Rotational excitation of methylidynium (CH+) by a helium atom at high temperature
aa r X i v : . [ a s t r o - ph . S R ] S e p astro-phMay 29, 2018 Rotational excitation of methylidynium (CH + ) by a helium atom athigh temperature K. Hammami , L. C. Owono Owono , , and P. St¨auber Laboratoire de Spectroscopie Atomique Mol´eculaire et Applications, D´epartement de Physique, Facult´e des Sciences,Universit´e Tunis El Manar, Campus Universitaire, 1060 Tunis, Tunisiae-mail: [email protected] Department of Physics, Advanced Teachers Training College, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon Centre for Atomic Molecular Physics and Quantum Optics, Faculty of Science, University of Douala, P.O. Box 8580, Douala,Cameroon Institute for Astronomy, ETH Zurich, 8093 Zurich, SwitzerlandReceived March 25, 2009; accepted March 16, 1997
ABSTRACT
Context.
The
Herschel Space Observatory with its high-resolution instrument HIFI on board will observe the CH + → → Aims.
We aim to obtain accurate rate coe ffi cients for the collisional excitation of CH + by He for high gas temperatures. Methods.
The ab initio coupled-cluster [CCSD(T)] approximation was used to compute the interaction potential energy. Cross sec-tions are then derived in the close coupling (CC) approach and rate coe ffi cients inferred by averaging these cross sections over aMaxwell-Boltzmann distribution of kinetic energies. Results.
Cross sections are calculated up to 10 000 cm − for J ranging from 0 to 10. Rate coe ffi cients are obtained at high temperaturesup to 2000 K. Key words.
ISM: molecules – molecular data – molecular processes – astrochemistry – radiative transfer
1. Introduction
The molecular ion CH + ( X Σ + ) is most commonly observedfrom the ground in the visible in absorption lines against nearbybright stars. The lowest rotational transitions in the submillime-ter and far-infrared band are blocked from the earth’s atmosphereand therefore di ffi cult to observe with ground-based telescopes.A tentative detection of the J = → CH + has been reported recently by Falgarone et al. (2005).Cernicharo et al. (1997) reported the first emission lines of the J = →
1, 3 → → Infrared Space Observatory .Although CH + has been observed frequently in the ISM overthe past few decades, its abundance is still an enigma. Chemicalmodels persistently fail to reproduce the large abundances, as itis not yet clear, what the main source for CH + is (Black et al.1978; Black 1998; Nehm´e et al. 2008). The highly endothermicreaction C + + H → CH + + H (- ∆ E / k = ffi cient route to CH + . Besides high gas temperatures,an important part of the energy needed to activate this reactionmay come from vibrationally excited H (Sternberg & Dalgarno1995). Since CH + is also abundantly found in the cold neutralmedium (CNM), this reaction may not be fast enough though tocompete with the destruction reactions. Attempts to resolve thispuzzle continue (Joulain et al. 1998; Godard et al. 2009). A thor-ough understanding of the CH + abundance is important since itprovides information about physical gas properties such as tem-perature and fraction of ionization. Furthermore, CH + is a fun-damental building block for more complex molecules and a rel-evant coolant in hot and dense interstellar gas. Related to the problem to model observed CH + abundancesis the uncertainty in the excitation mechanism. The CH + is likelyto be destroyed rather than excited in collisions with hydrogenand electrons - the most abundant species. In addition, the ex-citation of CH + is found to be sensitive to the dust continuumbackground of the CH + emitting source (Black 1998). To modelthe CH + abundance, the chemical reactions and radiative trans-fer equations may therefore need to be solved simultaneously.However, excitation calculations require accurate collisional ratecoe ffi cients.Observations with the Herschel Space Observatory mayshed some light on these problems. The high spectral reso-lution instrument HIFI onboard the satellite will observe the J = → J = → ff use ISM to dense and hotstar-forming regions. PACS, another instrument for Herschel,will cover the CH + transitions with J up = + abundance is sensitive to far-ultraviolet (FUV) photons and X-rays (Sternberg & Dalgarno 1995; St¨auber et al. 2004, 2005), itwill be searched for in typical photo-dissociation regions (PDRs)and X-ray dominated regions (XDRs). The gas temperature insuch areas can easily reach a few 1000 K (Tielens & Hollenbach1985; Maloney et al. 1996). Collisional rate coe ffi cients aretherefore needed for a large range of gas temperatures and upperenergy levels in order to interpret the observations properly.The aim of this paper is to obtain accurate rate coe ffi cientsfor the collisional excitation of CH + at high temperatures. As itis far more di ffi cult to determine rate coe ffi cients for excitationwith H , we focus on the rotational excitation by He, another K. Hammami et al.: Rotational excitation of CH + by He at high temperature major gas component. This study is thus an extension of an ear-lier published paper, where coe ffi cients for temperatures up to200 K were presented (Hammami et al. 2008).
2. Potential energy surface
Recently, we have computed the interaction PES for theCH + (X Σ + ) − He ( S) van der Waals system using the rigid ro-tor approximation and the Jacobi coordinate system in which r is the CH + internuclear distance, R the distance from the centerof mass (c.m.) of CH + to the He atom, and θ the angle betweenthe two distance vectors (Hammami et al. 2008). The collinearCH + ... He geometry corresponds to θ = ◦ while the CH + bonddistance was frozen at its value at the experimental equilibriumgeometry of the ground X Σ + state, i.e., r = r e = . s p d f ) set ofbond functions defined by Tao & Pan (1992) are added andplaced at mid-distance between the c.m. of CH + and He.The basis set superposition errors (BSSE) was corrected atall geometries following the Boys & Bernardi (1970) counter-poise procedure. Our PES has a global minimum of 537 cm − at R = .
05 Bohr and θ = ◦ . This value is consistent withthat obtained by Stoecklin & Voronin (2008) with the BCCD(T)method and a aug-cc-pVQZ basis set, i.e., 513 cm − at R = . θ = ◦ . Indeed, our well depth is lower than theirvalue.To perform the dynamical calculations, the basic inputs re-quired by the MOLSCAT package (Hutson & Green 1994) wereobtained by expanding the interaction potential in terms ofLegendre polynomials as V ( r = r e , R , θ ) = X λ V λ ( R ) P λ ( cos θ ) . (1)The calculated surface is well reproduced by the analytical po-tential over the entire grid of used coordinate points. The stan-dard deviation between the analytical and the calculated surfaceremains below 1 .
3. Cross sections
The quantum mechanical close coupling approach(Arthurs & Dalgarno 1960) implemented in the MOLSCATcode was used to calculate state to state rotational integral crosssections for values of J ranging from 0 to 10, and a total energyup to 10 000 cm − . The energy range was carefully spannedin order to account for resonances. The incremental stepswere chosen as follows: From 0 to 50 cm − , they were set to0 . − , from 50 cm − to 100 cm − to 0 . − , from 100 cm − to 200 cm − to 0 . − , from 200 cm − to 1200 cm − to 1 cm − ,from 1200 cm − to 1400 cm − to 2 cm − , from 1400 cm − to1500 cm − to 5 cm − , from 1500 cm − to 2000 cm − to 10 cm − ,from 2000 cm − to 2500 cm − to 25 cm − , from 2500 cm − to 3000 cm − to 50 cm − , from 3000 cm − to 5000 cm − to100 cm − , and finally from 5000 cm − to 10 000 cm − to200 cm − . To include all open channels and some closedchannels, we have set J max =
15 for E ≤ − , and Fig. 1.
Deexcitation cross sections of CH + by collision with Heas a function of the kinetic energy. C r o ss s ec ti on s ( Å ) Ec(cm -1 ) -1 J max =
17 for E > − . This corresponds to a rotationalbasis set of adequate size for a good accuracy in the calculatedcross sections. The input parameters required by MOLSCATare displayed in Table 1. These parameters were fixed after wehave performed some tests to ensure the convergence of crosssections for energies up to 10 000 cm − . The coupled equationswere conveniently solved using the propagator of Manolopoulos(1986).Figure 1 presents the energy variation of the CH + − He col-lisional deexcitation cross sections for the transitions J →
0, for J = − →
1. Up to almost 600 cm − ,resonances can be seen in the cross sections. These are due tothe global minimum located at 4 .
05 Bohr and whose well depthis ∼
537 cm − . As one can also see from the figure, the crosssections for the transition 1 → J > → → − and for E C > ∼ − . It isalso obvious from the figure that the cross sections for the tran-sition 2 → → − .An analysis of the excitation cross sections has been car-ried out by Hammami et al. (2008). It was pointed out in thatfor energies lower than 600 cm − , the cross sections for the tran-sition 0 → → → → ∼
150 cm − . Indeed, for energies greater than that value, thetransition 1 → ∆ J evenparity transitions for almost the entire range of energy. . Hammami et al.: Rotational excitation of CH + by He at high temperature 3 Table 1.
MOLSCAT parameters used in the present calculations. Be and De are from Huber & Herzberg (1979).
INTFLG = =
30 OTOL = = = − De = − JMAX =
17 RMIN = =
50 Bohr
Fig. 2.
Rotational quenching cross sections of CH + by collisionwith He as a function of the kinetic energy. C r o ss s ec ti on s ( Å ) Ec(cm -1 ) J = 1J = 2J = 3J = 4J = 5
4. Collisional rates
Downward rate coe ffi cients are obtained by averaging the crosssections σ J → J ′ over a Maxwell-Boltzmann distribution of kineticenergies, q J → J ′ ( T ) = β πµ ! Z ∞ E C σ J → J ′ ( E ) e − β E C dE C , (2)where T is the kinetic temperature, µ = . . u . is the re-duced mass of the CH + − He collision partners, β = k B T ( k B is the Boltzmann constant) and E C = E − E J is the relativekinetic energy. Table 2 displays the results at selected temper-atures. Additional numbers may be obtained upon request tothe authors. Figure 3 illustrates the variation of q J → J ′ ( T ) withthe kinetic temperature. The general trends observed earlier byHammami et al. (2008) for these numbers at low temperature re-main. As one can see from the figure, the magnitude of the ratecoe ffi cients for the transition 2 → → →
5. Summary
Using a previously computed PES (Hammami et al. 2008), wehave obtained results of a quantum mechanical close couplingcalculation of integral cross sections for transitions between thelower rotational levels of CH + induced by collisions with He.The cross sections were averaged over a Maxwell-Boltzmanndistribution of kinetic energies to determine downward rate coef-ficients. The kinetic temperature spans a wide range of values up Fig. 3.
Calculated downward rate coe ffi cients for selected transi-tions as a function of the kinetic temperature. r a t e c o e ff i c i e n t s ( c m s - )
00 400 800 1200 1600 2000 -11 -11 -11 -10
T(K) to 2000 K. The spectroscopic parameters obtained at high tem-perature exhibit the general trends for such quantities. The ratecoe ffi cients will help to understand and interpret astrophysicalobservations in connection with the Herschel Space Observatorymission. Acknowledgements.
Author LCOO acknowledges with thanks the financial sup-port of the Abdus Salam International Centre for Theoretical Physics, O ffi ce ofExternal Activities (ICTP-OEA ) under NET45 Programme. References
Arthurs, A. M. & Dalgarno, A. 1960, Royal Society of London ProceedingsSeries A, 256, 540Black, J. H. 1998, in Chemistry and Physics of Molecules and Grains in Space.Faraday Discussions No. 109, 257– + Black, J. H., Hartquist, T. W., & Dalgarno, A. 1978, ApJ, 224, 448Boys, S. F. & Bernardi, F. 1970, Molecular Physics, 19, 553Cernicharo, J., Liu, X.-W., Gonzalez-Alfonso, E., et al. 1997, ApJ, 483, L65 + Falgarone, E., Phillips, T. G., & Pearson, J. C. 2005, ApJ, 634, L149Godard, B., Falgarone, E., & Pineau Des Forˆets, G. 2009, A&A, 495, 847Hammami, K., Owono Owono, L. C., Jaidane, N., & Ben Lakhdar, Z. 2008,Journal of Molecular Structure: THEOCHEM, 853, 18Huber, K. P. & Herzberg, G. 1979, Molecular Spectra and Molecular StructureIV. Constants of Diatomic Molecules, Van Nostrand, New YorkHutson, J. M. & Green, S. 1994, MOLSCAT computer code, version 14,Collaborative Computational Project No. 6 of the Science and EngineeringResearch Council, United KingdomJoulain, K., Falgarone, E., Des Forets, G. P., & Flower, D. 1998, A&A, 340, 241Knowles, P. J., Hampel, C., & Werner, H.-J. 1993, J. Chem. Phys., 99, 5219Knowles, P. J., Hampel, C., & Werner, H.-J. 2000, J. Chem. Phys., 112, 3106Maloney, P. R., Hollenbach, D. J., & Tielens, A. G. G. M. 1996, ApJ, 466, 561Manolopoulos, D. E. 1986, J. Chem. Phys., 85, 6425Nehm´e, C., Le Bourlot, J., Boulanger, F., Pineau Des Forˆets, G., & Gry, C. 2008,A&A, 483, 485
K. Hammami et al.: Rotational excitation of CH + by He at high temperature Table 2.
Downward rate coe ffi cients of rotational levels of CH + in collision with He as a function of kinetic temperature (in unitsof cm s − ). Initial level Final level Rate Coe ffi cientsJ J’ 300 K 500 K 1000 K 1500 K 2000 K1 0 1.0892(-10) 1.0474(-10) 9.3755(-11) 8.6919(-11) 8.2199(-11)2 0 5.4712(-11) 6.0895(-11) 6.8789(-11) 7.1572(-11) 7.2620(-11)2 1 1.1448(-10) 1.1728(-10) 1.1262(-10) 1.0753(-10) 1.0347(-10)3 0 2.1373(-11) 1.9118(-11) 1.4804(-11) 1.2486(-11) 1.1121(-11)3 1 8.8407(-11) 1.0579(-10) 1.3046(-10) 1.4071(-10) 1.4494(-10)3 2 7.4955(-11) 8.2783(-11) 8.9974(-11) 9.1966(-11) 9.2355(-11)4 0 1.2349(-11) 1.6741(-11) 2.4127(-11) 2.7741(-11) 2.9365(-11)4 1 1.8445(-11) 1.8632(-11) 1.7816(-11) 1.7104(-11) 1.6602(-11)4 2 1.0329(-10) 1.2071(-10) 1.4877(-10) 1.6297(-10) 1.7013(-10)4 3 4.4272(-11) 5.3386(-11) 6.6319(-11) 7.2883(-11) 7.6588(-11)5 0 1.8893(-12) 2.1234(-12) 2.4700(-12) 2.6880(-12) 2.8346(-12)5 1 2.6285(-11) 3.2428(-11) 4.5476(-11) 5.3674(-11) 5.8207(-11)5 2 9.2178(-12) 1.1060(-11) 1.3795(-11) 1.5314(-11) 1.6233(-11)5 3 9.5934(-11) 1.1444(-10) 1.4622(-10) 1.6408(-10) 1.7437(-10)5 4 2.6196(-11) 3.4002(-11) 4.8021(-11) 5.6663(-11) 6.2226(-11)6 0 4.5865(-12) 5.5789(-12) 8.2688(-12) 1.0468(-11) 1.1906(-11)6 1 1.7118(-12) 2.3213(-12) 3.6910(-12) 4.7426(-12) 5.4934(-12)6 2 2.7322(-11) 3.4211(-11) 4.9685(-11) 6.0782(-11) 6.7859(-11)6 3 4.5014(-12) 6.5071(-12) 1.0643(-11) 1.3429(-11) 1.5279(-11)6 4 8.6611(-11) 1.0467(-10) 1.3755(-10) 1.5779(-10) 1.7065(-10)6 5 1.7971(-11) 2.4058(-11) 3.6692(-11) 4.5470(-11) 5.1550(-11)7 0 2.9225(-13) 4.2016(-13) 7.6333(-13) 1.0737(-12) 1.3164(-12)7 1 8.1105(-12) 1.0045(-11) 1.5631(-11) 2.0767(-11) 2.4497(-11)7 2 1.5290(-12) 2.2415(-12) 4.1090(-12) 5.7047(-12) 6.9146(-12)7 3 2.4418(-11) 3.1705(-11) 4.8123(-11) 6.0615(-11) 6.9304(-11)7 4 4.0769(-12) 5.6780(-12) 9.7084(-12) 1.2814(-11) 1.5021(-11)7 5 7.3617(-11) 9.2301(-11) 1.2673(-10) 1.4868(-10) 1.6326(-10)7 6 1.5508(-11) 2.0148(-11) 3.0772(-11) 3.8740(-11) 4.4530(-11)8 0 1.4359(-12) 1.8300(-12) 3.0413(-12) 4.3682(-12) 5.4488(-12)8 1 9.0915(-13) 1.2019(-12) 1.9538(-12) 2.6903(-12) 3.3078(-12)8 2 7.7361(-12) 1.0389(-11) 1.7651(-11) 2.4520(-11) 2.9880(-11)8 3 2.3727(-12) 3.1083(-12) 5.0438(-12) 6.8186(-12) 8.2340(-12)8 4 2.0407(-11) 2.8012(-11) 4.5156(-11) 5.8453(-11) 6.8144(-11)8 5 5.3564(-12) 6.7877(-12) 1.0395(-11) 1.3359(-11) 1.5551(-11)8 6 5.9379(-11) 7.8105(-11) 1.1414(-10) 1.3805(-10) 1.5436(-10)8 7 1.6382(-11) 1.9895(-11) 2.8486(-11) 3.5245(-11) 4.0284(-11)9 0 3.1749(-13) 4.1519(-13) 6.0967(-13) 7.9911(-13) 9.6908(-13)9 1 2.2637(-12) 3.2040(-12) 6.0848(-12) 9.2677(-12) 1.2005(-11)9 2 1.6464(-12) 2.1416(-12) 3.1582(-12) 4.1187(-12) 4.9529(-12)9 3 6.3038(-12) 9.4243(-12) 1.7913(-11) 2.5792(-11) 3.2072(-11)9 4 3.5314(-12) 4.4470(-12) 6.3778(-12) 8.1075(-12) 9.5331(-12)9 5 1.5792(-11) 2.3335(-11) 4.0997(-11) 5.4987(-11) 6.5343(-11)9 6 6.9032(-12) 8.4680(-12) 1.1802(-11) 1.4466(-11) 1.6473(-11)9 7 4.6271(-11) 6.3763(-11) 9.9734(-11) 1.2494(-10) 1.4252(-10)9 8 1.8795(-11) 2.1639(-11) 2.8624(-11) 3.4250(-11) 3.8524(-11)10 0 3.7206(-13) 5.7888(-13) 1.2592(-12) 2.0545(-12) 2.7827(-12)10 1 8.7732(-13) 1.2033(-12) 1.7474(-12) 2.2194(-12) 2.6452(-12)10 2 1.9744(-12) 3.2175(-12) 7.1936(-12) 1.1441(-11) 1.5137(-11)10 3 2.2421(-12) 3.0153(-12) 4.3132(-12) 5.3982(-12) 6.3360(-12)10 4 4.7195(-12) 7.8986(-12) 1.7002(-11) 2.5507(-11) 3.2383(-11)10 5 4.3451(-12) 5.6292(-12) 7.7712(-12) 9.4441(-12) 1.0795(-11)10 6 1.1332(-11) 1.8149(-11) 3.5276(-11) 4.9542(-11) 6.0390(-11)10 7 8.1395(-12) 1.0071(-11) 1.3538(-11) 1.6073(-11) 1.7923(-11)10 8 3.5458(-11) 5.1093(-11) 8.5561(-11) 1.1114(-10) 1.2939(-10)10 9 2.1861(-11) 2.4448(-11) 3.0405(-11) 3.5149(-11) 3.8707(-11) St¨auber, P., Doty, S. D., van Dishoeck, E. F., & Benz, A. O. 2005, A&A, 440,949St¨auber, P., Doty, S. D., van Dishoeck, E. F., Jørgensen, J. K., & Benz, A. O.2004, A&A, 425, 577Sternberg, A. & Dalgarno, A. 1995, ApJS, 99, 565Stoecklin, T. & Voronin, A. 2008, European Physical Journal D, 46, 259Tao, F.-M. & Pan, Y.-K. 1992, J. Chem. Phys., 97, 4989 Tielens, A. G. G. M. & Hollenbach, D. 1985, ApJ, 291, 722Werner, H.-J., Knowles, P. J., & Alm¨of, J., et al. 2002, MOLPRO, a packageof ab initio programs, University College Cardi ff Consultants Limited, seehttp: ////