2f 57.1558
- - 1 F2f 58.1881

- - 0 F2e 56.9666
- - 1 F2e 58.0008
- - 2 F2e 60.0692
- - 3 F2e 63.1718

Only a few entries are shown above; the complete file is available as CrOTableVIII.lin. Notes on producing this:

Given the settings above, table VIII can be reproduced exactly if minor adjustments by fitting are made to the Origin, B, o, gamma and the perturbations. The adjustments are mainly because the constants have not been quoted to sufficient precision to reproduce the calculated values.

The transitions given in the appendix can be similarly converted into a form that PGOPHER can read using the BranchTable format:

BranchTable sR54 rQ54  qP54    pQ34      oQ24     oP34     nP24
7 8021.3 8011.337 - - - - -
8 8023.748 8012.779 8002.799 - 7988.457 - -
9 8026.122 8014.157 8003.182 - 7986.855 - -
10 8028.406 8015.457 8003.513 - - - 7976.194
11 8030.687 8016.698 8003.742 - 7983.441 - 7973.451
12 8032.814 8017.876 8003.909 - 7981.63 - 7970.649
13 8034.913 8018.965 8004.007 - 7979.758 - 7967.778
14 8036.943 8020.003 8004.04 7992.332 7977.818 - 7964.822
15 8038.898 8020.967 8004.007 7991.327 7975.811 7976.352 7961.852
16 8040.787 8021.867 8003.909 7990.252 7973.74 7974.309 7958.792

Again, only a few sample lines are shown; the full list is available as CrOAppendix.lin. The agreement with the observed lines is good, but can be slightly improved (average error dropping from 0.011 to 0.0086 cm-1) by fitting the upper state Origin, B and LambdaD. The changes are minor, with the exception of LambdaD which changes from 4.6e-6 to 1.6e-6. This may reflect a misprint in the original paper, or possibly a more thorough treatment of blended transitions in the original fit.

CrO.pgo is set up using the standard constants available in PGOPHER, derived as follows:

CrOAdjusted.pgo is the same data file, with selected constants adjusted slightly to give the best fit with the energy levels and transitions values given in the paper.