Journal of Molecular Spectroscopy 228 (2004) 528–543
www.elsevier.com/locate/jms
Fourier transform spectroscopy of CH3OH:
rotation–torsion–vibration structure for the CH3-rocking
and OH-bending modes
R.M. Lees,a,* Li-Hong Xu,a J.W.C. Johns,b Z.-F. Lu,b B.P. Winnewisser,c,d
M. Lock,d and R.L. Samse
b
a
Department of Physical Sciences,University of New Brunswick, Saint John, NB, Canada E2L 4L5
Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ont., Canada K1A 0R6
c
Department of Physics, The Ohio State University, 174 W. 18th Avenue, Columbus, OH 43210, USA
d
Physikalisch-Chemisches Institut, Justus-Liebig-Universit€at, Heinrich-Buff-Ring 58, D-35392 Giessen, Germany
e
Pacific Northwest National Laboratory, P.O. Box 999, Mail Stop K8-88, Richland, WA 99352, USA
Received 5 April 2004; in revised form 13 June 2004
Available online 13 July 2004
Abstract
High-resolution Fourier transform spectra of CH3 OH have been investigated in the infrared region from 930 to 1450 cm1 in
order to map the torsion–rotation energy manifolds associated with the m7 in-plane CH3 rock, the m11 out-of-plane CH3 rock, and
the m6 OH bend. Upper-state term values have been determined from the assigned spectral subbands, and have been fitted to powerseries expansions to obtain substate origins and effective B-values for the three modes. The substate origins have been grouped into
related families according to systematic trends observed in the torsion–vibration energy map, but there are substantial differences
from the traditional torsional patterns. There appears to be significant torsion-mediated spectral mixing, and a variety of ‘‘forbidden’’ torsional combination subbands with jDtt j > 1 have been observed, where tt denotes the torsional quantum number
(equivalent to t12 ). For example, coupling of the ðt6 ; tt Þ ¼ ð1; 0Þ OH bend to nearby torsionally excited ðt7 ; tt Þ ¼ ð1; 1Þ CH3 -rock
and ðt8 ; tt Þ ¼ ð1; 1Þ CO-stretch states introduces ðt6 ; tt Þ ¼ ð1; 0Þ
ð0; 1Þ subbands into the spectrum and makes the m7 þ m12 m12
torsional hot band stronger than the m7 fundamental. The results suggest a picture of strong coupling among the OH-bending, CH3 rocking, and CO-stretching modes that significantly modifies the traditional energy structure and raises interesting and provocative
questions about the torsion–vibration identity of a number of the observed states.
Ó 2004 Elsevier Inc. All rights reserved.
Keywords: Methanol; CH3 OH; Infrared spectra; Methyl rock; OH bend; Internal rotation; Torsion–vibration term values; Torsion-mediated
vibrational coupling
1. Introduction
This paper reports identification and analysis of
spectral subbands in Fourier transform infrared (FTIR)
spectra of CH3 OH in the 930–1450 cm1 region, with
interesting implications for the question of intermode
coupling among the lower vibrations. The region of the
CH3 OH IR spectrum lying above the strong m8 COstretching fundamental at 1034 cm1 [1] has long pre*
Corresponding author. Fax: +1-506-648-5948.
E-mail address: [email protected] (R.M. Lees).
0022-2852/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jms.2004.06.004
sented significant problems and challenges. Viewed at
low-resolution, the IR absorption is broad and relatively weak and lacks distinct band structure, despite
the fact that it contains six vibrational fundamentals
[2,3] and a number of torsional combination bands.
However, progress in the analysis of the CH3 -rocking
and OH-bending bands for the O-18 [4,5], C-13 [6,7],
and normal 12 CH3 16 OH [8–10] methanol isotopomers
has shown that torsionally mediated interactions
among the modes strongly perturb the excited state
energy manifolds and thus the shapes and widths of the
infrared band profiles.
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
There is a rich body of literature on the spectroscopy
of methanol. The foundations for our understanding
that were originally laid in a classic series of papers by
Dennison and co-workers (see [11] as the latest of the
series) have now evolved to sophisticated multiparameter models [12–15] which can fit the microwave (MW)
and far-infrared (FIR) spectra of the ground vibrational
state to within experimental uncertainty. The accurate
CH3 OH ground-state torsion–rotation energies obtained from the MW and FIR studies [1,14] then provide
a platform from which to launch spectroscopic investigations of the excited vibrational modes. The strong m8
CO-stretching band has been extensively studied and
analyzed [1,16] along with several subbands of the m7 inplane CH3 -rocking band that are enhanced by Coriolis
resonance with the CO stretch [1,17]. In recent years,
analyses of optically pumped FIR laser emission [18–20]
as well as continuing FTIR studies have also provided
high-resolution information for the other low-frequency
modes, highlighted by the discovery of inverted torsional structure in the m11 out-of-plane CH3 -rocking
band [9] and the m4 in-plane asymmetric CH3 -deformation band [21,22]. These latter results paralleled the
original finding of inverted splitting for the m2 CHstretching mode of CH3 OH [23], and have helped to
stimulate a variety of recent approaches to the torsion–
vibration Hamiltonian that have had some striking
successes in modeling the observed structures [22,24–27].
In the current work we have analyzed high-resolution
Fourier transform (FTIR) spectra of 12 CH3 16 OH from
930 to 1450 cm1 , with particular interest in the regions
of the m7 and m11 CH3 -rocking and m6 OH-bending vibrational bands. Brief descriptions of some of the results
have been presented earlier in connection with related
studies [8–10]. Our spectroscopic goal was to map the
excited-state rotation–torsion–vibration (R–T–V) energy structure in as much detail as possible, identifying
individual torsion–K–rotation subbands and determining excited-state energy term values and effective substate parameters. The present paper reports assignments
and term-value analysis for a variety of CH3 -rocking
and OH-bending subbands, both allowed and perturbation-induced, for the vibrational fundamentals as well
as torsional hot and combination bands. First, we give
an overview of the spectral region from 1100 to
1450 cm1 , with a catalog of assigned subband origin
wavenumbers to show their distribution and grouping
across the spectrum. We then present a listing of the
upper substate origins and effective B-values obtained
from fitting the excited-state term values, classified into
related families as far as possible, together with a map of
the K-reduced energy manifold. The question of the
torsional and vibrational parentage of the substate
families is discussed next and systematic trends in
the substate energy patterns are illustrated for several of
the more complex and crowded regions. Finally, in the
529
concluding remarks, we point out the urgent need for
extension of the promising Hamiltonian model developed by Hougen [25] to bring in excited torsional states
and K-dependence of the energies to seek to meet the
interesting spectroscopic challenges and puzzles emerging from the torsion–vibration energies in this region.
2. Experimental details
For this work, a variety of CH3 OH FTIR spectra
were recorded under different conditions to optimize
different spectral features. The original spectra were
obtained at room temperature on the modified DA3.002
Bomem FTIR spectrometer at the National Research
Council of Canada (NRC). Our first spectrum covered
the 900–1100 cm1 region at 0.002 cm1 resolution in 75
coadded scans, with a pressure of 13.5 Pa and 2.0 m
pathlength. This was originally aimed at the strong COstretching fundamental, but several of the subbranches
of the weaker CH3 -rocking m7 þ m12 m12 torsional hot
band could also be identified. Thus, a further spectrum
from 930 to 1301 cm1 was recorded with 94 scans coadded at 0.002 cm1 resolution using a higher 100 Pa
pressure and 2.0 m path length in order to bring up the
weaker features and extend the rocking-band analysis.
Also, a spectrum of the 1245–1475 cm1 region containing the m6 OH-bending band was recorded at
0.003 cm1 resolution with 100 scans coadded using
72 Pa pressure and 2.0 m path length. The NRC spectra
were calibrated against known offsets for CH3 OH absorption lines pumped by CO2 lasers in the lower region
[1], and against standard wavenumbers [28] for the residual water lines observed in the spectrum in the higher
region.
To improve our coverage of the entire region with a
greater range of optical densities, three spectra were
recorded from 1100 to 1800 cm1 on the Bruker IFS 120
instrument at Giessen at a resolution (1/MOPD) of
0.00244 cm1 . The respective pressures were 7.0 Pa with
104 coadded scans, 25 Pa with 250 coadded scans, and
280 Pa with 321 coadded scans. A path length of 16.3 m
in a 1-m White cell operated at room temperature was
employed in each case. KBr optics, a globar source and
a Ge:Cu detector were used in the spectrometer. External calibration against a group of 15 standard OCS lines
in the 1690–1730 cm1 region [29] gave a standard error
of 0.000025 cm1 , representing the statistical uncertainty
in a single measurement.
As well, in order to isolate subbands of low quantum
number to assist the analysis and help confirm the assignments, cooled-beam spectra in the 1275–1648 cm1
region were recorded with the FTIR-jet spectrometer at
the Pacific Northwest National Laboratory. This system
was designed to produce the highest quality at the
highest resolution possible. A Bruker 120HR with a
530
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
maximum resolution of 0.0015 cm1 was coupled to a
12 cm by 50 lm slit nozzle pumped by a stack of four
Roots blowers combined to produce a pumping speed of
greater than 6 m3 /s. The light from the FTIR was coupled into the slit compartment by a Gregorian telescope,
producing a 6 mm diameter beam, and made a total of 5
passes through the molecular jet before exiting the slit
compartment to the detector. The low temperature
(about 10 K) spectrum of methanol was obtained at a
resolution (full width at half height) of 0.0025 cm1 by
expanding a 7% mixture of CH3 OH in helium at a total
backing pressure of 1.053 kPa.
ðt7 ; tt Þ ¼ ð1; 1Þ,
ðt8 ; tt Þ ¼ ð1; 1Þ,
ðt6 ; tt Þ ¼ ð1; 1Þ,
ðt8 ; tt Þ ¼ ð1; 2Þ, and ðt; tt Þ ¼ ð0; 4Þ torsionally excited
states, respectively. For levels of E symmetry, a signed K
is used, with K > 0 corresponding to levels often labeled
as E1 and K < 0 to E2 [30]. States of A symmetry with
K > 0 can also display K-doubling, hence an additional
superscript is added to distinguish resolved doublet
components as Aþ or A [31].
The energy term values of different J for a vibrational
substate of given tt , K, and r can normally be well
represented [1] as a series expansion in powers of
J ðJ þ 1Þ with state-specific coefficients
EðJ Þ ¼W0 þ BJ ðJ þ 1Þ DJ 2 ðJ þ 1Þ2 þ HJ 3 ðJ þ 1Þ3
3. Notation and torsion–vibration energy structure
The rotation–torsion–vibration energy levels of
methanol can conveniently be labeled by the set of
quantum numbers (r; t; tt ; K; J ), where r is the A or E
torsional symmetry, t is the vibrational state, tt is the
torsional quantum number (equivalent to t12 ), and K is
the a-component of the rotational angular momentum
J . The vibrational modes are denoted by t ¼ gr, co, ri,
ro, oh, sb, and ab for the ground, m8 CO-stretching, m7
in-plane and m11 out-of-plane CH3 -rocking, m6 OHbending, and m5 symmetric and m4 asymmetric CH3 -deformation modes, respectively [1]. In some of the figures
we have also used the convenient shorthand m6 þ m12 ,
m7 þ m12 , m8 þ m12 , m8 þ 2m12 , and 4m12 to label the
þ LJ 4 ðJ þ 1Þ4 þ MJ 5 ðJ þ 1Þ5 þ NJ 6 ðJ þ 1Þ6 þ ð1Þ
The first two series coefficients in Eq. (1), W0 and B,
represent the substate origin and the effective B-value,
respectively. By subtracting the K-rotational energy of
½A ðB þ CÞ=2K 2 from W0 , where A, B, and C are the
effective rotational constants, one obtains K-reduced
torsion–vibration energies. In the customary one-dimensional model of the torsional Hamiltonian, these
energies are periodic functions of K that can conveniently be plotted in s-curves of the form shown in Fig. 1
of [9]. The s index [11] is DennisonÕs useful alternative
specification for r defined by: ðK þ sÞ mod 3 ¼ 1 (A), 0
ðE1 or K > 0Þ, 2 ðE2 or K < 0Þ.
Fig. 1. Schematic calculated s-curves of K-reduced torsion–vibration energies for CH3 OH in the neighborhood of the OH-bending fundamental state,
showing predicted close proximity between ðt6 ; tt Þ ¼ ð1; 0Þ OH-bend [m6 ], ðt7 ; tt Þ ¼ ð1; 1Þ in-plane CH3 -rock [m7 þ m12 ], ðt8 ; tt Þ ¼ ð1; 1Þ CO-stretch
[m8 þ m12 ], and ðt11 ; tt Þ ¼ ð1; 1Þ out-of-plane CH3 -rock [m11 þ m12 ] levels and possible accidental near-degeneracies with ðt; tt Þ ¼ ð0; 4Þ ground-state
[4m12 ] levels. The K-reduced energy is given by subtracting K-rotational energy of 3.45K 2 from the K-rotation–torsion–vibration energy, where
3.45 cm1 is an effective value of the K-rotational constant, [A ðB þ CÞ=2]. The s ¼ 1 points are shown as open circles, s ¼ 2 as filled circles, and
s ¼ 3 as open triangles.
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
When the vibrational energies are added in, a complex torsion–vibration energy manifold results (see
Fig. 1 of [20], for example) in which there are numerous
possibilities for anharmonic or Coriolis resonances
between near-degenerate levels. Fig. 1 illustrates the
general situation to be expected in the region of the OHbending state, showing schematic K-reduced s-curves
obtained by simply adding calculated ground-state torsional energies onto predicted vibrational energies. The
region is evidently an interesting one, with multiple interactions likely to occur among the tt ¼ 0 OH-bending
and tt ¼ 1 CH3 -rocking and CO-stretching levels as well
as the possibility of Fermi resonance with tt ¼ 4 groundstate levels [32]. Because the overlapping ladders of
torsional states greatly enhance the possibilities for intermode resonance, torsionally mediated vibrational
coupling is undoubtedly an important factor in determining mechanisms and rates for intramolecular vibrational energy redistribution (IVR).
4. Subband assignments, torsion–vibration substate origins, and effective B values
4.1. Overview of the spectrum and torsion–vibration
energy structure
Because large energy changes occur in jDtt j ¼ 1 and
jDtt j ¼ 2 torsional combination transitions, the origin
wavenumbers of all of the observed subbands that are
associated with CH3 -rocking and OH-bending upper
states cover a very wide range from 958 up to 1418 cm1 .
Those origins lying below 1100 cm1 are reported in a
companion paper on the 10 lm spectrum dealing principally with assignments and analysis for the strong m8 COstretching band [33]. The region below 1100 cm1 contains, in addition to the m8 band, the tt ¼ 0 fundamental of
the m7 in-plane CH3 -rock plus several tt ¼ 1 m7 subbands
and a variety of Dtt ¼ 1 and Dtt ¼ 2 torsional combination subbands from the m5 , m6 , and m7 modes.
In the present work, we have moved our focus up to
the next region of the CH3 OH spectrum extending from
1100 to 1450 cm1 . Table 1 presents the origins of the
subbands identified so far in this region, illustrating the
extent and distribution of the spectral structure. In
general, the subbands fall into characteristic groupings
in different regions of the spectrum. Further tt ¼ 1 torsionally excited m7 subbands are found from 1100 to
1125 cm1 , mingling with subbands of the
ðt6 ; tt Þ ¼ ð1; 0Þ
ð0; 1Þ OH-bending torsional combination band that extends from 1107 to 1142 cm1 . The
m11 out-of-plane CH3 -rocking fundamental then takes
over from 1142 to 1165 cm1 . The region from 1180 to
1263 cm1 is an interesting but puzzling one containing
numerous DK ¼ 0 a-type subbands that originate from
531
Table 1
Observed subband origins (in cm1 ) in the spectral region from 1100 to
1450 cm1 for CH3 OHa
r
t0
t0t
t00t
K0
K 00
Originb
E
E
E
A
A
E
E
A
A
A
E
A
E
A
E
E
A
E
E
E
A
E
A
E
E
E
A
A
E
E
A
E
E
A
A
A
E
A
E
E
A
E
E
A
E
E
A
A
E
A
E
E
E
A
A
A
A
A
E
E
E
E
A
E
E
A
A
A
E
ri
co
oh
oh
ri
oh
ri
ri
co
ro
oh
ri
ri
ri
ri
oh
oh
ri
ri
oh
oh
oh
oh
oh
oh
oh
oh
oh
oh
oh
oh
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
ro
U1
U1
ro
ro
ro
U2
oh
U0
U2
U2
U2
U2
U2
U2
ro
U2
co
U0
1
1
0
0
0
0
1
0
0
0
0
1
1
1
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
2
1
1
2
2
2
2
2
2
0
2
2
0
1
1
1
1
0
1
1
0
0
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
0
0
0
2
2
1
2
2
2
2
2
2
0
2
2
1
1
2
)8
3
6
2
4
5
7
5
)2
2
8
9
)3
7
9
)4
8
)4
1
)6
5
0
4
5
2
10
1
)7
6
)7
8
6
9
5
)10
10
)4
)6
1
9
11
7
7
)8
8
4
)9
11
10
6
)5
6
4
5
0
3
)4
)5
6
1
7
2
3
8
4
5
5
1
1
)8
3
5
4
4
4
5
6
)2
2
8
9
)3
7
9
)3
7
)4
1
)6
5
0
4
5
2
10
1
)7
6
)7
8
6
9
5
)10
10
)4
)6
1
9
11
7
7
)8
8
4
)9
11
10
6
)5
5
3
4
0
3
)4
)5
6
1
7
2
3
7
4
4
5
1101.49
1102.55
1103.87
1106.535
1108.0
1108.75
1108.76
1108.84
1108.9
1109.02
1110.813
1111.78
1114.64
1114.8
1115.130
1115.91
1123.62
1123.7
1124.46
1124.481
1126.501
1128.97
1135.295
1135.409
1135.949
1136.26
1136.316
1137.96
1138.059
1141.396
1141.613
1142.206
1142.515
1144.246
1145.125
1147.835
1149.41
1149.92
1152.5
1152.889
1152.89
1154.18
1154.36
1157.53
1157.700
1158.180
1161.38
1163.447
1164.63
1165.32
1165.62
1166.30
1180.02
1183.04
1184.678
1187.833
1193.999
1195.28
1196.94
1202.20
1202.95
1208.283
1209.056
1211.624
1213.091
1214.41
1214.49c
1214.86c
1215.48
532
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
Table 1 (continued)
Table 1 (continued)
r
t
0
t0t
t00t
0
K
A
E
E
E
A
E
E
E
A
E
A
E
E
A
E
A
E
A
E
E
A
A
E
E
E
A
E
A
E
A
E
E
E
E
E
A
E
A
A
A
A
E
E
A
E
E
E
E
E
A
E
E
A
A
A
E
A
E
E
A
E
A
E
E
E
E
E
E
E
A
A
co
U1
U2
U1
U1
U1
U1
U2
U1
U1
U2
U2
U2
U2
U1
U1
U2
U1
U1
U2
sb
U2
U1
U1
U2
U1
U2
ri
ri
ri
ri
co
ri
ri
co
co
co
ri
ri
ri
co
ri
oh
ri
ri
ri
ri
co
co
co
ri
ri
ri
oh
ri
oh
oh
co
oh
oh
oh
ri
ri
ri
oh
ri
oh
ri
oh
oh
U0
2
1
2
1
1
1
1
2
1
1
2
2
2
2
1
1
2
1
1
2
0
2
1
1
2
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
0
1
0
0
1
0
0
0
1
1
1
0
1
0
1
0
0
0
1
1
2
1
1
1
1
2
1
1
2
2
2
2
1
1
2
1
1
2
1
2
1
1
2
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
)9
)2
8
1
3
)4
4
2
1
)2
0
3
)1
3
1
7
)8
7
1
5
6
)5
4
0
)5
10
5
6
)4
3
0
1
2
7
2
1
2
5
0
4
2
3
)6
)2
7
6
)1
4
2
3
7
0
8
)3
2
)5
8
0
4
4
)5
)1
)7
)9
1
6
)10
5
4
K
00
3
0
)9
)2
8
1
3
)4
4
2
1
)2
0
3
)1
3
1
7
)8
7
1
5
6
)5
4
0
)5
10
5
6
)4
3
0
1
2
7
1
1
2
5
)2
4
4
3
)6
)2
7
6
)1
4
2
3
7
1
8
)3
2
)5
8
0
4
4
)5
)1
)7
)9
1
6
)10
5
5
Origin
b
1215.8
1218.197
1220.96
1222.680
1225.61
1226.003
1226.20
1227.30
1228.353
1228.84
1230.4
1231.1
1232.31
1232.611
1232.789
1232.8
1233.846
1234.03
1235.41
1237.142
1238.47
1238.805
1239.88
1240.624
1240.634
1247.031
1262.80
1280.15
1283.89
1285.000
1286.830
1287.459
1288.345
1289.448
1289.643
1290.02
1290.47
1291.050
1293.161
1293.871
1293.9
1294.55
1294.63
1295.048
1296.54
1298.223
1303.031
1304.456
1305.800
1306.04
1308.55
1309.539
1309.992
1310.58
1313.228
1315.286
1317.69
1318.170
1319.044
1320.633
1321.795
1322.441
1322.450
1323.670
1324.33
1324.47
1326.017
1326.3
1330.15
1331.753
1332.296
r
t0
t0t
t00t
K0
K 00
Originb
A
E
E
A
E
E
A
A
E
E
E
E
A
E
E
A
E
E
A
A
E
E
A
E
E
A
E
A
E
A
A
A
A
A
E
E
E
A
E
A
A
E
A
E
E
E
oh
oh
oh
oh
oh
oh
UA
oh
oh
oh
ri
oh
oh
ri
oh
oh
oh
oh
oh
oh
oh
oh
UB
oh
oh
oh
oh
oh
oh
U0
oh
oh
oh
U0
oh
U0
U0
oh
U0
oh
U1
U1
UC
U0
U1
U0
0
0
0
0
0
0
?
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
?
1
1
1
1
1
1
0
1
1
0
0
1
0
0
1
0
1
1
1
?
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
0
1
1
0
0
1
0
0
1
0
1
0
0
2
0
0
0
6
0
5
4
)6
)1
2
1
6
)4
5
3
8
4
2
10
7
)2
3
7
9
)8
2
)3
4
5
1
2
0
0
1
6
5
4
)2
)1
6
3
2
4
2
1
3
)4
0
5
6
0
5
4
)6
)1
2
1
6
)4
3
3
8
2
2
10
7
)2
3
7
9
)8
2
)3
4
5
1
2
0
0
1
6
4
4
)2
)1
6
3
2
4
2
1
3
)4
0
5
1332.396
1335.199
1339.940
1340.151
1340.57
1341.440
1342.387
1342.462
1343.151
1345.155
1345.31
1345.56
1345.612
1345.82
1345.9
1345.97
1346.22
1346.233
1347.235
1347.77
1349.226
1349.848
1350.161
1355.047
1358.149
1360.53
1362.692
1364.909
1366.414
1369.691
1370.441
1370.625
1371.76
1372.296
1372.537
1373.456
1376.457
1377.36
1385.274
1399.449
1413.192
1413.971
1417.084
1417.60
1417.988
1419.16
a
The lower levels of the subbands are established from ground-state combination differences, but the vibrational and/or torsional labeling of the upper
levels is tentative in a number of cases. The U0 , U1 , and U2 entries for the upper
levels refer to groupings of substates that appear to follow systematic patterns
associated with tt ¼ 0, 1, and 2 torsional levels, respectively, but which have not
yet been vibrationally assigned. Labels UA , UB , and UC refer to individual unidentified torsion–vibration substates that are not associated with other substate
groupings.
b
Origin wavenumbers listed to only 1 or 2 decimal place have correspondingly greater uncertainties than those listed to 3 places. Origins marked with
asterisks indicate subbands for which one or more of the initial lines are observed in the 10 K cooled-beam slit-jet spectrum, confirming that the subband
originates from a tt ¼ 0 lower level of low energy and low quantum number.
c
The strongly interacting (A; U2 ; 2; 4) and (A,co,2,5) upper states of these
two hybridized subbands are the states labeled as (A,co,2,5), hd, and hu, respectively, in [20,33].
tt ¼ 1 and tt ¼ 2 levels of the ground vibrational state,
but whose upper states have not yet been confidently
labeled either vibrationally or torsionally. From 1280 to
1313 cm1 , tt ¼ 1
0 torsional combination subbands
of the CO stretch and in-plane CH3 rock dominate the
spectrum. The tt ¼ 0 subbands of the m6 OH-bending
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
fundamental then appear, extending over a wide range
from 1315 to 1350 cm1 , followed by torsionally excited
tt ¼ 1 OH-bending subbands up to 1400 cm1 . Six further mysterious subbands lie between 1400 and
1420 cm1 whose upper states are vibrationally unassigned. Beyond 1450 cm1 , one then enters the domain
of the m5 , m10 , and m4 CH3 -deformation modes that have
been reported previously [21,22].
Our overall data set currently includes about 4600
assigned absorption lines accessing torsion–rotation
levels of the in-plane CH3 rock, 900 for the out-of-plane
CH3 rock, 3400 for the OH bend, and 2800 whose upper
levels are still vibrationally unidentified. We will not
report the details of the spectra here, but plan to include
them in the near future as supplementary data to accompany a study in progress on the J -dependent level
patterns of the (r; t; tt ; K) substates and the numerous
level-crossing interactions among them.
From the spectroscopic data, upper-state term values
were obtained by adding ground-state energies from
Moruzzi et al. [1] to the wavenumbers of the assigned
line. For each identified (r; t; tt ; K) substate, the substate
origin and effective B value were then determined by
fitting the term values of the sequence of substate J levels to the power-series expansion of Eq. (1). In each
case, fits were performed with maximum powers of
J ðJ þ 1Þ ranging from 3 to 6, and the order giving the
minimum standard error in the substate origin Wo was
adopted as the optimum. Our results are collected in
Table 2, grouped in families of related substates. For
each substate, we give the origin Wo , the effective B value, the order of the optimum fit, the weighted standard
deviation of the optimum fit, and the difference dWo
between minimum and maximum origin values obtained
over the four fits from order 3 to 6. The dWo variation is
a measure of how well the power-series model fits the
term values for a substate, and we believe it gives a
useful and realistic estimate of the likely accuracy of the
substate origin. For substates of A torsional symmetry
with resolved asymmetry K-doubling, the Aþ and A
components were fitted separately and the resulting
constants, which were always very close, were averaged
to give the values in Table 2.
As mentioned above, an informative pictorial synthesis of the experimental information on the excited
energy manifold is given by plotting the K-reduced energies as a function of the K quantum number. Fig. 2
shows our current map of known CH3 OH torsion–vibration substate energies up to 1950 cm1 , in which the
new results from Table 2 are combined with previous
results for the m8 CO stretch [1,33], the m11 out-of-plane
rock [9,10] and the m4 , m5 and m10 CH3 -deformation
modes [21,22]. The challenge now is to ‘‘connect the
dots’’ in order to classify the substates into a consistent
picture with full torsional and vibrational labeling, and
this is discussed in Section 5 below.
533
Table 2
CH3 OH (r; t; tt ; K) substate Wo origins and effective B values (in cm1 )
from power-series fitting of experimental term valuesa
Substateb
Origin Wo
B Value
Ordc
SDd
dWo e
(A,ri,0,0)
(A,ri,0,1)
(A,ri,0,2)
(A,ri,0,3)
(A,ri,0,4)g
(A,ri,0,5)g
(A,ri,0,6)g
(A,ri,0,8)
(E,ri,0,0)
(E,ri,0,1)
(E,ri,0,2)
(E,ri,0,3)
(E,ri,0,)1)
(E,ri,0,)2)
(E,ri,0,)4)
(E,ri,0,)5)
(A,oh,0,0)
(A,oh,0,1)
(A,oh,0,2)
(A,oh,0,3)
(A,oh,0,4)
(A,oh,0,5)
(A,oh,0,6)
(A,oh,0,7)
(A,oh,0,8)
(A,oh,0,10)
(E,oh,0,0)
(E,oh,0,1)
(E,oh,0,2)
(E,oh,0,3)
(E,oh,0,4)
(E,oh,0,5)
(E,oh,0,6)
(E,oh,0,7)
(E,oh,0,8)
(E,oh,0,9)
(E,oh,0,)1)
(E,oh,0,)2)
(E,oh,0,)3)
(E,oh,0,)4)
(E,oh,0,)6)
(E,oh,0,)7)
(E,oh,0,)8)
(E,oh,0,)10)
(A,U0 ,0,0)
(A,U0 ,0,4)
(E,U0 ,0,2)
(E,U0 ,0,5)
(E,U0 ,0,6)
(E,U0 ,0,)1)
(E,U0 ,0,)4)
(A,U1 ,1,0)
(A,U1 ,1,2)
(A,U1 ,1,3)
(A,U1 ,1,4)
(A,U1 ,1,7)
(A,U1 ,1,8)
(E,U1 ,1,0)
(E,U1 ,1,1)
(E,U1 ,1,2)
(E,U1 ,1,3)
(E,U1 ,1,6)
1202.860
1209.354f
1223.014f
1235.993f
1257.721f
1292.652
1332.493
1422.527
1207.403
1212.073
1217.629
1234.311
1206.527
1218.239
1261.439
1288.745
1448.609
1480.543f
1471.877f
1509.823f
1523.969f
1555.574
1595.014
1645.829
1696.685
1825.902
1472.296
1468.620
1489.358
1506.694
1516.492
1562.493
1595.414
1647.781
1679.941
1761.378
1473.293
1492.207
1486.846
1534.227
1599.891
1633.170
1701.270
1812.527
1497.667
1556.114
1528.710
1641.717
1628.719
1505.308
1606.668
1669.458
1567.376f
1636.079f
1690.696
1799.792
1828.116
1555.085
1556.573
1634.758
1644.291
1779.522
0.8036
0.8034f
0.8029f
0.8035f
0.8104f
0.8012
0.8086
0.8021
0.8032
0.8028
0.8034
0.8041
0.8036
0.8036
0.8033
0.8015
0.8032
0.8040f
0.8039f
0.8039f
0.8012f
0.8034
0.8027
0.8020
0.8021
0.8017
0.8005
0.8033
0.8049
0.8026
0.8042
0.8032
0.7994
0.8032
0.8022
0.8022
0.8047
0.8040
0.8032
0.8033
0.8080
0.8023
0.8025
0.8016
0.8028
0.8031
0.8022
0.8006
0.8022
0.8025
0.8038
0.8017
0.8015f
0.7986f
0.8031
0.8012
0.8002
0.8019
0.8016
0.7972
0.7962
0.8040
5
3
3
4
5
3
5
3
6
4
5
4
6
5
3
4
6
5
6
4
3
4
3
4
3
3
4
5
6
6
4
4
3
4
3
3
3
5
4
4
5
4
3
3
5
3
3
5
3
3
5
5
3
4
5
3
3
3
4
6
6
5
0.37
0.44
0.40
0.98
6.17
0.73
11.19
0.43
1.03
0.72
1.65
1.12
1.23
0.41
0.62
0.74
0.48
0.25
1.29
0.41
0.29
0.31
0.50
0.45
0.89
0.88
0.32
1.20
8.18
2.97
0.21
0.19
0.51
0.33
1.24
0.51
0.64
0.94
1.26
0.50
1.40
0.69
0.51
0.79
0.30
0.81
0.60
0.43
0.25
1.64
2.60
0.34
2.07
23.64
0.55
0.23
0.28
0.72
0.63
3.71
5.94
2.07
0.004
0.002
0.083
0.002
0.155
0.005
0.263
0.001
0.022
0.049
0.025
0.009
0.017
0.035
0.000
0.004
0.002
0.001
0.010
0.002
0.001
0.003
0.001
0.013
0.004
0.007
0.001
0.015
0.035
0.033
0.001
0.003
0.001
1.543
0.013
0.011
0.001
0.019
0.005
0.001
0.059
0.009
0.009
0.028
0.001
0.001
0.001
0.015
0.002
0.001
0.023
0.001
0.024
0.067
0.004
2.554
0.004
0.001
0.001
0.045
0.091
0.055
534
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
Table 2 (continued)
b
Table 2 (continued)
Substate
Origin Wo
B Value
Ord
(E,U1 ,1,)1)
(E,U1 ,1,)2)
(A,oh,1,1)
(A,oh,1,2)
(A,oh,1,3)
(A,oh,1,4)
(A,oh,1,5)
(A,oh,1,6)
(A,UA ,1,2)
(A,UB ,1,2)
(A,ro,0,4)
(A,ro,0,5)
(A,ro,0,6)
(A,ro,0,7)
(A,ro,0,8)
(A,ro,0,9)
(A,ro,0,10)
(A,ro,0,11)
(E,ro,0,7)
(E,ro,0,8)
(E,ro,0,9)
(E,ro,0,10)
(E,ro,0,11)
(E,ro,0,)6)
(E,ro,0,)7)
(E,ro,0,)8)
(E,ro,0,)9)
(E,ro,0,)10)
(A,ri,1,1)
(A,ri,1,2)
(A,ri,1,3)
(A,ri,1,4)
(A,ri,1,5)
(A,ri,1,6)
(A,ri,1,7)
(A,ri,1,8)
(A,ri,1,9)
(A,ri,1,10)
(E,ri,1,0)
(E,ri,1,1)
(E,ri,1,2)
(E,ri,1,3)
(E,ri,1,4)
(E,ri,1,5)
(E,ri,1,6)
(E,ri,1,7)
(E,ri,1,8)
(E,ri,1,)1)
(E,ri,1,)2)
(E,ri,1,)3)
(E,ri,1,)4)
(E,ri,1,)5)
(E,ri,1,)6)
(E,ri,1,)9)
(E,U1 ,1,)5)
(E,U1 ,1,)8)
(A,U2 ,2,0)
(A,U2 ,2,1)
(A,U2 ,2,3)
(A,U2 ,2,4)h
(A,U2 ,2,5)
(A,U2 ,2,7)
(E,U2 ,2,0)
(E,U2 ,2,1)
1648.076
1604.074
1724.483f
1700.474f
1780.782
1861.792
1780.807
1824.027
1677.953f
1685.726f
1347.265
1371.656
1406.864
1455.579
1512.463
1563.819
1629.846
1709.957
1459.314
1503.412
1566.333
1638.078
1706.683
1412.207
1451.045
1509.604
1572.595
1631.795
1429.130f
1447.344f
1457.636f
1506.258
1517.691
1547.619
1608.045
1664.301
1717.932
1760.080
1425.442
1432.051
1451.985
1470.676
1489.264
1506.457
1578.567
1604.645
1656.331
1455.522
1444.198
1467.346
1475.904
1536.623
1555.863
1732.441
1748.188
1832.833
1675.192
1832.984f
1817.984
1769.582
1971.766
1892.129
1870.608
1917.128
0.8018
0.7987
0.8019f
0.8011f
0.8004
0.8001
0.8017
0.8015
0.8014f
0.8000f
0.8037
0.8037
0.8036
0.8019
0.8035
0.8041
0.8036
0.8038
0.8038
0.8037
0.8032
0.8037
0.8062
0.8037
0.8036
0.8035
0.8036
0.8048
0.8009f
0.8026f
0.7999f
0.7981
0.8027
0.8006
0.7959
0.8001
0.7989
0.8009
0.8011
0.7954
0.7979
0.7992
0.7995
0.8000
0.7961
0.8019
0.8018
0.7978
0.8003
0.8024
0.8005
0.8003
0.7998
0.7996
0.8012
0.7995
0.8017
0.8019f
0.8007
0.7956
0.8036
0.8006
0.8007
0.8007
6
6
3
3
6
3
4
3
3
3
4
7
4
4
5
4
4
4
4
4
4
4
4
4
4
4
4
4
3
5
3
5
4
4
3
3
4
3
3
6
4
6
6
4
5
4
3
7
4
6
4
5
5
3
6
4
6
3
6
6
4
5
6
4
c
d
e
SD
dWo
1.13
0.50
0.55
0.67
0.51
0.94
0.77
0.29
0.79
0.42
0.61
0.47
0.89
2.00
1.24
0.79
1.93
1.04
1.74
0.43
2.17
2.75
1.21
1.04
0.46
0.68
0.88
0.51
0.45
1.95
1.13
0.80
0.34
0.60
0.32
0.45
0.62
0.55
0.29
0.96
0.30
1.18
3.89
1.63
1.15
0.32
0.56
0.47
1.17
0.46
0.73
0.46
0.25
0.47
0.31
1.10
0.94
43.01
0.36
4.28
0.60
0.54
4.21
0.39
0.002
0.004
0.001
0.001
0.015
0.004
0.035
0.001
0.001
0.001
0.001
0.015
0.002
0.029
0.025
0.002
0.051
0.105
0.021
0.015
0.079
0.169
0.063
0.006
0.009
0.028
0.014
0.192
0.001
0.022
0.003
0.005
0.002
0.002
0.001
0.001
0.098
0.005
0.001
0.014
0.001
0.024
0.029
0.036
0.180
0.003
0.002
0.001
0.002
0.004
0.003
0.002
0.014
0.003
0.008
0.048
0.001
0.085
0.018
0.052
0.005
0.008
0.008
0.001
Substateb
Origin Wo
B Value
Ordc
SDd
dWo e
(E,U2 ,2,2)
(E,U2 ,2,3)
(E,U2 ,2,4)
(E,U2 ,2,6)
(E,U2 ,2,7)
(E,U2 ,2,)1)
(E,U2 ,2,)2)
(E,U2 ,2,)4)
(E,U2 ,2,)5)
(E,U2 ,2,)9)
(E,oh,1,0)
(E,oh,1,1)
(E,oh,1,4)
(E,oh,1,)2)
(E,oh,1,)3)
(A,UC ,2,3)
(A,sb,0,1)
1747.054
1772.988
1984.599
1816.170
1975.927
1706.036
1807.313
1872.760
1770.366
2005.622
1703.301
1693.252
1738.692
1753.931
1707.264
2002.458
1592.517f
0.8013
0.8012
0.8033
0.8011
0.8004
0.8019
0.8131
0.8000
0.8035
0.8003
0.8017
0.8019
0.8019
0.8014
0.8008
0.8004
0.7973f
3
6
6
3
3
3
5
6
6
3
3
3
3
3
3
6
6
0.75
0.51
1.08
0.38
0.40
2.95
46.55
0.67
0.97
0.86
0.74
0.46
0.57
0.49
0.90
0.79
1.23
0.001
0.012
0.025
0.006
0.001
0.001
0.194
0.007
0.031
0.002
0.001
0.001
0.001
0.001
0.003
0.002
0.018
a
Term values were determined by adding ground-state energies
from Moruzzi et al. [1] to observed wavenumbers. Thus, they are
referenced to the values of 127.97549 and 137.09724 cm1 from [1] for
the (r; t; tt ; K; J ) ¼ (A,gr,0,0,0) and (E,gr,0,0,0) levels. Note that these
torsional zero-point energies are model dependent, and are slightly
lower than the values of 128.10687 and 137.22891 cm1 calculated with
the global fit parameters of Xu and Hougen [14].
b
The torsional and vibrational labeling is not yet fully established
for substates other than the tt ¼ 0 CH3 -rocking ri and ro modes.
Substates of a given classification are believed to belong to a group
following a systematic pattern, but the patterns do not always conform
to the traditional torsional model. The unknown vibrational states
labeled as U0 , U1 , and U2 are the upper states for families of subbands
originating from tt ¼ 0, tt ¼ 1, and tt ¼ 2 ground-state levels, respectively, although a number of the upper substates are also accessed
by nominally forbidden subbands with jDtt j > 0.
c
Order of the fit, i.e., the maximum power of J ðJ þ 1Þ included.
Fits of order 3 to 6 were compared for each substate; the optimum
order shown here is the one giving the minimum standard error in the
substate origin.
d
Overall unitless weighted standard deviation of the fit. The default
uncertainty for a single term value was 0.0005 cm1 .
e
dWo is the (max ) min) spread of Wo values over the four fits of
order 3 to 6. This is a measure of how well the substate term values are
represented by a power series, and thus of the reliability of the Wo
value.
f
Average of values from separate fits to Aþ and A substates.
g
Substate affected by strong Coriolis resonance with (K þ 1)
partner in the t8 ¼ 1 CO-stretching state.
h
The (A,U2 ,2,4) substate is strongly hybridized with the (A,co,2,5)
substate lying just above it, and is labeled as the (A,co,2,5) hd state in
[20,33].
In the following subsections, we consider the assignments and analyses for the various subband groupings
in approximate order of increasing upper-state energy.
We have endeavored to arrange the excited substate
origins into systematic families as far as possible on the
basis of their positions and patterns in the energy map of
Fig. 2. We will discuss the rationale for this classification
below in Section 5. However, we note that the ordering
and vibrational labeling are tentative in a number of
cases, as there are significant differences between the
observed patterns and the regular oscillating s-curves
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
535
Fig. 2. K-reduced torsion–vibration energy map showing the locations of currently known substates associated with the lower vibrational modes of
CH3 OH. Each point represents a K-reduced substate origin established from one or more identified subbands in the spectrum. The labeling on the
right-hand side represents the expected approximate energy ordering of the indicated torsion–vibration states.
expected from the traditional model. The existence of so
many nominally forbidden a-type subbands with
jDtt j > 0 implies a large degree of torsion-mediated
mixing among the modes. Furthermore, the dramatic
departures from the traditional model, notably torsional
inversion, that have been observed for the tt ¼ 0 ground
torsional levels of certain of the CH3 -rocking, CH3 -deformation, and CH-stretching modes [9,10,21,23] imply
that similar changes may be expected for their excited
torsional states. Such states have not yet been explored
for these modes within the framework of the new torsion–vibration formalisms [22,24–27]; we hope that the
present results may serve as a stimulus to do so.
4.2. The m7 in-plane CH3 -rocking fundamental
The m7 in-plane CH3 -rocking fundamental (in the
tt ¼ 0 torsional state) is a hybrid a=b band centered
around 1072 cm1 . This is just 38 cm1 above the much
stronger m8 CO-stretching band, so that much of the P
and Q branch structure of the m7 band is heavily obscured by the m8 R branch. However, a number of subbands having (r; t; tt ; K) ¼ (A,ri,0,4), (A,ri,0,5), and
(A,ri,0,6) upper states had been identified previously [1]
due to intensity enhancement arising from strong Coriolis resonances between those levels and the corresponding (K þ 1) m8 states [17]. The Coriolis-induced
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R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
mixing is almost 50:50 for the {4ri /5co } and {6ri /7co }
pairs of states, hence intensity borrowing leads to a
variety of readily observable DK ¼ 0 and DK ¼ 1
subbands accessing the hybridized levels.
Our present assignments for new and weaker m7
subbands were based principally on ground-state combination differences using known ground-state energies
[1]. The CH3 OH m7 band differs from those of the O-18
and C-13 isotopomers [5,6] in having significant b-type
character. The DK ¼ 1 subbands are as strong as the
a-type DK ¼ 0 subbands in some cases, although the
relative intensities do not follow a consistent pattern.
The presence of both a-type and b-type subbranches
then permits valuable combination loop checks of the
assignments. A sample diagram is given in Fig. 3 for
R(9) transitions of the (E,ri,0,)2), (E,ri,0,)1), and
(E,ri,0,0) subbands. The closure defects for the numerous loops that can be formed among the transitions and
the ground state levels are all well below the experimental uncertainty, confirming our identifications. Another interesting test comes from the mean term values
of 1294.8039 and 1295.6438 cm1 obtained for the upper
K ¼ 1 and 0 levels from the data in Fig. 3. The dif-
ference of 0.8399 cm1 corresponds to a frequency of
25.18 GHz, in agreement with the microwave value of
25.1786 GHz for the Q(10) line of the K ¼ 0
1 E Qbranch measured directly in the t7 ¼ 1 excited state
many years ago [34].
With the substantial change in torsional splitting in
going from the ground state to the ðt7 ; tt Þ ¼ ð1; 0Þ excited state [10], the origins of the a-type subbands are
distributed over a relatively broad range from 1068.8
to 1074.9 cm1 . The b-type subbands are spread more
widely, with origins from 1003.8 up to 1124.5 cm1 . As
mentioned above, most of the m7 subband origin
wavenumbers lie below 1100 cm1 so were listed in our
companion paper on the FTIR spectrum in the 10 lm
region [33]. The remainder are included here in
Table 1.
The narrow band of ðt7 ; tt Þ ¼ ð1; 0Þ K-reduced energies around 1200 cm1 in Fig. 2 follows a well-behaved
oscillating s-curve pattern [10]. The notable feature is
that the E–A splitting between the K ¼ 0 E and A levels
in Table 2 is only 4.543 cm1 , just under half of the
ground-state value of 9.122 cm1 [1]. When interpreted
according to the traditional one-dimensional torsional
Hamiltonian, this would imply a much higher torsional
barrier height [4,6] for the t7 ¼ 1 state than for the
ground state. However, HougenÕs recent model now
suggests the reduction in splitting is more likely the result of torsion–vibration interaction [10,25]. Note that
the trend continues with a further sharp reduction in
torsional splitting down to 2.29 cm1 for the t7 ¼ 2
second excited rocking state [10].
4.3. The m11 out-of-plane CH3 -rocking fundamental
Fig. 3. Sample transition diagram for the m7 in-plane (ri) CH3 -rocking
band of CH3 OH. FTIR wavenumbers are shown in the boxes on the
transitions; term values from [1] are shown for the ground-state levels.
Closure of combination loops on the diagram with near-zero wavenumber defects confirms a- and b-type assignments of R(9) transitions
to K ¼ 0, )1, and )2 E levels. For example, starting at the lower left
and going up and around, we find 1088.0373 ) 1102.2361 +
1091.2350 ) 1086.0318 + 1085.1918 – 1076.1965 ¼ )0.0003 cm1 , well
within the experimental tolerance.
The m11 fundamental was recently identified as a relatively weak and predominantly parallel band centered
around 1154 cm1 [9]. Analysis of the torsional structure
showed that the E–A splitting for tt ¼ 0 was inverted (A
level above the E level) compared to the ground and
CO-stretching states, as can be seen for the band of
energies around 1285 cm1 in Fig. 2. This feature was
shown by Hougen [25] to arise naturally in his coupled
torsion–vibration formalism for those CH3 OH modes,
such as the in-plane and out-of-plane CH3 rock, that
correlate to degenerate E vibrations of the limiting
symmetric top with a linear COH group. In a recent fulldimensional ab initio reaction-path treatment, Fehrensen et al. [26] interpret the inversion within the context
of an adiabatically projected one-dimensional torsional
Hamiltonian as arising from a torsional geometric phase
for these E-like modes. However, they predict that all
such modes should display inversion, unlike our above
observations for m7 . Another recent full-dimensional ab
initio treatment [27], on the other hand, calculates the
splittings to be normal for m7 and inverted for m11 as we
observe experimentally, so that there are still interesting
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
differences between theoretical models and the need for
additional precise spectroscopic data.
Because the strongest m11 spectral features are the
parallel DK ¼ 0 Q-subbranches, for which the relative
intensities are proportional to K 2 , our information on
this band is primarily for levels of medium to high K, as
seen in Table 2. So far, we have not identified transitions
to the K ¼ 0 levels, so do not have a direct experimental
E–A splitting. However, the earlier Fourier fit of the
substate origins to the periodic s-curve model gave a
splitting of )7.50 cm1 , with the negative sign indicating
the torsional inversion [9,10].
4.4. The m6 OH-bending fundamental and m7 and m8
torsional combination and hot bands
As seen from the s-curves in Fig. 1, the calculated
levels of the ðt6 ; tt Þ ¼ ð1; 0Þ fundamental OH-bending
state lie in the same energy region as those of the
ðt7 ; tt Þ ¼ ð1; 1Þ and ðt11 ; tt Þ ¼ ð1; 1Þ in-plane and out-ofplane torsionally excited CH3 -rocking states and also
the higher levels of the (t8 ; tt Þ ¼ ð1; 1Þ CO stretch. The
close proximity of many of these levels results in strong
mixing between them. The levels are then best described
as hybridized eigenstates, and forbidden jDtt j ¼ 1,
DK ¼ 0 subbands appear in the spectrum through intensity borrowing [5,7,8]. This allows useful checks of
the spectral assignments from combination relations
537
among the interlocking subbands that connect the
ground and excited states. A combination-loop diagram
was presented in [8], for example, that confirmed assignments for K ¼ 2A K-doublet transitions and also
showed the K ¼ 2 asymmetry splitting to be inverted for
the ðt6 ; tt Þ ¼ ð1; 0Þ OH bend as compared to the ground
state.
From Fig. 1, the highest-lying ðt8 ; tt Þ ¼ ð1; 1Þ COstretching levels most likely to mix with ðt6 ; tt Þ ¼ ð1; 0Þ
and ðt7 ; tt Þ ¼ ð1; 1Þ partners are those with jKj ¼ 1–2 for
s ¼ 1 and jKj ¼ 3–7 for s ¼ 3. (The K ¼ 0, s ¼ 1 level is
even higher, but does not mix with the OH bend because
the A selection rules forbid tt ¼ 1 Aþ levels from interacting with tt ¼ 0 Aþ levels for vibrational states of
the same symmetry.) This mixing lends intensity to
ðt6 ; tt Þ ¼ ð1; 0Þ
ð0; 1Þ, ðt7 ; tt Þ ¼ ð1; 1Þ
ð0; 0Þ and
ðt8 ; tt Þ ¼ ð1; 1Þ
ð0; 0Þ forbidden subbands that we
have seen in the spectrum. It can also give rise to farinfrared laser (FIRL) emission into the three different
vibrational states from a single optically pumped upper
level, as was illustrated in Fig. 2 of [20] for the FIRL
system pumped by the 9P (24) CO2 laser line.
An example of the three-way vibrational coupling,
again with FIRL emission involved, is shown in Fig. 4
for the K ¼ 7A, s ¼ 3 substates. In this system, interlocking transitions are observed from both (A,gr,0,7)
and (A,gr,1,7) lower levels up to each of the three interacting (A,oh,0,7), (A,ri,1,7) and (A,co,1,7) upper
Fig. 4. CH3 OH transition diagram showing allowed and forbidden K ¼ 7A transitions to hybridized ðt6 ; tt Þ ¼ ð1; 0Þ, ðt7 ; tt Þ ¼ ð1; 1Þ, and
ðt8 ; tt Þ ¼ ð1; 1Þ levels coupled by anharmonic interactions. Solid arrows represent allowed transitions; dashed arrows are forbidden. Far-infrared
laser emission from optical pumping by the 18–9P (34) CO2 laser line is also shown with associated spectroscopic data [20]. FIR laser wavenumbers
are derived from the reported wavelengths in [35]; ground-state energies are from [1].
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R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
substates, consistent with mixing and hybridization of
those states. The 18–9P (34) isotopic C18 O2 laser line
coincides with the R(9) transition of the allowed
(A,ri,1,7) (A,gr,1,7) subband and pumps FIRL emission down to (A,ri,1,6) and (A,co,1,6) levels [20]. From
the line wavenumbers and ground-state energies given in
Fig. 4, all loop closure relations are found to be satisfied
to well within our measurement uncertainty, confirming
the assignments. Furthermore, the spectroscopic FIRL
line wavenumbers calculated by combination relations
from the data in Fig. 4 are 59.9166, 75.9445, and
136.2805 cm1 for lines La –Lc , respectively, in good
agreement with the values shown in Fig. 4 that are derived from the reported wavelength measurements [35].
In Table 2, the origin wavenumbers and effective B
values for the substates labeled as ðt6 ; tt Þ ¼ ð1; 0Þ and
ðt7 ; tt Þ ¼ ð1; 1Þ are presented. (Those for the ðt8 ; tt Þ ¼
ð1; 1Þ substates were included in our companion COstretching paper [33].) The ðt7 ; tt Þ ¼ ð1; 1Þ CH3 -rocking
levels lie from about 15 to 50 cm1 below their
ðt6 ; tt Þ ¼ ð1; 0Þ OH-bending counterparts, with the topmost ðt8 ; tt Þ ¼ ð1; 1Þ CO-stretching levels a further
5–20 cm1 below. Thus, the strength of the coupling
interactions must be of comparable magnitude, given
the substantial mixing that occurs. In general, this anharmonic coupling among the ðt6 ; tt Þ ¼ ð1; 0Þ, ðt7 ; tt Þ ¼
ð1; 1Þ, and ðt8 ; tt Þ ¼ ð1; 1Þ states acts to perturb and
spread the OH-bending energy structure, contributing to
the lack of observable detail in the low-resolution
spectrum of the m6 OH-bending fundamental.
In addition to the main group of tt ¼ 0 subbands of
the m6 fundamental from 1310 to 1350 cm1 , there are
eleven identified a-type subbands lying slightly higher
that have tt ¼ 1 lower states. These are tentatively assigned in Table 1 to the m6 þ m12 m12 OH-bending
torsional hot band on the basis of their spectral positions and the location and patterns of the upper state
energies.
4.5. Rotationally assigned subbands with unidentified
upper torsion–vibration states
Lying between the m11 and m6 fundamentals in the
spectral region from 1194 to 1247 cm1 is a substantial
group of a-type subbands with assigned lower levels
belonging to the tt ¼ 1 torsional state and a second
group with lower levels belonging to tt ¼ 2. As well,
there is a small cluster of apparently related subbands
with tt ¼ 0 lower states lying near 1370 cm1 , about
50 cm1 above their OH-bending m6 counterparts.
Knowing the torsion–rotation assignments of the lower
levels for these groups of subbands, we could accurately
determine the upper-state term values and obtain the
substate origins and B values. However, the torsion–vibration identity of the excited states is not obvious, and
we have chosen simply to label the three groups as U0 ,
U1 , and U2 in Tables 1 and 2. The subscripts refer to the
tt values of the lower levels of the main defining subband groups, but we note that in a number of cases there
are jDtt j > 1 torsional combination subbands accessing
the upper levels from other torsional states, so that there
is evidently considerable torsional mixing.
The entries in Table 2 include three individual U
substates and one that we assign to the m5 symmetric
CH3 -deformation mode. Substates UA and UB both derive from K ¼ 2A subbands with tt ¼ 1 lower levels and
show resolved asymmetry K-doubling, but neither appears to be related to other families. They lie close below
the (A,oh,1,2) torsionally excited OH-bending substate,
however, so there may be substantial interaction and
mixing. The UC substate belongs to a K ¼ 3A subband
with a tt ¼ 2 lower level, and lies at a high energy for
which we have little information as yet. The final
(A,sb,0,1) substate is interesting in that its origin of
1592.517 cm1 is encouragingly close to the prediction
of 1591.29 cm1 from the CH3 -deformation model in a
previous work [22], raising the hope that further m5
subbands may be located with the help of that model.
4.6. Guidance towards low-K substates from jet-cooled
spectra
Since the subband assignments frequently depend on
observation of the Q subbranches as significant clues,
and the a-type Q-branch relative intensities vary as K 2 ,
identification of the low-K subbands in the crowded
spectrum can be particularly challenging. It is here,
therefore, that the jet-cooled supersonic beam spectra
were of great value in simplifying the spectrum and exposing only those subbands originating from the low-K,
low-J states populated in the cold 10 K beam. In the
cooled spectra from 1275 to 1648 cm1 , we were able to
identify a substantial number of low-energy subbands,
as marked with asterisks in Table 1. While some of these
had previously been seen and tentatively assigned, their
observation in the jet provided important confirmation
of the identification. In other cases, the jet results were
crucial in locating the initial lines and determining the
assignments of new low-K subbands that could then be
followed to higher J in the room-temperature FTIR
spectra. Currently, the cooled-beam observations support the assignments for all ðt6 ; tt Þ ¼ ð1; 0Þ OH-bending
subbands up to K ¼ 4 plus K ¼ 5A, all ðt7 ; tt Þ ¼
ð1; 1Þ
ð0; 0Þ forbidden CH3 -rocking subbands up to
K ¼ 3 with the exception of the still-missing K ¼ 3E
subband, and the K ¼ 1E, 2E, and 4A
ðt8 ; tt Þ ¼ ð1; 1Þ
ð0; 0Þ subbands whose upper states
are closest to the highly mixed peak regions of the tt ¼ 1
CO-stretching s-curves. The K ¼ 0A, 1E, 2E, and 4A
subbands of the U0 family are also seen. Thus, nearly all
of the subbands that we could expect to find in the 10 K
cooled spectrum have indeed been detected.
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
5. Systematic s-curve energy patterns and substate
grouping into families
As seen in the energy map of Fig. 2, there are numerous overlapping torsion–vibration states for CH3
OH and individual s-curves can be difficult to pick out.
In this section, we will focus more closely on specific
regions of the energy manifold to look for systematics in
the substate distributions. By isolating the energies for
specific classes of substate, we find interesting regularities in behavior that form the basis for our choice of
substate grouping and our torsion–vibration labeling in
Tables 1 and 2.
5.1. The (t6 ; tt )¼(1; 0), (t7 ; tt )¼(1; 1), and (t8 ; tt )¼
(1; 1) region
The energy region from 1350 to 1500 cm1 contains
the ðt6 ; tt Þ ¼ ð1; 0Þ, ðt7 ; tt Þ ¼ ð1; 1Þ, and ðt8 ; tt Þ ¼ ð1; 1Þ
trio of coupled states. In Fig. 5, we have plotted the
experimental K-reduced substate origins for each value
Fig. 5. K-reduced torsion–vibration m6 , m7 þ m12 , and m8 þ m12 energy scurves for the coupled ðt6 ; tt Þ ¼ ð1; 0Þ, ðt7 ; tt Þ ¼ ð1; 1Þ, and
ðt8 ; tt Þ ¼ ð1; 1Þ states of CH3 OH, plotted separately for each value of
s. The dashed line in (A) for s ¼ 1 indicates a tentative connection of
the ðt6 ; tt Þ ¼ ð1; 0Þ curve to the K ¼ 0Aþ levels, for which the repulsive
interaction with the ðt7 ; tt Þ ¼ ð1; 1Þ and ðt8 ; tt Þ ¼ ð1; 1Þ states becomes
forbidden and is switched off.
539
of s separately in order to expose possible patterns in the
energies. In general, apart from a few irregularities, the
substates indeed appear to lie along systematic curves,
drawn in as solid lines in Fig. 5 in our proposed
grouping. The shapes of these s-curves differ markedly
from the regular oscillation seen for the lower states
[1,10,33], and in fact have a strong flavor of avoided
crossings at the points near K ¼ 0 and 8 for s ¼ 1, near
K ¼ 3 for s ¼ 2 and near K ¼ 5 for s ¼ 3. It is interesting that those are the very points in Fig. 1 where
calculated curves from other states approach closely
tangent to the OH-bending curves. The patterns suggest
a general picture of mutually repelling states, with the
ðt7 ; tt Þ ¼ ð1; 1Þ CH3 -rocking curves being squeezed in
between the ðt6 ; tt Þ ¼ ð1; 0Þ OH-bending and ðt8 ; tt Þ ¼
ð1; 1Þ CO-stretching curves, and yet-to-be-determined
ðt11 ; tt Þ ¼ ð1; 1Þ CH3 -rocking curves possibly playing a
role from above. Downward shifts of those
ðt8 ; tt Þ ¼ ð1; 1Þ levels located near the peaks of the COstretching curves have already been noted previously in
the literature [36], and account for an apparent reduction in torsional barrier height for the ðt8 ; tt Þ ¼ ð1; 1Þ
state when treated according to the traditional one-dimensional torsional model [37]. In general, it appears
that there is new perturbation physics at work in this
energy region that is creating new energy patterns. The
interesting thing will be to see whether these can be
explained by treating the ðt6 ; tt Þ ¼ ð1; 0Þ, ðt7 ; tt Þ ¼
ð1; 1Þ, ðt8 ; tt Þ ¼ ð1; 1Þ, and ðt11 ; tt Þ ¼ ð1; 1Þ states together as a coupled system with appropriate interaction
terms in a unified torsion–vibration Hamiltonian
[22,24–27].
The irregularities in Fig. 5 are of significance, particularly for the K ¼ 0Aþ states in the s ¼ 1-curves of
Fig. 5A. The (Aþ ,co,1,0) state lies at 1447.81 cm1 [1,33],
just underneath a second K ¼ 0Aþ state at 1448.45 cm1 ,
with a third lying rather higher at 1497.51 cm1 .
K ¼ 0Aþ states have definite parity, so torsion-mediated
coupling to certain other levels can be symmetry-forbidden. The rule for A level interactions between vibrational states of the same symmetry is that an Aþ level
must couple to another Aþ level for even changes Dtt in
torsional state, but can only couple to an A level for
interactions with odd Dtt . This was the origin of the
‘‘giant K-doubling’’ of the (A,ri,1,2) levels lying just
below the (Aþ ,co,1,0) levels, for example, with strong
interaction between the (Aþ ,co,1,0) and (Aþ ,ri,1,2) levels
repelling the latter downwards but leaving the (A ,ri,
1,2) levels untouched [8,38]. Similarly, while the
(Aþ ,co,1,0) and (Aþ ,ri,1,0) states will repel each other,
neither can influence the (Aþ ,oh,0,0) state which is thus
free to find its own location unaffected by perturbation.
Now given the apparent strong repulsion between the scurves, it seems very unlikely that the close pair of
K ¼ 0Aþ states in Fig. 5A could both be tt ¼ 1, implying
that the upper level should be the (Aþ ,oh,0,0)
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R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
OH-bending state (as drawn with a dashed line in
Fig. 5A to indicate the tentative nature of the assignment). The sharp rise in the ðt6 ; tt Þ ¼ ð1; 0Þ s ¼ 1-curve
when the repulsive interaction is switched on at K ¼ 1
would then suggest an upward perturbation of order 10–
20 cm1 in the OH-bending energy. This would be of
some significance in seeking to fit the methanol vibrational energies to a harmonic force field [2,3].
The question remaining, of course, is the whereabouts
of the (Aþ ,ri,1,0) substate. While the higher K ¼ 0Aþ
level at 1497.51 cm1 might look like a plausible candidate in Fig. 5A, the selection rules for observed FIR
laser lines rule it out [20], and furthermore the higher
level appears to belong to the U0 family discussed below.
Thus, there is yet another critical substate still to find
that is carrying crucial information about torsion–vibration coupling.
For the s ¼ 3-curves around K ¼ 5 in Fig. 5C, the
(E,co,1,)5) and (E,ri,1,)5) levels seem anomalously
close, and the (E,oh,0,)5) substate is still unaccounted
for. Note that our present (E,ri,1,)5) identification for
the upper level of the close K ¼ 5 pair in Fig. 5C represents a very recent vibrational reassignment from the
(E,oh,0,)5) labeling proposed earlier [20,33] which itself
was a reassignment of the (E,co,1,)5) labeling reported
in [1], illustrating the difficulty in interpreting this
complex energy region. Our choice for the (E,co,1,6)
substate has also just been revised from the previous one
[1,18,33], on the basis of our latest s-curve patterns, and
changed to the lower of the mixed K ¼ 6E levels in
Fig. 5C. However, the subbands to the two K ¼ 6E
levels are of very similar intensity, suggesting the levels
are highly mixed. For the subbands accessing the
K ¼ 4A levels, the relative intensity patterns and
K-doublet splittings are peculiar, and the (A,oh,0,4)
subband seems anomalously weak. Thus, further spectroscopic detective work and detailed analysis are still
needed for these key K ¼ 4A, 5E, and 6E s ¼ 3 states,
since they are expected to be the most sensitive to the
effects of the vibrational coupling.
Fig. 6. K-reduced torsion–vibration m6 þ m12 energy s-curves for the
CH3 OH substates assigned to the ðt6 ; tt Þ ¼ ð1; 1Þ torsionally excited
OH-bending family. The vibrational identity is not yet established for
the lower two extra K ¼ 2A substates for s ¼ 2, designated as UA and
UB , which likely perturb the (A,oh,1,2) substate upwards.
through this region at a sharp angle, as seen from Fig. 1,
and our predicted energy of 1686.05 cm1 for the
K ¼ 2A substate is virtually coincident with the UB energy in Table 2.
5.3. The U0 , U1 , and U2 groupings
The three families of U curves present one of the
more interesting and provocative assignment challenges
in the present energy region. Their K-reduced energies
are shown in isolation in Fig. 7 with our proposed scurve connections drawn in on the plots. On the face of
it, the energy patterns appear to be both highly systematic and closely related, with the tt ¼ 1 and tt ¼ 2
curves being strikingly similar in form and amplitude to
the corresponding curves for the CO-stretching state
[33]. We show the comparison for tt ¼ 2 in Fig. 8, and
the likeness is very persuasive, motivating our choice of
5.2. The (t6 ; tt )¼(1; 1) torsionally excited OH-bending
state
There is a clear family of s ¼ 2 substate origins just
below 1700 cm1 in Fig. 2 that we associate tentatively
with the ðt6 ; tt Þ ¼ ð1; 1Þ torsionally excited OH bend.
The pattern is shown in Fig. 6, together with related
origins from the s ¼ 1 and s ¼ 3 substates. The UA and
UB K ¼ 2A substates are also in the same vicinity, not
far below the s ¼ 2-curve, and may account for an apparent upward perturbation to the K ¼ 2A point on that
curve. Given the apparent isolation of the UA and UB
states, it seems very likely that one of them originates
from the K ¼ 2A, tt ¼ 4 level of the vibrational ground
state. The calculated tt ¼ 4 curve for s ¼ 2 rises up
Fig. 7. K-reduced torsion–vibration energy s-curves for the proposed
U0 , U1 , and U2 family groupings of CH3 OH substates. On the basis of
the close similarity of their patterns to the corresponding COstretching s-curves, the U1 and U2 families are believed to be associated
with tt ¼ 1 and 2 torsional states, respectively, but the vibrational
parentage is not yet established.
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
541
we believe there is a strong avoided crossing between the
two substates that switches their K character early in the
level sequences at low J . In Fig. 8, the K-reduced energy
for the (A; U2 ,2,4) s ¼ 3 substate lies about 31 cm1
above that of the (A,co,2,5) s ¼ 2 substate, and this
difference is almost exactly compensated by the extra
contribution for the latter from the K-rotational energy
of 3.45K 2 , creating the near-degeneracy. In Tables 1 and
2, the relevant subbands and U2 substate are labeled
according to their initial starting K values but one
should keep in mind the highly mixed character of the
actual levels.
6. Discussion and conclusions
Fig. 8. Comparison between the CH3 OH tt ¼ 2 K-reduced torsion–
vibration energy s-curves for the U2 substate family (solid lines) and
the ðt8 ; tt Þ ¼ ð1; 2Þ CO-stretching state ðm8 þ 2m12 dashed lines),
showing the strong similarity in form and amplitude.
the tt ¼ 2 label for the U2 family. However, the separation between the U1 and U2 s ¼ 1-curves at K ¼ 0 is
only 5.4 cm1 in Fig. 7, very much smaller than the
corresponding separation of 68.6 cm1 observed for the
CO stretch between the K ¼ 0A points for tt ¼ 1 and
tt ¼ 2. Thus, the torsional and vibrational identity of
the U substates is not clear, despite the apparently well
defined and systematic s-curves. The U1 family lies in the
same region as predicted for tt ¼ 0 levels of the CH3 deformation modes that have not yet been identified
[21,22], so might be associated with tt ¼ 0
1 torsional
combination subbands to those states.
The separation between the U0 and U1 curves in
Fig. 7 is also very small, and the amplitude of oscillation
of the U0 curves is substantially greater than expected
for a tt ¼ 0 torsional state. Another difficulty with
identification of the U0 grouping as a tt ¼ 0 family is the
lack of obvious vibrational candidates near that energy.
Thus, it is possible that ðt11 ; tt Þ ¼ ð1; 1Þ torsionally excited levels of the out-of-plane CH3 -rocking mode are
involved, for which the torsional structure might differ
substantially from the traditional picture judging from
the tt ¼ 0 behavior [9].
The U2 grouping does give partial insight now into
one previous spectral puzzle, namely the presence of the
(A; U2 ,2,4) state lying very close to the (A,co,2,5) level
and mixing with it to give the hybridized pair of states
labeled earlier as K ¼ 5A hd and hu [20,33]. Subbands
are observed to each of the mixed states from both
K ¼ 4A and 5A tt ¼ 2 ground-state levels [20] with unusual intensity patterns whose variations indicate an
interchanging of the K character at low J . The initial
lines of the Q-subbranches show clearly that the hd and
hu substates start out as K ¼ 4 and K ¼ 5, respectively,
but the relative subbranch intensities at higher J are
equally clear in implying the opposite K labeling! Thus,
In this work, we have investigated the FTIR spectrum of CH3 OH in the spectral regions associated
principally with the CH3 -rocking and OH-bending
modes and their excited torsional states. At present, we
are close to the culmination of the ‘‘data-gathering’’
phase, in which the great majority of the stronger
spectral features have been arranged into subbands
with assigned lower states. Analysis of the upper state
term values has yielded a detailed map of the torsion–
vibration energy manifold spanned by the subband
upper levels, in which the excited substates have largely
been classified into families. While the energy patterns
for the ðt7 ; tt Þ ¼ ð1; 0Þ in-plane and ðt11 ; tt Þ ¼ ð1; 0Þ
out-of-plane CH3 -rocking modes follow well-defined
oscillatory curves as a function of K, this is not the
case for the excited torsional levels. We find that the
tt ¼ 1 states are coupled to the OH bend by significant
resonances, as also has been predicted from ab initio
calculations [27]. The ðt7 ; tt Þ ¼ ð1; 1Þ in-plane CH3 rocking, ðt8 ; tt Þ ¼ ð1; 1Þ CO-stretching, ðt6 ; tt Þ ¼ ð1; 0Þ
OH-bending, and possibly ðt11 ; tt Þ ¼ ð1; 1Þ out-of-plane
CH3 -rocking states interact and mix together in a
complex energy region in which the patterns are quite
unlike the traditional picture. Instead of regular oscillations, the energies appear to lie along mutually repelling curves with strongly avoided crossings. Thus,
determination of the precise torsion–vibration identities
of the upper states is still an open question that represents an interesting challenge.
The present paper concerns what might be called the
‘‘macroscopic’’ view of the energy manifold giving an
overall picture of the torsion–vibration structure and
substate groupings. A further significant phase of the
investigation will be a more ‘‘microscopic’’ view in
which the J quantum number is brought in and all of the
individual rotation–torsion–vibration levels are mapped
as a function of J . As this extended study is actively in
progress, we have chosen not to report the experimental
term values and detailed spectral measurements in the
present paper, but will defer them for listing as supple-
542
R.M. Lees et al. / Journal of Molecular Spectroscopy 228 (2004) 528–543
mentary data to a subsequent report. An interesting
aspect of this ongoing work is the observation of numerous spectral perturbations to specific levels that arise
from intermode interactions. In addition to the overall
coupling among the ðt6 ; tt Þ ¼ ð1; 0Þ, ðt7 ; tt Þ ¼ ð1; 1Þ, and
ðt8 ; tt Þ ¼ ð1; 1Þ states discussed here, there are other
resonances localized in J and K that arise through accidental degeneracies within the dense network of
overlapping states when different series of levels cross
over each other as K or J increases. It is hoped that
cataloging and analysis of these intermode resonances
will bring additional insight to bear on the torsion–vibration character of the observed families of substates.
In particular, the observation of so many nominally
forbidden tt ¼ 1
0, 2
0, 2
1, and 0
1 subbands indicates that there is considerable torsional
mixing among the levels, analysis of which should aid
the understanding of torsion-mediated enhancement of
intramolecular vibrational energy redistribution (IVR)
processes.
The fact that the CH3 OH torsion–vibration energy
patterns differ so markedly from the expected behavior
in the energy region of the OH bend suggests that we
still have much to learn about torsionally excited vibrational states. There is a clear need to extend the scope
of the recent expanded torsion–vibration models [22,24–
27] to bring in higher torsional states as well as the Kdependence of the energies. The success of the Hougen
model [25] in reproducing the qualitative features observed for the two CH3 -rocking modes in their tt ¼ 0
torsional ground states suggests similar important advances will be achieved by extending the model to higher
values of tt . We look forward to forthcoming insights
from such extension in the future.
Acknowledgments
It is a great pleasure to acknowledge the many contribution of J.T. Hougen to the understanding of largeamplitude torsional motions in molecular spectra, and
to thank him for his interest in and encouragement of
the present investigation. A significant part of the work
was in fact carried out while R.M.L. and L.-H.X. were
guest researchers with him at the National Institute of
Standards and Technology (NIST) in Gaithersburg.
R.M.L. and L.-H.X. acknowledge financial support for
this research from the Natural Sciences and Engineering
Research Council of Canada. The United States Department of Energy supported part of the research, both
at NIST and at the W.R. Riley Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of EnergyÕs Office of
Biological and Environmental Research located at the
Pacific Northwest National Laboratory (PNNL).
PNNL is operated for the United States Department of
Energy by Battelle under Contract DE-AC06-76RLO
1830.
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