Download On the physical interpretation of torsion

JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 8
22 FEBRUARY 1999
On the physical interpretation of torsion-rotation parameters in methanol
and acetaldehyde: Comparison of global fit and ab initio results
Li-Hong Xua)
Department of Physical Sciences, University of New Brunswick, Saint John, New Brunswick E2L 4L5, Canada
Ronald M. Lees
Department of Physics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada
Jon T. Hougen
Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
~Received 19 October 1998; accepted 16 November 1998!
Equilibrium structural constants and certain torsion–rotation interaction parameters have been
determined for methanol and acetaldehyde from ab initio calculations using GAUSSIAN 94. The
substantial molecular flexing which occurs in going from the bottom to the top of the torsional
potential barrier can be quantitatively related to coefficients of torsion–rotation terms having a
(12cos 3g) dependence on torsional angle g. The barrier height, six equilibrium structural constants
characterizing the bottom of the potential well, and six torsion–rotation constants are all compared
to experimental parameters obtained from global fits to large microwave and far-infrared data sets
for methanol and acetaldehyde. The rather encouraging agreement between the Gaussian and global
fit results for methanol seems both to validate the accuracy of ab initio calculations of these
parameters, and to demonstrate that the physical origin of these torsion–rotation interaction terms in
methanol lies primarily in structural relaxation with torsion. The less satisfactory agreement
between theory and experiment for acetaldehyde requires further study. © 1999 American Institute
of Physics. @S0021-9606~99!00308-6#
(231024 cm21) to picohartree ~6 kHz! range, quantum
chemistry results on internal rotor systems should be tested
by comparison with experimental values in order to gauge
their degree of accuracy before proceeding further. Such
comparison is the main focus of the present paper.
In recent years, ab initio methods have been applied extensively to internal rotor molecules. For methanol, Florian
et al.14 have compared performance at different levels of
theory for determination of the equilibrium structure and harmonic force constants, while Chung-Phillips and Jebber15
have calculated the molecular geometries at the bottom and
top of the torsional barrier, in addition to the barrier height.
For acetaldehyde, a series of papers by Goodman and
co-workers16–19 serves to review the extensive literature on
barrier calculations for this molecule, and demonstrates the
important effects of structural flexing during torsion on both
the barrier height and barrier shape. Current ab initio treatments are thus well developed in terms of torsional barrier
calculations, and they have clearly established that the largeamplitude torsion is accompanied by significant structural
relaxation for both methanol and acetaldehyde. Here we wish
to push comparison with experiment to the next level beyond
the torsional barrier potential, and investigate the degree to
which ab initio results for torsional flexing can lead to quantitative agreement with experimentally determined torsion–
rotation distortional parameters.
Global least-squares fits of several thousand torsion–
rotation levels for acetaldehyde6,7 and for a number of isotopomers of methanol,8,20,21 essentially to within measurement
uncertainty, require the use of 55 to 65 molecular parameters
I. INTRODUCTION
This work is part of an ongoing frequency domain study
of torsionally assisted intramolecular vibrational energy redistribution in methanol and acetaldehyde, which includes as
one of its intermediate goals the ability to carry out global
fits approaching experimental accuracy for high resolution
infrared spectra of low-lying vibrational fundamentals embedded in the manifold of torsional bath states. Current theoretical formalisms in the literature which discuss in any
detail the various interactions among the small-amplitude vibrations, the large-amplitude internal rotation, and overall
rotation for an N-atom molecule ~for example, Refs. 1–3! all
contain large numbers of interaction parameters. Even for the
purely vibrational problem, there appears to be little possibility of determining from experimental data alone all interaction parameters @e.g., the ~3N-6!~3N-7!/2 on-diagonal and
off-diagonal small-amplitude vibrational force constants at
the bottom, middle, and top of the torsional potential barrier#. Thus, theoretical input is needed, and ab initio quantum
chemistry calculations are an obvious source of help to explore structural changes and variation of the vibrational potential surface as a function of torsional angle. These structural changes can be related to certain of the interaction
parameters
appearing
in
current
torsion–rotation
Hamiltonians,4–13 giving insight into the physical origins of
these parameters. Because modern high resolution spectroscopy yields data with relative accuracies in the nanohartree
a!
Author to whom correspondence should be addressed.
0021-9606/99/110(8)/3835/7/$15.00
3835
© 1999 American Institute of Physics
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3836
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Xu, Lees, and Hougen
each, but the ratio of energy levels to parameters is often as
high as 30 or 40, indicating that the effective onedimensional large-amplitude Hamiltonian used is a rather
good model for the quantum mechanical torsion–rotation
system. However, because the full torsion–rotation Hamiltonian must be reduced in practice by eliminating or constraining certain terms, there may be significant extraneous
contributions to various parameters which vary with the
choice of reduction scheme. The question then arises of
whether a given parameter returned by the least-squares procedure indeed has the meaning directly associated with the
operator it multiplies, or does it act to some extent as a
catchall containing contributions from a large number of unknown sources?
In the present paper we investigate this question by comparing values for some of the parameters obtained from global fits of large experimental data sets with values obtained
from ab initio calculations for methanol and acetaldehyde,
both of which contain methyl rotors with relatively low torsional barriers. In particular, we consider the barrier height
V 3 and the six structural terms entering the effective onedimensional large-amplitude Hamiltonian in the form
@ C 0 1C 3 ~ 12cos 3 g !#@ P 2 , P 2a , P 2b 2 P 2c , P a P b
1 P b P a , P g2 , or P a P g # ,
~1!
i.e., in the form of a g-dependent coefficient C( g ) multiplying a quadratic product of the rotational ( P a , P b , P c ) and/or
torsional ( P g ) angular momentum operators. If the minimum
of the torsional potential corresponds to g 50° and the maximum corresponds to g 560°, we see that C 0 gives the value
of the coefficient of the operator at the bottom of the well,
while C 0 12C 3 gives the value for that same coefficient at
the top of the barrier. ~We note in passing that an alternative
expansion of the form @ C 80 1C 83 cos 3g# @quadratic operator#
gives a value of C 80 1C 83 at the bottom of the well and C 80
2C 83 at the top. Such an expansion seems less appropriate
than Eq. ~1! for a discussion primarily of levels below the
top of the barrier, and will not be considered further here.!
The principal reason for limiting consideration to V 3 and
parameters of the form of Eq. ~1! lies in the fact that experimental comparison is possible using quantum chemistry information for only the two stationary points corresponding to
the minimum and maximum of the torsional potential curve.
This eliminates all complications associated with calculations at nonstationary points along some path connecting
these two extrema, and bypasses the question of the precise
relationship between the intrinsic reaction coordinate ~IRC!
of quantum chemistry and the internal rotation angle ~g! of
spectroscopic fitting Hamiltonians.16 Such calculations will,
however, be necessary to obtain information on parameters
which vary as sin 3g, (12cos 6g), etc.
II. COMPUTATIONAL DETAILS
The quantum chemistry calculational tool selected for
use in this work is the GAUSSIAN 94 commercial ab initio
package,22,23 referred to as G94 in the remaining text, installed on the NIST Cray Y-MP-C90 computer. G94 commands used in the command-line argument for the various
calculations of the present work are shown in bold face below. In both the methanol and acetaldehyde calculations, we
allowed full structural relaxation with no symmetry constraints applied to the internal coordinates in our Z-matrices.
Full geometry optimization was carried out using the very
tight ~VTight! convergence criterion. The G94 commands
Opt and Opt5„QST3… were used, respectively, to obtain
geometries at the bottom and top of the torsional potential
well. The basis set was 6-3111G„3df,2p… and the theoretical
level was MP2. Single-point, SP, energy calculations at the
MP4 level were also performed using geometries optimized
at the MP2 level. For both molecules, the two stationary
points on the torsional potential curve corresponding to the
bottom and the top of the well were examined with three
types of calculations, namely: the geometry optimizations at
MP25Full/6-3111G„3df,2p)
Opt5„VTight…
and
Opt5„QST3,VTight…, the single-point energy calculations
MP45Full/6-3111G„3df,2p… SP with MP2 optimized geometries, and the harmonic frequency calculations MP25„Full,FullDirect…/6-3111G„3df,2p… Freq
5HPModes.
In addition, structural changes due to methyl-group torsion were carefully monitored at approximately 1.5° steps in
the internal rotation angle g along the intrinsic reaction path
~IRC! at the MP2/6-3111G„3df,2p… level using the VTight
convergence criterion. Such information is needed in order to
determine those torsion–rotation parameters having a sin 3g
or (12cos 6g) torsional dependence. Although visual
graphical examination cannot distinguish between the IRC in
G94 and the g in traditional internal rotation formalism, we
will not discuss information along the IRC path any further
in the present paper, postponing such discussion until we
fully understand the precise mathematical relationship between the IRC and g.
III. MOLECULAR STRUCTURES AT THE BOTTOM
AND THE TOP OF THE TORSIONAL WELL
For both methanol and acetaldehyde, we used internal
coordinates ~12 for methanol, 15 for acetaldehyde! to set up
the Z-matrices. Full geometry optimizations with no symmetry restrictions applied ~minimization for the three equivalent
local minima at the bottom of the potential well, and optimization for the three equivalent transition states at the top of
the well! were performed in the G94 default system of redundant internal coordinates. These optimized geometries at the
bottom and the top of the torsional barrier were output in
internal coordinates and also in Cartesian coordinates for
both input and standard orientations. Table I, which shows
the internal coordinates for methanol and acetaldehyde at
one minimum and one transition-state configuration, with the
three redundant methyl-top HCH angles included for completeness, illustrates the following three points. ~i! Our results are essentially identical to those for methanol in Refs.
14 and 15, where the same basis set was used with an earlier
version of Gaussian, but differ slightly from those for acetaldehyde in Ref. 18, where a slightly different basis set was
investigated. ~ii! In general, our calculated bond lengths vary
by less than 0.5% from the bottom to the top of the torsional
barrier, while the methyl deformation angles change by up to
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
TABLE I. Ab initio molecular structures from
torsional potential barrier.
Xu, Lees, and Hougen
GAUSSIAN 94
in internal coordinates ~Å and deg! for methanol and acetaldehyde at the bottom and top of the
Methanol
a
Coordinate
Definition
b
r OH
r CO
r CH4
r CH5
r CH6
R~2,1!
R~3,2!
R~3,4!
R~3,5!
R~3,6!
a COH
A~3,2,1!
a H4CO
a H5CO
a H6CO
d4
d5
d6
A~4,3,2!
A~5,3,2!
A~6,3,2!
D~4,3,2,1!
D~5,3,2,1!
D~6,3,2,1!
b H4CH5
b H5CH6
b H6CH4
A~4,3,5!
A~5,3,6!
A~6,3,4!
Bottom
0.957 554 5
1.414 694 6
1.089 637 0
1.084 066 1
1.089 637 3
3837
Acetaldehyde
Top
0.955 491 0
1.418 248 2
1.087 233 9
1.087 233 9
1.087 439 4
108.585 52
109.047 13
112.054 11
106.705 69
112.054 08
61.448 99
179.998 91
261.451 16
109.529 00
109.529 00
112.217 74
120.441 09
2120.441 25
20.000 08
108.440 8
108.440 8
109.002 5
108.692 1
108.400 2
108.400 2
Coordinate
r CH
r CC
r CH4
r CH5
r CH6
r CO
a CCH
a CCO
a H4CC
a H5CC
a H6CC
d4
d5
d6
d
b H4CH5
b H5CH6
b H6CH4
a
Definition
b
R~2,1!
R~3,2!
R~3,4!
R~3,5!
R~3,6!
R~2,7!
A~3,2,1!
A~3,2,7!
A~4,3,2!
A~5,3,2!
A~6,3,2!
D~4,3,2,1!
D~5,3,2,1!
D~6,3,2,1!
D~7,2,3,6!
A~4,3,5!
A~5,3,6!
A~6,3,4!
Bottom
Top
1.104 747 3
1.493 761 5
1.089 147 1
1.089 147 1
1.084 162 1
1.209 586 6
115.503 25
124.502 72
109.355 97
109.355 98
110.802 84
258.359 10
158.359 07
180.000 00
0.000 00
106.879 7
110.179 7
110.179 7
1.103 606 1
1.500 619 0
1.085 045 6
1.087 457 2
1.087 457 2
1.209 596 9
116.903 08
123.210 88
112.004 21
109.111 33
109.111 35
10.000 12
1121.441 55
2121.441 34
58.558 64
109.530 2
107.443 0
109.530 2
Bond lengths, interatomic angles, and dihedral angles are denoted by r, a, and d, respectively. Bond lengths are in Å; bond and dihedral angles are in degrees.
The methyl top HCH angles b are redundant coordinates used by GAUSSIAN 94. Note the small departures from C s symmetry for the methyl group, which give
a measure of the Gaussian output accuracy for our basis set, calculation method, convergence criteria, etc.
b
Atom numbering for methanol is H1, O2, C3, H4, H5, and H6; that for acetaldehyde is H1, C2, C3, H4, H5, H6, O7. Calculated structures with the particular
methyl rotor configurations shown here were chosen to best illustrate G94 deviations from C s symmetry.
a
5%. The COH bending angle in methanol changes by less
than 0.5% between the two conformations, indicating that
the water-like structure is relatively persistent in CH3 –O–H.
Nevertheless, these Gaussian results show that there are substantial structural changes between the bottom and the top of
the torsional potential barrier, and that the methyl tops deviate significantly from threefold symmetry. This then raises
the question in the present context of how properly to define
the direction of the ‘‘methyl-top axis,’’ which is required for
both the internal-axis ~IAM! and rho-axis ~RAM! formulations of the torsional problem given in the literature.24,25 ~iii!
The absolute accuracy of the G94 structural information presented in Table I is still an open question since the accuracy
of the experimental structures26,27 is uncertain. Some indication of the precision of the calculations can be obtained,
however, by examining deviations from C s symmetry for the
calculated structures shown in Table I, which were deliberately selected to illustrate this point. Examination of symmetry related bonds and angles in Table I indicates that MP2
structures under the very tight convergence criterion using a
6-3111G(3d f ,2p) basis set should have a precision of
6331027 Å for bond lengths and 6131023 degrees for
bond angles. This translates into relative precisions of 1023
to 1024 for changes in interatomic distances upon going
from the bottom to the top of the barrier, which should be
adequate for the torsion–rotation parameter comparison purposes of this paper.
GAUSSIAN 94 also provides the optimized structures to six
decimal places in Cartesian coordinates in axis systems designated as ‘‘input’’ and ‘‘standard’’ orientations. While the
actual physical definition of the input and standard orientations is not clear to us, it should be noted that neither is a
center of mass system. Therefore, in order to proceed further,
we first transformed to the principal axis system ~PAM!. In
contrast to Table I, PAM coordinates for our most fully converged ~and therefore presumably our best! calculated configurations are presented in Table II. For later convenience,
the configurations of both molecules at the bottom of the
well have been oriented so that the approximate methyl-top
axis, i.e., a vector from the center of mass of the H3 plane at
the base of the methyl pyramid to the C atom at the apex,
points nearly in the direction of the positive a axis, and lies
in the (a.0, b.0! quadrant. The PAM moments of inertia
for the bottom and top of the torsional potential barrier were
then further rotated to an appropriate coordinate system in
which comparison could take place between the G94 ab initio
results and our experimental global fit parameters, as discussed below in Sec. V.
IV. COMPARISON OF THE V 3 TORSIONAL
POTENTIAL BARRIER HEIGHTS
In the Gaussian calculation, geometry optimization is
terminated when all forces ~the first derivatives of the energy
with respect to the internal coordinates! and all atom displacements calculated for the next iteration are zero to within
the convergence limits chosen. When this criterion is
reached, the optimized structural parameters together with
the stationary-point energy are output. At these MP2
stationary-point geometries additional single-point energy
calculations were performed at the MP4 level, as shown in
Table III. As in Table II ~but again in contrast to Table I!, the
most symmetrical of the three equivalent bottom and top
geometries were used here for maximum accuracy.
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output. Atomic masses used
GAUSSIAN 94
Cartesian coordinates in the center of mass principal axis system were obtained by appropriate translation and rotation of the input or standard coordinates given directly in the
in the transformations were 1.007 825, 15.994 915, and 12.0 u for hydrogen, oxygen, and carbon, respectively.
a
11.532 399
10.431 102
20.159 961
20.792 852
20.792 853
10.604 595
20.238 156
0.000 000
0.000 000
0.000 000
10.874 838
20.874 838
0.000 000
0.000 000
11.522 370
10.418 637
20.166 138
10.188 584
10.188 585
21.249 470
20.230 395
10.825 629
20.063 938
10.013 431
10.517 743
10.517 743
21.006 285
11.041 489
10.687 116
20.725 461
21.107 591
21.107 591
21.093 401
H1
O2
C3
H4
H5
H6
0.000 000
0.000 000
0.000 000
10.887 104
20.887 104
0.000 000
11.049 604
10.688 759
20.727 358
21.118 706
21.082 765
21.118 706
10.819 466
20.065 268
10.012 450
20.486 014
11.040 172
20.486 015
0.000 000
0.000 000
0.000 000
10.883 435
20.000 001
20.883 434
H1
C2
C3
H4
H5
H6
O7
10.168 385
10.121 031
21.253 509
21.796 713
21.796 714
21.211 063
11.141 745
10.197 279
10.125 926
21.253 387
21.369 531
21.369 531
22.023 308
11.133 507
b
c
b
b
b
a
Atom
Bottom
c
a
Top
c
Atom
a
a
Top
Bottom
Acetaldehyde a, b, and c coordinates
Methanol a, b, and c coordinates
TABLE II. Atomic Cartesian coordinates ~in Å! in the a, b, c principal-axis systems for methanol and acetaldehyde at the bottom and top of the torsional potential barrier.a
0.000 000
0.000 000
0.000 000
20.876 654
10.876 654
0.000 000
0.000 000
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
c
3838
Xu, Lees, and Hougen
At first glance, one might think that the G94 barrier
height V 3 value would just be equal to the difference between energy values for the two stationary points on the
torsional potential. However, in order to ensure a valid comparison between the G94 results and the global fit experimental values,7,8 the zero-point energy ~ZPE! corrections for all
of the small-amplitude vibrations must be taken into
account19 since the vibrational frequencies will change with
the variation of potential energy surface from bottom to top
of the barrier. The ZPE corrections equal one-half the sum of
the 3N-7 small-amplitude vibrational wavenumbers. With
the inclusion of the ZPE corrections, as shown in Table III,
the agreement between the G94 and global fit barrier height
values is excellent for both methanol and acetaldehyde. The
values differ for methanol by only 0.3%, for acetaldehyde by
2.5%. ~Note that the ZPE corrections in Table III were derived directly from the harmonic frequencies calculated by
G94, and were not scaled.!
V. COMPARISON OF STRUCTURAL AND
TORSION–ROTATION PARAMETERS
There are six important structural parameters in our
RAM torsion–rotation global fit calculations, namely: the
three diagonal rotational constants A, B, and C, generally
occurring in the Hamiltonian as the linear combinations (B
1C)/2, A2(B1C)/2, and (B2C)/2; the off-diagonal rotational constant D ab ; the reduced torsional constant for the
methyl top F; and a moment of inertia ratio usually called
r.24 If we imagine expanding each of these six terms according to Eq. ~1!, then the six symbols above would correspond
in the customary Hamiltonian to the C 0 coefficients, i.e., to
the values of the terms at the bottom of the torsional well.
The associated C 3 coefficients of the (12cos 3g) terms in
the Fourier expansions are denoted in the Hamiltonian as
F v , k 5 , c 2 , d ab , k 7 , and k 6 , respectively.4,5,7,8
Parameter comparison must take place in the same axis
system. In our case the six structural parameters of interest
from the G94 calculations must be properly transformed to
the RAM system in order to compare with the global fit
values. Traditional formulas in the literature24 define the A,
B, C, D ab , F, and r parameters in the RAM system in terms
of the moments of inertia I a , I b , I c , and I ab calculated in a
top-axis system ~TAM! in which the a axis is defined to be
parallel to the symmetry axis of the methyl top. However, it
is not obvious how to define this top-axis system when, as
the G94 results clearly show, the methyl top is no longer
symmetric and the unique in-plane methyl hydrogen differs
from the two out-of-plane hydrogens. We sidestepped this
important question for the present work and simply tried two
reasonable approximations for the TAM system in which: ~i!
the top axis was assumed to lie along the vector pointing
from the center of mass of the three methyl hydrogen atoms
to the methyl carbon atom, and ~ii! the top axis was assumed
to lie along the normal to the plane of three methyl hydrogen
atoms erected at their center of mass. With the coordinates of
Table II, the two schemes differ by only 0.03° at the bottom
and 0.2° at the top of the barrier for methanol and by about
2.5° and 1.5° for acetaldehyde, so either scheme should give
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
Xu, Lees, and Hougen
3839
TABLE III. MP4-level total energies and zero-point vibrational energies ~ZPE! at the bottom and top of the torsional potential barrier and comparison between
GAUSSIAN 94 and global fit barrier heights for methanol and acetaldehyde.
Methanol
Acetaldehyde
a
b
Parameter Units
E MP4
ZPE
~ZPE
Barrier
c
Hartree
Hartreec
cm21!
cm21
a
Global Fit
GAUSSIAN 94
Global Fitb
GAUSSIAN 94
Bottom
Top
Bottom
Top
2115.595 543 46
0.051 599 80
~11 324.847!
2115.593 934 82
0.051 702 75
~11 347.441!
375.649
2153.656 547 33
0.055 842 88
~12 256.096!
2153.654 796 17
0.055 902 66
~12 269.215!
397.455
374.6~2!
407.530~12!
barrier height5(E MP41ZPE) top2(E MP41ZPE) bottom .
Global fit barrier height5V 3 1V 9 . For methanol, V 3 5373.594(7) cm21, V 9 51.0(2) cm21 ~Ref. 8!; for acetaldehyde, V 3 5407.716(10) cm21, V 9 5
20.186(2) cm21 ~Ref. 7!.
c
1 Hartree5219 474.631 cm21.
a
GAUSSIAN 94
b
an acceptable inertia tensor for the purposes of comparison
with global fit results.
In Table IV we compare our experimental global fit parameters with values obtained from the ab initio G94 calculations for choice ~i! of the TAM axes above. Agreement in
the first six rows of Table IV, at the few percent level or
better for five of the structural parameters, is rather encouraging, although the 60% disagreement for the small D ab
value in methanol is disappointing. It should also be recalled,
however, that the G94 results are strictly for the equilibrium
structure with no small-amplitude vibrational effects incorporated, while the global fit results represent averages over
the zero-point motions of the small-amplitude vibrations.
Contributions of vibrational averaging to observed rotational
constants ~i.e., the a rotation–vibration constants! are typically negative, implying that agreement between the G94 and
global fit results would be even closer if we knew the a
values for all of the vibrational modes and could convert our
global fit parameters to equilibrium values. A comparison of
three common linear combinations of the rotational constants
is given in the last three rows of Table IV. As might be
expected, agreement for the small difference parameter (B
2C)/2 is significantly worse than for the two larger linear
combinations. Nevertheless, apart from the off-diagonal rotational constant D ab , the agreement achieved here for the
equilibrium geometrical parameters is comparable to that obtained for molecules of similar size which do not exhibit any
large-amplitude vibrational motions, giving some confidence
in the reliability of the G94 results.
Agreement is not so encouraging for the higher-order
parameters of the global fitting Hamiltonian shown in rows
7–12 of Table IV, which correspond to the C 3 coefficients of
the (12cos 3g) operator in Eq. ~1!. From the perspective of
G94 calculations, these parameters are ascribed solely to
changes in the molecular structure as the methyl top rotates
from the bottom to the top of the torsional barrier, e.g., F v is
TABLE IV. Comparison between ab initio GAUSSIAN 94 results and experimental global fit values of structural and torsion–rotation interaction parameters for
methanol and acetaldehyde.
Methanol
Parameter
a
Operator
Global fit
A
B
C
D ab
F
r
P 2a
P 2b
P 2c
$ P a , P b%
P g2
PgPa
4.253 724~2!
0.823 5767~2!
0.792 5390~3!
20.004 171~4!
27.646 82~2!
0.810 20601~1!
4.320 664
0.833 551
0.804 531
20.006 590
27.684 707
0.806 529
Fv
k5
c2
d ab
k7
k6
(12cos 3g)P2
(12cos 3g)P2a
(12cos 3g)(P2b2P2c )
(12cos 3g)$Pa ,Pb%
$ (12cos 3g),Pg2 %
$ (12cos 3g),PgPa%
20.002 38796~8!
0.011 183~1!
20.000 0760~3!
0.009 048~2!
0.0 ~fixed!
0.0 ~fixed!
20.002 061
0.010 455
0.000 055
0.012 370
0.042 237
0.080 024
A2(B1C)/2
(B1C)/2
(B2C)/2
P 2a
P2
P 2b 2 P 2c
3.445 666
0.808 058
0.015 519
GAUSSIAN 94
3.501 623
0.819 041
0.014 510
Acetaldehyde
Ratio
b
1.016
1.012
1.015
1.58
1.001
0.995
0.86
0.93
20.72
1.37
1.016
1.014
0.93
Global fit
c
GAUSSIAN 94
1.885 1594~68!
0.348 7056~3!
0.303 18096~4!
20.122 636~2!
7.599 7~28!
0.331 6~1!
1.907 776
0.347 362
0.305 615
20.099 959
7.907 150
0.338 445
0.558 91(6)31023
20.038 0~3!
0.208 78(8)31023
0.209 96(6)31022
20.016 3~14!
20.033 512 ~fixed!
2.444 5131023
20.025 714
1.959 1831023
21.957 4831022
20.083 583
20.148 280
1.559 216
0.325 943
0.022 762
1.581 288
0.326 488
0.020 873
Ratiob
Ratiod
1.012
0.996
1.008
0.82
1.040
1.021
1.010
1.005
1.008
1.000d
1.018
0.994
4.4
0.68
9.4
29.3
5.1
4.4
1.014
1.002
0.92
1.35
0.54
1.28
1.000d
0.29
0.86
1.011
1.007
0.988
Values of the parameters from Ref. 8, in cm21, except for r, which is unitless. F and r enter the Hamiltonian as F( P g 1 r P a ) 2 . $ A,B % wAB1BA.
Ratio5GAUSSIAN 94/global fit.
c
Values of the parameters from Ref. 7 in cm21, except for r, which is unitless.
d
This ratio5Adjusted GAUSSIAN 94/global fit, where the angle between the principal a axis and the r axis has been empirically adjusted to give perfect
agreement with the global fit values of D ab and d ab ~see text!.
a
b
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3840
J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
viewed as arising entirely from structural corrections to (B
1C)/2 as internal rotation occurs, and any contributions to
F v from centrifugal distortion corrections to the barrier
height as overall rotation occurs are ignored. The relatively
good agreement between the experimental and ab initio G94
values for F v , k 5 , and d ab in methanol suggests both that
the G94 results are realistic and that the physical origin of
these parameters is indeed found primarily in flexing of the
molecular structure with torsion. In particular, the large
value of d ab for methanol, which at first glance appears surprising for a supposedly higher-order parameter, is well
modeled by G94 and shown to originate from a substantial
variation, and in fact a change in sign, of the sensitive I ab
product of inertia upon going from the bottom to the top of
the barrier. The wrong sign obtained for c 2 in methanol may
be a warning that even rough estimates of higher-order parameters of such small magnitude cannot yet be obtained
from ab initio calculations.
The general validation of the Gaussian results for methanol in Table IV has the important implication that G94 values
should be usable with some confidence for parameters such
as k 6 or k 7 which need to be constrained in the global fits of
experimental data under certain Hamiltonian reduction
schemes.9,28 Fixing k 6 and k 7 to values more physically
meaningful than zero should in turn improve the physical
significance of other parameters returned by the fit.
For acetaldehyde, agreement for the six higher-order parameters in Table IV is not really at a usable level. The
obvious possible generic explanations are: ~i! the global fit
numbers are not reliable; ~ii! the GAUSSIAN 94 numbers are
not reliable; ~iii! the global fit and Gaussian numbers represent physically different quantities. Comparison of global fit
parameters in Ref. 7 from a fit restricted to v t <2 with those
from a fit with v t <4, suggests that these parameters are
reliable, since they are stable with respect to introduction of
new energy level information. The GAUSSIAN 94 results seem
reliable for methanol, so any failure in acetaldehyde would
have to be related to special problems associated with the
additional six electrons or the carbonyl double bond. As
mentioned above, however, values for all six of these higherorder parameters will be determined both from internalrotation-induced structural changes in various moments of
inertia and from overall-rotation-induced centrifugal-force
contributions to the barrier height, so that large contributions
of the latter type could account for the large discrepancies in
Table IV.
As a purely empirical observation, it is possible to arbitrarily change the angle between the principal a axis and the
r axis ~i.e., between the principal a axis and the z axis actually used in a r axis method treatment! at both the bottom
and the top of the barrier for these two molecules in such a
way that the GAUSSIAN 94 D ab and d ab values agree exactly
with the global fit values. When this is done for methanol,
involving a decrease in the magnitude of the r axis angle of
0.04° at the barrier minimum and 0.07° at the barrier top, no
significant improvement in the other parameters occurs.
When a similar procedure is carried out for acetaldehyde,
involving an increase in the magnitude of the r axis angle of
0.83° at the barrier minimum and a decrease of 0.78° at the
Xu, Lees, and Hougen
barrier top, a dramatic improvement occurs. As shown in the
last column of Table IV, agreement for the other five equilibrium parameters after this adjustment is better than 2%;
agreement for the other five higher-order parameters ranges
from 15% to 72%, with all signs correct. We do not yet
know if this empirical observation has any physical significance.
VI. DISCUSSION AND CONCLUSIONS
In this work, we have employed ab initio calculations for
methanol and acetaldehyde to explore changes in energy and
molecular structure as the methyl top undergoes internal rotation. In particular, by taking differences in the values for
six molecular parameters calculated from GAUSSIAN 94 structures at the minimum energy and maximum energy stationary points of the barrier, the significant molecular flexing
with torsion observed here and in quantum chemistry studies
prior to our own could be quantitatively converted into Fourier coefficients suitable for comparison with experiment.
For methanol ~though not for acetaldehyde! values for the
larger terms match well with those from global fits in the
literature, strongly suggesting that the physical origin of
these higher-order terms lies primarily in structural relaxation of the molecule with torsion. This comparison is quite
encouraging, and the good correspondence between ab initio
and experimental values for the barrier height, five of the six
equilibrium structural parameters, and five of the corresponding higher-order torsion–vibration interaction parameters multiplying operators of the form (12cos 3g)
3(a quadratic angular momentum operator) implies that
GAUSSIAN 94 may be a useful predictive tool for studies of
torsionally mediated intramolecular vibrational energy redistribution in methanol.
In order to make further progress in comparing ab initio
and experimental results for terms involving (12cos 6g),
(12cos 9g) and sin 3g torsional dependences and to extend
the study to other internal rotor molecules, at least three
questions must be further investigated. ~i! What is the precise
mathematical relation between the intrinsic reaction coordinate ~IRC! used by GAUSSIAN 94, which is defined by properties of the potential energy surface, and the internal rotation angle g found in traditional fitting Hamiltonians, which
is defined by a set of constraint equations arising from a
desire to simplify the kinetic energy operator? ~ii! What is
the precise direction of the top axis ~and therefore how is the
top axis system, and then subsequently the r axis system,
defined! when the methyl top is clearly distorted ~as here!
from C 3 v symmetry? ~iii! How are the fitting parameters at
each order affected by the implicit choices for various contact transformations associated with setting undeterminable
fitting parameters to zero?
The first two questions can perhaps best be dealt with by
abandoning the traditional internal rotation formalism, where
the qualitative thinking is limited by its strong ties to geometrical ideas associated with the simple model of two rigid
blocks rotating about a rod connecting them, and replacing it
with a more powerful large-amplitude motion formalism,3,29
where, for example, the direction of the rod ~i.e. the top axis!
is replaced by the direction of the angular momentum vector
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J. Chem. Phys., Vol. 110, No. 8, 22 February 1999
actually generated during internal rotation of the flexing
body. ~Such a large-amplitude motion treatment would presumably also lead to a slightly altered direction for the r
axis, which might in turn justify the empirical adjustment
scheme discussed above in connection with the last column
of Table IV.! While it is appealing to imagine tackling the
third ~contact transformation! question algebraically, it may
in the end be simpler ~as confidence in ab initio values
grows! to fix a suitably chosen set of undeterminable parameters to their ab initio values ~rather than to zero! before
beginning any spectroscopic fitting procedures.
While the precise relationship between the IRC coordinate and internal rotation angle g remains to be worked out,
we can assume that these two quantities must be almost identical based on their linear graphical dependence. This assumption has motivated us to move beyond the two stationary points on the torsional potential in methanol and begin a
detailed investigation of the variation in molecular structure
and vibrational force field along the full IRC path from the
top to the bottom of the barrier. Such information will then
permit determination of further torsion–vibration interaction
terms involving sin 3g, (12cos 6g), etc., torsional dependences. By combining these results with calculations of contributions to the higher-order interaction constants from the
kinetic energy operators, we hope to develop much fuller
theoretical insight into the physical interpretation of the
ground-state Hamiltonian, and to begin modeling those
small-amplitude vibrational interactions which are significantly influenced by the internal rotation motion.
ACKNOWLEDGMENTS
This research was financially supported by the Natural
Sciences and Engineering Research Council of Canada, the
University of New Brunswick Research Fund, and the Division of Chemical Sciences, Office of Basic Energy Sciences,
Office of Energy Research, U.S. Department of Energy. The
authors are grateful to Drs. J. E. Bertie, L. B. Harding, R. D.
Johnson III, and M. A. Mekhtiev for numerous helpful discussions at various stages during the course of this work.
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Certain commercial products are identified in this paper in order to specify
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