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 a r X i v : 1 0 0 1 . 3 3 7 9 v 1

 [ p h y s i c s . a t o m - p h ] 1 9 J a n 2 0 1 0

Mon. Not. R. Astron. Soc. 000, 1–8 (2010) Printed 19 January 2010 (MN LATEX style file v2.2)

Calculation of rate constants for vibrational and rotational

excitation of the H+3 ion by electron impact

Viatcheslav Kokoouline1⋆, Alexandre Faure2, Jonathan Tennyson3, Chris H. Greene41Department of Physics, University of Central Florida, Orlando, Florida 32816, USA

Laboratoire Aime Cotton, CNRS, Universite Paris 11, 91405 Orsay France;2 Laboratoire d’Astrophysique, UMR 5571 CNRS, Universite Joseph-Fourier, B.P. 53, 38041 Grenoble cedex 09, France;3 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK;4 Department of Physics and JILA, University of Colorado, Boulder, Colorado 80309, USA

19 January 2010

ABSTRACTWe present theoretical thermally-averaged rate constants for vibrational and rotational(de−)excitation of the H+

3 ion by electron impact. The constants are calculated usingthe multi-channel quantum-defect approach. The calculation includes processes thatinvolve a change |∆J | 2 in the rotational angular momentum J  of H+

3 . The rateconstants are calculated for states with J  5 for rotational transitions of the H+

3

ground vibrational level. The thermal rates for transitions among the lowest eightvibrational levels are also presented, averaged over the rotational structure of thevibrational levels. The conditions for producing non-thermal rotational and vibrationaldistributions of H+

3 in astrophysical environments are discussed.

Key words: ISM: molecules, plasmas, molecular processes, molecular data

1 INTRODUCTION

Rotational and vibrational excitation of small polyatomicions by electron impact is one of the important processesoccurring in a neutral molecular plasma. In particular, theprobability of rovibrational (de-)excitation in electron-ioncollisions can be relatively high. The high probabilities andcorrespondingly high rate constants are driven by the non-Born-Oppenheimer coupling between electronic and rovibra-tional motions of the ion-electron system. In certain smallpolyatomic molecules, the coupling is particularly strong, asis the case for the H+

3 ion.Owing to its importance in interstellar space (Oka

2006), planetary ionospheres (Miller et al. 2000) and lab-oratory experiments (Larsson 2000; Plasil et al. 2002;Johnsen 2005), the H+

3 ion has been studied for manyyears. In particular, processes involving electron scat-tering from the ion have been recently studied ex-perimentally and theoretically. Such processes includeelectron-impact rovibrational excitation (Faure & Tennyson2002; Faure et al. 2006a, 2009), dissociative recombina-tion (Larsson 2000; Kokoouline et al. 2001; Johnsen 2005;Fonseca dos Santos et al. 2007), electronic excitation andionisation (Gorfinkiel & Tennyson 2004) and photoioniza-tion (Bordas & Helm 1991; Stephens & Greene 1994, 1995;Mistrık et al. 2000; Kokoouline & Greene 2004a) of the

⋆ E-mail: slavako@mail.ucf.edu

metastable neutral H3 molecule. Astrophysically, electronimpact excitation of molecular ions has been observed to bethe dominant collisional excitation process in some environ-ments (e.g. Jimenez-Serra et al. 2006).

Previously, cross sections and rate constants fora few rotational transitions in H+

3 have been calcu-lated (Faure & Tennyson 2002, 2003; Kokoouline & Greene2003b; Faure et al. 2006a,b). Vibrational rate constantshave apparently not been studied extensively for this fun-damental ion. Here, we present thermal rate constants fortransitions between different rotational states of the ground

vibrational level of H+3 with low angular momentum, J  5.

We also present thermal rate constants for rotationally-averaged transitions between different vibrational levels.

The next section of the article briefly discusses thetheoretical approach used in the present calculation. Adetailed description of the approach is lengthy and hasalready been published elsewhere (Kokoouline & Greene2003b; Fonseca dos Santos et al. 2007). Therefore we onlysketch here the main ideas of the approach. In section 3,we present the rates for vibrational (de-)excitation of H+

3 .Section 4 is devoted to the calculation of rotational rateconstants for transitions within rotational manifold of theground vibrational level. Astrophysical implications are dis-

cussed in Section 5. Finally, Section 6 presents our conclu-sions.

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2 V. Kokoouline, A. Faure, J. Tennyson, and C. H. Greene

2 THEORETICAL APPROACH

The theoretical model employed in the present study isbased on quantum defect theory. It is discussed in de-tail by Kokoouline & Greene (2003a), Kokoouline & Greene(2003b), Faure et al. (2006a) and Fonseca dos Santos et al.(2007). Here, we only mention the main ideas used in themodel.

The energy-dependent theoretical rate constantαi′←i(E ) for a transition from the initial rovibrational leveli to a final one i′ is obtained from the corresponding matrixelement S physi′i (E ) of the energy-dependent scattering

operator, S phys(E ). The main difficulty in the theoreticalapproach is the construction of the scattering matrixS physi′i (E ), (the indices i and i′ refer to specific rovibrationalstates of the H+

3 ion; the electron angular momentum andits coupling with the ion to form a total angular momentumeigenstate are implied as well, but these will be suppressed

in our notation since they are diagonal quantum numbersin the present approximation). For our discussion, it isconvenient to represent the index i as rv, where r = (JK )specifies the rotational quantum numbers, i.e. ionic angularmomentum J  and its projection K  on the molecularaxis; and v = {v1, vl22 } specifies the vibrational quantumnumbers in a normal mode classification. We note that ourmodel neglects the explicit coupling between rotationaland vibrational angular momenta which occurs for excitedvibrational states with l2 > 0 and which leads to a morecomplicated set of quantum numbers (Lindsay & McCall2001).

The construction of the scattering matrix S physi′i(E ) be-

gins from the ab initio potential surfaces of the ground elec-

tronic state of the ion and several excited states of the neu-tral H3 molecule (Mistrık et al. 2000), (U +(Q) for the ionand U n(Q) for the neutral molecule). We will use the sym-bol Q to specify collectively the three internuclear distances.For a given geometry Q, the electronic wave function of theouter electron of the H3 excited states resembles the elec-tronic wave function of the hydrogen atom. However, due tofact that at short distances from the ionic core the electron-ion interaction is different from that in the hydrogen atom,the electron binding energy U +(Q)−U n(Q) is generally dif-ferent than the corresponding binding energy 1/(2n2) in thehydrogen atom (n is the principal quantum number). Thedeparture of the U +(Q) − U n(Q) difference from 1/(2n2)is written as 1/(2(n

−µ)2), where the quantum defect µ is

only weakly dependent on the principal quantum number n.When the energy E  of the electron+ion system approachesthe ionization limit (n →∞) and becomes larger, the prin-cipal quantum number n looses its physical meaning, butthe quantum defect µ does not: it gives the collisional phaseshift δ(E ) = πµ in terms of the scattering phase in electron-ion collisions at energies above the ionization limit (Seaton1966). The phase shift also depends weakly on the energy E and it determines the scattering matrix S (E ) = exp(2iδ).This is the reason why the energies of excited electronicstates in the neutral molecule can be used to obtain thequantum defect and to describe collisions between the ionand the electron.

The preceding discussion assumes that the electron

scatters from (or is bound to) the molecular ion, which staysat a given geometry, i.e. the nuclei remain fixed through-

out. This approximation is only reliable on a time scalemuch shorter than the period of ionic vibrational motion.In this limit, the quantum defects are functions of geome-try Q, µ(Q). Generally speaking, the dependence of  µ(Q) is

much smoother than the dependence of  U +

(Q) and U (Q).Although the quantum defect depends only weakly on theprincipal quantum number n, it usually depends stronglyon the angular momentum l and on its projection Λ on themolecular axis of the ionic core. For a nonlinear triatomicion such as H+

3 , the body-frame quantization axis is chosenas the normal to the plane containing the nuclei. At largel 2, the quantum defects become small, because the elec-tronic wave function of high l approaches the unperturbedwave function in the hydrogen atom.

Therefore, for the description of an electron-ion scat-tering process involving low values of  l, one must obtain thequantum defect functions µlΛ(Q) for various l and Λ andthe corresponding scattering matrices S (Q) = exp(2iδlΛ).

The calculated potential energy surfaces are obtained in theBorn-Oppenheimer approximation (Mistrık et al. 2000), i.e.in which the coupling between electronic and nuclear mo-tion is neglected. The quantum defects and the scatteringmatrices obtained from the potential surfaces as describedabove, therefore, fail to account for the coupling. However,in the H+

3 + e− case a strong non-adiabatic Jahn-Teller in-teraction must be accounted for in order to appropriatelydescribe the scattering process. It mixes the two π elec-tronic states of the same orbital angular momentum l of the incoming electron and can be included in the scatteringmatrix built from quantum defects using the formalism sug-gested in Staib & Domcke (1990). The resulting scatteringmatrix is now not diagonal over the Λ quantum numbers.

The strongest coupling is between the pπ electronic states.In our treatment we include only p-states (π and σ). Thisimplies that this body-frame scattering matrix S BF 

Λ′,Λ(Q) isa 3× 3 matrix (Kokoouline & Greene 2003b).

The scattering matrix constructed in this way repre-sents the electron-ion scattering only if the ion stays at thesame configuration Q during the entire process, which isnot a valid description. The physically meaningful scatter-ing matrix must describe the amplitude of scattering froma particular rovibrational state of the ion to another one,including the possibility of nuclear motion during the fullcollision process. We denote this as the space-fixed scatter-ing matrix S SF s′,s, where s and s′ refer to initial and finalstates of the ion. The formalism of the rovibrational frame

transformation (Atabek et al. 1974; Jungen & Atabek 1977;Fano 1975) allows us to use the matrix S BF (Q) to constructthe S SF s′,s matrix. In this formalism, the two matrices are con-sidered as two equivalent forms of the same scattering opera-tor in two different representation bases. The representationbasis of  S SF s′,s = s′|S |s is the set of rovibrational energy

eigenstates |s. The basis for the S BF (Q) matrix is made of tensor products |b = |Q|Λ, where |Q represents the vi-brational position eigenstates of the ion, |Λ is the angularstate vector of the electron in either the pπ or pσ state.

The two representations are connected by the standardbasis transformation formula

S SF s′;s =b′

,b

s′|b′b′|S |bb|s , (1)

where the summation indicates a sum over discrete indices

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Calculation of rate constants for vibrational and rotational excitation of the H+3 ion by electron impact  3

and an integration over the continuous coordinatesQ and ro-tational coordinates (three Euler angles). Because the scat-tering matrix in the |b basis is diagonal over |Q, it is con-venient to write b′|S |b as S BF 

Λ′,Λ(Q)δ(Q−Q′), the notation

that has been already used in the above discussion. Theexplicit form of the matrix elements s|b for the unitarytransformation is given in Kokoouline & Greene (2004a,b).

The S BF  matrix is diagonal with respect to the rota-tional quantum numbers J tot, K tot and M tot of the wholemolecule and the continuous coordinate Q. J tot, K tot andM tot are the angular momentum of the neutral molecule,and the two projections of the angular momentum on themolecular symmetry axis and on the space-fixed z-axis, re-spectively. To completely define the BF basis functions |b,in addition to Λ and Q, the rotational quantum numbersJ tot, K tot and M tot must be also specified; for brevity theyare omitted in the above equation although their presenceis implied.

To specify the vibrational states in the SF representa-tion, we will use the normal mode approximation, i.e. speci-fying ionic vibrational eigenstates by the quantum numbers{v1, vl22 }. The rotational part of the total wave function isspecified by rotational quantum numbers J tot,J,K,M,l,m,where J, K , and M  are the angular momentum of the ion,and the two projections of the angular momentum on themolecular symmetry axis and on the space-fixed z-axis, mis the projection of electronic angular momentum l on thespace-fixed z-axis. In the following, we will not specify anyother conserved quantum numbers that are the same in bothbases, such as the total nuclear spin and the irreducible rep-resentation of the total wavefunction.

The matrix S SF s′;s obtained by the above procedure

does not yet represent the physical scattering matrix(Aymar et al. 1996; Seaton 1966). In fact, it represents theactual scattering matrix S phys(E ) only for energies highenough such that all  of the channels |s are open for elec-tron escape, i.e. where the total energy of the system ishigher than the energy of the highest relevant ionizationchannel |s. When at least one channel is closed, the physicalscattering matrix S phys(E ) is obtained from S SF  using thestandard MQDT channel-elimination formula (see Eq. (2.50)in Aymar et al. (1996) or Eq. (38) in Kokoouline & Greene(2004a)).

In terms of the energy-dependent scattering matrixS phys(E ), the cross-section for rovibrational (de-)excitationof the ion from the initial state

|s

is written (in atomic

units, a.u.) as:

σRFTs′←s(E el) =

π

2E el(2J  + 1)

J totKtot

(2J tot + 1)|S (J tot,Ktot)s′;s |2,

(2)where E el is the relative kinetic energy of the ion and theelectron before the collision. In the above expression it hasbeen assumed that the initial |s and final |s′ states aredifferent.

3 RATE CONSTANTS FOR VIBRATIONAL(DE-)EXCITATION

If one is not interested in the rotational structure of ini-tial and final vibrational states, the final cross-section (or

2600 2800 3000 3200

total energy, with respect to {000}(00) (cm

-1)

10-3

10-2

10-1

  e  x  c   i   t  a   t   i  o  n  p  r  o   b  a   b   i   l   i   t  y

{011}(11) {01

1}(21) {01

1}(31) {01

1}(33) {10

0}(11)

rotational excitation

{000}(11)->{00

0}(31)

{000}(21)->{00

0}(31)

{000}(11)->{00

0}(21)

{000}(21)->{01

1}(11)

{000}(11)->{01

1}(33)a

{000}(11)->{10

0}(11)

{000}(11)->{01

1}(21)

{000}(11)->{01

1}(31)

{000}(21)->{01

1}(31)

{000}(21)->{01

1}(21)

{000}(11)->{01

1}(11)

Rovibrational excitation (all curves below){00

0}(11)->{01

1}(33)b

Figure 1. (Color online) Probabilities of rovibrational excitationof the H+3 ion calculated using the full rovibrational frame trans-

formation. Only transitions from the ground vibrational level

{000} are shown. The {000} → {011} probabilities oscillate a

lot below 3000 cm−1 and become less energy-dependent above.The oscillations are due to the strong rotational coupling between

individual rotational levels of the initial and final states of theion. When averaged over the initial and summed over the finalrotational states and averaged over the appropriate energy dis-

tribution, the resulting probabilities are similar in magnitude tothe probabilities shown in Fig. 2. The labels on top of the figure

indicate different rovibrational ionization limits. Notice that thezero of energy in the figure is set to the energy of the forbiddenrovibrational level {000}(00).

thermal rate constant) has to be averaged over the ini-tial rotational levels and summed up over the final rota-tional levels. This can be done using the full rovibrationalframe transformation technique described above. However,the cross-section averaged over initial rotational levels andsummed up over the final rotational levels is very similarto the one obtained neglecting the rotational structure of the ion. The cross-section obtained accounting for the rota-tional structure has more resonances due to interaction be-tween rotational states but the averaged value is close to thevalue obtained without the rotational structure. The quan-tity of interest in astrophysical applications is the thermally-averaged rate constant. Because the thermally-averaged rate

is not sensitive to the position of individual resonances in theenergy-dependent cross-section, the calculations with andwithout the rotational structure give the same result.

Figs. 1 and 2 compare our calculations with and withoutrotational structure included, for the probabilities of rovi-brational and vibrational (de-)excitation of the ion. Fig. 1shows in detail the rovibrational transitions from the groundto the first excited vibrational level {011} with J tot = 2 inpara-H+

3 . In order to compare with the results on Fig. 2,one would need to take a sum over final quanta and averageover initial J  and K  and account for all possible J tot andK tot similar as it is done in Eq. 2. This would mean thatto achieve a converged result at reasonably high energy (∼2000 cm−1) calculations would be needed for all J tot up to

10. This would require a tremendous numerical effort if thefully quantum approach were to be applied.

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4 V. Kokoouline, A. Faure, J. Tennyson, and C. H. Greene

2000 3000 4000 5000 6000 7000 8000 9000 10000

electron scattering energy (cm-1

)

10-5

10-4

10-3

10-2

10-1

  e  x  c   i   t  a   t   i  o  n  p  r  o   b  a   b   i   l   i   t   i  e  s

   {   0   1

   1   }

   {   1   0

   0   }

   {   0   2

   0   }

   {   1   1

   1   }

   {   0   2

   2   }

   {   2   0

   0   }

   {   0   3

   1   }

{000}->{01

1}

{000}->{02

2}

{000}->{02

0}

{000}->{10

0}

{000}->{11

1}

{000}->{03

1}

{000}->{20

0}

Figure 2. (Color online) Probabilities of vibrational excitation

from the ground vibrational level {000} to several excited vibra-tional levels calculated using the vibrational frame transformation

only. Energies of vibrational thresholds are labeled with arrowsand the corresponding vibrational quantum numbers.

100 1000temperature (K)

10-13

10-12

10-11

10-10

10-9

10-8

10-7

   t   h  e  r  m  a   l   l  y  a  v  e  r  a  g  e   d   (

   d  e   )  e  x  c   i   t  a   t   i  o  n  r  a   t  e   (  c  m

   3   /  s   )

{000}->{01

1}

{000}->{10

0}

{011}->{00

0}

{000}->{02

0}

{022}->{00

0}

Figure 3. (Color online) Thermally-averaged rate constants forseveral (de-)excitation transitions obtained by direct integration

using Eq. (4) (solid lines) and the approximate formula of Eq.(7) (dashed line). The averaged probabilities for vibrational (de-)excitations are listed in Table 1.

Neglect of the rotational structure of the initial and finalvibrational state simplifies considerably the numerical cal-culation. The complete rovibrational frame transformationof Eq. (1) is reduced to the vibrational frame transforma-tion if the rotation is neglected, i.e it is carried out using thefollowing formula

S SF (v′Λ′)(vΛ) =

v′|S BF Λ′;Λ(Q)|v

, (3)

where the brackets imply an integration over the vibrationalcoordinates only. Many elements among S SF (v′Λ′)(vΛ) are zerobecause of the symmetry of vibrational wave functions andmatrix elements S BF 

Λ′Λ(

Q). The vibrational (de-)excitation

v → v′ cross-section obtained from the scattering matrix of Eq. (3) should be averaged over Λ and summed over Λ ′.

The thermally averaged rate constant αth(T ) (in a.u.)is obtained from the energy-dependent cross-section σ(E ) as

αth(T ) =8π

(2πkT )3/2

 ∞

0

σ(E el)e−EelkT  E eldE el , (4)

where T  is the temperature. Temperature dependenciesαth(T ) for different (ro-)vibrational transitions v → v′ ob-tained using Eq. (4) are shown in Fig. 3 as solid lines.

For further discussion, it is convenient to represent thecross-section σ(E el) in the form

σ(E el) =π

k2P (E el) , (5)

where k is the wave vector of the incident electron, P (E el)is the probability for vibrational (de-)excitation at collisionenergy E el. Figures 1 and 2 suggest that on average (here wemean a running average taken over a few intervals betweenresonances), the probability behaves approximately as a step

functionP (E el) = P 0 θ(E el −∆v′,v) , (6)

where ∆v′v = E v′ − E v is the threshold energy for (ro-)vibrational excitation (if  E v′ − E v > 0); ∆v′v = 0 for de-excitation (if  E v′ −E v < 0), θ is the Heaviside function, P 0is a constant. The above approximation for the probabilityis accurate enough to calculate the thermally-averaged rateconstant that is not sensitive to the detailed resonance struc-ture of the energy-dependent (de-)excitation cross-section.Using Eq. (6) the thermally rate constant of Eq. (4) be-comes

αth(T ) =

 2π

kT e−

∆v′vkT  P 0 . (7)

The above formula with only one parameter P 0 provides avery good approximation for the actual thermally averagedrate constant. It is demonstrated in Fig. 3 that comparesthe thermal rate constants for different (de-)excitation tran-sitions obtained with the direct numerical integration usingEq. (4) and with the approximate formula of Eq. (7). There-fore, for practical applications, it is convenient to provide just averaged probabilities P 0 and the energies of vibrationalthresholds E v for each pair of vibrational (de-)excitations.These parameters are listed in Table 1 for all combinationsof the first eight vibrational states of H+

3 . Note that theconversion factor from a.u. to cm3/s is 6.126×10−9.

4 RATE CONSTANTS FOR ROTATIONAL(DE-)EXCITATION

If the temperature T  of the H+3 +e− plasma is not very high,

such that only the ground vibrational level {000} of H+3 is

significantly populated, knowledge of rate constants for tran-sitions r → r′ between individual rotational levels r and r′

of {000} may be important for the analysis of experimentalor astronomical spectra. For this purpose, we have made adetailed analysis of transitions between individual rotationalstates of the ground vibrational level. The calculation of ro-tational (de-)excitation rate constants was carried out usingthe cross-section of Eq. (2) and numerical integration of Eq.

(4). Examples of the thermally-averaged rate constants forthe rotational (de-)excitation are shown in Fig. 4.

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Calculation of rate constants for vibrational and rotational excitation of the H+3 ion by electron impact  7

T  {011} {100} {020} {022} {111} {200} {031}10 1.3(9) 2.4(7) 3.1(9) 2.6(9) 9.5(8) 4.8(7) 1.3(9)

100 4.0(9) 7.7(7) 9.7(9) 8.1(9) 3.0(9) 1.5(8) 4.2(9)

1000 1.1(10) 2.0(8) 2.3(10) 2.1(10) 8.9(9) 4.8(8) 1.3(10)

Table 6. Critical electron density, ncr in cm−3, at 10, 100 and 1000 K, forvibrational levels of H+

3 . Powers of 10 are given in parentheses.

effects, a practical definition is to refer to a specific level s:

ncr(s, T ) =

s′ A(s → s′)

s′ α(s → s′, T ), (10)

where A(s → s′) are the Einstein coefficients for sponta-neous emission, α(s → s′, T ) are the collisional rates andthe sums run over all possible transitions s → s′. Consid-ering electron collisions only, these latter will maintain thelevel s in local thermodynamic equilibrium (LTE) for elec-tron densities ne

≫ncr(s, T ), while deviations from LTE in-

cluding population inversions are expected for densities ne ncr(s, T ). For ne ≪ ncr(s, T ), electron collisions will be neg-ligible. Non-LTE effects caused by H2 collisions have beeninvestigated both in interstellar clouds (Oka & Epp 2004;Oka et al. 2005) and in the Jovian atmosphere (Melin et al.2005). We note in this context that microcanonical sta-tistical calculations have been performed recently to esti-mate thermal state-to-state rate coefficients for the H+

3 +H2

reaction and its deuterated variants (Park & Light 2007;Hugo et al. 2009). Electron-impact rotational excitation hasbeen considered by Faure et al. (2006b) but, to the best of our knowledge, electron-impact vibrational excitation hasbeen so far ignored in non-LTE modelling.

Eq. (10) was computed with the vibrational and ro-

tational rates presented in the previous sections. EinsteinA coefficients were taken from Dinelli et al. (1992a,b) forvibrational transitions and from Pan & Oka (1986) for ro-tational transitions. Results are presented in Tables 6 and7. It can be noticed that critical densities for vibrationaland rotational levels differ by typically 6 orders of magni-tude: they range between 107 and 1010cm−3 for the for-mer and between 10−1 and 104cm−3 for the latter. Inthe diffuse interstellar medium, the electron density is∼0.1 cm−3 (Black & van Dishoeck 1991) while it can reachabout 106 cm−3 in planetary atmospheres (see for exam-ple Lystrup et al. 2008). As a result, non-LTE rotationalpopulations are expected in interstellar clouds whereas ro-tational levels should be at or close to LTE in the jovian at-

mosphere. A non-thermal rotational distribution of H+3 wasactually observed towards galactic center clouds where themetastable (3, 3) level has a population comparable to thatin (1, 1) despite being 361.5 K higher (Oka et al. 2005). Onthe other hand, electrons are expected to be negligible in vi-brationally exciting H+

3 in the interstellar medium but theycould establish a non-LTE vibrational population of H+

3 inplanetary environments, as observed in the jovian thermo-sphere (Kim et al. 1992).

Table 6 shows that the two levels {100} and {200} havecritical densities significantly lower than the other levels.This directly reflects the lower Einstein A coefficients of ∼1 s−1 (Dinelli et al. 1992a). We note that the values arehowever not low enough to ensure LTE vibrational popula-

tion in planetary atmospheres where ne < 107 cm−3.Table 7 shows that at 10 K the levels (22) and (44)

have much higher critical densities than the other levels.This actually reflects the fact that these two levels can de-populate collisionally through excitation  only since rates forrotational transitions with ∆K  = 0 are null within ourtreatment. These rates, as those with |∆J | > 2, were actu-ally estimated by Faure & Tennyson (2003) and Faure et al.(2006a) and were found to be 3 to 4 orders of magnitudesmaller than those with ∆J  = ±1,±2 and ∆K  = 0. The(44) level is particularly interesting because an astrophysicalmaser is predicted in the (44)

→(31) transition of H+

3 (Black

2000). It should be noted that the selection rules for the(forbidden) rotational radiative transitions are ∆J  = 0,±1and ∆K  = ±3 (Pan & Oka 1986; Miller & Tennyson 1988).Critical densities suggest that electrons might contribute, insome environments, to create and maintain the necessarypopulation inversion. Note that in Table 7 the (33) level isnot listed since it is metastable. In this case, the conceptof critical density is meaningless. Finally we emphasize thatcritical densities provide guidance at the order of magnitudelevel and that a detailed non-LTE modelling, including allrelevant colliders, is necessary to properly quantify devia-tions from LTE.

6 CONCLUSION

In this study we have performed calculations of thermallyaveraged rate constants for rotational and vibrational tran-sitions in H+

3 caused by an electron impact. The calculationswere made from the first principles using the quantum de-fect approach. The rotational rate constants are calculatedfor the ground vibrational level of the ion in the initial andfinal states. The rate constants for transitions between differ-ent vibrational levels are calculated neglecting the rotationalsubstructure of each vibrational level, which corresponds toaveraging over initial rotational states and summing over thefinal rotational states. The obtained thermally-averaged rateconstants are well described by the analytical formula of Eq.

(7) with the parameter P 0, that can be considered as tem-perature independent for vibrational transitions, and weaklydependent on temperature for the rotational transitions. Forthe rotational transitions, we have made a numerical fit of the parameter by a cubic polynomial. The numerical valuesof the fitting procedure are provided in Tables 1-5. The pre-sented thermally-averaged rate constants can be useful ininterpretation of hydrogen-dominated plasma experimentsas well as for modelling interstellar clouds and planetary at-mospheres, where the H+

3 ion is present. The computation of critical densities suggests in particular that electrons couldestablish non-LTE rotational populations of H+

3 in diffuseinterstellar clouds and non-LTE vibrational populations inplanetary atmospheres.

This work has been supported by the National ScienceFoundation under Grant No. PHY-0855622, by the Depart-

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