This example is based on A. SC. 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:
 The RSquaredH parameter is set to true, as the
Hamiltonian specified in the paper is closest to this form.
 LimitSearch
is set to true to give quantum number assignments consistent
with the paper.
 The origins of the individual components of the
ground X^{5}Π state are specified individually,
rather than in several spin orbit coupling constants. This
effect can be achieved in PGOPHER using perturbations, in this
case homogeneous contributions with OmegaSelect set.
For example <v=0Om=3v=0> shifts the Ω = 3
component (only) by the amount of the perturbation Value.
For correct assignment of the quantum numbers PGOPHER needs to
know whether the state is regular or inverted. One approach is
to set an approximately correct value for the spinorbit
coupling constant, A. in which case not all the components can
be positioned in this way. For the file given here, the
Ω = 0 and 1 components are positioned by setting the
state Origin and A values, and
perturbations are used to position the other spinorbit
components. In this case it is possible to derive the values
by comparing the matrix elements used in PGOPHER and
the paper, though in general it can be simpler to simply
refit data from the literature.
As an example of how this can be done, consider the
diagonal Ω = 0 matrix elements given in the paper for the
X^{5}Π state (Table I):
T0
+ (BAD2LambdaD+/3*oD)*(J*(J+1)+5)
 D*(J*J*(J+1)*(J+1)+20*J*(J+1)+25)
+/3*(o+p+q)
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 T_{O} = OriginA6*gamma. Repeating
the process with Ω = 1 shows that T_{1} =
Origin6*gamma. The difference between these, T_{1} 
T_{0} = A, so given that the paper fixes T_{0}
= 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) = Origin2*A4*gamma+<v=0Om=1v=0>
T2 = Origin+A4*gamma+<v=0Om=2v=0>
T3 = Origin+2*A+<v=0Om=3v=0>
Note that the homogeneous perturbations such as
<v=0Om=3v=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=0Om=2v=0> = T2T0 
(2A+2*gamma).
 The lambda doubling constants are given in an alternative
form that naturally involves J rather than N
(see Λ doubling for
more on this). The term involving o (in PGOPHER) has the same
operator form as o+p+q as used in the
paper, so no extra term is required for this. This also
applies to its centrifugal distortion, oD, as confirmed by
comparing the matrix elements above. To reproduce the p+2q
term requires a perturbation, <v=0J+L2S+v=0>; this
uses Op = Luncouple, n = 2, Srank
= 1 and Scomp = 2. To match the definition given
in the paper with the PGOPHER perturbation requires a scale
factor of 1/2; this can be worked out by comparing the
operator forms (see perturbations
for the PGOPHER definitions) or by comparing matrix elements.
Similarly the <v=0J+2L2v=0> perturbation is used to
reproduce the q term; this time the scale factor is
+1/2.
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(Ω1) 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
NQN 2
  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:

The e and f levels have
been swapped, to give energy levels consistent with the
signs of the published lambda doubling constants, which were
in turn chosen on the assumption that the upper state is a ^{5}Σ^{
} state. The true sign, and thus the absolute parity
assignments, was not known and the tabulated energy levels
seem to have been worked out with the opposite signs of the
constants. In general swapping + and  parity assignments,
and swapping Σ^{ } and Σ^{ +} states will
make no change to the spectrum, provided the signs of all
the lambda doubling constants are also changed.

The FrequencyOffset directive
is required as the energy zero for this list is taken as the
lowest energy level, J = 1 of the Ω = 1 component.
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 4.6e6
to 1.6e6. 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:
 The RSquaredH parameter left at false, the
default.
 LimitSearch
is set to true to give quantum number assignments consistent
with the paper.
 The Origin for the ground X^{5}Π state is left at
zero, the default, and values for A, LambdaSS and eta are
taken from equation (20) of the paper.
 The lambda doubling constants o, p and q are worked out from
the combined values given in Table V
 The Origin of the upper A^{5}Σ^{ } state
needs to be offset by approximately the gap between the Ω = 0
and Ω = 1; the exact value is taken to give the best fit to
the tabulated values. This arises from the different choice of
energy zero for the ground state used by PGOPHER.
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.
 For the ground state table VIII can be reproduced exactly if
minor adjustments by fitting are made to the Origin, B, A,
LambdaSS, eta, o, theta, AD and eta of the ground state. The
adjustments are mainly because the constants have not been
quoted to sufficient precision to reproduce the calculated
values, and minor differences to the matrix elements. Note
that fitting requires a FrequencyOffset of
119.72578 as the choice of energy zero is different.
 For the excited state Origin, B and LambdaD are fitted. The
changes are minor, with the exception of LambdaD which changes
from 4.6e6 to 1.6e6. This may reflect a misprint in the
original paper, or including blended transitions in the fit.
The average error drops from 0.011 to 0.0086 cm^{1}
on fitting.