Mon. Not. R. Astron. Soc. 406, 95–101 (2010)
doi:10.1111/j.1365-2966.2010.16671.x
The rotational excitation of methanol by molecular hydrogen
Djamal Rabli and D. R. Flower
Physics Department, The University, Durham DH1 3LE
Accepted 2010 March 11. Received 2010 March 11; in original form 2010 February 3
ABSTRACT
We have computed cross-sections and rate coefficients for the rotational excitation of A- and
E-type methanol by molecular hydrogen. Calculations were performed for rotational transitions
within the torsional ground state, ν = 0, and within the first and second excited torsional states,
ν = 1 and ν = 2. For collisions of methanol with para-H2 in its rotational ground state, j2 = 0,
the methanol basis included rotational states j1 ≤ 15, thereby extending previous calculations,
which included states j1 ≤ 9 only. For the first time, calculations have also been performed
for ortho-H2 in its rotational ground state, j2 = 1, although it was necessary to revert to the
smaller methanol basis, j1 ≤ 9, owing to the coupling to states of molecular hydrogen with
j2 > 0. The coupled states approximation was used in the production calculations, to generate
the thermal rate coefficients at temperatures 10 ≤ T ≤ 200 K, but some limited comparisons,
at a few collision energies, of cross-sections obtained using the full coupled channels (CC)
method have been made. The propensities of the collisions, with respect to changes in the
rotational quantum number, j1 , and its projection, K, on the symmetry axis of the methanol
molecule, were investigated. There are qualitative differences between the K propensities for
collisions with ortho- and para-H2 , which relate to the fact that the inelastic cross-sections
tend to be significantly larger when ortho-H2 is the perturber.
Key words: molecular data – molecular processes – ISM: molecules – submillimetre: ISM.
1 I N T RO D U C T I O N
The rotational excitation of methanol by para-H2 in its ground rotational state, j = 0, was studied by Pottage, Flower & Davis (2004a,
hereafter PFD04a). These authors derived the cross-sections for rotational transitions and thence the thermal rate coefficients in the
range of kinetic temperatures 5 ≤ T ≤ 200 K.
In low-temperature interstellar gas (T 100 K), molecular hydrogen is expected to exist primarily in its lowest energy level, i.e.
as para-H2 (j = 0). However, when the temperature of the gas increases, owing to the passage of a shock wave or some other violent
event, ground state (j = 1) ortho-H2 will be produced in reactions of para-H2 with H+ , H+
3 and at sufficiently high temperatures,
T 1000 K, with H (there is a barrier to this reaction). The orthoH2 that is formed can remain in the gas even when it has cooled
to its equilibrium temperature if the number densities of H+ and
H+
3 ions are sufficiently low. In addition, H2 is believed to form, on
grain surfaces, as ortho and para in the statistical abundance ratio of
3:1. Thus, collisions with ortho-H2 may be a significant contributor
to the rotational excitation of methanol, even in cold molecular gas.
Indeed, it might be anticipated that collisions with ortho-H2 contribute to the total rate of excitation of methanol disproportionately
E-mail: [email protected]
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2010 The Authors. Journal compilation to the ortho:para-H2 number density ratio. In the case of water,
which, like methanol, is an asymmetric top, the recent work of
Dubernet et al. (2006) has confirmed the results of an earlier study
by Phillips, Maluendes & Green (1996), which showed that the
cross-sections for excitation by ortho-H2 (j = 1) are generally
larger – by an order of magnitude, for some transitions – than by
para-H2 (j = 0).
In Section 2, we outline our methodology. Section 3 contains our
results and their discussion. We make our concluding remarks in
Section 4.
2 METHODOLOGY
Methanol (CH3 OH) is an asymmetric top molecule – in practice,
an ‘almost symmetric’ top, in which the difference between the
rotational constants about the x- and y-axes is only 3.7 per cent
of their mean value (Pei, Zeng & Gou 1988). The CH3 group performs internal torsional motion, relative to the OH radical, about
the molecular symmetry (z-) axis. Thus, the starting point when
treating collisions between methanol and H2 j > 0 is the seminal
study by Phillips, Maluendes & Green (1995) of the scattering of
H2 on an asymmetric top molecule – H2 O, in their application. We
first consider the form of the interaction potential that applies in this
case and then the CH3 OH–H2 collision process.
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D. Rabli and D. R. Flower
Rotationally inelastic collisions of methanol with He 1s2 1 S have
been recently considered by Rabli & Flower (2010, hereafter R&F),
who refer to previous work on this topic. Collisions of methanol with
para-H2 X1 g+ , constrained to its v = 0 = j rovibrational ground
state, are formally identical to those with ground-state helium. The
H2 and He interaction potentials differ, of course, but the form of
their series expansions, for the purposes of a collision calculation,
is the same, namely
1 Vλμn (R)Yλμ (θ, φ)einω ,
(1)
V (R, θ, φ, ω) = (2π)− 2
λμn
where Yλμ is the normalized spherical harmonic, (R, θ, φ) are the
spherical polar coordinates of the helium atom relative to a coordinate system fixed in the molecule and ω is the internal torsion
angle. It follows that subject to modifying the reduced mass of the
perturber and target and providing the appropriate numerical values of the expansion coefficients, Vλμn (R), procedures identical to
those described previously may be followed when calculating crosssections and hence thermal rate coefficients. On the other hand, collisions with ground-state ortho-H2 X1 g+ , v = 0, j = 1 or, more
generally, with H2 j > 0, are intrinsically more complex, as the
rotational degrees of freedom of the H2 molecule must be treated
explicitly. In a quantum mechanical approach, adopted here, the
coupling of the rotational angular momenta of the H2 and CH3 OH
molecules leads to an increase in the total number of coupled differential equations that have to be solved numerically in order to
derive the collision cross-sections.
Figure 1. The nuclear coordinates of CH3 OH, expressed in bohr and relative
to an origin at the molecular centre of mass (cf. Davis & Entley 1992).
interaction potential can be expanded as
Vλ1 μ1 λ2 μ2 (R)
V (R, θ, φ, θ , φ ) =
λ1 μ1 λ2 μ2
× Yλ1 μ1 (θ, φ)Yλ2 μ2 (θ , φ ).
(2)
For the purposes of their scattering calculations, Phillips et al. (1995)
used an alternative expansion of the interaction potential, of the form
vp1 q1 p2 p (R)tp1 q1 p2 p (θ, φ, θ , φ ),
(3)
V (R, θ, φ, θ , φ ) =
p1 q1 p2 p
2.1 The CH3 OH–H2 interaction potential
where
The energy of interaction between CH3 OH and H2 was computed
in the methanol (body-fixed) coordinate system. The z-axis of this
reference frame is directed towards and passes through the centre
of mass of the methanol molecule from the plane containing the
three hydrogens of the methyl (CH3 ) group, to which the z-axis is
perpendicular. The symmetry axis of the methyl group is also perpendicular to the equilateral triangle formed by the three hydrogen
atoms, passing through its centre and through the carbon atom. It
follows that the z-axis is parallel to, but slightly displaced from, the
methyl symmetry axis. The xz plane is taken to be the symmetry
plane, in which lies the C–OH part of the molecule, together with
one of the hydrogens of the methyl group. In this body-fixed frame,
the centre of mass of the H2 molecule has spherical polar coordinates (R, θ, φ) and the H2 internuclear axis has angular coordinates
(θ , φ ). The H2 internuclear distance was kept fixed at its equilibrium value of 1.4 bohr. In addition, the atoms comprising CH3 OH
were assumed to be fixed at their equilibrium positions (see Davis &
Entley 1992), corresponding to a fixed value of the internal torsion
angle of the CH3 relative to the OH radical, ω = 0, and a minimum
of the potential with respect to the torsional motion (the so-called
staggered conformation of methanol). The geometry of the CH3 OH
molecule is shown in Fig. 1.
The calculations of the interaction potential between methanol
and H2 were summarized by PFD04a. Second-order, many-body
perturbation theory was used. The grid of 17 values of R extends
from 4 to 20 bohr, which is further than the outer limit (of 14 bohr)
in the work of PFD04a. There are 13 equally spaced values of θ ,
seven values of φ, six values of θ and seven values of φ ; each of
the angles was incremented in equal steps, of 15◦ for θ and 30◦ for
the other angles. The number of non-equivalent geometries at each
value of R is 2508. In these coordinates, the molecule–molecule
tp1 q1 p2 p (θ, φ, θ , φ ) = (1 + δq1 0 )
−1
p1
p2
p
r1
r2
r
r1 r2 r
× Yp2 r2 (θ , φ )Ypr (θ, φ) δq1 r1 + (−1)p1 +q1 +p2 +p δ−q1 r1 ,
(4)
defined by equation (8) of Phillips et al. (1995), is a rotationally
invariant function of the angular variables.
p1 p2 p
r1
r2
r
is a Wigner 3j angular momentum coupling coefficient, and
p = p1 + p2 . In equations (3) and (4), the subscript 1 refers to
CH3 OH and 2 to H2 . The definition of tp1 q1 p2 p in (4) implies that
the summation in (3) includes only those values of q1 ≥ 0.
For para-H2 j = 0, it is appropriate to average the interaction
potential over the orientation of the H2 internuclear axis, which
leads to the simpler form, for the staggered conformation (ω = 0),
stg
Vλμ (R)Yλμ (θ, φ),
(5)
V (R, θ, φ, ω = 0) =
λμ
where
1
Vλμ (R) = (2π)− 2
stg
Vλμn (R)
(6)
n
relates expansions (1) and (5). Applying this averaging to equation (2), it may be shown that
1
Vλμ (R) = (−1)λ [4π(2λ + 1)]− 2 vλμ0λ (R),
stg
(7)
which is equivalent to equation (9) of Phillips et al. (1994).
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2010 The Authors. Journal compilation The rotational excitation of methanol by H2
2.2 CH3 OH–H2 collisions
As the CH3 OH–H2 interaction potential was computed only for
the staggered conformation of methanol, its variation with the
torsion angle, ω, is not known. Hence it is not possible to determine the cross-sections for torsionally inelastic transitions, involving changes in the torsional quantum number, ν. In practice, it would
be difficult to do so even if the ω dependence of the interaction potential were known, given the size of the collision calculations that
would be involved. It is likely that further approximations of the
rotational motion of the methanol molecule would be unavoidable,
if the torsional degree of freedom were included. Nevertheless, it
is possible, using the currently available potential, to compute rotationally inelastic cross-sections within a given torsional manifold.
Each of the torsional manifolds ν = 0, ν = 1 and ν = 2 is treated
independently, with only the energies and the wavefunctions of the
rotational states changing with ν. The justification for this approach,
other than its feasibility, is that the cross-sections for torsionally inelastic transitions are likely to be much smaller than for rotational
transitions within a given torsional manifold, as was found to be the
case of collisions of methanol with He (Pottage, Flower & Davis
2004b).
R&F have described an approach to calculating the rotational
structure of the methanol molecule that is well adapted to being incorporated into collision calculations. Their study related to
CH3 OH–He collisions. As mentioned above, CH3 OH–para-H2 collisions may be treated in a completely analogous manner, providing
that the para-H2 is constrained to its j = 0 rotational ground state.
A similar approach may be adopted to collisions of methanol with
H2 j > 0 and specifically to collisions with ortho-H2 j = 1.
The modifications of the MOLSCAT computer program (Hutson &
Green 1995), necessary to treat CH3 OH–H2 collisions, were made to
the asymmetric top – linear rotor mode of the code (collision type 4).
The interaction potential is assumed to take the form of equation (3),
and numerical values of the expansion coefficients, vp1 q1 p2 p (R),
have to be provided for the grid of R values. In this study, the ranges
of the indices were 0 ≤ p1 ≤ 10, 0 ≤ q1 ≤ p1 , 0 ≤ p2 ≤ 4 and
0 ≤ p ≤ 6; we recall that subscript 1 refers to CH3 OH and 2 to H2 .
The rotational basis of the H2 molecule was taken to be j = 0
(para-H2 ) or j = 1 (ortho-H2 ). In the former case, an internal
consistency check was possible. The MOLSCAT code was run in its
asymmetric top – atom mode of operation (collision type 6), using the potential (5) and the values of the expansion coefficients,
stg
Vλμ (R), derived by averaging over the orientation of the H2 internuclear axis. These two calculations are formally equivalent and
should yield the same values of the cross-sections for rotational transitions in methanol. The test was performed on transitions within
the torsional ground state, ν = 0, but is independent of the torsional
state considered. At a collision energy E = 40 cm−1 , we obtained
inelastic cross-sections that agreed to within a median deviation of
3 per cent. This small difference is probably attributable mainly to
the numerical procedure used to average the interaction potential
over the orientations of the H2 internuclear axis.
Two sets of calculations were carried out, using different rotational bases for the methanol molecule. First, we performed calculations for para-H2 j2 = 0 with the rotational basis j1 ≤ 15 for each of
the torsional states ν = 0, 1 and 2; these computations mirror those
undertaken previously for collisions with He (R&F). Secondly, we
performed calculations for both para-H2 j2 = 0 and ortho-H2 j2 = 1
with a reduced methanol basis, j1 ≤ 9. The reduction in the basis
size was imposed by the additional demands of the computations for
ortho-H2 , which involve three times as many coupled channels (CC)
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2010 The Authors. Journal compilation 97
for a given methanol basis; the computing time increases approximately in proportion to the cube of the number of CC. Furthermore,
the earlier study of CH3 OH–para-H2 j2 = 0 collisions by PFD04a
made use of a methanol basis j1 ≤ 9, in ν = 0.
In all our production calculations, we used the coupled states
(CS) approximation and a grid of collision energies extending to
E = 2000 cm−1 . A comparison of CS and CC calculations, for
collisions of A- and E-type methanol with para-H2 , at energies 10 ≤
E ≤ 100 cm−1 , using a methanol basis j1 ≤ 9, in ν = 0, showed
the median deviation of the CS from the CC cross-sections to be
typically 20–30 per cent. The median deviation tends to decrease as
the collision energy increases. The analogous comparison for orthoH2 collisions, using a smaller methanol basis, j1 ≤ 5, because of
the larger computation time involved, showed similar discrepancies,
which were larger for E than for A type. Of course, discrepancies for
individual transitions can be greater than the median value quoted
above.
3 R E S U LT S A N D D I S C U S S I O N
3.1 CH3 OH ( j1 ≤ 15)–para-H2 ( j2 = 0)
These results are probably the most significant, from the perspective
of their astrophysical applications, as ground-state para-H2 is likely
to be the most abundant perturber in many interstellar environments.
The rotational basis of CH3 OH, 0 ≤ j1 ≤ 15, is complete up
to the opening of the first excited torsional state, ν = 1, at an
energy of approximately 200 cm−1 above the ground state. However,
the cross-sections connecting to levels approaching the limit of
this basis are not fully converged. Test calculations performed by
R&F for the excitation of E-type methanol by He, at a collision
energy E = 200 cm−1 , in which the rotational basis was extended
to j ≤ 17, showed mean and median deviations of the cross-sections
obtained with j ≤ 15 of 4.2 and 1.5 per cent, respectively, but a
maximum deviation of 63.5 per cent for a transition to the level with
j = 15. We believe that similar discrepancies are to be expected in
the case of excitation by H2 .
The calculations for para-H2 (j2 = 0) were performed in a completely analogous manner to those for He. As in that case, thermal
rate coefficients are available for rotational transitions within each
of the torsional manifolds ν = 0, 1 and 2 of both A- and E-type
methanol, on a grid of kinetic temperatures 10 ≤ T ≤ 200 K.1
We wish to emphasize that the previous calculations and results of
PFD04a were limited to ν = 0 and j1 ≤ 9.
The methanol basis 0 ≤ j1 ≤ 15 gives rise to 256 rotational states
and over 32 000 possible collisional de-excitation transitions. It is
impractical to discuss such a large data set in other than statistical
terms. Accordingly, we follow the approach adopted by PFD04a
and present the rate coefficients for inelastic transitions, at a kinetic
temperature T = 200 K, as scatter plots in which the independent
variable is either the change in the rotational quantum number, j ,
or the change in its projection on the molecular symmetry axis, K.
The results of this analysis for E-type methanol in the torsional state
ν = 0 are shown in Fig. 2.2
1 http://massey.dur.ac.uk/drf/methanol_H2/An-type_rates.out
and http://
massey.dur.ac.uk/drf/methanol_H2/En-type_rates.out, where n = 0, 1
and 2.
2 The projection quantum number, K, is nearer to being a good quantum
number in E-type than in A-type methanol. Accordingly, we analyse our
results for E type, rather than A type, but recall that thermal rate coefficients
are available for both modifications.
98
D. Rabli and D. R. Flower
3 10
-10
2.5 10
E0; T = 200K; para-H
E2; T = 100K; para-H
-1
Rate coefficient (cm s )
3
2 10-10
1.5 10-10
1 10
5 10
-11
1.5 10-10
1 10
-10
5 10-11
0
0
0
4
8
0
4
| K|
3 10
2.5 10-10
-10
E1; T = 100K; para-H 2
2
-10
3 -1
Rate coefficient (cm s )
-10
3
Rate coefficient (cm s-1)
2.5 10
1.5 10-10
1 10
8
| K|
E0; T = 200K; para-H
2 10
2
2 10-10
3
Rate coefficient (cm s-1)
2.5 10-10
-10
-10
2
-10
2 10
-10
1.5 10
-10
1 10-10
5 10
-11
5 10-11
0
0
0
4
8
| j|
12
0
16
8
| K|
Figure 2. Scatter plots showing the values of the rate coefficients at T =
200 K for inelastic, de-excitation transitions in E-type CH3 OH (ν = 0, j1 ≤
15), induced by para-H2 (j2 = 0), as functions of |
K| and |
j1 |.
2.5 10-10
E0; T = 100K; para-H
2 10
-10
1.5 10
-10
1 10
-10
5 10
-11
2
3
-1
Rate coefficient (cm s )
The two panels in Fig. 2 are similar but not identical to the
corresponding figs 1 and 5 of PDF04a. We find, as was the case
previously, a strong propensity for transitions in which K = 0 and
hence for intra-ladder3 transitions, but the rate coefficients extend to
values that are approximately twice those found previously. There
is a secondary peak in the |
K| distribution, at |
K| = 3, which
reflects the three-fold symmetry of the methyl group, but this peak
is lower than that found previously, by a factor of about 2. Thus,
the propensity for transitions K = 0 is more pronounced in the
present calculations, shown in Fig. 2. The extension of the rotational
basis in the present calculations, from j ≤ 9 to j ≤ 15, results in
more rotational levels being embedded in the torsional continuum.
As we shall see below, an enhancement of the propensity towards
transitions in which K = 0 might then be expected.
Radiative transitions in methanol follow the electric dipole selection rules |
j |, |
K| = 0, 1. The collisional propensities K = 0
and |
j | = 1, apparent in Fig. 2, are compatible with the electric
dipole selection rules. The small values of the rate coefficients for
transitions with j = 0 have been remarked upon previously by
Pottage, Flower & Davis (2001, hereafter PFD01). It is attributable
to the fact that j = 0 corresponds, by definition, to inter-ladder
transitions, whose rate coefficients are smaller than those for intraladder transitions, in which |
j | > 0.
3A
4
0
0
4
8
| K|
Figure 3. Scatter plots showing the values of the rate coefficients at T =
100 K for inelastic, de-excitation transitions in E0-, E1- and E2-type CH3 OH
(j1 ≤ 15), induced by para-H2 (j2 = 0), as functions of |
K|.
Calculations of cross-sections and rate coefficients were also
performed for the first and second excited torsional states; the samesize rotational basis, j1 ≤ 15, was employed. In Fig. 3 the values
of the rate coefficients for rotational transitions within all three
torsional states, ν = 0, 1 and 2 of E-type methanol (denoted E0,
E1, and E2), at a kinetic temperature T = 100 K are compared.
As can be seen from Fig. 3, the propensity towards K = 0
transitions amplifies with increasing torsional excitation. The lowest E2 state, j = 1, K = −1, has an energy of 499 cm−1 ,
ladder comprises levels with j ≥ K, for a given value of K.
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which may be compared with the torsional well depth of 373
cm−1 (http://ftp.monash.edu.au/pub/chem/1998a/energy.d; Cragg,
Sobolev & Godfrey 2005). Thus, the rotational levels with ν = 2
are embedded in the torsional continuum, and it might be anticipated that the behaviour of the rate coefficients would approach
that observed in the ‘free-rotor’ limit, which is attained when the
interaction potential is pre-averaged over the torsion angle, ω (see
PFD01). Calculations that we have undertaken for CH3 OH–He collisions, using the corresponding torsionally averaged interaction po-
tential, have confirmed that the same trend, towards predominance
of transitions in which K = 0, is observed in this case.
3.2 CH3 OH ( j1 ≤ 9)–para-H2 ( j2 = 0) and ortho-H2 ( j2 = 1)
Although they were performed with a smaller methanol rotational
basis, j1 ≤ 9, the calculations for ortho-H2 (j2 = 1) and the
equivalent computations for para-H2 (j2 = 0) enable the effects
of non-zero values of j2 to be established. From the viewpoint of the
-10
-10
2.5 10
2.5 10
E2; T = 100K; ortho-H
-10
2 10
1.5 10-10
1 10
E2; T = 100K; para-H
2
3 -1
Rate coefficient (cm s )
3 -1
Rate coefficient (cm s )
2 10
-10
5 10-11
1.5 10-10
1 10
-10
5 10-11
0
0
4
8
0
4
| K|
2.5 10-10
E1; T = 100K; ortho-H 2
E1; T = 100K; para-H 2
-10
-10
-1
Rate coefficient (cm3 s )
2 10
-1
Rate coefficient (cm3 s )
8
| K|
2.5 10-10
-10
1.5 10
1 10-10
5 10
-11
-10
1.5 10
1 10-10
5 10
-11
0
0
0
4
8
0
4
| K|
8
| K|
2.5 10-10
2.5 10-10
E0; T = 100K; ortho-H
E0; T = 100K; para-H
2
2
2 10-10
3
3
Rate coefficient (cm s-1)
2 10-10
Rate coefficient (cm s-1)
2
-10
0
2 10
99
-10
1.5 10
1 10
-10
5 10
-11
0
-10
1.5 10
1 10
-10
5 10
-11
0
0
4
8
| K|
0
4
8
| K|
Figure 4. Scatter plots showing the values of the rate coefficients at T = 100 K for inelastic, de-excitation transitions in E0-, E1- and E2-type CH3 OH (j1 ≤ 9),
induced by ortho-H2 (j2 = 1) and para-H2 (j2 = 0), as functions of |
K|.
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2010 The Authors. Journal compilation 100
D. Rabli and D. R. Flower
involving the multipole moments of the H2 molecule, of which the
quadrupole moment is the leading term, vanish identically. However, this is not the case when j2 > 0 and, in particular, when
j2 = 1.
In Fig. 4 the scatter plots of the values of the thermal rate coefficients, at T = 100 K, as functions of K, for collisions with
both ortho-H2 and para-H2 , are presented; the corresponding plots
as functions of |
j1 | are shown in Fig. 5. It may be seen from Fig. 4
that there are qualitative differences between the K propensities
interaction of H2 with another molecule, such as methanol, introducing non-zero values of the rotational quantum number, j2 , has the
consequence of probing the dependence of the interaction potential
on the orientation of the H2 internuclear axis. In the rotational state
j2 = 0, whose eigenfunction is the normalized spherical harmonic
1
Y00 (θ , φ ) = (4π)− 2 , the rotational wavefunction is independent
of the orientation (θ , φ ) of the internuclear axis. In other words,
all orientations of the H2 internuclear axis have an equal probability
of |Y00 |2 = (4π)−1 . As a consequence, the long-range interactions
-10
2.5 10
2.5 10
E2; T = 100K; ortho-H
E2; T = 100K; para-H
-1
3
1.5 10-10
1 10
-10
5 10-11
1.5 10-10
1 10
-10
5 10-11
0
0
0
4
8
0
4
| j|
2.5 10-10
E1; T = 100K; ortho-H 2
E1; T = 100K; para-H 2
2 10
-10
1.5 10
-10
3
-1
Rate coefficient (cm s )
-10
-1
Rate coefficient (cm3 s )
2 10
8
| j|
2.5 10-10
-10
1.5 10
1 10-10
5 10
-11
1 10-10
5 10
0
-11
0
0
4
8
0
4
| j|
8
| j|
2.5 10-10
2.5 10-10
E0; T = 100K; ortho-H
E0; T = 100K; para-H
2
2
2 10-10
3
3
-1
Rate coefficient (cm s )
2 10-10
Rate coefficient (cm s-1)
2
2 10-10
Rate coefficient (cm s )
2 10-10
3 -1
Rate coefficient (cm s )
-10
2
1.5 10-10
1 10
-10
5 10-11
1.5 10-10
1 10
-10
5 10-11
0
0
0
4
8
0
| j|
4
8
| j|
Figure 5. Scatter plots showing the values of the rate coefficients at T = 100 K for inelastic, de-excitation transitions in E0-, E1-, and E2-type CH3 OH
(j1 ≤ 9), induced by ortho-H2 (j2 = 1) and para-H2 (j2 = 0), as functions of |
j1 |.
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Table 1. The mean and median values of the rate coefficients (cm3 s−1 ), at T = 100 K, for de-excitation transitions
in both E- and A-type methanol in their torsional ground
states, ν = 0. Numbers in parentheses are powers of 10.
Ortho-H2
Para-H2
Ortho:Para
E0 type: mean
E0 type: median
6.2 (−12)
2.3 (−13)
4.4 (−12)
3.7 (−14)
1.4
6.2
A0 type: mean
A0 type: median
8.0 (−12)
2.0 (−12)
5.8 (−12)
3.1 (−13)
1.4
6.5
for ortho- and para-H2 . When ortho-H2 is the perturber, transitions
in which |
K| > 0 and j1 = 0 have larger rate coefficients than
when para-H2 is the perturber; this is true of all three torsional
manifolds, ν = 0, 1 and 2. These differences are attributable to
the rate coefficients for inter-ladder transitions, in which K changes
its value, being larger for ortho-H2 than for para-H2 collisions. On
the other hand, the j1 propensities are qualitatively similar for
collisions with both ortho- and para-H2 and almost independent of
the value of the torsional quantum number, ν.
A statistical analysis of the rate coefficients, at T = 100 K, for
de-excitation transitions in both E0- and A0-type methanol, yields
the mean and median values listed in Table 1. It may be seen that
the rate coefficients for transitions induced by ortho-H2 are systematically larger than for para-H2 , with the difference being greater
for the smaller values of the rate coefficients (compare the mean
and the median values); these trends are apparent in both E0- and
A0-type methanol. In A0 type, the mean and particularly the median values of the rate coefficients are larger than in E0 type, whilst
the variations with respect to the perturber (ortho- or para-H2 ) are
practically the same. These conclusions, in the case of methanol,
are qualitatively similar to those reached previously, in studies of
the collisional excitation of water by ortho- and para-H2 (Phillips
et al. 1996; Dubernet et al. 2006).
At a kinetic temperature T = 100 K, the ratio of the population
densities of the levels j1 = 1 and j1 = 0 of H2 is 1.6, under conditions of thermodynamic (Boltzmann) equilibrium. Then, ortho-H2
collisions would dominate the excitation of methanol molecules.
4 CONCLUDING REMARKS
The current study of collisions between methanol and para-H2
molecules is a significant extension of previous work in that the
rotational basis of the methanol has been increased, from j1 ≤ 9 to
j1 ≤ 15, and that rotational transitions within the first two excited
torsional manifolds have been investigated more systematically.
Furthermore, the representation of the rotational–torsional eigenfunctions of E-type methanol has been improved, as described by
C
C 2010 RAS, MNRAS 406, 95–101
2010 The Authors. Journal compilation 101
R&F. Thus, this work represents not only the most extensive but
also the most accurate analysis of CH3 OH–para-H2 scattering.
In addition to calculating cross-sections and low-temperature rate
coefficients for rotational transitions in methanol, induced by paraH2 , we have computed thermal rate coefficients for transitions induced by ortho-H2 . We found substantial qualitative and quantitative differences between the results for para- and ortho-H2 , with
rate coefficients for rotationally inelastic scattering being larger
when ortho-H2 is the perturber. In this respect, our results accord
with those of Phillips et al. (1996) and Dubernet et al. (2006),
based on their studies of the excitation of the asymmetric top H2 O
molecule by ortho- and para-H2 . Regarding the influence of rotational states of para-H2 with j2 > 0, Phillips et al. (1995) concluded
that states j2 = 2 need to be included to achieve a precision of better than 10 per cent in the rotationally inelastic cross-sections of
H2 O. Furthermore, Dubernet & Grosjean (2002) have shown that
the inclusion of j2 = 2 modifies the resonance structure of these
cross-sections at low collision energies.
The present data are being incorporated in a model of molecular
emission from interstellar shock waves, and the analysis of the
predictions of this model will be the topic of a future publication.
AC K N OW L E D G M E N T S
This work was supported by a research grant from the STFC (UK).
We thank the referee, Marie-Lise Dubernet, for comments which
helped to improve our paper.
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