MNRAS 455, 276–281 (2015)
doi:10.1093/mnras/stv2329
Reactive collisions of very low-energy electrons with H+
2 : rotational
transitions and dissociative recombination
M. D. Epée Epée,1 J. Zs Mezei,2,3,4 O. Motapon,1,5‹ N. Pop6 and I. F. Schneider2,3‹
1 LPF,
UFD Mathématiques, Informatique Appliquée et Physique Fondamentale, University of Douala, P. O. Box 24157, Douala, Cameroon
Ondes et Milieux Complexes, UMR 6294 CNRS and Université du Havre, 25, rue Philippe Lebon, BP 540, F-76058 Le Havre, France
3 Laboratoire Aimé Cotton CNRS-UPR-3321, Université Paris-Sud, Orsay F-91405, France
4 Laboratoire des Sciences des Procédés et des Matériaux, UPR 3407 CNRS and Univ. Paris 13, 99 avenue Jean-Baptiste Clément, F-93430 Villetaneuse,
France
5 Faculty of Science, University of Maroua, PO Box 814 Maroua, Cameroon
6 Department of Physical Foundation of Engineering, University Politechnica of Timisoara, Bv Vasile Parvan No 2, 300223, Timisoara, Romania
2 Laboratoire
ABSTRACT
A new series of computations has been performed to obtain cross-sections and rate coefficients
for state-to-state rotational transitions in the H+
2 ion, induced by collisions with very low-energy
electrons. Following our recent work on the HD+ ion (Motapon et al. 2014), and using the same
molecular structure data sets, excitations Ni+ → Ni+ + 2 for Ni+ = 0 to 10, and de-excitations
Ni+ → Ni+ − 2, for Ni+ = 2 to 10, in the energy range 0.01 meV–0.3 eV, have been explored.
The calculated cross-sections have been convolved in order to obtain Maxwell rate coefficients
for electronic temperatures up to a few hundred of Kelvin. Moreover, Maxwell rate coefficients
for dissociative recombination have been calculated for the same initial rotational levels.
Key words: Molecular processes – Scattering – ISM: abundances.
1 I N T RO D U C T I O N
In diffuse interstellar media and planetary atmospheres, electrons
are expected to be the most important exciting species for molecular
ions. At very low energy, the cross-sections for rotational transitions
are indeed several orders of magnitude larger than those for atomic
and molecular impact excitation, as one could learn from the numerous previous computations concerning diatomic and polyatomic
molecular ions (Faure & Tennyson 2001, 2002, 2003; Faure et al.
2006; Kokoouline et al. 2010).
In particular, the electron-impact induced transitions:
+
+
+
+
+
−
− H+
2 (Ni , vi = 0) + e (ε) −→ H2 (Nf , vi = 0) + e (ε ),
(1)
competes with dissociative recombination (DR):
+
+
−
H+
2 (Ni , vi = 0) + e (ε) −→ H(1s) + H(n ≥ 2),
of vibrationally relaxed (vi+
(2)
0) H+
2 . These are important reactions
=
in a variety of contexts (Black & Dalgarno 1976; Van Dishoeck &
Black 1986; Black & Van Dishoeck 1987; Le Petit, Roueff & Le
Bourlot 2002; Agundez et al. 2010; Coppola et al. 2011; Gay et al.
2012). Here, Ni+ /Nf+ is the initial/final rotational quantum number of the cation, ε/ε the kinetic energy of the incident/emergent
E-mail: [email protected] (OM); ioan.schneider@univ-lehavre
.fr (IFS)
electron, and n the principal quantum number of the excited atom
resulting from dissociation.
Recently, Motapon et al. (2014) have reported cross-sections
and rate coefficients for state-to-state rotational transitions in HD+
–inelastic collisions (IC, Ni+ < Nf+ ) and superelastic collisions
(SEC, Ni+ > Nf+ )–and DR, within the framework of our stepwise
method based on the multichannel quantum defect theory (MQDT)
(Giusti-Suzor 1980; Motapon et al. 2008). The comparison of our
scattering cross-sections with the results of the R-matrix method
(Tennyson 2010), combined with the Coulomb–Born approximation (Chu 1975), and the adiabatic-nuclei rotation (ANR) approximation (Rabadán & Tennyson 1998; Faure & Tennyson 2002), was
found to be very satisfactory. These results were compatible with
those for fast rotational relaxation of HD+ observed at the Heidelberg Test Storage Ring (Shafir et al 2009; Schwalm et al 2011).
On the other hand, the DR measurements performed in this latter
device agree well with our computed data.
The aim of the present article is to provide astrophysicallyrelevant results coming from a similar work on H+
2 , based on the
molecular-structure data sets used in our previous study (Motapon
et al. 2014). Contrary to the R-matrix based computations already
carried out for two transitions in this system (Faure & Tennyson
2001), the MQDT has the advantage to treat simultaneously the direct and indirect processes, which result in rich resonance structures
in the cross-sections. The cross-sections of rotational transitions for
this system are compared with those recently reported for HD+ in
C 2015 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Downloaded from http://mnras.oxfordjournals.org/ at Stockholms Universitet on November 2, 2015
Accepted 2015 October 5. Received 2015 October 5; in original form 2015 September 8
H2+ rotational excitation and recombination
order to estimate the effect of isotopic substitution on the collisional
processes.
The paper is organized as follows. In Section 2, we briefly describe our theoretical approach. Cross-sections, rate coefficients and
their comparison with previous results are presented in Section 3,
and the conclusions follow in Section 4.
2 THEORETICAL METHOD
The main steps of our current MQDT treatment (Motapon et al.
2008) are described below.
2.1 Construction of the interaction matrix V
N
|Vd(e)
|χN+ ,v+ ,
VdNM
+ + (E, E) = χd
j ,l
j
j ,lN v
(3)
where E is the total energy, χdN
and χN+ ,v+ are the nuclear wave
j
functions corresponding to a dissociative state and to an ionization
channel, respectively.
This procedure applies in each -subspace and results in a blockdiagonal global interaction matrix. The block-diagonal structure,
corresponding to the symmetries, propagates to the matrices
invoked below.
2.2 Construction of the reaction matrix K
Starting from the interaction matrix V, we build the K-matrix,
which satisfies the Lippmann–Schwinger integral equation:
K=V +V
1
K.
E − H0
(4)
This equation has to be solved once that V - whose elements are
given by equation (3) - is determined. Here H0 is the zero-order
Hamiltonian associated with the molecular system neglecting the
interaction potential V. It has been proven that, due to the energyindependent electronic couplings, the perturbative solution of equation (4) is exact to the second order (Ngassam et al. 2003).
2.3 Building of the eigenstates
In order to express the result of the short-range interaction in terms
of long-range phase-shifts, we perform a unitary transformation
of our initial basis into a new one, consisting of eigenstates of the
system. The columns α of the corresponding transformation unitary
and Udj ,α - are the eigenvectors of the
matrix U - of elements Ulv,α
K-matrix:
1
KU = − tan(η)U,
π
(5)
and its eigenvalues, expressed as the elements of a diagonal matrix
−π −1 tan(η), provide the diagonal matrix of the phase shifts ηα ,
induced in the eigenstates by the short-range interactions.
2.4 Frame transformation to the external region
In the external zone - the ‘B-region’ (Jungen & Atabek 1977) - characterized by large electron-core distances, the Born–Oppenheimer
representation is no longer valid for the whole molecule, but
only for the ionic core. Here, is no longer a good quantum number and a frame transformation (Fano 1970; Chang &
Fano 1972; Vâlcu et al. 1998) is performed between coupling
schemes corresponding to the incident electron being decoupled
from the core electrons (external region) or coupled to them (internal region). The frame transformation coefficients involve angular
coupling coefficients, electronic and ro-vibronic factors, and are
given by
2N + + 1 1/2 +
+
l − + N + + |lN + N ClN v ,α =
2N + 1
+
1 + τ + τ (−1)N−l−N
× 1/2
2 2 − δ+ ,0 1 + δ+ ,0 δ,0
+
×
Ulv,α
χN+ v+ | cos(π μ
l (R) + ηα )|χNv ,
(6)
v
and
Cdj ,α = Udj α cos ηα ,
(7)
as well as SlN + v+ ,α and Sdj ,α , which are obtained by replacing
cosine with sine in equations (6) and (7). In these formulas, the
+
vibrational wave functions of the molecular ion χN+ v+ are coupled
via the variation of the
to the those of the neutral system χNv
with
the
internuclear
distance
R. The quantities
quantum defect μ
l
τ + and τ are related to the reflection symmetry of the electronic ion
and neutral wave functions, respectively, and take the values +1 for
symmetric states and −1 for antisymmetric ones. The middle term
MNRAS 455, 276–281 (2015)
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The construction of the interaction matrix V is performed in the
outer shell of the region of small electron-ion and nucleus–nucleus
distances, that is, in the ‘A-region’ (Jungen & Atabek 1977), where
the Born–Oppenheimer approximation gives an appropriate description of the collision system. The good quantum numbers in this
region are N, M, and , associated respectively with the total angular
momentum and its projections on the z-axis of the laboratory-fixed
and of the molecule-fixed frame.
Within a quasi-diabatic representation (Bardsley 1968; Sidis &
Lefebvre-Brion 1971; Giusti-Suzor 1980), the relevant molecular
states are organized in channels, according to the type of fragmentation which they are meant to describe.
An ionization channel is built starting from one of the rovibrational level N+ v + of the ground electronic state of the ion,
and is completed by gathering all the one electron states of the
optical electron with a given orbital quantum number l.
Within an open ionization channel, these one electron states describe, with respect to the N+ v + threshold, a ‘free’ electron - in
which case the molecule experiences (auto)ionization. Within a
closed ionization channel, they correspond to a bound electron - in
which case, the neutral state corresponds to a temporary capture
into a Rydberg state. In the A-region, these states may be modelled
reasonably well with respect to the hydrogenic states in terms of the
quantum defect μ
l , dependent on the internuclear distance R, but
assumed to be independent of energy.
A dissociation channel, labelled dj , accounting for the atom–atom
scattering, consists of a valence state characterized by a potential
energy curve (PEC) whose asymptotic limit lies below the total
energy of the neutral collisional system.
The ionization channels are coupled to the dissociation ones by
the electrostatic interaction 1/r12 . In the molecular-orbital picture,
the states corresponding to the coupled channels must differ by at
least two orbitals, the dissociative states being doubly-excited in the
present case. We account for this coupling at the electronic level
first, through an R-dependent scaled ‘Rydberg-valence’ interaction
, assumed to be independent of the energy of the electerm, Vd(e)
j ,l
tronic states pertaining to the ionization channel. Subsequently, the
integration of this electronic interaction on the internuclear motion
results in the elements of the interaction matrix V:
277
278
M. D. Epée Epée et al.
in the right-hand side of equation (6) contains the selection rules
for the rotational quantum numbers.
The projection coefficients given in equations (6) and (7) include
both types of couplings that control the process: the electronic
coupling, expressed by the elements of the matrices U and η, and the
non-adiabatic coupling between the ionization channels, expressed
by the matrix elements involving the quantum defect μ
l .
3 C RO S S - S E C T I O N S A N D R AT E
COEFFICIENTS
2.5 Construction of the generalized matrix X
The matrices C and S with the elements given by equations (6) and
(7) are the building blocks of the ‘generalized’ scattering matrix X:
X=
C + iS
.
C − iS
(8)
where o and c label the lines or columns corresponding to open and
closed channels, respectively.
2.6 Elimination of closed channels
The building of the X matrix is performed independently of the
asymptotic behaviour of the different channel wavefunctions. Eventually, imposing physical boundary conditions leads to the ‘physical’ scattering matrix, restricted to the open channels (Seaton
1983):
1
X co .
X cc − exp(−i2πν)
(10)
It is obtained from the sub-matrices of X appearing in equation
(9) and from a further diagonal matrix ν formed with the effective
quantum numbers νN + v+ = [2(EN + v+ − E)]−1/2 (in atomic units)
associated with each vibrational threshold EN + v+ of the ion situated
above the current energy E (and consequently labeling a closed
channel).
2.7 Evaluation of the cross-section
Given a molecular ion initially on the level Ni+ vi+ and an electron
having the kinetic energy ε, the cross-section of recombination into
all the dissociative states dj of the same symmetry (‘sym’: total
electronic spin quantum number, gerade/ungerade) of the neutral is
given by
π 2N + 1 sym N
N,sym
|Sd ,lN + v+ |2 ,
(11)
ρ
σdiss←N + v+ =
j
i i
i i
4ε 2Ni+ + 1
l,,d
j
while the cross-section for the ro-vibrational transition into the final
level Nf+ vf+ , i.e. collisional (de-)excitation is:
2
π 2N + 1 sym N
N,sym
S
ρ
σN + v+ ←N + v+ =
(12)
Nf+ vf+ l ,Ni+ vi+ l .
f f
i i
4ε 2Ni+ + 1
l,l ,
Here, ρ sym is the ratio between the multiplicities of the neutral
and the target ion. After performing the MQDT calculation for
all the accessible total rotational quantum numbers N and for all
MNRAS 455, 276–281 (2015)
The MQDT treatment of rotational transitions and DR requires data
for PECs of the ion ground state and the relevant doubly excited
states of the neutral molecule, as well as for the Rydberg series of
mono-excited states, namely the quantum defects and their electronic couplings to the dissociative continuum. Besides considering the neutral state having the most favorable crossing with the
ion − the lowest dissociative 1 g+ state − one has to include other
molecular symmetries, since the rotational couplings are important in the low-energy collisions (Motapon et al. 2014). Therefore,
the molecular states of the neutral (H2 ) system taken into account
are of 1 g+ , 1 g , 1 g , 3 g+ , 3 g ,3 g , 3 u+ , and 3 u symmetries.
Most of these data were extracted from ab initio molecular structure calculations (Kolos & Wolniewicz 1969; Ross & Jungen 1987;
Wolniewicz & Dressler 1994; Detmer, Schmelcher & Cederbaum
1998; Orlikowski, Staszewska & Wolniewicz 1999; Staszewska &
Wolniewicz 2002), completed by R-matrix calculations (Tennyson
1996; Telmini & Jungen 2003; Bezzaouia, Telmini & Jungen Ch.
2004; Bezzaouia et al. 2011). This latter method produces with
high accuracy on one hand the Rydberg states close to the ionization threshold and consequently the R−dependent quantum defects,
and one the other hand the positions and widths of doubly excited states above the ionization threshold, providing the electronic
couplings.
For each of the symmetries involved, only the lowest dissociative
state, relevant for low-energy collisions, is considered. Accordingly,
the partial waves considered for the incident electron were s and
d for the 1 g+ states, d for 1 g , 1 g , 3 g+ , 3 g and 3 g , and p
for 3 u+ , and 3 u . The resulting DR and rotational transition crosssections and rate coefficients of H+
2 on the lowest 11 rotational
levels of the ground and first excited vibrational levels (vi+ = 0, 1
and Ni+ = 0 − 10) are given below.
In Fig. 1 we compare the cross-sections of rotational transitions
0 → 2 and 1 → 3 from this work with those of Faure & Tennyson
(2001), based on the R-matrix method combined with the ANR
approximation. One may notice the very good overall agreement
with those of the ANR calculations, but also the presence of a rich
resonance structure in the MQDT results, unlike in the ANR ones.
These resonances are due to the indirect process, that is the temporary capture on the Rydberg states of H2 , a mechanism neglected
within the ANR approach. In the MQDT treatment of H2 the gerade
symmetry is completely uncoupled from the ungerade one and, due
to selection rules inherent to the relevant partial waves (see equation
6), only transitions with even N+ (0, 2 and 4) are allowed.
The influence of the vibrational excitation on the rotational excitation at low energy can be seen in Fig. 2, where we illustrate
2 +
the changes in the cross-sections for H+
2 (X g ) initially in the vi+
brational levels vi = 1 with respect to those obtained in the case
vi+ = 0. One can see that the resonances are slightly shifted, and that
the shape of several resonances is changed, but the overall values
of the cross-sections are almost the same.
In Fig. 3, we represent the Maxwell rate coefficients for IC Ni+ →
2 +
Ni+ + 2, with Ni+ = 0 to 10 for H+
2 (X g ) on its ground vibrational
Downloaded from http://mnras.oxfordjournals.org/ at Stockholms Universitet on November 2, 2015
It involves all the channels, open and closed. Although one may
expect that only the open channels are relevant for a complete collisional event, the participation of the closed channels can strongly
influence the cross-section, as shown below.
The X matrix relies on four block sub-matrices:
X oo X oc
,
(9)
X=
X co X cc
S = X oo − X oc
the relevant symmetries, one has to sum up the corresponding
cross-sections in order to obtain the global cross-section for DR
or ro-vibrational transitions, as a function of the electron collision
energy ε.
H2+ rotational excitation and recombination
2 +
Figure 4. Maxwell rate coefficients for the DR of H+
2 (X g ) on its ground
vibrational level vi+ = 0, as a function of its initial rotational level, Ni+ = 0
to 10.
Figure 5. Isotopic effects in rotational excitation: rate coefficients for
Ni+ → Ni+ + 2 transitions, Ni+ = 0 and 1, for the vibrationally relaxed
+
X 2 g+ H+
2 and HD systems.
Figure 2. Cross-sections for rotational excitation Ni+ → Ni+ + 2, with
+
2 +
Ni+ = 0 to 7, of H+
2 (X g ) on its ground (vi = 0, black solid curves) and
+
on its lowest excited (vi = 1, red dashed curves) vibrational levels.
Figure 3. Maxwell rate coefficients for rotational excitation Ni+ → Ni+ +
+
2 +
2, with Ni+ = 0 to 10 of H+
2 (X g ) on its ground vibrational level vi = 0.
level vi+ = 0. One may notice that the magnitudes of the IC rate
coefficients are decreasing as Ni+ is increasing.
The DR Maxwell rate coefficients computed for the lowest 11
rotational levels of the ground vibrational level, i.e. vi+ = 0, Ni+ =
0 − 10, are represented in Fig. 4. It can be noticed that even for temperatures exceeding 50 K the rate coefficients for different initial
rotational levels vary by an order of magnitude. Up to 100 K those
for the lowest rotational quantum number gives the major contribution, while above this temperature the higher rotational quantum
numbers become more and more important. The Maxwell rate coefficients for the excitation from the lowest two rotational levels of
+
H+
2 and HD are represented in Fig. 5, illustrating the weak isotopic
effect (notice the use of the linear scale). This effect can also be
noticed in Table 1.
The rate coefficients for rotational de-excitation (superelastic collisions) with N+ = 2 are given in the Tables 2 and 3. In all transitions, the de-excitation process is significant under 1000 K.
In order to facilitate the use of our de-excitation rate coefficients
for kinetic modelling, we have fitted their temperature dependence
by using an Arrhenius-type formula of the form:
α(T ) = a(T /300)b exp(−c/T ),
(13)
MNRAS 455, 276–281 (2015)
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Figure 1. Cross-sections for rotational excitation Ni+ → Ni+ + 2, with
+
2 +
Ni+ = 0 and 1, of H+
2 (X g ) on its ground vibrational level (vi = 0).
Black solid curves: MQDT computations; red dashed curves: ANR approximation based computations of Faure & Tennyson (2001).
279
280
M. D. Epée Epée et al.
Table 1. Maxwell rate coefficients (in cm3 s−1 ) for IC Ni+ → Ni+ + 2 of
+
H+
2 and HD with electrons at room temperature (T = 300 K). Powers of
10 are given in parentheses.
Ni+
0
1
2
3
4
5
6
7
8
H+
2
HD+
4.27473(−7)
1.47448(−7)
7.30450(−8)
4.0060(−8)
2.30159(−8)
1.36503(−8)
8.54116(−9)
5.56954(−9)
3.69408(−9)
5.20549(−7)
2.06564(−7)
1.16444(−7)
7.26357(−8)
4.70647(−8)
3.12933(−8)
2.15608(−8)
1.49720(−8)
1.04648(−8)
Ni+
a(cm3 s−1 )
b
c(K)
2
3
4
5
6
7
8
9
10
1.97306(−7)
2.52735(−7)
2.77163(−7)
2.96279(−7)
3.06114(−7)
2.99066(−7)
3.22690(−7)
3.30095(−7)
3.36555(−7)
−0.50214
−0.50444
−0.49740
−0.50263
−0.50396
−0.50505
−0.50254
−0.50159
−0.49761
0.07878
0.07630
0.00330
0.07353
0.07703
0.00094
0.07028
0.06233
0.06603
where T is given in Kelvin and α in cm3 s−1 . The fitting parameters
are summarized in Table 2. For all the transitions, the fitted values reproduce well the data of this work in the temperature range
10 < T < 1000 K.
The rate coefficients for rotational transitions with N+ = 4 have
been obtained from our computations. They are about two orders
of magnitude smaller than those with N+ = 2 for the same initial
rotational quantum number. We have not reported their values in
the present paper, but they are available as supplementary data.
The cross-sections between 0.01 meV and 0.3 eV, and the Maxwell
rate coefficients between 10 and 1000 K, for DR and rotational
2 +
excitation/de-excitation of electrons with H+
2 (X g ) ions on their
lowest 11 rotational levels and on their vibrational levels vi+ = 0 and
vi+ = 1 have been computed, graphically represented and tabulated.
The numerical data, ready to be used in the kinetic modelling in
astrochemistry and cold plasma physics are available upon request.
The rate coefficients for HD+ are close to those of H+
2 as expected,
but the isotopic differences are clearly put in evidence, in order to
be used for fine modelling of the environments where deuterated
species are available.
The account of the major electronic states within all the relevant
symmetries and of their interactions, using our MQDT method efficiently accounting for both direct and indirect mechanisms, makes
the current treatment more accurate than the previous ones. We are
taking profit of this accuracy in order to address other systems, as
CH+ and SH+ , and the corresponding results will be reported in
future publications.
AC K N OW L E D G E M E N T S
The authors warmly thank Evelyne Roueff, Jonathan Tennyson and
Alexandre Faure for numerous fruitful discussions.
We acknowledge the support of the European COST Actions
‘Our Astrochemical History’ CM1401 and ‘Molecules in motion’
(MOLIM) CM1405. We are grateful to the International Atomic
Energy Agency (IAEA, Vienna) for scientific and financial support
through the Contract No. 16712 with the University of Douala
(CAMEROON) and through the Coordinated Research Projects
‘Atomic and Molecular Data for State-Resolved Modelling of Hydrogen and Helium and their Isotopes in Fusion Plasmas’ and ‘Light
Element Atom, Molecule and Radical Behaviour in the Divertor and
Edge Plasma Regions’.
We thank the financial support from the French CNRS via the
programmes ‘Physique et Chimie du Milieu Interstellaire’, the
‘Physique théorique et ses interfaces’ projects TheMS, TPCECAM
and ‘InPhynity MEXAT’.
Table 3. Maxwell rate coefficients (in cm3 s−1 ) for superelastic collision (Ni+ → Ni+ − 2) of H+
2 with low-energy electrons. Powers of 10 are given in
parentheses.
T(K)
Ni+ =2
Ni+ =3
Ni+ =4
Ni+ =5
Ni+ =6
Ni+ =7
Ni+ =8
Ni+ =9
Ni+ =10
10
20
30
40
50
60
70
80
90
100
200
300
400
500
600
700
800
900
1000
1.0745(−6)
7.6593(−7)
6.2688(−7)
5.4332(−7)
4.8607(−7)
4.4371(−7)
4.1072(−7)
3.8410(−7)
3.6203(−7)
3.4335(−7)
2.4211(−7)
1.9731(−7)
1.7066(−7)
1.5240(−7)
1.3877(−7)
1.2794(−7)
1.1893(−7)
1.1120(−7)
1.0442(−7)
1.3883(−6)
9.8531(−7)
8.0514(−7)
6.9703(−7)
6.2298(−7)
5.6825(−7)
5.2572(−7)
4.9145(−7)
4.6309(−7)
4.3913(−7)
3.0983(−7)
2.5274(−7)
2.1873(−7)
1.9540(−7)
1.7796(−7)
1.6409(−7)
1.5255(−7)
1.4264(−7)
1.3395(−7)
1.4747(−6)
1.0458(−6)
8.6031(−7)
7.4905(−7)
6.7245(−7)
6.1547(−7)
5.7093(−7)
5.3484(−7)
5.0484(−7)
4.7938(−7)
3.4034(−7)
2.7816(−7
2.4097(−7)
2.1538(−7)
1.9621(−7)
1.8094(−7)
1.6823(−7)
1.5731(−7)
1.4773(−7)
1.6186(−6)
1.1480(−6)
9.3848(−7)
8.1334(−7)
7.2776(−7)
6.6446(−7)
6.1518(−7)
5.7539(−7)
5.4240(−7)
5.1448(−7)
3.6321(−7)
2.9628(−7)
2.5644(−7)
2.2908(−7)
2.0863(−7)
1.9235(−7)
1.7881(−7)
1.6718(−7)
1.5697(−7)
1.6791(−6)
1.1917(−6)
9.7422(−7)
8.4376(−7)
7.5432(−7)
6.8813(−7)
6.3664(−7)
5.9514(−7)
5.6077(−7)
5.3172(−7)
3.7504(−7)
3.0611(−7)
2.6516(−7)
2.3705(−7)
2.1599(−7)
1.9923(−7)
1.8525(−7)
1.7325(−7)
1.6270(−7)
1.6239(−6)
1.1479(−6)
9.4624(−7)
8.2597(−7)
7.4319(−7)
6.8151(−7)
6.3316(−7)
5.9389(−7)
5.6118(−7)
5.3336(−7)
3.8073(−7)
3.1217(−7)
2.7100(−7)
2.4256(−7)
2.2118(−7)
2.0410(−7)
1.8984(−7)
1.7758(−7)
1.6679(−7)
1.7653(−6)
1.2486(−6)
1.0191(−6)
8.8249(−7)
7.8920(−7)
7.2029(−7)
6.6673(−7)
6.2357(−7)
5.8783(−7)
5.5763(−7)
3.9468(−7)
3.2269(−7)
2.7970(−7)
2.5006(−7)
2.2783(−7)
2.1009(−7)
1.9531(−7)
1.8260(−7)
1.7145(−7)
1.8081(−6)
1.2717(−6)
1.0325(−6)
8.9195(−7)
7.9727(−7)
7.2799(−7)
6.7447(−7)
6.3148(−7)
5.9595(−7)
5.6591(−7)
4.0306(−7)
3.3009(−7)
2.8619(−7)
2.5584(−7)
2.3304(−7)
2.1485(−7)
1.9968(−7)
1.8667(−7)
1.7523(−7)
1.8051(−6)
1.2856(−6)
1.0547(−6)
9.1637(−7)
8.2146(−7)
7.5106(−7)
6.9612(−7)
6.5167(−7)
6.1474(−7)
5.8341(−7)
4.1268(−7)
3.3656(−7)
2.9106(−7)
2.5978(−7)
2.3638(−7)
2.1776(−7)
2.0228(−7)
1.8901(−7)
1.7738(−7)
MNRAS 455, 276–281 (2015)
Downloaded from http://mnras.oxfordjournals.org/ at Stockholms Universitet on November 2, 2015
Table 2. Fitting parameters for the formula (13), corresponding to the rate
+
+
coefficients for superelastic collisions of H+
2 with electrons, Ni → Ni − 2.
The powers of 10 are given in parentheses.
4 CONCLUSIONS
H2+ rotational excitation and recombination
We are very grateful for the generous support of Agence Nationale de la Recherche via the projects ‘SUMOSTAI’ (ANR-09BLAN-020901) and ‘HYDRIDES’ (ANR-12-BS05-0011-01), of
IFRAF-Triangle de la Physique via the project ‘SpecoRyd’, and
of Fédération de Recherche Fusion par Confinement Magnétique
- ITER. We acknowledge the constant financial support from La
Région Haute-Normandie via the CPER ‘THETE’ project, and the
GRR Electronique, Energie et Matériaux. Part of this work has
been performed in the frame of the ‘Fédération de Recherche Energie, Propulsion, Environnement’, and of the LabEx EMC3 , via the
project PicoLIBS (ANR-10-LABX-09-01).
Finally, MDEE and OM thank the Laboratoire Ondes et Milieux
Complexes of the University of Le Havre for hospitality.
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Additional Supporting Information may be found in the online version of this article:
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