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Rotational Quenching Rate Coefficients for H 2 in Collisions with H 2 from 2 to 10,000 K

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a r X i v : 0 8 0 5 . 1 6 2 3 v 1 [ a s t r o - p h ] 1 2 M a y 2 0 0 8
Rotational quenching rate coeﬃcients for H
2
in collisions with H
2
from 2 to 10,000 K
T.-G. LEE
1
,
2
, N. BALAKRISHNAN
3
, R. C. FORREY
4
,P. C. STANCIL
5
, G. SHAW
6
, D. R. SCHULTZ
2
, AND G. J. FERLAND
1
ABSTRACT
Rate coeﬃcients for rotational transitions in H
2
induced by H
2
impact arepresented. Extensive quantum mechanical coupled-channel calculations basedon a recently published (H
2
)
2
potential energy surface were performed. Thepotential energy surface used here is presumed to be more reliable than surfacesused in previous work. Rotational transition cross sections with initial levels
J
≤
8 were computed for collision energies ranging between 10
−
4
and 2.5 eV, and thecorresponding rate coeﬃcients were calculated for the temperature range 2
≤
T
≤
10,000 K. In general, agreement with earlier calculations, which were limitedto 100-6000 K, is good though discrepancies are found at the lowest and highesttemperatures. Low-density-limit cooling functions due to para- and ortho-H
2
collisions are obtained from the collisional rate coeﬃcients. Implications of thenew results for non-thermal H
2
rotational distributions in molecular regions arealso investigated.
Subject headings:
molecular processes —- molecular data — ISM: molecules
1. INTRODUCTION
Collision-induced energy transfer involving H
2
molecules plays an important role in manyareas of astrophysics since hydrogen is the most abundant molecule in the interstellar medium
1
Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506
2
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
3
Department of Chemistry, University of Nevada–Las Vegas, Las Vegas, Nevada 89154
4
Department of Physics, Penn State University, Berks Campus, Reading, PA 19610
5
Department of Physics and Astronomy and Center for Simulational Physics, University of Georgia,Athens, GA 30602
6
Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai-400005,India
– 2 –(ISM), dominating the mass of both dense and translucent molecular clouds. Because H
2
isa homonuclear molecule and lacks a dipole moment, its infrared emission is weak and it isdiﬃcult to observe directly. Furthermore, because of the low mass of the hydrogen atom,the rotational levels of molecular hydrogen are widely spaced and require relatively hightemperatures,
T
≥
100 K, to excite even the lowest levels appreciably via thermal collisions.Nevertheless, with the current sensitivity of detectors deployed on space-based observatories,such as the
Spitzer Space Telescope
and the former
Infrared Space Observatory (ISO)
, thedetection and mapping of the pure rotational lines of H
2
in a signiﬁcant sample of galaxies isnow possible (Neufeld et al. 2006; Fuente et al. 1998). These observations, as well as othersfrom the ultraviolet (Meyer et al. 2001; Boiss´e et al. 2005; France et al. 2005, 2007) to thenear infrared (Burton et al. 1998; McCartney et al. 1999; Allers et al. 2005) probe shock-induced heating and cooling and UV and x-ray irradiation of gas in the ISM of galactic andextragalactic sources and star-forming regions.Being the simplest diatom-diatom collision system and one in which all constituentsare quantum mechanically indistinguishable, early work, notably by Green and coworkers,carried out a succession of theoretical studies on rotational relaxation in H
2
+H
2
collisions(Green 1975; Ramaswamy et al. 1977, 1978; Green et al. 1978). The calculation of the re-laxation cross sections involve the quantum mechanical scattering of the heavy particles ona potential energy surface (PES) which is computed using quantum chemistry techniques,accounting for the electronic motion for all relevant nuclear conﬁgurations of the constituentcolliders. The availability of more realistic (H
2
)
2
potential surfaces has enabled consider-able progress in obtaining a more reliable set of state-to-state collisional cross sections andrate coeﬃcients. To date, the most comprehensive study has been made by Flower (1998)and Flower & Roueﬀ (1998). Within the rigid-rotor approximation, Flower & Roueﬀ havemade use of the (H
2
)
2
interaction potential of Schwenke (1988) in a quantal coupled-channelmethod to determine the rate coeﬃcients for rotational transitions in H
2
+ H
2
collisions.Rotational levels
J
≤
8 and kinetic temperatures
T
≤
1000 K were employed and the resultswere compared with those reported by Danby, Flower, & Monteiro (1987) based on an olderpotential surface.Accurate determinations of rate coeﬃcients for state-to-state transitions in small molec-ular systems such as the H
2
molecule require a quantum mechanical description of the scat-tering process. A full quantum calculation of rovibrational transitions in the H
2
-H
2
systemis computationally challenging. So far, only a few studies have been reported that includeboth rotational and vibrational degrees of freedom of the H
2
molecules. Pogrebnya and Clary(2002) reported vibrational relaxation of H
2
in collisions with H
2
using a coupled-states ap-proximation implemented within a time-independent quantum mechanical approach. Pandaet al. (2007) and Otto et al. (2008) employed a time-dependent quantum mechanical ap-
– 3 –proach and reported rotational transition cross sections in ortho-H
2
+para-H
2
and para-H
2
collisions. More recently, Quemener et al. (2008) reported a full-dimensional quantum scat-tering calculation of rotational and vibrational transitions in the H
2
-H
2
system that does notinvolve any dynamics approximation. All these calculations adopted the six-dimensional H
4
PES developed by Boothroyd et al. (2002). While this PES is useful for benchmarking full-dimensional quantum scattering codes for the H
2
-H
2
system, comparisons with experimentaldata have shown (Lee et al. 2006; Otto, Gatti, & Meyer 2008) that the PES predicts rota-tional transition rate coeﬃcients between the
J
= 0 and
J
= 2 levels that are too small andthat it is not appropriate for accurate determination of rotational transition rate constantsin the H
2
-H
2
system.Since full-dimensional quantum calculations of four-atom systems are computationallydemanding, especially for excited rotational levels, the rigid-rotor approximation is oftenemployed. The adequacy of the rigid-rotor approximation for pure rotational transitions inH
2
-H
2
collisions was recently demonstrated by Otto et al. (2008) who reported cross sec-tions for rotational excitations in para-H
2
using the rigid-rotor model and a full-dimensionalquantum calculation. While these calculations have been performed using the BMKP PES,we believe that the conclusions are valid for the H
2
-H
2
system in general.The accuracy of collisional data for astrophysical applications is limited by the uncer-tainty in the PES and the approximations employed in the dynamics calculations. It is anon-trivial task to judge the impact of the uncertainties in the collisional data, associatedwith the choice of the PES. Accurate determination of the potential energy in the interactionregion to the requisite level of accuracy of
∼
10
−
3
hartrees (or
∼
10 K) remains a challenge, asmentioned above. This is especially important at low temperatures where small uncertaintiesin the interaction potential translate into larger errors in the cross sections and rate con-stants. A further diﬃculty arises from ﬁtting a limited number of ab initio potential energypoints to obtain a surface over all conﬁgurations which are sampled in the scattering calcula-tions. Unphysical behavior in the ﬁtted PES can result, particularly when bridging explicitlycalculated energies at intermediate internuclear separations to long-range asymptotic forms.Recently, a new and improved (H
2
)
2
rigid-rotor PES was computed by Diep & Johnson(2000). The accuracy of this PES for rotational transitions in H
2
has recently been demon-strated by comparing computed rate coeﬃcients for
J
= 0
→
2 rotational excitation againstexperimental results (Mat´e et al. 2005; Lee et al. 2006; Otto, Gatti, & Meyer 2008). Thus,we believe that the new PES by Diep & Johnson (2000) could be employed to provide re-liable values of rate constants for rotational transitions in the H
2
+H
2
system. Since theH
2
+H
2
collision system is of astrophysical signiﬁcance, it is crucial to establish whether thecross sections and rate coeﬃcients for rotational energy transfer based on this PES are in
– 4 –agreement with the earlier data. Therefore, we adopted this PES in the present quantummechanical close-coupling calculations to obtain the rotational transition cross sections withinitial levels
J
≤
8 for collision energies ranging from 10
−
4
to 2.5 eV. The correspondingrate coeﬃcients were computed for the temperature range of 2
≤
T
≤
10,000 K. The presentrate coeﬃcients are compared with the results of Flower (1998) and Flower & Roueﬀ (1998),which are to the best of our knowledge, the only comprehensive calculations available forrotational de-excitation rate coeﬃcients. We also present H
2
-H
2
cooling functions in thelow-density limit and test the new rate coeﬃcients in simulations of UV irradiated moleculargas. Atomic units are used throughout unless otherwise speciﬁed, i.e., distances are in bohrs(a
o
) and energy in hartrees (
E
h
). Recall that 1 a
o
= 0.529177 ˚A, while 1
E
h
= 27.2114 eV= 219474.635156 cm
−
1
= 627.51 kcal/mole.
2. COMPUTATIONAL DETAILS
We carried out quantal coupled-channel calculations for collisions of H
2
with H
2
us-ing a ﬁxed bond length of 1.449 a
o
(0.7668 ˚A). The rigid-rotor H
2
target and projectileenergy levels were calculated using a rotational constant of
B
= 60.853 cm
−
1
. To solvethe coupled-channel equations, we used the hybrid modiﬁed log-derivative Airy propaga-tor (Alexander & Manolopoulos 1987) in the general purpose non-reactive scattering codeMOLSCAT developed by Hutson & Green (1994). The log-derivative matrix is propagatedto large intermolecular separations where the numerical results are matched to the knownasymptotic solutions to extract the physical scattering matrix. This procedure is carried outfor each partial wave. We have checked that the total integral cross sections are convergedwith respect to the number of partial waves, as well as the asymptotic matching radius (e.g.,
R
= 40 a
o
) for all channels included in the calculations.In addition to the partial wave convergence, we have checked that the results are op-timized with respect to other parameters that enter into the scattering calculations. Inparticular, the parameters used for the analytical expression for the Diep & Johnson PES,
V
(
R,θ
1
,θ
2
,φ
12
) =
l
1
,l
2
,l
V
l
1
,l
2
,l
(
R
)
G
l
1
,l
2
,l
(
θ
1
,θ
2
,φ
12
)
,
(1)where
V
l
1
,l
2
,l
(
R
) are radial expansion coeﬃcients and
G
l
1
,l
2
,l
(
θ
1
,
θ
2
,
φ
12
) are bispherical har-monics. The angles
θ
1
,
θ
2
denote the two in-plane angles and
φ
12
is the relative torsionalangle. We used 10 quadrature points each for integration along angular coordinates
θ
1
, θ
2
,and
φ
12
. From the Diep & Johnson ﬁt to their numerical PES, the expansion coeﬃcients
V
l
1
,l
2
,l
(
R
) are determined. They noted that only the
V
0
,
0
,
0
(
R
),
V
2
,
0
,
2
(
R
),
V
0
,
2
,
2
(
R
) and
V
2
,
2
,
4
(
R
)terms make signiﬁcant contribution to the potential.
– 5 –In Table 1, we show a comparison between the PES adopted by Flower & Roueﬀ, theSchwenke (1988) potential, and the Diep & Johnson (H
2
)
2
PES for linear [
θ
1
,
θ
2
,
φ
12
]=[0,0, 0] and parallel [
θ
1
,
θ
2
,
φ
12
]=[
π
/2,
π
/2,0] conﬁgurations, respectively. Although the twopotentials show close agreement for both conﬁgurations, the analytic PES of Schwenke (1988)for (H
2
)
2
is known to be accurate only for pairs of hydrogen molecules with intermolecularseparations not greater than
∼
5 a
o
. The small discrepancy may be attributed to theimproved accuracy in the Diep & Johnson PES which incorporates the correct long-rangebehavior.The present calculations consider the hydrogen molecules to be indistinguishable. Sym-metric basis sets were used for para-para and ortho-ortho collisions, whereas, for para-orthocollisions, asymmetric basis sets were chosen based on the order of the energy spectrum.The scattering cross sections for rotational transitions were computed for collision energiesranging between 10
−
4
and 2.5 eV. Four de-excitation cases with ∆
J
1
=
−
2 were considered(i.e., para-para and para-ortho; ortho-para and ortho-ortho). Note that a suﬃciently large,but truncated basis set has been used to optimize the computation time and minimize theloss of numerical accuracy (i.e., to
5%) in the scattering cross section calculations. Furtherdetails about the calculations can be found in Lee et al. (2006).Rate coeﬃcients for state-to-state rotational transitions were obtained by averaging theappropriate cross sections over a Boltzmann distribution of the relative kinetic energy
E
k
of the H
2
molecules at a given temperature
T k
J
1
J
2
→
J
′
1
J
′
2
(
T
) =
G
(1 +
δ
J
1
J
2
)(1 +
δ
J
′
1
J
′
2
)
∞
0
dE
k
σ
J
1
J
2
→
J
′
1
J
′
2
(
E
k
)
E
k
e
(
−
βE
k
)
,
(2)where
G
=
8
µπβ
β
2
,
β
= (
k
B
T
)
−
1
with
k
B
being the Boltzmann constant, and we haveadopted the deﬁnition of the cross section for indistinguishable particles as given by Green(1975). This deﬁnition diﬀers from that adopted by Flower & Roueﬀ which was givenby Zarur & Rabitz (1974). As discussed by Danby, Flower, & Monteiro (1987), for singlerotational excitation transitions, the cross section deﬁned by Green (1975) must be dividedby two if
J
1
=
J
2
or
J
′
1
=
J
′
2
to prevent double counting in the determination of theproduction rate of
J
′
1
or
J
′
2
. The factor (1 +
δ
J
1
J
2
)(1 +
δ
J
′
1
J
′
2
) in the denominator of eq. (2)accounts for it.

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