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1. A computer assisted procedure of assignments of vibration–rotation
bands of asymmetric and symmetric top molecules
Š. Urbana,b,*, J. Behrenda,c
, P. Pracnaa
a
J. Heyrovský Institute of Physical Chemistry, Dolejškova 3, Academy of the Sciences of the Czech Republic, CZ-182 23 Prague 8, Czech Republic
b
Department of Analytical Chemistry, Institute of Chemical Technology, CZ-166 28 Prague 6, Czech Republic
c
I. Physikalisches Institut, Universität Köln, D-50937 Köln, Germany
Received 30 July 2003; revised 5 November 2003; accepted 17 November 2003
Abstract
An advanced graphically oriented interactive program package for assignments of complex (perturbed) vibration–rotation spectra of
asymmetric and symmetric top molecules has been developed. In addition to the well known Loomis–Wood algorithm, the new procedure
takes advantage of a precise knowledge of the lower (e.g. ground) vibrational state energies, works with a realistic approximation of effective
Hamiltonians for lower as well as upper vibrational states, and allows an instant combination difference inspection of spectral lines by the
graphical representation of the appropriate parts of the analyzed experimental spectrum. Being constrained to the combination difference
checking, the new algorithm can directly assign the correct rotational quantum numbers as well as ‘quality weights’ estimating relative
accuracies of the identified lines.
q 2003 Elsevier B.V. All rights reserved.
Keywords: Assignments of vibration–rotation spectra; Combination differences; Loomis–Wood algorithm
1. Introduction
The assignments of high-resolution spectra to
vibration–rotation transitions are often a difficult task
and usually the most time consuming part of spectroscopic
analyses. From numerous methods simplifying this tedious
work, the best known is a Loomis–Wood (LW) method
where successive parts of the spectrum are ordered one
under another in order to create visually recognizable
patterns, which would be otherwise difficult to find because
of a large frequency separation and/or overlaps with other
spectral features [1]. At the present time, the LW method
can take an advantage of personal computers where
corresponding interactive assignment programs are
implemented and frequently used [2,3]. This standard
LW algorithm, however, does not provide explicit assign-
ments of lines to rotational quantum numbers and often
breaks down in cases of weak lines and strongly perturbed
vibration–rotation levels.
In the present paper we describe a substantially
extended algorithm, which combines several modified
LW diagrams with the method of combination differences.
Three selected LW diagrams, which are displayed
simultaneously, correspond to three different spectral
branches and are mutually constrained by lower state
combination differences (LSCD). For a symmetric top
molecule, the branches displayed in the individual LW
windows are always P; Q; and R; due to a strict selection
rule for the K quantum number. The triad of transitions
PðJ þ 1; KÞ; QðJ; KÞ; and RðJ 2 1; KÞ related by one
common upper level appears on the same horizontal line
of the triple LW diagram. The rotational quantum number
J is the only one that changes between subsequent lines in
the vertical direction in the LW diagrams. For asymmetric
top molecules, where there are more than three transitions
pertaining to one common upper level, three of them are
selected to be displayed at the same time. The advantage of
displaying LW diagrams of three branches simultaneously
consists in that the branches belonging to one set of upper
state quantum numbers have to be of the same shape in all
0022-2860/$ - see front matter q 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.molstruc.2003.11.019
Journal of Molecular Structure 690 (2004) 105–114
www.elsevier.com/locate/molstruc
* Corresponding author. Address: J. Jeyrovský Institute of Physical
Chemistry, Dolejškova 3, Academy of Sciences of Czech Republic, CZ-182
23 Prague 8, Czech Republic. Tel.: þ420-2-6605-3635; fax: þ420-2-
8582307.
E-mail address: urban@jh-inst.cas.cz (Š. Urban).

2. three diagrams. This facilitates and accelerates greatly the
search for the complete branches of transitions and
together with the LSCD checking provides an immediate
assignment of all quantum numbers to the whole triad of
branches.
2. Spectroscopic background of the algorithm
The standard LW algorithm enables a visual recognition
of J-sequences of transitions with constrained rotational
quantum numbers Ka and Kc or K and l (for asymmetric or
symmetric top molecules, respectively) with the quantum
number J changing throughout the J-sequences. The
separation of adjacent transitions of such sequences on the
frequency scale (with the value of J differing by one) can be
expressed in the first approximation as
D ~
nðJ þ 1; JÞ ¼ ~
nðJ þ 1Þ 2 ~
nðJÞ
¼ 2B0
m þ ðB0
2 B00
ÞJðJ þ 1Þ þ …; ð1Þ
where m equals 2J; 0; J þ 1 for P; Q; and R branches,
respectively, and B0
and B00
are the upper and lower state
rotational constants, respectively. Thus if the spectrum is
split into appropriate segments and these are ranged one
under another so that wavenumbers differing by D ~
nðJ þ
1; JÞ are aligned vertically, the J-sequences of transitions
appear as smooth patterns, when the corresponding upper
and lower state energies are estimated close enough to their
actual values. Once this alignment is adjusted in a way
corresponding to the lower and upper state rotational
structure, the J-sequences in all three LW diagrams become
vertically aligned (see Fig. 1).
Fig. 1. The upper window shows from left to right the triad of Loomis–Wood diagrams for the P; Q; and R branches, respectively. The lower window is an
expansion of one of the three windows from the above. The arrows in the windows point to the peaks whose features are displayed in the status line. The status
line contains from left to right: assignment status, transmittance/absorbance, peak wavenumber, difference between this wavenumber and the calculated peak
position, lower state J; and K quantum numbers, respectively. For further description see text.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114
106

3. The case with constrained quantum numbers other than J
(e.g. K and l or Ka and Kc) can be regarded as a special case
of a ‘linear’ molecule, for which the transition frequencies
can be written as a function of the J quantum number as the
only variable. This approach was used in Refs. [1–3], but it
does not provide direct information about the correct
assignment of angular momentum quantum numbers.
In our algorithm, this is achieved by constraining the LW
diagrams of the individual P; Q; and R branches by LSCD.
The rotational structure of the lower state is often well
known with a very good accuracy, which is usually higher
than the accuracy of experimental spectra to be analyzed.
These lower state rotational energies are used to calculate
the combination differences for vibration–rotation
transitions to a common upper vibration–rotation level. In
Fig. 2, the combination differences, i.e. the three possible
differences between the wavenumbers of the P; Q; and R
transitions going to a common upper level, are shown as the
thick lines connecting the levels J00
¼ 1; 2; and 3 of K00
¼ 1;
chosen here as an example for a symmetric top molecule.
It should be noted that in our program the LSCD can be
calculated either from the constants of a polynomial
expansion or from a table of rotational energies. It makes
our assignment algorithm far more flexible for use in
asymmetric top as well as symmetric top molecules, even in
cases of strongly perturbed levels. In addition to this, the
arrangement of the program is not restricted in fact to LSCD
checking, but can equally well work also with upper state
Fig. 2. An example of constraining the triads of the P; Q; R transitions with common upper levels by the lower state combination differences for K00
¼ 1 and the
sequence J0
¼ 2; 3; 4; …: For further description see text.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114 107

4. combination differences. In further discussions, however,
we will refer only to the concept of LSCD.
To take the advantage of the LSCD, the three LW
diagrams are simultaneously created on the screen (see
Fig. 1). Each of these LW diagrams represents one of the
transition branches (e.g. P; Q; and R in the symmetric top
case) and all three diagrams (triad) are mutually constrained
by the LSCD. It means that each row of this triple Loomis
diagram represent three segments of the analyzed exper-
imental spectrum corresponding to the triad of transitions
going to a common upper level. These three segments are
mutually shifted by the appropriate combination differ-
ences. In Fig. 1, the triple LW diagram of the symmetric top
rovibrational spectrum is shown for the beginning of the
sequence J0
¼ 2; 3; 4; … of transitions with K00
¼ 1: The
J-sequences in the individual diagrams are formed from
segments of the analyzed spectrum shifted according to
Eq. (1) and the correspondence to the P; Q; and R branches,
respectively. The peak corresponding to the PðJ00
¼ 3;
K00
¼ 1Þ transition marked with an arrow in Fig. 1
corresponds to the P-branch transition in the triad of
transitions, which have their LSCD indicated in Fig. 2.
Each row of the triple LW diagram can interactively be
displayed in the form of classical spectrum plots where the
three segments of the diagram are plotted to the three
corresponding horizontal spectrum parts (see Fig. 3), so that
the peak positions are aligned vertically according to the
LSCD and marked with a vertical cursor line. The three
spectrum segments in these ‘plot windows’ can be
simultaneously scrolled to higher or lower wavenumbers,
with the LSCD alignment conserved. By moving the aligned
cursor lines in all three windows simultaneously,
a corresponding coincidence of three PðJ þ 1; KÞ; QðJ; KÞ;
and RðJ 2 1; KÞ branch transitions (in the symmetric top
case) can be usually easily identified.
The three LW diagrams, which are simultaneously
displayed on the screen, are generated using the following
expression that provides a residual value d ~
nðJ00
; J0
Þ; which
determinesaspacebetweentheASCIIcharactersymbolofthe
given experimental transition and the center of the diagram:
d ~
nðJ0
; J00
Þ ¼ ½EupperðJ0
; Kp
þ DKp
Þ 2 ElowerðJ00
; Kp
Þ=hc
þ ~
ncorrðJ0
Þ 2 ~
nðJ00
; J0
; Kp
; DKp
Þ ð2Þ
where EupperðJ0
; Kp
þ DKp
Þ and ElowerðJ00
; Kp
Þ are the upper
and lower state energies, respectively, both generated either
from Hamiltonian parameters or from tables of energies,
~
nðJ00
; J0
; Kp
; DKp
Þ is the experimental wavenumber of the
transition whose symbol is considered, J00
; J0
; Kp
; and DKp
are quantum numbers that determine the transition to be
assigned in the given row and column. Here Kp
denotes the
set of constrained quantum numbers which are common to
the three (or more) LW diagrams as described above (e.g. K
and l for a symmetric top molecule, or Ka and Kc for an
asymmetric top molecule) and DKp
denotes the appropriate
selection rule for these quantum numbers. The residual value
Fig. 3. Three sections of the Fourier transform spectrum constrained by combination differences corresponding to one row of the Loomis–Wood diagram of
Fig. 1.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114
108

5. d ~
nðJ00
; J0
Þ is minimized by fitting the empirical coefficients
a0; a1; a2; …; b1; b2; … of the correction function
~
ncorrðJÞ ¼ a0 þ a1JðJ þ 1Þ þ a2½JðJ þ 1Þ2
þ · · ·
þ b1½JðJ þ 1Þ1=2
þ b2½JðJ þ 1Þ3=2
þ · · · ð3Þ
This polynomial expression serves as a correction
function to the estimation of the upper state energy levels.
It can be noted that in the case when all upper state energies
are equal to zero, the first part of the polynomial expression
(Eq. (3)) with the an parameters provides the same
approximation as that used in the classical LW algorithm
[1,2]. The additional bn terms in the expression (Eq. (3)) are
particularly useful for strongly perturbed rovibrational levels
when the perturbation approach does not converge well. It is
obvious that the correction function ~
ncorr is identical for all
three diagrams, which pertain the common sequence of
upper levels. When the residual d ~
nðJ00
; J0
Þ is very small, i.e.
when the assigned line is found very close to the predicted
wavenumber, its corresponding character line marker
appears on the highlighted vertical line in the center of the
LW diagram. Successive fitting of the correction function
with the appropriate number of its parameters varied should
align the complete J-sequences along the central vertical
lines in all three diagram windows as shown in Fig. 1. The
relation of parameters of the correction function to
parameters of the effective rotational Hamiltonians is briefly
discussed in Appendix A.
The assignments and fitting of ~
ncorr are done and stored
separately for distinct values of Kp
and DKp
: Thus a
complete alignment of all three sequences should be always
achieved because the effective parameters of the correction
function can absorb the effects of nonresonant perturbations.
Therefore, sporadic displacements of assigned lines from
the predicted positions can be usually explained by
accidental line overlaps and these lines should be given
reduced statistical weights in the least-square fitting. Large
systematic and discontinuous residuals can reveal local
resonances as shown further.
3. Input data and assignment procedures
The assignment procedure requires appropriately pre-
pared input data. These data consist of the following items
(out of which the first two are compulsory):
† Peak list of line positions and intensities.
† Lower state energies (represented by a table of energies
or corresponding parameters).
† Digitized spectrum.
† Estimated upper state energies (table or corresponding
parameters).
The last two items are optional. If the digitized spectrum
is not available, a special program creates a simulated
spectrum from data given in the peak list. In this case,
however, important information about the line quality
(overlapping lines, line shoulders, shape distortions, line
asymmetry, etc.) is not available and, therefore, the
experimental digitized spectrum is strongly recommended.
The estimation of upper state energies simplifies starting
assignments and it defines appropriate spectral regions for a
search of the lines to be assigned. The qualified guess of
these energies is out of the scope of this paper but it can be
mentioned, for example, that a reasonable estimation of the
upper state energies can be obtained by adding
the corresponding vibrational difference ðE0
0 2 E
00
0Þ to the
lower state rotational energies, in the case when the
rotational structures are the same (as in asymmetric top
molecules, parallel bands of symmetric top molecules, etc.).
It is obvious that the upper state energy estimation can be
step by step improved during the assignment procedure by
fitting ~
ncorr:
Detailed descriptions of the input data, their formats,
configuration files and working environments are given in a
guide of this program package [4], which is available upon
request from the authors.
Depending on the information in the configuration file,
the program works either in a symmetric or asymmetric top
mode. Distinctions between both modes are small, only in
the definition of transition selection rules, parameterization
of rotational levels and the number of considered transitions
to the common upper levels. In the case of the symmetric
top molecules, there are transitions of three P; Q; and R
branches to the same upper level but in asymmetric top
molecules, there are more subbranches of the transitions to
the same upper level. Therefore, in the asymmetric top
mode, the program offers an additional menu to select only
three subbranches to be assigned simultaneously. The
assignments of transitions of remaining subbranches are
then trivial, since the correction function is known, and can
be done in a following step calling this menu again.
The program works basically in two modes, which we
call diagram and plot modes and which can be toggled by
pressing a hot key. The assignment procedure always starts
with a definition of the subbranches, according to the type of
the molecule as described above. The program then creates
the triple LW diagram for three selected subbranches of
transitions in which the horizontal alignment among the
three LW windows is due to the LSCD constrain, while the
vertical alignment of the J-sequences inside each of the LW
window is defined by Eq. (2). At the beginning of the
assignment procedure the LW diagrams are based only on
the upper state energies and the corresponding combination
differences (lower state energies), because the correction
expression (Eq. 3) has all parameters equal to zero. There
are two basic possibilities how the first image of the triple
LW diagram can look like. If a good estimate of the upper
state levels has been provided, typical LW diagram contours
are immediately obvious (see Fig. 4). In the opposite case,
no such patterns are visible.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114 109

6. In the first case, the diagram window can be used directly
for marking the assignments. First of all it should be
mentioned, that the LSCD constrain of the three LW
diagrams results in a similar shapes of the J-sequences of
transitions with the common upper levels in all three
diagram windows. This can be further interactively checked
in the following way. When the correction function is not
yet refined, the J-sequences need not be vertically aligned at
the center of the diagram windows. However, in every row
of the triple diagram, the displacements from the center of
Fig. 4. Example of several strongly perturbed sequences with the K ¼ 5 transitions aligned in the upper window and the K ¼ 6 transitions in the middle
window. The part of the ‘g’ sequence ðK ¼ 7Þ in the ellipse in the middle window shows the effect of a weak local resonance which breaks the smooth
continuous Loomis–Wood pattern. The effect of such resonance is shown with an expanded wavenumber scale in the lower window. For more details see text.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114
110

7. the window should be more or less equal (i.e. being less than
the estimated accuracy of the LSCD), provided that the
assignment of the set of Kp
quantum numbers of these
transitions is correct.
The continuous patterns do not have to appear necess-
arily in all three LW windows. For example, in a parallel
band of a symmetric top molecule and low K values, the
intensities in the Q branch are weak at low J values, so that
the assignments are usually started with LSCD checking
only in the P and R branches. In a perpendicular band, the
assignment procedure is usually begun with the pairs of
stronger branches, i.e. r
Q and r
R or p
Q and p
P:
When the first triad of lines is assigned (for example the
P(3,1), Q(2,1), and R(1,1) transitions of the symmetric top,
shown in Figs. 1 and 2), the cursor should be moved to
another row (for example to the next one) in the triple
diagram and this self-checking procedure can be repeated to
assign the next triad of transitions [P(4,1), Q(3,1), and
R(2,1)]. When a sufficient number of transitions are
assigned, the leading parameters of the correction
expression ~
ncorr can be fitted to get a more accurate
description of the upper state levels and thus improve the
predictions of the continuation of the J-sequences of
analyzed transitions. It should be noted that the number of
the fitted parameters must not be higher than the number
of the assigned rows (a minimal number of parameters is
recommended). The assignment procedure can be repeated
step by step to higher quantum number J until the transition
intensities are too weak.
When one group of subbranches of transitions to the
common upper levels is completely identified, another one
(e.g. with successive quantum numbers Kp
) can be selected
using the interactive menu and the procedure continues
analogously as described above. An example can be seen in
Fig. 1 where the K00
¼ 1 transitions, whose assignments are
marked with the characters ’a’, are aligned along the central
line of all three LW diagrams after the least-squares fit of the
coefficients of the correction function ~
ncorr has been
performed. To the right of the ‘a’-sequences, several other
continuous sequences are already are visible. The next ones
to the right, marked with characters ‘b’, correspond to
sequences of the K00
¼ 2 transitions, which are here already
assigned. These have been assigned after the next group of
subbranches was defined (choosing the appropriate set of
quantum numbers Kp
), these transitions checked by LSCD
corresponding to the new set of quantum numbers Kp
; and
finally aligned by another fitting of the correction function
~
ncorr:
The absence of recognizable LW sequences is most often
due to strong perturbations of the structure of rotational
levels resulting in a large variation of the B0
rotational
constant (and higher order centrifugal distortion constants)
so that the initial alignment of the LW sequences is far from
vertical. The strong rovibrational perturbations spawn also
often significant intensity anomalies. It may happen that the
intensity of one or more of the branches is depleted so much
that the advantage of LSCD checking of assignments is
completely lost.
An example of LW sequences, which become less and
less recognizable due to a perturbation of the upper
vibrational state and consequent large variation of the B0
constant is shown in Fig. 4, which corresponds to the same
spectrum as used in Figs. 1 and 2, taken from the analysis of
the n2 band of 13
CH3
37
Cl [5] with a strong Coriolis resonance
having a crossing between K0
¼ 8 and 9. Here in the upper
diagram, the sequence marked with ‘e’ corresponds to
assigned and vertically aligned P branch transitions K ¼
K00
¼ K0
¼ 5 of this band. The sequences to the left from the
aligned ‘e’ sequence correspond to lower K values down to
K ¼ 1; which is marked with ‘a’ like in Fig. 1. The next
sequence to the right, K ¼ 6 marked with ‘f’, is already on
the verge of visual recognition in the forest of other lines
and should be searched for by the procedure exploiting the
plot window as described further. The absence of a
recognizable pattern is even more pronounced for the next
sequence ‘g’ ðK ¼ 7Þ; even after a vertical alignment of the
preceding ‘f’ series, shown in the middle diagram of Fig. 4.
One subtler feature of a local resonance in the ‘g’ ðK ¼ 7Þ;
sequence is shown in the lower window of Fig. 4. With the
wavenumber scale expanded with respect to the two
windows above, the effect of a resonant level crossing
becomes evident. It should be noted that the shown vertical
alignment of the low- and high-J parts of this sequence is
possible only when several transitions around the crossing
are given zero weights in fitting of ~
ncorr: The correctness of
assignments of the transitions, which cannot be brought to
the vertical alignment by fitting the correction function, can
still be undoubtedly checked by LSCD. It is also apparent
that the continuation of the Q-branch (in the middle part of
the lower window) beyond the resonant crossing vanishes
due to intensities being depleted by the perturbation.
In such cases of strong and/or local perturbations, it is
recommended to start the search for the sequences using the
plot window. The cursor is first moved in the diagram mode
to the row corresponding to the J rotational number with the
expected highest intensities of lines and then the program is
switched to the plot mode where a coincidence of three lines
to the common upper level can be searched by moving the
cursors in all the three spectrum segments simultaneously
(see Fig. 3). If the coincidence is not found on the current
screen, the spectrum parts can be scrolled to higher or lower
wavenumbers for the next ‘trial and error‘ search. When the
first three appropriate lines are found (in this graphical
search intensities and shapes of lines can be considered), the
program can be switched back to the diagram mode where
the corresponding line symbols can be assigned. Then the
search can be repeated for the subsequent triads of P; Q; and
R branch transitions, switching back and forth between the
two modes of the program. It is advisable at some point to
use the assigned transition wavenumbers to refit the
parameters of the upper level (or the corresponding energy
table) and replace the previously used ones. This makes
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114 111

8. the correction function ~
ncorr smaller and better convergent
and thus improves predictions for further assignments. This
procedure is facilitated by storing the list of assigned
transitions in an ASCII file in a format, which can be
directly used by a fitting program to refine the parameters of
the upper state.
This approach proved to be extremely useful in studies of
light nonrigid molecules like ammonia or disulfan [6–8]
where in the perturbed levels the regular LW patterns are
not recognizable from the very beginning since the initial
estimation of the rotational structure of the upper vibrational
level is not accurate enough. In that case it was strongly
recommended to start the search for sequences of tran-
sitions, which are assumed to have the highest intensity (like
K0
2 l0
¼ 3 in the case of C3v molecules).
In the case of asymmetric top molecules, because of
selection rules, there are more transitions to a common
upper level, which form more then three subbranches. Using
this interactive program, it is recommended to assign the
most intense subbranches first. Afterwards, the assignments
of the remaining subbranches are trivial because of the
precise knowledge of the correction function (3).
4. Options and advanced features of the program
In the diagram mode, always one of the three LW
windows is a working one with an active cursor in it.
Switching between the working windows is done by
pressing a hot key. The diagram mode represents the
spectral lines from the peak list by integer digits (from 0 to
9, which reflect peak intensities of unassigned lines) in
positions corresponding to their wavenumbers. The position
of each character (line marker) in the diagram row with
respect to its center represents the difference between the
wavenumber of this peak and the predicted wavenumber,
whose position is by definition at the row center. This
prediction is calculated from the actual J value (correspond-
ing to the particular row of the diagram), the actual DJ
(corresponding to the selected working window), and the
currently selected set of remaining quantum numbers Kp
:
When the currently selected peak (under cursor) is assigned
by pressing a hot key to a transition with the set of quantum
numbers described above, the numerical line marker
changes to an ASCII character, which corresponds to the
currently selected set of quantum numbers Kp
: Together
with that its color changes to that chosen in the definition of
the character to be used for marking the series belonging to
the set of quantum numbers Kp
: By moving the cursor
around the line markers in the working diagram window,
one gets information about the displayed peaks in the status
line at the bottom of the screen. From left to right, it contains
the following items (see Fig. 1)
† The assignment status of the currently selected peak.
If the peak is assigned to a particular sequence,
the corresponding identification character is displayed
(‘a’ in the example in Fig. 1). If the peak represents an
overlap of two or more transitions, all the identification
symbols appear in this place and the identification
character in the diagram window reads ‘@’. In addition
to this, in the leftmost position of the status line, one of
the assignment options expressing the statistical weight
of the assigned peak (character ‘ þ ’ in Fig. 1).
† Relative transmittance/absorbance intensity of the cur-
rently selected peak (for a detailed description see the
next paragraph).
† Wavenumber of the currently selected peak (in cm21
).
† The shift of the currently selected peak from the center of
the working LW diagram (in cm21
). As described above,
for a triad of transitions to the same upper level, these
shifts should be more or less identical once the assign-
ment of J and Kp
is correct.
† The values of lower state J and Kp
quantum numbers.
When the cursor is moved from one working LW
diagram to another, the status line information changes
accordingly. The diagram mode has the following options
and selections:
† Definition of a statistical weight of the line, which used
in the least-square fitting of the correction function ~
ncorr:
There are three assignment options available on pressing
different assignment hot keys that correspond to a
full/intermediate/zero weight and are estimated from
the quality of the line being assigned (according to the
exactness of the LSCD check, the line shape, intensity,
etc.).
† Canceling the assignment of lines if necessary.
† Change of the scale factor determining the wavenumber
range of diagram segments (the same scale factor
determines the wavenumber range also in the plot mode).
† Selection of the intensity range to be displayed. By
default, it is set to the full intensity range between 0 and
100% transmittance/absorbance. In that case, intensities
from 0 to 10% are represented by the digit ‘0’ in the
diagram windows, intensities from 10 to 20% are marked
with the digit ‘1’, etc. As an option, the displayed
intensities can be given other lower and upper limits. For
example, only peaks with the intensities from 50 to 100%
can be requested. Then lines with peak intensities
50–55% are represented by the digit ‘0’, 55–60% by
‘1’, etc.).
† Zooming the working LW diagram to a full screen
diagram and restoring it back to the triple LW diagram.
† Selection of the letter character used for the assigned
lines as well as of its color for any subbranch of
transitions.
† Scrolling the diagram in the horizontal as well as vertical
directions.
† Writing a backup copy of assigned data to disk anytime
by pressing a hot key.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114
112

9. † Moving the spectrum to a given wavenumber (all three
diagrams are moved simultaneously, preserving the
combination differences constrain).
† Changing, constraining, releasing for fitting, or clearing
any of the parameters of the correction expression
(Eq. (3)).
† Free selection of subbranches to be displayed in the triple
diagram (in the case of asymmetric top molecules).
When the program is switched to the plot mode, the
displayed portions of the spectrum have the same color
as the letter characters used in the corresponding
diagrams. There is one common vertical cursor line
which is moved in all three spectrum segments
simultaneously in this mode (preserving the combination
differences, see Fig. 3), and an additional symbol
marking the spectrum segment which corresponds to
the working diagram window and controls movements of
the cursor line to a peak position. If the three parts of the
spectrum are scrolled to higher or lower wavenumbers, a
return back to the diagram mode can be performed either
to the scrolled (changed) wavenumber range or to the
original one (unchanged).
The options in the plot mode are the following:
† Switching the cursor control from one segment of the
spectrum (branch) to another.
† Moving the cursor to a next left (right) peak maximum or
minimum (in the current portion of the spectrum); the
cursor moves in all the segments of the spectrum
simultaneously.
† Scrolling the spectrum to higher or lower wavenumbers.
† Changing the intensity scale in the current segment of the
spectrum.
† Zooming the intensity in the current segment.
† Moving to a given wavenumber in the current segment of
the spectrum (the remaining two parts of spectra are also
scrolled with LSCD constrain preserved).
5. Conclusions
The program has been found very useful especially in
cases when rotation–vibration bands are strongly perturbed
[5–7] as well as for extremely weak bands [7,8]. The use
of this program for ‘well behaved’ bands significantly
accelerates the assignments [9,10] and makes possible a
direct preparation (assigning quantum numbers and
statistical weights) of the experimental data for succeeding
analyses.
The source code of the program has been programmed by
J. Behrend in the ‘C’ (Borland, Turbo C) programming
language. The programs have been debugged and tested
under the MS DOS and MS Windows (95/98/NT/2000/XP)
operation systems. The minimal hardware requirements are
CPU INTEL 386 with a mathematical coprocessor, 2 MB
RAM and about 20–50 MB disk memory (depends on
spectra analyzed), VGA with 512 kB and a corresponding
color monitor.
Acknowledgements
We acknowledge the support through the Grant Agency
of the Czech Republic (#203/01/1274) and the German
Federal Ministry of Education and Science at the DLR
together with the Czech Ministry of Education, Youth, and
Sport via common grant CZE 00/030 (ME445). All support
is greatly appreciated.
Appendix A
The coefficients of the correction function (3) can be
related to the parameters of the effective rotational
Hamiltonians of the lower and upper rovibrational states.
We give here an example for a parallel and perpendicular
fundamental bands of an oblate symmetric top molecule,
where the energy terms, truncated to fourth-order, can be
expressed in a closed form as
EvrðJ; K;lÞ ¼ E0 þ BJðJ þ 1Þ þ ðC 2 BÞK2
2 DJJ2
ðJ þ 1Þ2
2 DJKJðJ þ 1ÞK2
2 DKK4
2 2CzKl
þ hJJðJ þ 1ÞKl þ hKK3
l þ ·· ·; ðA1Þ
obviously with l ¼ 0 for nondegenerate vibrational levels.
Let us assume, that we take the rotational and centrifugal
distortion constants identical in the estimation of the lower
ElowerðJ00
;K00
;…Þ and upper EupperðJ0
;K0
;…Þ levels in Eq. (2),
with an estimated upper state vibrational energy E0
0 and
E00
0 ¼ 0: In that case, for example, the a1 coefficient of the
correction function corresponds to DB ¼ B0
2 B00
for K ¼ 0
and for K – 0 absorbs the K-dependent terms of the
effective rotational Hamiltonian (A1). The explicit form of
the ai coefficients, truncated like (A1) to fourth-order, is
given as
a0 ¼ DE þ DðC 2 BÞðK00
þ DKÞ2
2 2CzðK00
þ DKÞ
l 2 DDKðK00
þ DKÞ4
þ hKðK00
þ DKÞ3
l þ ·· ·; ðA2Þ
a1 ¼ DB 2 DDJKðK00
þ DKÞ2
þ hJðK00
þ DKÞl þ ···; ðA3Þ
a2 ¼ 2DDJ þ ·· ·; ðA4Þ
etc:;
where the definition of the parameter difference between the
upper and lover vibrational states is analogous to DB: The
selection rule DK ¼ Dl ¼ ^1 for a perpendicular funda-
mental band means that for transitions with DK ¼ ^1 the
upper state value l ¼ ^1 has to be taken throughout Eqs.
(A2)–(A4). For a parallel fundamental band DK ¼ 0 and
the terms containing l vanish.
Š. Urban et al. / Journal of Molecular Structure 690 (2004) 105–114 113

10. It should be noted that a polynomial expansion of the
effective rotational Hamiltonian (A1) is valid only for
vibration–rotation interactions, which can be treated by
perturbation theory. In case of slow convergence of the
perturbation treatment, the fit of the correction function and
the vertical alignment can be considerably improved by
using empirical parameters bi standing with the non-integer
powers of JðJ þ 1Þ; which algebraically approximate the
role of variational diagonalization of the Hamiltonian
matrix. It is, however, obvious, that this is of no help in
case of a local resonance with a level crossing which results
in a discontinuity of the LW pattern as shown in Fig. 4.
The correction function is unique for each value of K00
and is stored internally once a fit of this function is done.
This means that at the beginning of assignment of a new
series, the ai coefficients should be given realistic values so
that the corresponding LW diagrams are displayed with
visually recognizable patterns and the assignment procedure
can be started. In case of not much perturbed rovibrational
energies one can usually use the coefficients of the nearest K
series, otherwise corrections based on estimations of
constants in Eqs. (A2)–(A4) should be done.
References
[1] F.W. Loomis, R.W. Wood, Phys. Rev. 32 (1928) 223.
[2] B.P. Winnewisser, J. Reinstädtler, K.M.T. Yamada, J. Behrend,
J. Mol. Spectrosc. 136 (1989) 12.
[3] R. Brotherus, J. Comput. Chem. 20 (1999) 610.
[4] J. Behrend, Program guide.
[5] P. Pracna, F.L. Constantin, H. Bürger, L. Féjard, in preparation.
[6] M. Stamova, S. Anders, J. Jonuscheit, H.W. Schrötter, P. Pracna, Š.
Urban, S. Klee, M. Winnewisser, K. Sarka, J. Mol. Struct. 482/483
(1999) 481.
[7] Š. Urban, J. Behrend, K.M.T. Yamada, G. Winnewisser, J. Mol.
Spectrosc., in preparation.
[8] Š. Urban, J. Behrend, Qing-Li Kou, G. Guelachvili, unpublished
results.
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H. Bürger, J. Mol. Struct. 517/518 (2000) 119.
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114