A&A 432, 369379 (2005)
DOI: 10.1051/00046361:20041729
An atomic and molecular database for analysis of submillimetre line
observations^{}^{,}^{}
F. L. Schöier^{1,3} 
F. F. S. van der Tak^{2} 
E. F. van Dishoeck^{3} 
J. H. Black^{4}
1  Stockholm Observatory, AlbaNova, 106 91 Stockholm, Sweden
2  MaxPlanckInstitut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
3  Leiden Observatory, PO Box 9513, 2300 RA Leiden, The Netherlands
4  Onsala Space Observatory, 439 92 Onsala, Sweden
Received 26 July 2004 / Accepted 4 November 2004
Abstract
Atomic and molecular data for the transitions of a
number of astrophysically interesting species are summarized,
including energy levels, statistical weights, Einstein
Acoefficients and collisional rate coefficients. Available
collisional data from quantum chemical calculations and
experiments are extrapolated to higher energies (up to
K).
These data, which are made publically available through the WWW at
http://www.strw.leidenuniv.nl/~moldata,
are essential input for nonLTE line radiative transfer
programs. An online version of a computer program for performing statistical
equilibrium calculations is also made available as part of the database.
Comparisons of calculated emission lines using different sets of
collisional rate coefficients are presented.
This database should form an important tool in analyzing
observations from current and future (sub)millimetre and infrared
telescopes.
Key words: astronomical data bases: miscellaneous  atomic data  molecular data
 radiative transfer  ISM: atoms  ISM: molecules
A wide variety of molecules has been detected in space to date
ranging from simple molecules like CO to more complex organic
molecules like ethers and alcohols. Observations of molecular lines
at millimetre and infrared wavelengths, supplemented by careful and
detailed modelling, are a powerful tool to investigate the physical
and chemical conditions of astrophysical objects
(e.g., Black 2000; Genzel 1991).
To constrain these conditions,
lines with a large range of critical densities and excitation
temperatures are needed, since densities typically range from
10^{2}10^{9} cm^{3} and temperatures from 101000 K in
the interstellar and circumstellar environments probed by current and
future instrumentation.
In recent years, different molecules have been developed as tracers
for different physical and
chemical conditions (see van Dishoeck & Hogerheijde 1999, for a review).
For example, CO is used as a tracer of the
total gas mass whereas readily observed molecules with large dipole
moments, such as CS, HCO^{+} and HCN constrain the density
structure.
The wide variety of H_{2}CO and CH_{3}OH lines accessible at millimetre and submillimetre wavelengths trace both the temperature
and density structure (e.g., Mangum & Wootten 1993). Organic
molecules like CH_{3}OCH_{3} and CH_{3}CN probe the chemical
complexity. Deuterated molecules contain a record of
the conditions and duration of the cold prestellar phase. Si and
Sbearing molecules, in particular SiO and SO_{2}, probe shocks. Lines
of the main species as well as the (generally) optically thin isotopes
are needed to determine accurate abundances and line profiles. High
frequency lines and vibrationally excited lines are particularly
valuable for probing the warm and dense inner parts of the
circumstellar envelopes (e.g., Boonman et al. 2001; Ziurys & Turner 1986).
To extract astrophysical parameters, the excitation and radiative
transfer of the lines need to be calculated. Indeed, it is becoming
increasingly clear that more information  including chemical gradients
throughout the source  can be inferred from the
data if a good molecular excitation model is available
(e.g., Maret et al. 2004; Schöier et al. 2002). The simplest models adopt the
"local'' approximation, for example in the widely used large velocity
gradient (LVG) method. A number of more sophisticated, nonlocal radiative
transfer codes have been developed for the interpretation of molecular
line emission (e.g., Ossenkopf et al. 2001; Juvela 1997; Hogerheijde & van der Tak 2000; Schöier & Olofsson 2001; Bernes 1979, see van Zadelhoff et al. 2002, for a review). The
application of these codes ranges from protostellar environments to
the circumstellar envelopes of latetype stars. The radiative
transfer analysis requires accurate molecular data in the form of energy
levels, statistical weights and transition frequencies as well as the
spontaneous emission probabilities and collisional rate coefficients.
The JPL^{} catalog
(Pickett et al. 1998),
HITRAN^{} database
(Rothman et al. 2003), and the CDMS^{}
catalogue (Müller et al. 2001) contain energy levels and transition
strengths for a large number of molecular species. Detailed summaries
of the theoretical methods and the uncertainties involved in
determining collisional rate coefficients are given by
Green (1975a), Roueff (1990) and Flower (1990). In this
paper, these and other literature data on the rotational transitions
of 23 different molecules are summarized and extrapolations of
collisional rate coefficients to higher energy levels and temperatures
are made. The molecular data files can be found at the webpage
http://www.strw.leidenuniv.nl/~moldata and is the first effort
to systematically collect and present the data in a form easily used
in radiative transfer modelling of interstellar regions. The focus is
on rotational transitions within the ground vibrational state, but the
lowest vibrational levels are included for a few common species where such
data are available. Many of the data files presented here were adopted
by Schöier et al. (2002) to model the circumstellar environment of the
protostar IRAS 162932422. In addition, data files for three atomic species
are presented.
The excitation of atomic fine structure levels plays an important role in cooling of a wide variety of astrophysical objetcs.
An online version of RADEX^{}, a statistical equilibrium radiative transfer code using an escape probability formalism,
is made available for public use as part of the database. RADEX is
comparable to the LVG method and provides a useful tool for rapidly
analysing a large set of observational data providing constraints on
physical conditions, such as density and kinetic temperature
(Jansen et al. 1994; Jansen 1995). RADEX provides an alternative to the widely used
rotation temperature diagram method (e.g., Blake et al. 1987) which
relies upon the availability of many optically thin emission lines and
is useful only in roughly constraining the excitation temperature in
addition to the column density.
A guide for using the code in practice is provided
at the RADEX homepage.
RADEX will be presented in more detail in a forthcoming paper (van der Tak et al., in preparation) at which point the source code will be made publically available.
In this section the molecular structure is briefly reviewed.
This serves merely to provide
some basic information needed to properly use the data files.
Detailed
discussions on molecular (and atomic) structure can be found in, e.g.,
Townes & Schawlow (1975).
The energy levels are obtained from the JPL, HITRAN,
and CDMS catalogues. The energy levels and the corresponding line
frequencies are thus of spectroscopic quality and may be used for the
purpose of line identification, unless stated otherwise.
Generally, we retain only the ground vibrational state and include
energy levels up to
K. Vibrationally excited levels are
usually not well populated in the regions probed by current
(sub)millimetre telescopes. Moreover, little is known about
collisional rate coefficients for vibrational transitions (e.g., Chandra & Sharma 2001).
However, for some specific molecules, e.g. CO and CS, vibrationally excited
levels are also included. In the prolongation of this project, data
files including vibrational levels will be added for more molecular
species.
Molecules with ortho and para versions (or A and Etype as in the
case of e.g. CH_{3}OH) are treated as separate species.
The energy levels for diatomic and linear polyatomic molecules in the
electronic state are quantified, to first order, according
to
where B is the rotational constant and related to the moment of
inertia I, around axes perpendicular to the internuclear axis,
through
B=(2I)^{1}. Heavy linear molecules, like HC_{3}N,
have more densely spaced energy levels than diatomic molecules like,
e.g., CO. These pure rotational energy levels are classified
according to the rotational quantum number J and their statistical
weights are
Note that to obtain state energies of
spectroscopic accuracy, Eq. (1) must be augmented with centrifugal
distortion (
J^{2}(J+1)^{2}) and higherorder terms.
The majority of molecular species presented here have a electronic ground state, i.e., the sum of the orbital angular momenta
of their electrons and the sum of the electron spins are
both zero. However, there are some exceptions where either can be
nonzero.
For a molecule in a
electronic ground state, e.g., SO and
CN, the sum of the electron spins is 1/2. The nonzero spin
creates a splitting of the levels due to coupling between the electron
spin and the total angular momentum of the molecule. The total angular
momentum is quantified according to N and includes the rotation of
the molecule. Molecules like, e.g., O_{2} have a total electron
spin of 1 in a
electronic ground state.
Some important molecules such as NO, NS, and OH have a
ground state
with a total electronic orbital momentum of 1 and total spin of 1/2.
Spectroscopically, such molecules show "Doubling'', with
and
ladders.
The various molecular angular momenta may couple together in many
different ways, such as spinorbit and spinspin coupling. Ideally,
these fall in one of five different classes, known as Hund's coupling
cases. In practice, intermediate cases often occur; see
Townes & Schawlow (1975) for details.
The structure of nonlinear molecules, such as e.g. H_{2}CO
and CH_{3}OH, is more complex. Rotation can take place around
different axes of inertia, characterized by the rotational constants A, B and C which, in absence of any symmetry, involve different
amounts of energy, A>B>C. The degree of asymmetry is
measured by Ray's parameter



(3) 
and is 1 for a prolate symmetric top (B=C, e.g. CH_{3}CN) and
+1 for an oblate symmetric top (B=A, e.g. NH_{3}).
Asymmetric rotors such as H_{2}O have
.
The energy levels of symmetric top molecules, such as NH_{3} and
CH_{3}CN, are described by the quantum numbers J and K, where Kis the projection of the total angular momentum J on the symmetry
axis. For a prolate symmetric top molecule, the energy
of a rotational level is given (to first order) by
E = BJ(J+1) + (AB)K^{2}. 


(4) 
The energy levels for a slightly asymmetric prolate top such as H_{2}CO
can be calculated from



(5) 
where
with corrections due to the slight asymmetry
(Townes & Schawlow 1975, Appendix III).
A further complication arises when the nuclear spin couples to the
rotation producing what is known as hyperfine splitting. The
astrophysically most relevant cases are when the molecule contains a
^{14}N or D nucleus. When the lines are spectroscopically resolved, hyperfine
structure provides information on the optical depths, which
is otherwise hard to obtain (e.g., SchmidBurgk et al. 2004).
Hyperfine splitting can be important in line transfer and
introduce nonlocal effects for lines overlapping in frequency, e.g., the
lines of
N_{2}H^{+} and HCN. The common assumption is that the hyperfine components each have the
same excitation temperature. However, exceptions to this rule
("hyperfine anomalies'') have been observed and more detailed
treatments developed (Stutzki & Winnewisser 1985; TruongBach & NguyenQRieu 1989; Lindqvist et al. 2000).
Often the splitting
between individual hyperfine components is small, producing lines
which are separated in frequency by a small amount compared to the
linebroadening, so that this splitting can be safely neglected and
treated as a single level for the purpose of excitation analysis.
The first release of the database includes hyperfine splitting for some of the most
relevant molecules, such as HCN and OH. Future releases
will present data files with hyperfine splitting included for additional species.
The radiative rates for dipole transitions from an upper state u to
a lower state l can be calculated from



(6) 
where
is the electric dipole moment and
is the
transition strength. The transition strength depends on the complexity
of the molecule and is explained below in some detail. Strictly
speaking, the dipole moment should be averaged over the vibrational
wavefunction(s) of the transition involved (
), but in practice
often the dipole moment appropriate for the equilibrium geometry is
taken (
). The electric dipole moments are assumed to be the
same for all isotopes of a particular molecule, even though small
differences exist for
.
Because of the factor, the
resulting Einstein Avalues can still differ considerably for
isotopes, especially for deuterated species.
For transitions with
in linear molecules, the transition strength is



(7) 
whereas for symmetric top molecules, the transition strength from level J,K to J1,K is given by



(8) 
In the general case of asymmetric tops, simple expressions for
do not exist.
3.2 Dipole moments
Transition strengths are available from the spectroscopic databases
mentioned above (JPL, CDMS, HITRAN). There is some inconsistency in
the astrophysical literature regarding the choice of values of
electric dipole moments, however. This often manifests itself as an
apparent bias against results of ab initio theoretical
calculations, even when experimental results for transient species are
merely estimated or wholly absent. A case in point concerns the pair
of ions HCO^{+} and HOC^{+}: the widely cited JPL catalogue
offers
D
for HCO^{+} and
D for HOC^{+} based on lowlevel theoretical
estimates of Woods et al. (1975) and Gudeman & Woods (1982),
respectively, whereas accurate ab initio values from
Botschwina et al. (1993) give (HCO^{+}) =
D and
(HOC^{+}) = 2.74 D.
Ziurys & Apponi (1995) adopted a similar value,
(HOC^{+}) = 2.8 D, from an ab initio computation of
Defrees et al. (1982). Because the inferred column densities scale as
,
these discrepancies in dipole moments can result
in errors of factors of two in derived abundances.
Table 1 collects values of dipole moments for a
(nonexhaustive) sample of molecules of astrophysical interest. Users
are encouraged to remain aware of the original literature.
Unless otherwise indicated, all entries refer to the electronic and
vibrational ground states.
For small dipoles, centrifugal corrections to the dipole moment are
appreciable. In the case of CO, rotational effects reduce the
Avalue by 1% for J=7 and by 10% for J=22. The JPL and CDMS
catalogues consider this effect and so do our datafiles.
Table 1:
Summary of adopted dipole moments^{a,b}.
4.1 General considerations
The rate of collision is equal to



(9) 
where
is the number density of the collision
partner and
is the downward collisional rate coefficient (in
cm^{3} s^{1}). The rate coefficient is the Maxwellian average of
the collision cross section, ,



(10) 
where k is the Boltzmann constant,
is the reduced mass of
the system, and E is the centerofmass collision energy. The upward rates are obtained through detailed balance



(11) 
where g is a statistical weight.
The collisional rate coefficients
usually pose
the largest source of uncertainty of the molecular data input to the
radiative transfer analysis (however, see discussion on dipole moments
in Sect. 3.2). The dominant collision partner is often
H_{2} except in photon dominated regions (PDRs) where collisions with
electrons and H can become important. The collisional rate
coefficients presented here are mainly with H_{2} and only in a few
cases (in particular the atoms) are collisions with H and electrons also treated. Where
available, the data files include collisions with ortho and
paraH_{2}, e.g., in the case of CO.
If only data for collisions with He are available, a first order correction can
be made by assuming H_{2} to have the same cross sections. This
approximation is strictly only valid for very cold sources, where most
H_{2} is in the ground J=0 state without angular momentum.
Then from Eq. (10) the rate coefficient for
collisions between a molecular species X and H_{2}



(12) 
If the mass of the molecule is much larger than that of He and H_{2},
the scaling factor is 1.4.
Some molecules of significant
interest lack calculated collisional rate coefficients. In these cases the rates
for a similar molecule have been adopted and only scaled for the
difference in reduced mass following Eq. (12). This procedure
works best for OS substitutions (for example, scaling HCO^{+}rates for the case of HCS^{+}) since such molecules have a similar molecular structure.
For most species, only rate coefficients with He or H_{2} J=0 are
available. Values with H_{2} J=1 can be larger by factors of 25
due to supplementary terms in the interaction potential (e.g. Green
1977, H_{2}O example). This additional uncertainty is often not
considered in astrophysical analyses. In the case of CO and H_{2}O, separate
rate coefficients are available for collisions with ortho and
paraH_{2}. The online version of RADEX weighs these coefficients by
the thermal value of the H_{2} o/pratio at the kinetic temperature.
The o/pratio is approximated as the J=1 to J=0 population ratio with
a maximum of 3.0, which is an overestimate by at most 20% (at
T=155 K).
In the datafiles available for download the collisional rate
coefficients for collisions with orthoH_{2} and paraH_{2} are kept
separate.
To obtain the collision rate, RADEX simply multiplies the collisional
rate coefficients with the H_{2} density. To include the effect of
collisions with He, the user must multiply the density by 1.14 (to first order) for a He
abundance with respect to H_{2} of 20%.
The adopted collisional rate coefficients are presented in Tables 2 and 3 for atomic and molecular species, respectively. For isotopomers the same set of
collisional rate coefficients as for the main isotope was adopted, unless otherwise stated.
Tables 2 and 3 show the temperature range and
maximum energy (
)
for which calculations are available.
Also, the collision partner is indicated.
Only the downward values are given in the data
files; the upward rate coefficients are obtained through detailed
balance using Eq. (11).
Table 2:
Summary of atomic collisional data from the literature.
Table 3:
Summary of molecular collisional data from the literature and new extrapolated rate coefficients.
Most of the collisional data summarized in Table 2 have been obtained
from theoretical calculations, with experimental crosschecks possible
for only a few cases. Most experiments reflect the average of many
collisional events, with comparisons typically done for relaxation
rates and collisioninduced pressure line broadening. Statetostate
measurements have been possible for only a few systems and they often
report relative rather than absolute cross sections. Also,
experiments with H_{2} are usually done for nH_{2} (i.e., 75%
oH_{2} and 25% pH_{2}), rather than for H_{2} J=0 or 1.
Nevertheless, such comparisons between theory and experiment, as well
as those between different theoretical methods, have given some
indication of the uncertainties in the collisional rate
coefficients. Excellent accounts of the methods involved and details
on individual systems are given by Green (1975a), Flower (1990) and
Roueff (1990); recent developments are reviewed by
Roueff et al. (2004). Here only a brief summary is given.
The theoretical determination of collisional rate coefficients
consists of two steps: (i) determination of the interaction potential
V between the colliding systems; and (ii) calculation of the
collision dynamics. Significant progress in the second part has been
made in the last decades, aided by the increased computer speed. The
most accurate method is the CloseCoupling (CC) method, in which the
scattering wave function is expanded into a set of basis
functions. This method is exact if an infinite number of basis
functions or "channels'' is taken into account. In practice a finite
number of channels is used, resulting in a set of coupled secondorder
differential equations. The absolute accuracy of the results can
easily be checked by increasing the basis set, and is of order a few
% for a given interaction potential. This method works very well for
low collision energies and relatively light species, although care
should be taken at the lowest energies whether resonances are properly
sampled (e.g., Dubernet & Grosjean 2002). However, the method becomes
increasingly computationally demanding at high energies and for heavy
polyatomic molecules with small splittings between the rotational
energy levels resulting in many channels.
The most popular approximate dynamical methods are the Coupled States
(CSt) or "centrifugal decoupling'' method and the Infinite Order Sudden
(IOS) approximation. In the CSt method, the centrifugal potential is
assumed to conserve the projection of the angular momentum on the axis
perpendicular to the plane of the collision partners. This
approximation is often valid at higher energies if the collision is
dominated by the repulsive part of the potential. In the IOS
approximation, an additional assumption is that the molecule does not
rotate during collisions. This may be appropriate for heavy rotors at
energies much larger than the rotational energies. From comparisons
with the more exact CC results, it is found that absolute
uncertainties for the CSt method range from 10% to a factor of
2, with lesser uncertainties in the relative values. The propensities
in the collisions are recovered correctly. In contrast, the IOS method
can have uncertainties up to an order of magnitude. Computer programs
which include the CC, CSt and IOS options are publically available
(see Hutson & Green 1994^{}; Flower et al. 2000^{}, Manolopoulos 1986; Alexander & Manolopoulos 1987^{}).
The above quoted ranges of uncertainties assume that the interaction
potential is perfectly known. Often, this is not the case and the
potential surfaces form the largest source of error in the collisional
rates with uncertainties that are difficult to assess. The interaction
potential consists of a shortrange repulsive part, an
intermediaterange interaction part where a weak molecular bond is
formed, and a longrange part dominated by electrostatic
interaction. The intermediate part is most difficult to determine and
requires highlevel quantum chemical models. The most accurate method
is that of Configuration Interaction (CI), but it can become very
costly in computer time. Other methods include HartreeFock
SelfConsistentField (SCF) and perturbation methods, and more
recently Density Functional Theory (DFT), but each of these methods
has its drawbacks. An old approximate method, the Electron Gas model,
is now obsolete, but some dynamics calculations for astrophysical
systems still use these potentials
(e.g., CSH_{2}, Turner et al. 1992).
The following selected examples serve to illustrate the range of
absolute errors in the adopted collisional rate coefficients. It
should be noted that relative values often have less uncertainty and
that these are most relevant for astrophysical applications: small
absolute errors can often be compensated by small adjustments in the
abundance of the species.

Figure 1:
Predicted CO line intensities, using different sets of calculated
collisional rate coefficients, for an isothermal homogeneous sphere with
a kinetic temperature 10 K, a H_{2} density of 10^{3} cm^{3} and a CO
column density of
cm^{2}.
The line intensities are shown in relation to the values
obtained using the COpH_{2} rate coefficients from
Flower (2001a). The upper rotational quantum number
is indicated on the xaxis.
The rotational transitions are out of thermal equilibrium and, for transitions below
,
optically thick. 
Open with DEXTER 
Early calculations by Green & Thaddeus (1976),
Schinke et al. (1985) and Flower & Launay (1985) illustrate the
sensitivity of the results to different potential energy
surfaces. Absolute differences in individual collisional rate
coefficients range from a few % up to 40%, with the relative values
usually having less scatter. Comparison of computed cross sections
using a new COH_{2} potential by Jankowski & Szalewicz (1998)
with pressure broadening and scattering experiments by Mengel et al. (2000)
suggests an overall average absolute accuracy of better than 10%
at K, but somewhat less good at the lowest temperatures
where the deviations can increase to 3050%. No information is
available on the accuracy of the larger
transitions (e.g.,
), which become important at high temperatures such as
found in dense shocks. The same potential surface has been used in the latest set of rate coefficients given by Flower (2001a) which are adopted here.
The following simple test problem illustrates the consequences of using different sets of collisional rate coefficients. Line intensities were calculated for
the lowest 5 rotational transitions of CO for a molecular cloud of constant temperature
and density using RADEX. The model has a temperature of 10 K, H_{2} density of
cm^{3} and a total CO column density of
cm^{2} over a
line width (fullwidth at halfmaximum) of 1 km s^{1}. All lines are out of thermal equilibrium and the three lowest rotational transitions are optically thick. As is shown in Fig. 1, differences of up to % are found, especially for collisions with paraH_{2}compared with orthoH_{2}.
The H_{2}COH_{2} rate coefficients given in our
database are obtained from Green (1991), who calculated values for
the H_{2}COHe system using a very old potential energy surface by
Garrison et al. (1975) based on SCF and limited CI calculations.
These rate
coefficients and the adopted surface have recently been tested against
pressure broadening and timeresolved doubleresonance studies for
three lowlying transitions (Mengel & De Lucia 2000). Satisfactory
agreement is found for the H_{2}COHe system, with differences in
cross sections ranging from a few % up to 20%. The deviations are
largest at the lowest temperatures, <10 K, as was also found for
COH_{2}. For H_{2} as the collision partner, the cross sections are
found to be up to a factor of two higher, significantly more than the
value of 1.4 expected from the difference in masses, illustrating that
simple scaling from He collisions may introduce errors up to 50%.
One of the computationally most challenging systems
is OHH_{2}, since OH is an open shell molecule with a
ground
state so that two potential surfaces and hyperfine splitting are
involved. Results for collisions with both oH_{2} and pH_{2} are
presented by Offer & van Dishoeck (1992) using an old potential
surface based on SCF calculations (Kochanski & Flower 1981), and by
Offer et al. (1994) using a new surface computed using CI (Offer & van Hemert 1993).
The differences due to the potential energy surface
range from 10% to more than an order of magnitude for individual
rate coefficients. Comparison with statetostate experimental cross
sections with both nH_{2} and pH_{2} at one specific energy gives
surprisingly good agreement, usually within 50% but with occasional
excursions up to an order of magnitude (Schreel & ter Meulen 1996).
Moreover, all the propensities for individual hyperfine transitions
are well reproduced.

Figure 2:
Calculated and extrapolated collisional deexcitation rate coefficients for
CO in collisions with paraH_{2}. Open triangles indicate extrapolation in temperature to the
rate coefficients of Flower & Launay (1985) (filled triangles). Open squares show the extrapolation to higher temperatures and energy levels of the recent rate coefficients calculated by Flower (2001a) (filled squares). For comparison the rate coefficients presented by Schinke et al. (1985) (filled circles) and the extrapolation performed by Larsson et al. (2002) (filled stars) are shown. 
Open with DEXTER 
Below follows a summary of the collisional rate coefficients adopted
in the first release of the database.
Molecules for which only one set of calculated collisional rate
coefficients is available and where no extrapolation was performed
are not described further here. The principle method for extrapolating the downward
collisional rate coefficients (
J_{l},
J_{u} > J_{l}) in temperature in the case of a linear molecule is
(de Jong et al. 1975; Bieging et al. 1998)



(13) 
where
and the three parameters a, b, and care determined by leastsquares fits to the initial set of rate
coefficients for each .
This reproduces most of the calculated rate coefficients
to within 50% and typically within 20%.
For more details on the extrapolation
scheme, including extrapolation in energy levels, see Sect. 6 (only available in the online version of this journal).
When the specified kinetic temperature falls outside the region where collisional rate
coefficients are available, i.e. from
to
,
RADEX makes no further extrapolation and assumes the downward rate coefficients at
and
,
respectively.
4.3.1 CO
For CO the collisional rate coefficients calculated by Flower (2001a) have
been adopted as a starting point. These computations cover
temperatures in the range from 5 K up to 400 K and
include rotational levels up to J=29 and J=20 for collisions with
paraH_{2} and orthoH_{2}, respectively. Both sets of rate
coefficients were then extrapolated to include energy levels up to
J=40 (using Eq. (18)) and temperatures up to 2000 K (using Eq. (13)), as described in
Sect. 6.1. In the datafile available for download,
the collisional rate coefficients for collisions with orthoH_{2} and
paraH_{2} are kept separate. However, in RADEX they are weighted
together as described in Sect. 4.1.
Figure 2 shows the extrapolation of CO collisional deexcitation
rate coefficients for collisions with paraH_{2}.
It is clear that extrapolated rate
coefficients are uncertain and depend on both the original data set
from which the extrapolation is made and the method adopted. However,
the extrapolated values typically agree within 50% in the case of CO.
The largest discrepancies, up to an order of magnitude, naturally
arise in the region where extrapolation in both temperature and energy
levels are performed.
Thus, in the parts of parameter space where extrapolated rates are being used to infer
physical conditions, care should be taken as to any astrophysical conclusions drawn
from the modeling.
For CS the rate coefficients calculated by Turner et al. (1992) have been
adopted as a starting point. These values have been computed
for temperatures in the range 20 300 K and include
rotational levels up to J=20 for collisions with H_{2}. This set
was then extrapolated to include energy levels up
to J=40 (using Eq. (18)) and temperatures up to 2000 K
(using Eq. (13)), as described in
Sect. 6.1. No extrapolation to temperatures
lower than 20 K was attempted.
For SiO the rate coefficients calculated by Turner et al. (1992) have been
adopted, computed for temperatures in the range 20300 K and including
rotational levels up to J=20 for collisions with H_{2}. This set
was then extrapolated to include energy levels up
to J=50 (using Eq. (18)) and temperatures up to 2000 K (using Eq. (13)), as described in
Sect. 6.1. No extrapolation to temperatures lower
than 20 K was attempted.
No calculated rate coefficients are available for SiS. Instead, the
same set of collisional rate coefficients as for SiO has been adopted.
The rate coefficients for HCO^{+} in collisions with H_{2} have been calculated by
Flower (1999) for temperatures in the range 10400 K and
rotational levels up to J=20. This set of rate
coefficients was then extrapolated to include energy levels up to
J=30 (using Eq. (18)) and temperatures up to 2000 K (using Eq. (13)), as described in
Sect. 6.1.
The rate coefficients for HC_{3}N in collisions with He
have been calculated by Green & Chapman (1978) for temperatures in the range 1080 K and
rotational levels up to J=20. This set of rate coefficients was then
extrapolated to include energy levels up to J=50 (using Eq. (18)) and
temperatures up to 2000 K (using Eq. (13)), as described in
Sect. 6.1. The rate coefficients were then scaled
by 1.39 to represent collisions with H_{2} instead of He.
The rate coefficients for HCN in collisions with He
have been calculated by Green & Thaddeus (1974) for temperatures in the range 5100 K and
rotational levels up to J=7. This work has subsequently been extended by S. Green (unpublished data) to include rotational levels up to J=29 and temperatures from
1001200 K. Extrapolation of the rate coefficients to include energy levels up to
J=29 for temperatures below 100 K (using Eq. (18)), as described in
Sect. 6.1, has been made. The rate coefficients were subsequently
scaled by 1.37 to represent collisions with H_{2} instead of He.
The collisional rate coefficients between various
hyperfine levels have been calculated by Monteiro & Stutzki (1986) for the lowest ()
rotational levels
and temperatures from 1030 K in collisions with He. A datafile
based on these collisional rate coefficients is
also made available separately.
No calculated rate coefficients are available for HNC. Instead, the
same set of collisional rate coefficients as for HCN has been adopted.
The rate coefficients for N_{2}H^{+} in collisions with He
atoms have been calculated by Green (1975b) for temperatures in the range 540 K and
rotational levels up to J=6. Given the limited range in temperature
and energy levels, we have instead adopted the same rate
coefficients as for HCO^{+}. This is motivated by the discussion in
Monteiro (1984) where the rate coefficients for these two species in collisions with He are found to be very similar, typically within 10%.
The rate coefficients for HCS^{+} in collisions with He
atoms have been calculated by Monteiro (1984) for temperatures in the
range 1060 K and rotational levels up to J=10. This set of rate coefficients was then
extrapolated to include energy levels up to J=23 (using Eq. (18)) and
temperatures up to 1000 K (using Eq. (13)), as described in
Sect. 6.1. The rate coefficients were subsequently scaled
by 1.38 to represent collisions with H_{2} instead of He.
In RADEX the rate coefficients for H_{2}O in collisions with He calculated by Green et al. (1993)
are used as default. The rates were computed for temperatures in the range from 20 to 2000 K including energy levels up to about 1400 cm^{1}. These rate coefficients were subsequently scaled
by 1.35 to represent collisions with H_{2} instead of He.
In addition, a datafile containing the recent rate coefficients for H_{2}O in collisions with pH_{2} (Grosjean et al. 2003) and
oH_{2} (Dubernet & Grosjean 2002) calculated for low temperatures (520 K) has been constructed.
In the datafiles available for download,
the rate coefficients for collisions with orthoH_{2} and
paraH_{2} are kept separate. However, in RADEX they are weighted
together as described in Sect. 5.1.
For nonlinear molecules there are no simple scaling relations such as Eq. (13).
In Sect. 6.2 (only available in the online version of this paper) the procedure adopted to extrapolate rate coefficients for SO_{2} is presented. As starting point the calculated rate coefficients for SO_{2} in collisions with He calculated by Green (1995) were used. These rates were computed for temperatures in the range from 25 to 125 K including energy levels up to about 62 cm^{1}.
Extrapolation was made to include energy levels up to 250 cm^{1} and
temperatures in the range from 10 to 375 K. The rate coefficients were subsequently scaled
by 1.4 to represent collisions with H_{2} instead of He.
A compilation of atomic and molecular data in
a homogeneous format relevant for radiative transfer modelling is presented. The
data files are made available through the WWW and include energy
levels, statistical weights, Einstein Acoefficients and collisional
rate coefficients. Extrapolation of collisional rate coefficients are
generally needed and different schemes for this are reviewed.
In addition to the atomic and molecular database, an online version of a computer code for performing
statistical equilibrium calculations is made available for use through
the WWW. The program, named RADEX, is an alternative to the widely
used rotation diagram method and has the advantage of supplying the
user with physical parameters such as density and temperature.
Databases such as these depend heavily on the efforts by the chemical physics community to provide the relevant atomic and molecular data. We strongly encourage further efforts in this direction, so that the current extrapolations of collisional rate coefficients can be replaced by actual calculations in future releases.
Acknowledgements
The authors are grateful to D. J. Jansen and F. P. Helmich for
contributions to the data files and programs. B. Larsson is thanked
for providing his collisional rate coefficients for CO.
The referee A. Markwick is thanked for a constructive report that helped improve both the paper as well as the online database.
This
research was supported by the Netherlands Organization for
Scientific Research (NWO) grant 614.041.004 and a NWO Spinoza grant.
F.L.S. and J.H.B. further acknowledge financial support from the Swedish Research
Council.
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Online Material
6 Extrapolation of collisional rate coefficients
6.1 Linear molecules
An often adopted starting point when fitting and extrapolating
collisional rate coefficients is to take advantage of the IOS approximation
in which the entire matrix of
statetostate rate coefficients can be calculated from the basic
rate coefficients (e.g. Goldflam et al. 1977)

(14) 
where

(15) 
is the Wigner 3j symbol. This expression is valid only in the limit
where the kinetic energy of the colliding molecules is large compared
to the energy splitting of the rotational levels. Since the
energy splitting increases with J this expression becomes less
accurate for higher rotational levels. DePristo et al. (1979) show
that by multiplying Eq. (14) (within the summation) with



(16) 
where



(17) 
one can approximately correct for this deficiency. Here B_{0} is
the rotational constant in cm^{1}, l is the scattering length in Å (typically
Å),
is the reduced mass of the
system in amu and T is the kinetic temperature in K.
Extrapolation of the rate coefficients down to the lowest J =0level can be made both in temperature as well as in J allowing the
general statetostate coefficients to be extended (e.g., Larsson et al. 2002; Albrecht 1983).

Figure 3:
The solid lines are fits to the COpH_{2} collisional rate coefficients from
Flower (2001a) (squares ) for transitions down to the ground state from upper
energy levels
using a second order polynomial. 
Open with DEXTER 
Alternatively, and in line with the IOS approximation, the downward
collisional rate coefficients (
,
J_{u} > J_{l}) can be extrapolated in temperature using Eq. (13).
Given its simplicity we have
adopted this procedure for extrapolation of the rate coefficients in
temperature.
Extrapolation to include higher rotational
levels was carried out by fitting the collisional rate coefficients connecting to the ground rotational state, at a
particular temperature, to a second order polynomial



(18) 
where a, b and c are parameters determined from the fit. Figure 3 illustrates the fit to collisional rate coefficients down to the ground rotational state for COH_{2} using Eq. (18).
Similar extrapolations can be made in temperature. However, here we have adopted the approach by
de Jong et al. (1975) and Bieging et al. (1998) and used Eq. (13)
for the extrapolation in temperature. This extends the fit over a
larger range of energies.
The IOS approximation (Eq. (14)) was then used to calculate the entire matrix of
statetostate rate coefficients. The CO molecule is used in Sect. 4.3.1 to illustrate the above mentioned schemes.
6.2 Nonlinear molecules
For nonlinear species there are no simple scaling relations and one has to
resort to custommade fitting formulae for each case. The only case considered here is that
of SO_{2} used by van der Tak et al. (2003). In the prolongation of this project extrapolated collisional rate coefficients will be presented for additional nonlinear molecules.
As the starting point for inelastic collisional data for SO_{2}, the results of
Green (1995) were used. However, those data only cover the lowest 50
states, up to 62 cm^{1} (
), while states up to J=25 are
commonly observed.

Figure 4:
Collisional deexcitation rate coefficients for the lowest 50
states of SO_{2} summed over all lower levels,
calculated from the data by Green (1995) for various temperatures. 
Open with DEXTER 
Figure 4 plots Green's downward rate coefficients,
summed over all final states, as functions of initial state. These
sums approach asymptotic values for
cm^{1}.
Deviations from this behaviour due to detailed quantum mechanical
selection rules are seen not to exceed 20%.
The figure also shows that the rate coefficients increase
approximately as T^{1/2}, again to 20% accuracy. This
behaviour indicates that the total rate coefficients only depend on
temperature through the collision velocity, while the deexcitation
cross sections are constant.
Figure 5 shows that most collisions lead to
deexcitation into states that are not far down in energy. The
thick black curve is our fit to this behaviour: it is the normalized mean of
the various thin light curves which represent Green's data. Transitions by
more than 15 states are considered negligible.

Figure 5:
Statetostate deexcitation rate coefficients for SO_{2} as fractions of the total downward rate coefficient
(Fig. 4), as a function of the number of levels
by which the transition is changed. The light (coloured) curves
are values from Green (1995) at T=25 K for the 10th,
20th, 30th, 40th and 50th
state above ground. The thick black curve is the normalized mean
of the light (coloured) curves, adopted here to extrapolate
Green's rate coefficients to higherlying levels. The states are
labelled in order of increasing energy. 
Open with DEXTER 
Based on these trends, the rate coefficients for deexcitation of SO_{2} in inelastic
collisions with He are extrapolated as follows. For the 50 lowest
states, Green's values at 25<T<125 K are used and multiplied by
at temperatures up to 375 K and down to
10 K. For states between 62 and 250 cm^{1} above ground, a total
deexcitation rate coefficient of
cm^{3} s^{1} is assumed, shown by Fig. 4 to be a good
zerothorder description for other levels. The statetostate rate
coefficients are derived by multiplying these totals by the mean
propensities from Green (1995), given by the black curve in
Fig. 5. All results are multiplied by 1.4 to account
for the mass difference between H_{2} and He. While this procedure is
admittedly crude and does not take the detailed quantum mechanics of
the interaction into account, it catches the spirit of more detailed
calculations.
Copyright ESO 2005