Pair Identity and Smooth Variation
Rules Applicable for the Spectroscopic
Parameters of H2O Transitions
Involving High J States
Q. Ma
NASA/Goddard Institute for Space Studies & Department of
Applied Physics and Applied Mathematics, Columbia University
2880 Broadway, New York, NY 10025, USA
R. H. Tipping
Department of Physics and Astronomy, University of Alabama,
Tuscaloosa, AL 35487, USA
N. N. Lavrentieva
V. E. Zuev Institute of Atmospheric Optics SB RAS, 1,
Akademician Zuev square, Tomsk 634021, Russia
I. Basic Idea in Analyzing Spectroscopic
Parameters of H2O Lines

A whole system consists of one absorber H2O molecule, bath
molecules, and electromagnetic fields.

By considering this system as a black box, its inputs are the
H2O lines of interest and its outputs are the spectroscopic
parameters.
outputs
A Black Box
inputs

Basic assumptions: (1) The outputs depend on the inputs. (2)
Identical inputs should yield identical outputs. (3) Similar
inputs should yield similar outputs.

The inputs are the energy levels and wave functions associated
with the initial and final states of the H2O lines.
II. Properties of the energy levels and wave
functions of H2O
(1) One categories the H2O states into different sets of paired states {J0,J,
J1,J}, {J1,J-1, J2,J-1}, {J2,J-2, J3,J-2}, ···, {JJ-2,3, JJ-2,2}, {JJ-1,2,JJ-1,1}, {JJ,1,JJ,0}.
(2) Within each of the sets, the energy levels of two paired states with
high J are almost identical. For different pairs, their energy levels vary
smoothly as J varies and these variation patterns are well organized.
(3) With respect to the H2O wave functions, they are given in terms of
expansion coefficients U J over the symmetric top wave functions
K
|JKM>,
| J M   U KJ | JKM  .
K
(4) Within each of the sets, the coefficients of two paired states with high
J have almost identical magnitudes. For different pairs, patterns of their
coefficients are very similar and the placements of the patterns shift
smoothly as J varies.
(5) The above are useful features of the inputs. By exploiting them, one
can access important conclusions on the outputs without requiring to
know what really happens inside the box.
II-1 Properties of the Energy Levels of H2O
Fig. 1 A plot to show energy levels of H2O states with J = 11 thru 20 in the
vibrational ground state. For states with J + Ka – Kc = even, their energy
levels are plotted by × and their values of Ka – Kc are presented on the
right side of the symbols. Meanwhile, for states with J + Ka – Kc = odd,
their energy levels are plotted by ∆ and values of Ka – Kc are on the left
side of the symbols.
II-2 Properties of the H2O Wave Functions
Fig. 2 A plot to show properties of H2O wave functions in the I R
representation for three sets of pairs of states: {JJ,0,JJ,1}, {JJ-1,1,JJ-1,2}, and
{JJ-2,2,JJ2,3}.
II-2 Properties of the H2O Wave Functions
Fig. 3 A plot to show properties of H2O wave functions in the III R
representation for three sets of pairs of states: {J0,J,J1,J}, {J1,J-1,J2,J-1}, and
{J2,J-2,J3,J-2}.
:.
II-3 Boundaries for individual sets of paired states
set
JJ,0
JJ,1
JJ-1,1
JJ-1,2
JJ-2,2
JJ-2,3
JJ-3,3
JJ-3,4
JJ-4,4
JJ-4,5
JJ-5,5
JJ-5,6
JJ-6,6
JJ-6,7
···
J4,J-4
J5,J-4
J3,J-3
J4,J-3
J2,J-2
J3,J-2
J1,J-1
J2,J-1
J0,J
J1,J
Jbd
3
5
7
9
10
12
14
···
19
16
13
10
7
ε 0.63 0.66 0.49 0.32 1.08 0.60 0.32
(%)
···
1.28 0.55 1.13 1.00 0.60
In each of the sets, the pair identity and the smooth variation break down
for J is below certain boundaries.
By introducing a numerical measure ε defined by
   ||U KJ |2  | U KJ |2 | /  | U KJ | ,
2
2
K
1
1
K
where J, 1 and J, 2 are paired states and  = Ka – Kc, one can calculate
how ε varies with J.
The higher the J, the smaller the ε. By choosing ε is about 1 %, one can
determine a boundary Jbd for each of the sets. The results is listed in
Table.
III. Categorizations of H2O lines and discovery of two
rules applicable for all spectroscopic parameters

To categorize H2O lines such that within individual groups, the
inputs of lines have similarities. Then, one expects their
outputs have similarities too.

The procedure is carried out by dividing all lines into the P, Q,
and R branches first and then, by categorizing them into sets of
paired lines. As example, a group consists of paired lines J′0,J’
← J″1,J" and J′1,J′ ← J″0,J" and a group of J′J′,0 ← J″J″,1 and J′J′,1 ←
J″J″,0.

It turns out that the similarities of the outputs do exist. Thus,
two rules can be established. These rules hold within certain
accuracy tolerances. Corresponding to 1 % accuracy of the
inputs, we choose 5 % as the accuracy tolerances.

The pair identity rule: Two paired lines whose J values are
above certain boundaries in the same groups have almost
identical spectroscopic parameters.

The smooth variation rule: For different pairs of states in the
same groups, values of their spectroscopic parameters vary
smoothly as their J values vary.
IV. Demonstrations of the two rules by Measurements
Fig. 4 Demonstrations of the pair identity and smooth variation rules for line
strengths, air- and N2-broadened half-widths, and induced shifts in a group of
paired lines J′3,J’-2 ← J″0,J" and J′2,J’-2 ← J″1,J" in the R branch. The measured values
by Toth are plotted by × and ∆, respectively. The spin degeneracy factor is
excluded for the strength. The boundary of this group is Jbd = 13.
IV. Demonstration of the two rules by Measurements
Fig. 5 Demonstration of the pair identity and smooth variation rules in a
group of paired lines of J'0,J' ← J"1,J"-1 and J'1,J' ← J"2,J"-1 in the Q branch of
the ν2 band. The air-broadened half-widths (in cm-1/bar) are from
measurements by Yamada. The boundary of this group is about Jbd = 10.
V. Comparison between HITRAN H2O 2006 and 2009
Fig. 6 Comparisons between air-broadened half-widths in HITRAN H2O
2006 and 2009. They are plotted by × and ∆, respectively. The 1639 lines
of the H2O pure rotational band are arranged according to the ascending
order of the half-width values of HITRAN H2O 2009.
V. Comparison between HITRAN H2O 2006 and 2009
Fig. 7 The same as Fig. 6 except for values of the temperature exponent
(T exponent) n. There are dramatically differences between these two
versions of n. Among 1639 lines, there are 245 lines whose T exponent n
become negative.
V. Comparison between HITRAN H2O 2006 and 2009
Number of lines
difference
Intensity
Air-width
Self-width
shift
T-expon.
> 50 %
25
143
246
572
630
30 - 50 %
8
165
280
121
360
10 – 30 %
13
369
601
420
421
< 10 %
1593
962
512
526
228
unchanged
1585
150
107
158
20
Table 2. Relative differences of parameters in HITRAN H2O 2006 and 2009.
Rough estimations of uncertainties associated with all the spectroscopic
parameters.
(1) For the line position and intensity, their uncertainties are less than the
accuracy tolerance.
(2) For the half-widths, shift, and T exponent, their uncertainties are larger
than the accuracy tolerance.
Conclusions: For positions and intensities of lines, two rules enable one
to identify errors. For other parameters, they enable one to identify errors
and to improve accuracies.
VI-1. Screening HITRAN H2O
2009 (Example 1 in the R branch)
Parameter
Questionable Lines
Position
None
Intensity
None
AirHalf-width
99,0 ← 88,1, 99,1 ← 88,0,
1010,0 ← 99,1, 1010,1 ← 99,0,
1111,0 ← 1010,1, 1111,1 ← 1010,0
SelfHalf-width
1313,0 ← 1212,1,1313,1 ← 1212,0
Induced
shift
99,0 ← 88,1, 99,1 ← 88,0,
1010,0 ← 99,1, 1010,1 ← 99,0,
1111,0 ← 1010,1, 1111,1 ← 1010,0,
1212,0 ← 1111,1, 1212,1 ← 1111,0,
1313,0 ← 1212,1, 1313,1 ← 1212,0,
1414,0 ← 1313,1, 1414,1 ← 1313,0
T
exponent
99,0 ← 88,1, 99,1 ← 88,0,
1010,0 ← 99,1, 1010,1 ← 99,0,
1111,0 ← 1010,1, 1111,1 ← 1010,0
Fig. 8 Six spectroscopic
parameters for a group of
paired lines J′J′,1 ← J″J″,0 and
J′J′,0 ← J″J″,1 in the R branch.
Their values in HITRAN H2O
2009 are plotted by × and ∆,
respectively. The boundary of
this group is Jbd = 3.
VI-1. Suggested values of the air-broadened half-width for lines
in the example 1 of the R branch
Lines
HITRAN
2009
Suggested
values
89,0 ← 78,1
89,1 ← 78,0
0.0272
0.0272
0.0264
99,0 ← 88,1
99,1 ← 88,0
0.0134
0.0134
0.0195
109,0 ← 98,1
109,1 ← 98,0
0.0124
0.0124
0.0161
119,0 ← 108,1
119,1 ← 108,0
0.0203
0.0203
0.0156
Fig. 9 Based on the two rules, suggested air-broadened half-widths for the group of
paired lines J′J′,1 ← J″J″,0 and J′J′,0 ← J″J″,1 in the R branch are plotted by +.
Meanwhile, the original values in HITRAN H2O 2009 are given by × and ∆,
respectively.
VI-2. Screening HITRAN H2O
2009 (Example 2 in the R branch)
Parameter
Questionable lines
Positions
None
Intensity
213,19 ← 200,20, 212,19 ← 201,20
AirHalf-width
173,15 ← 160,16, 172,15 ← 161,16,
183,16 ← 170,17, 182,16 ← 171,17,
193,17 ← 180,18, 192,17 ← 181,18
SelfHalf-width
183,16 ← 170,17, 182,16 ← 171,17,
193,17 ← 180,18, 192,17 ← 181,18,
203,18 ← 190,19, 202,18 ← 191,19
Induced
shift
143,12 ← 130,13, 142,12 ← 131,13,
153,13 ← 140,14, 152,13 ← 141,14,
163,14 ← 150,15, 162,14 ← 151,15,
183,16 ← 170,17, 182,16 ← 171,17,
193,17 ← 180,18, 192,17 ← 181,18
T
exponent
153,13 ← 140,14, 152,13 ← 141,14,
163,14 ← 150,15, 162,14 ← 151,15,
213,19 ← 200,20, 212,19 ← 201,20
Fig. 10 The same as Fig. 8
except for a group of paired
lines J′3,J′-2 ← J″0,J″ and J′2,J′-2 ←
J″1,J″ in the R branch. The
boundary of this group is Jbd =
13.
VI-2. Suggested values of the induced shift for lines in
the example 2 of the R branch
Fig. 11 Based on the two rules, suggested induced shifts
(in units of 10 -3 × cm-1 atm-1) for the group of paired lines
J′3,J′-2 ← J″0,J″ and J′2,J′-2 ← J″1,J″ in the R branch are plotted
by +. Meanwhile, the original values in HITRAN H2O 2009
are given by × and ∆, respectively.
Lines
HITRAN
2009
Suggested
values
143,12 ← 130,13
142,12 ← 131,13
3.00
1.14
2.11
153,13 ← 140,14
152,13 ← 141,14
3.12
1.74
2.34
163,14 ← 150,15
162,14 ← 151,15
2.91
1.61
2.46
173,15 ← 160,16
172,15 ← 161,16
2.66
2.64
2.50
183,16 ← 170,17
182,16 ← 171,17
3.35
6.98
2.03
193,17 ← 180,18
192,17 ← 181,18
0.48
0.47
1.22
203,18 ← 190,13
202,18 ← 191,19
0.86
0.86
1.02
213,19 ← 200,20
212,19 ← 201,20
1.55
1.55
1.41
VI-3. Screening HITRAN H2O
2009 (Example 3 in the R branch)
Parameter
Questionable lines
Positions
None
Intensity
193,16 ← 182,17, 194,16 ← 181,17,
203,17 ← 192,18, 204,17 ← 191,18
AirHalf-width
183,15 ← 172,16, 184,15 ← 171,16,
193,16 ← 182,17, 194,16 ← 181,17,
203,17 ← 192,18, 204,17 ← 191,18
SelfHalf-width
183,15 ← 172,16, 184,15 ← 171,16
Induced
shift
173,14 ← 162,15, 174,14 ← 161,15,
183,15 ← 172,16, 184,15 ← 171,16,
193,16 ← 182,17, 194,16 ← 181,17,
203,17 ← 192,18, 204,17 ← 191,18
T exponent
193,16 ← 182,17, 194,16 ← 181,17
Fig. 12 The same as Fig. 8
except for a group of paired
lines J′3,J′-3 ← J″2J″-1 and J′4,J′-3
← J″1,J″-1 in the R branch. The
boundary of this group is Jbd
= 16.
VI-4. Screening HITRAN H2O 2009 (Example 1 in the Q branch)
Fig. 13 The same as Fig. 8 except for a group of paired lines J′J′,0 ← J″J″-1,1 and J′J′,1
← J″J″-1,2 in the Q branch. The boundary of this group is Jbd = 5.
VI-5. Screening HITRAN H2O 2009 (Example 2 in the Q branch)
Fig. 14 The same as Fig. 8 except for a group of paired lines J′2,J′-2 ← J″1,J″-1 and
J′3J′-2 ← J″2,J″-1 in the Q branch. The boundary of this group is Jbd = 13.
VI-6. Screening HITRAN H2O 2009 (Example 1 in the P branch)
Fig. 15 The same as Fig. 8 except for a group of paired lines J′2,J′-2 ← J″1,J″ and J′3,J′2 ← J″0,J″ in the P branch. The boundary of this group is Jbd = 13.
VI-7. Screening HITRAN 2009
(Example 1 in the Q branch of the v2 band)
Parameter
Questionable lines
Positions
None
Intensity
None
AirHalf-width
150,15 ← 151,14, 151,15 ← 152,14,
160,16 ← 161,15, 161,16 ← 162,15,
170,17 ← 171,16, 171,17 ← 172,16,
180,18 ← 181,17, 181,18 ← 182,17,
190,19 ← 191,18, 191,19 ← 192,18
SelfHalf-width
140,14 ← 141,13, 141,14 ← 142,13,
150,15 ← 151,14, 151,15 ← 152,14,
160,16 ← 161,15, 161,16 ← 162,15,
190,19 ← 191,18, 191,19 ← 192,18
Induced
shift
150,15 ← 151,14, 151,15 ← 152,14,
160,16 ← 161,15, 161,16 ← 162,15,
170,17 ← 171,16, 171,17 ← 172,16,
180,18 ← 181,17, 181,18 ← 182,17
T exponent
190,19 ← 191,18, 191,19 ← 192,18
Fig. 16 The same as Fig. 8
except for a group of paired
lines J′0,J′ ← J″1,J″-1 and J′1,J′ ←
J″2,J″ -1 in the Q branch of the
v2 band. The boundary Jbd ≈
10.
VI-7. Suggested values of the air-broadened half-width
for lines in the example 1 of the Q branch of the ν2 band
Lines
HITRAN
2009
Suggested
values
150,15 ← 151,14
151,15 ← 152,14
0.0140
0.0170
0.0150
160,16 ← 161,15
161,16 ← 16215
0.0144
0.0090
0.0130
170,17 ← 171,16
171,17 ← 172,16
0.0092
0.0125
0.0115
180,18 ← 181,17
181,18 ← 182,17
0.0108
0.0108
0.0099
190,19 ← 191,18
191,19 ← 192,18
0.0064
0.0064
0.0073
Fig. 17 Based on the two rules, suggested air-broadened half-widths (in cm-1 atm-1)
for the group of paired lines J′0,J′ ← J″1,J″-1 and J′1,J′ ← J″2,J″ -1 in the Q branch of the
v2 band are plotted by +. Meanwhile, the original values in HITRAN H2O 2009 are
given by × and ∆, respectively.
VI-8. Screening HITRAN 2009
(Example 1 in the P branch of the v2 band)
Parameter
Questionable lines
Positions
None
Intensity
None
AirHalf-width
121,12 ← 130,13, 120,12 ← 131,13,
131,13 ← 140,14, 130,13 ← 141,14,
141,14 ← 150,15, 140,14 ← 151,15,
161,16 ← 170,17, 160,16 ← 171,17,
171,17 ← 180,18, 170,17 ← 181,18,
191,19 ← 200,20, 190,19 ← 201,20
SelfHalf-width
71,7 ← 80,8, 70,7 ← 81,8,
81,8 ← 90,9, 80,8 ← 91,9,
91,9 ← 100,10, 90,9 ← 101,10,
131,13 ← 140,14, 130,13 ← 141,14,
161,16 ← 170,17, 160,16 ← 171,17,
181,18 ← 190,19, 180,18 ← 191,19,
Induced
shift
131,13 ← 140,14, 130,13 ← 141,14,
141,14 ← 150,15, 140,14 ← 151,15,
181,18 ← 190,19, 180,18 ← 191,19,
191,19 ← 200,20, 190,19 ← 201,20
T exponent
181,18 ← 190,19, 180,18 ← 191,19,
191,19 ← 200,20, 190,19 ← 201,20
Fig. 18 The same as Fig. 8 except
for a group of paired lines J′1,J′ ←
J″0,J″ and J′0,J′ ← J″1,J″ in the P
branch of the v2 band. The
boundary Jbd ≈ 7.
VI-9. Screening HITRAN 2009 (Example 1 in the R branch of the v2 band)
Fig. 19 The same as Fig. 8 except for a group of paired lines J′1,J′ ← J″0,J″ and J′0,J′
← J″1,J″ in the R branch of the v2 band. The boundary Jbd ≈ 7.
VI-10. Screening HITRAN 2009 (Example 2 in the R branch of the v2 band)
Fig. 20 The same as Fig. 8 except for a group of paired lines J′J′,0 ← J″J″,1 and J′J′,1
← J″J″,0 in the R branch of the v2 band. The boundary Jbd ≈ 3.
VII. Predicting spectroscopic parameters for HITEMP
Fig. 21 A plot to show N2-broadened half-widths for two groups of paired lines with
J˝ = 26 ··· 50 predicted from an extrapolation method. One is a group of J′0,J’ ←
J″1,J" and J′1J’ ← J″0J“ in the R branch and another is a group of J′2,J’-2 ← J″1,J"-1 and
J′3J’-2 ← J″2J"-2 in the Q branch. For the former, theoretically calculated values with
J″ = 6, ∙∙∙, 25 and those predicted ones are given by × and ∆. For the latter,
calculated values with J″ = 12, ∙∙∙, 25 and predicted ones are given by + and □.
VIII. Conclusions

Two basic rules (i.e., the pair identity and the smooth variation rules)
are applicable for all the spectroscopic parameters of H2O lines whose
J values are above certain boundaries in the same groups.

Different groups of lines have different boundaries. The latter can be
calculated from the identity requirements for the energy levels and
wave functions of paired H2O states.

The rule hold within certain accuracy tolerance. The latter could vary
as the parameter of interest varies. In general, the 5 % accuracy
tolerance is suggested for the half-widths, shifts, and T exponents.

The 5 % accuracy tolerance is poorer than the accuracies associated
with line positions and intensities, but is better than those with other
parameters. In addition, the current theoretical calculations and
measurements for high J lines cannot reach such high accuracies.

The present work can be extended for lines in vibrationaly excited
states. Of course, one needs to check the properties of the energy
levels and wave functions of vibrationaly excited states first and
recalculate corresponding boundaries.

The idea of the present work is simple and general. One can applied it
for other molecules whose energy levels and wave functions have
similarities.