Photodissociation of small carbonaceous molecules of astrophysical interest
aa r X i v : . [ a s t r o - ph ] M a y Photodissociation of small carbonaceousmolecules of astrophysical interest
M.C. van Hemert a E.F. van Dishoeck b , ∗ a Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O.Box 9502, 2300 RA Leiden, The Netherlands b Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, TheNetherlands
Abstract
Astronomical observations have shown that small carbonaceous molecules canpersist in interstellar clouds exposed to intense ultraviolet radiation. Current astro-chemical models lack quantitative information on photodissociation rates in orderto interpret these data. We here present ab initio multi-reference configuration-interaction calculations of the vertical excitation energies, transition dipole momentsand oscillator strengths for a number of astrophysically relevant molecules: C , C ,C H, l − and c − C H, l − and c − C H , HC H, l − C H and l − C H. Highly excitedstates up to the 9’th root of each symmetry are computed, and several new stateswith large oscillator strengths are found below the ionization potentials. These dataare used to calculate upper limits on photodissociation rates in the unattenuatedinterstellar radiation field by assuming that all absorptions above the dissociationlimit lead to dissociation.
Key words:
Molecular data, Excited electronic states, Oscillator strengths,Photodissociation, Interstellar molecules
PACS:
Of the more than 130 different molecules found in interstellar space, an impor-tant class is formed by the unsaturated carbonaceous species. Carbohydrides ∗ Corresponding author
Email address: [email protected] (E.F. van Dishoeck).
Preprint submitted to Elsevier 21 November 2018 anging from small molecules such as C H (1) to long chains like C H (2) andHC N (3) have been detected through their millimeter transitions in cold darkclouds like TMC-1 for several decades. Some of these molecules have also beenseen in their cyclic form, with c-C H as the best-known example (4). Evennegative ions, in particular C H − , have now been detected (5). Bare carbonchains are likely present as well but do not have a permanent dipole momentand can therefore not be observed through their pure rotational transitions inemission. Instead, the smallest members of this family, C and C , have beendetected in diffuse interstellar clouds, –i.e., clouds which are not completelyopaque to visible and ultraviolet radiation– through their electronic absorp-tions against bright background stars (6; 7; 8). c − C H has been seen in diffuseclouds as well, through absorption at radio wavelengths against backgroundquasars (9). The relatively large abundances of these non-saturated molecules,in spite of the fact that there is 10 times more hydrogen than carbon in inter-stellar clouds, are a vivid demonstration that interstellar chemistry is not inthermodynamic equilibrium. Instead, the kinetics of the reactions that formand destroy these molecules need to be taken into account explicitly in orderto explain their abundances. In cold dark clouds, these traditionally involve aseries of ion-molecule and neutral-neutral reactions (10; 11).More recently, several of the carbohydrides C n H m have also been observed inso-called Photon-Dominated Regions (PDRs) (12), i.e., clouds which are ex-posed to intense ultraviolet (UV) radiation. Species like C H and c-C H havebeen detected in the Orion Bar PDR (13) and even more complex moleculeslike C H and C H have been seen in the Horsehead nebula, M38 (14; 15).Together with the above mentioned observations of diffuse clouds, the datademonstrate that these molecules can exist in UV-exposed environments withabundances comparable to those in cold dark clouds. Traditional PDR chemi-cal models cannot explain the high abundances of these molecules, leading tospeculations that they are perhaps produced by fragmentation of even largercarbonaceous molecules such as polycyclic aromatic hydrocarbons (PAHs)(15). However, a major uncertainty in these and other models are the pho-todissociation rates of the carbohydrides, which are basically unknown.Other regions in which ultraviolet photons play a role in the chemistry in-clude the envelopes of evolved stars and cometary atmospheres. The samecarbohydride molecules have been detected in the envelope of the carbon-richlate-type star IRC+10216, where their similar distributions are a puzzle forphotochemical models (16). In cometary atmospheres, species as complex asC H have been inferred, whose presence is not readily explained by the stan-dard parent-daughter photochemical models (17).Many theoretical studies of the electronic structure of small carbohydridemolecules exist in the literature (18; 19; 20; 21; 22; 23, e.g.,). However, most ofthem are limited to the ground and lowest few excited electronic states. Only2ew studies of higher states exist, most of them performed by Peyerimhoff andher co-workers starting more than 30 years ago, e.g., (24; 25) As will be shownhere, those results are still highly relevant and provide an excellent startingpoint for further studies.The aim of this paper is to provide insight into the photodissociation processesof carbohydride molecules through ab initio quantum chemical calculations ofthe vertical excitation energies and oscillator strengths for as large a numberof excited electronic states as feasible with current programs. By assumingthat most of the absorptions with energies above the dissociation limit leadto destruction, estimates of the (upper limits of) photodissociation rates un-der interstellar conditions can be obtained. Even though such calculations arenecessarily limited to the smaller members of the carbo-hydride family, theydo provide quantitative constraints to test the basic interstellar chemical net-works. Specifically, C H, C , C , l − and c − C H, l − and c − C H , HC H, l − C H and l − C H are studied here. Except for HC H, all these species havebeen chosen to have been observed in interstellar space.
For small diatomic and triatomic molecules, quantum chemical calculations ofthe potential energy surfaces and transition dipole moments combined withdynamical calculations of the nuclear motions can provide photodissociationcross sections and oscillator strengths that agree with experiments to betterthan 20–30%, e.g., (26; 27; 28) (see (29) for an early review). For the poly-atomic molecules considered here, calculations of the full potential surfacesincluding all degrees of freedom become very time consuming, as do the multi-dimensional dynamics. Moreover, such detail is not needed in order to computephotorates, since those are largely determined by the potentials and transitionmoments in the Franck-Condon region. The simplest alternative is thereforeto only compute the vertical excitation energies and transition dipole momentof the molecule at its equilibrium position and assume a certain probabilitythat absorption into each excited state above the dissociation limit leads todissociation. For the examples cited above, such an approach leads to similarrates within the accuracy of the calculations. The focus of our calculations istherefore on electric dipole-allowed transitions to states lying above the low-est dissociation limit but below 13.6 eV. The interstellar radiation field hasa broad spectrum from visible to extreme ultraviolet wavelengths, peaking inintensity around 7 eV and cutting off at the atomic H ionization potential at13.6 eV (30). 3ll calculations presented here were performed with the Wuppertal-BonnMRDCI set of programs as implemented in the GAMESS-UK program pack-age version 7.0 (31). For C, the TZVP atomic orbital basis set was used, andfor H, the DZP basis set (32). To allow for a proper description of molecularRydberg states, two diffuse p and two diffuse d functions were put on specificsites. For C and C H, this was the middle of the CC bond; for C and C Hthe middle of the C1-C2 and C3-C4 bonds; for C , l − C H, l − C H and HC Hthe middle C; for C on C2 and C4; for c − C H on the C connected to H; andfor c − C H on the lone C. These sites were chosen to avoid linear dependen-cies in the atomic orbital basis set due to large overlap of diffuse functionspositioned on adjacent atoms. Cartesian d functions were used, of which thespherical component served as the 3 s basis function.Molecular orbitals were generated using up to the maximum number of ref-erence states, 255. The selection threshold was generally set at 0.5 µ Hartree.The total number of configurations included in the configuration interaction(CI) calculation ranged from 200000–400000 per symmetry. The sum overthe coefficients-squared in the final CI wave function is typically 0.95 for thesmaller species, dropping to 0.9 for the larger molecules. The CI energies wereextrapolated and corrected using the Davidson extrapolation (33). The aimwas to compute as many excited electronic states as feasible, up to 9 per sym-metry. For the lower states, comparison with existing calculations and experi-ments indicates accuracies within 0.3 eV, generally better. Oscillator strengthsto the lower states agree within 30% or better. For the higher states, typicallythe 5’th root and higher per symmetry, the accuracy decreases because manystates and orbitals can mix. Nevertheless, such calculations should still pro-vide insight into the location of those states, in particular whether they areabove or below the ionization potential and below the 13.6 eV cutoff of theinterstellar radiation field. Moreover, the magnitude of the oscillator strengths(strong or weak) should be reliable. Note that the precise values of the excita-tion energies are not so important for the purposes of calculating interstellarphotorates because of the broad range of incident energies. The only exceptionis possible overlap with Lyman α radiation at 10.2 eV, which is important forcertain astrophysical environments (34).In all calculations, the largest Abelian subgroup of the full C ∞ v group, C v ,was used, with the molecule put along the z − axis. The doubly degeneratestates are then split into the B and B irreducible representations for Π andΦ states, and into A and A for the ∆ states. In general, the degeneracies forthe ∆ states are recovered within a few hundreds of an eV in the calculations.The average of the two A and A values is tabulated here. The A irreduciblerepresentation also contains the Σ + states, and the A the Σ − states. Someexcited states with electronic angular momentum higher than 2 (in particularΦ states) are also found in our calculations but they are not tabulated heresince the electric dipole transition moments from the ground state to these4tates are zero, implying that they do not contribute to photodissociation. Thevalence or Rydberg character of the states is determined from calculations ofthe x , y and z expectation values.Calculations have been limited to the equilibrium geometry corresponding tothe lowest energy. For C , C H, C , C H and C H this is the linear geometry.C H and C H have been detected in various isomeric forms so both cyclicand (near-)linear forms have been studied.The dissociation energies for C n H are calculated as the difference betweenthe ground state energy of C n H at its equilibrium geometry and the energy ofground state C n , also at its equilibrium geometry, calculated in the presence ofan H atom positioned at a distance of 20 Bohr from the center of mass. In thisway the errors associated with the lack of size consistency are significantlyreduced. Note that the spatial symmetry of the molecule and the fragmentgenerally differs. Oscillator strengths to all excited states have been calculated according to f elul = 23 g ul ∆ E ul µ ul (1)where all quantities are in atomic units (a.u.), µ ul is the transition dipolemoment from lower state l to upper state u , and g ul is a degeneracy factorwhich is 2 for a Π ← Σ transitions and 1 for any other transition. For linearmolecules, the dipole moment operator contained in µ ul corresponds to − P z j for transitions between states with the same value of the electronic angularmomentum projection quantum number Λ and to − P ( x j + iy j ) / √ ±
1. The computed excitation energies ∆ E ul were used inthis formula, not any experimental values. Hence, differences with other workcan stem from differences in both µ ul and in ∆ E ul .The photodissociation rate of a molecule can be computed from k contpd = Z σ ( λ ) I ( λ ) dλ s − (2)where σ is the photodissociation cross section in cm and I is the meanintensity of the radiation in photons cm − s − ˚A − (34). Under interstellarconditions, only single photon processes are important. For photodissociation5nitiated by line absorptions (e.g., predissociation), the rate becomes k linepd = πe mc λ ul f ul η u I ul s − (3)where η u the dissociation efficiency of state u , which lies between 0 and 1. Thenumerical value of the factor πe /mc is 8 . × − in the adopted units with λ in ˚A. The total photodissociation rate of a molecule is obtained by summingover all channels. In this work, no dynamical calculations are performed toobtain continuous cross sections for dissociative states. Hence, all photoratesare computed with Eq. (3). It is furthermore assumed that the dissociationefficiency η u =1, either through direct dissociation in a repulsive state or bypredissociation. The motivation for this choice is that for larger molecules,internal conversion to a lower (dissociative) electronic state is usually muchmore rapid than any radiative decay rates owing to the high density of states(35). Specific experimental evidence of high dissociation efficiencies will bepresented in the results section for individual molecules. H The C H radical, detected in interstellar clouds since 1974 (1), is an impor-tant step in the formation of longer carbon chains. In comets, it could bea photodissociation product of C H and a precursor of the widely observedC molecule. Given also its importance in combustion processes, this radicalhas received ample theoretical attention, starting with the papers by Shih,Peyerimhoff and co-workers (24; 36) and culminating with the more recenttwo-dimensional potential surfaces by Duflot et al. (37).As a test of our computational procedure, we present in Table 1 our computedvertical excitation energies, oscillator strengths and the main configurationsof each state. Our energies generally agree well within 0.3 eV with those ofKoures and Harding (38) (their CI + DV2 results) and lie in between thoseof Duflot et al. (37) and Shih et al. (24). Close inspection shows that thelarger differences are usually for states with Rydberg character (which arenot included in Duflot et al. (37)) or for states with a strong interaction witha neighboring state, leading sometimes even to a switch in the character ofthe state. For example, the 4 Σ + state in our calculation has Rydberg 3 p π character whereas it has ... 4 σ π σ π in Duflot et al., resulting also in avery different transition dipole moment (see below).6ur computed A Π − X Σ + transition dipole moment of 0.22 a.u. is veryclose to that of 0.23 a.u. computed by Duflot et al. (37) and Peric et al. (39).Because of the slightly higher excitation energy in our work, the oscillatorstrengths show somewhat larger differences. Comparison for other states isdifficult since Duflot et al. do not give any numerical values. However, theirFigure 7 shows that neither the 3 and 4 Σ + , nor the 2 and 3 Π stateshave significant transition dipole moment. This is generally consistent withour results. For the 5 Σ + state we find a huge transition dipole moment of1.1 a.u. due to the Rydberg character of our wavefunction. Thus, this statearound 10 eV will dominate the interstellar C H photodissociation but thehigher Π states in the 8.5–10.5 eV range can also contribute significantly upto the ionization potential of ∼ The C molecule was detected in cometary spectra in 1882 through its A − X system at 4050 ˚A (40) and in interstellar clouds more than a century later(41; 8). The molecule is also seen in the atmospheres of cool carbon starsthrough its mid-infrared (42) and far-infrared (43) transitions (see (23) forreview).C is a linear molecule with a ground X Σ + g state. Its dissociation energy iscomputed to be about 4.6 eV, whereas its ionization energy is around 12 eV(Table 11). Quantum chemical studies range from the early work by Cha-balowski et al. (44) to the recent calculations by Terentyev et al. (45). Table 2summarizes our computed excitation energies, together with the oscillatorstrengths and main configurations. For the low-lying valence states, our re-sults agree to better than 0.3 eV with the MR-AQCC values of Monninger etal. (46). These results also show that the Σ + u ← X Σ + g (1 π u → π g ) tran-sition at 8.17 eV has by far the largest oscillator strength, as predicted firstby Pitzer and Clementi (47). This is confirmed by the combined experimentaland theoretical study by Monninger et al. (46) whose 1100–5600 ˚A spectrumdemonstrates that the Σ + u ← X Σ + g transition around 1700 ˚A is indeed thestrongest band. Their experiments in Ne and Ar matrices show a broad bandbut with some progressions superposed, which are evidence for interstate vi-bronic coupling with adjacent Π g states. Indeed, the Σ + u state is predictedto be unstable to bending, leading to avoided crossings with the lower-lyingΠ g states (46). These interactions can also lead to dissociative channels toproduce C + C. 7 .3 C H Both linear and cyclic C H have been detected in the interstellar medium bytheir transitions at millimeter wavelengths (48; 49). The Π ground state of l − C H lies about 0.4–0.6 eV above the B ground state of c − C H (50), witha small barrier toward the cyclic state. Hence, when C H is produced in itslinear form by some sequence of ion molecule or other chemical processes, itcan be stable, and both isomers are therefore considered in this work. l − C H The geometry of the 1 Π state was taken from the experimental work of Mc-Carthy and Thaddeus (69). Table 3 summarizes our computed vertical ex-citation energies, oscillator strengths, and the corresponding configurations.Comparison with the CASSCF results of Ding et al. (51) shows good agree-ment for the lowest states. The dissociation energy of l − C H to C + H iscomputed at 3.3 eV. Of the dipole-allowed states above this dissociation limit,the higher Π states around 7.8 eV have the largest oscillator strengths, inparticular the 8 Π state, which will dominate the photodissociation of themolecule. c − C H The c − C H radical has C v geometry in its 1 B ground state, with theequilibrium structure for our calculations taken from Yamamoto and Saito(52). Table 4 summarizes our computed vertical excitation energies, oscillatorstrengths and configurations. Comparison with the CASSCF results of Dinget al. (51) shows again good agreement for the lowest states to 0.1–0.2 eV.The dissociation energy of c − C H to C + H is around 4.3 eV whereas itsionization potential is computed to lie at 9.6 eV (Table 11). Of the dipole-allowed states above the dissociation limit, the 2 A state around 5 eV andthe higher A states around 7.5 eV have the largest oscillator strengths, butthe sum over the other states is comparable. Thus, there appear to be manypotential routes to photodissociation for c − C H. H Cyclopropenylidene, c − C H , was the first cyclic molecule to be detected inthe interstellar medium (4) and subsequently found to be ubiquitous through-out the Galaxy, even in diffuse gas (53; 9). The near-linear form HCCCH(propargylene), denoted here as HC H, lies 0.4–0.8 eV higher in energy (54;80), depending whether the zero-point vibrational energy is included. Sinceit has a near-zero dipole moment, it has not yet been observed in interstellarclouds through radio transitions but is likely present as well. There are variousother stable isomers of C H , of which another linear form, H CCC (vinyli-denecarbene or propadienylidene), denoted here as l − C H , was discovered ininterstellar space in 1991 (55). It lies ∼ H isomers dating back to 1976 (56; 18; 57),the excited states are largely unexplored.The equilibrium structures of the electronic ground states are taken fromSeburg et al., their Figure 5 (54). Our calculated relative ordering of the C H isomers is consistent with previous findings but due to different amounts ofrecovered correlation energy, our energy splittings are larger than in otherstudies.The adiabatic ionization potentials of the various C H isomers lie at 8.96(HC H), 9.2 ( c − C H ) and 10.4 eV ( l − C H ), respectively (Table 11). Abovethese thresholds, the photoionization efficiency of all three C H isomers in-creases rapidly so that photoionization will become the main pathway (58).Some of these ionizations are likely to be dissociative, but this option is notconsidered here. H The HC H radical has C geometry with a B ground state. In this case,the C axis was put through the middle C atom and the middle of theH1–H2 line. Table 5 summarizes our computed vertical excitation energies,oscillator strengths and configurations. For the lowest four excited states,our results are in good agreement with those of Mebel et al. (59) (theirMRCI+D(4,8)/ANO(2+) results). The dissociation energy of HC H to l − C H+ H is computed to be around 3.1 eV, with the dissociation energy to C +H perhaps even less (59). Of the dipole-allowed states above the dissociationand below the ionization limit at 8.96 eV, the higher A states at 6.2 and 7.5eV, both of which have π → π ∗ character, have the largest oscillator strengths. c − C H The c − C H molecule has a A ground state in its C v geometry, with a largedipole moment of 3.4 Debye owing to the two unpaired electrons on one ofthe three carbon atoms. Table 6 summarizes our computed vertical excitationenergies and oscillator strengths, which again agree well in terms of energieswith those of Mebel et al. (59), although our oscillator strengths are some-9hat lower. Compared with c − C H, c − C H has only few low-lying electronicstates, and all of the dipole-allowed transitions lie above the dissociation en-ergy of 4.4 eV (59). Those to the 6 A ( π → π ∗ ) at 9.4, 7 B at 10.6 and9 B state at 11.2 eV have the largest oscillator strengths. However, none ofthese states lie below the ionization potential of c − C H at 9.2 eV. Thus, itsphotodissociation rate will differ substantially whether or not these states areincluded (see § l − C H The vinylidenecarbene or propadienylidene isomer of C H , denoted here as l − C H , also has A symmetry in its C v ground state geometry. Our com-puted vertical excitation energies (Table 7) of 1.86, 2.44 and 5.54 eV to the˜ A A , ˜ B B and ˜ C (2) A states are consistent with the experimental adiabaticvalues (60) of 1.73, 2.00 and 4.84 eV, respectively, and agree within 0.1–0.2 eVwith the vertical values computed by Mebel et al. (59) The dissociation energyof l − C H to C H + H is computed to lie around 3.9 eV, so that only the˜ C (2) A and higher states can lead to photodissociation. The ˜ C state showsa well-resolved progression in its electronic spectrum (60), but the resolutionof those data is not high enough to measure predissociation rates. Thus, weconsider the photodissociation both with and without taking the ˜ C state intoaccount.Of the dipole-allowed transitions below the ionization potential at 10.4 eV,those to the higher A states around 9 eV have the strongest oscillatorstrengths. These states all have 2 b → n ∗ b character and likely belong tothe Rydberg series converging to the lowest ionization potential. Similarly,the higher B states belong to the 2 b → na Rydberg series. The lowest energy X Σ − g ground state of C occurs for the linear geometry.C has been searched for in diffuse interstellar clouds through its Σ − u ← Σ − g transition around 3789 ˚A but not yet detected (61). However, a pattern ofbands at 57 µ m observed with the Infrared Space Observatory toward a handfulof objects (62) is consistent with transitions in the ν bending mode (63).Rhombic C , which is almost isoenergetic with l − C , has not yet been detectedin interstellar space.Our calculations use the l − C equilibrium values computed by Botschwina(64), which are consistent with experiments (23). The results are presentedin Table 8. Our excitation energies of the lower states are in good agreementwith experiments and with the calculations of Mass´o et al. (65), except for10he 2 ∆ u state. Note that the experimental values refer to T rather than T e ,which can differ by a few tenths of eV. Our computed dissociation energy D e is 4.7 eV, in good agreement with the experimentally inferred D value of 4.71 ± Π u state at 5.1 eV. Indeed, photofragment yieldspectra in the 2.22–5.40 eV range by Choi et al. (66) show significant single-photon dissociation into C + C at ≥ +C . Their spectra are well reproduced by phase space theory models in whichthe product state distributions are statistical. This implies that absorptioninto the excited state is most likely followed by rapid internal conversion tothe ground state potential energy surface with no barriers present along thedissociation coordinates.By far the strongest absorption occurs into the 2 Σ − u state around 6.95 eV,which has an oscillator strength of 1.56. Even though higher Σ − u and Π u stateswill contribute, the 2 Σ − u channel will dominate the interstellar photodissoci-ation of C , if indeed every absorption is followed by dissociation. l − C H The l − C H (butadiynyl) radical was detected in the envelopes of carbon-richevolved stars nearly 30 years ago (67). It was subsequently found to be veryabundant in cold dark clouds like TMC-1 (68) and even detected in comets(17). It is one of the carbon-bearing molecules found at the edges of PDRs(15), where it can photodissociate into smaller species.C H has linear symmetry with a Σ + ground state. The equilibrium coordi-nates in our calculations were taken from Ref. (69). Table 9 summarizes ourcomputed vertical excitation energies, oscillator strengths and configurations.Comparison with the results of Graf et al. (70) shows good agreement in bothenergies and transition dipole moments for the lowest states, but poor agree-ment for the higher Π states, where the Graf et al. energies are generallylower by up to 1 eV. There are two main resons for this. First, Graf et al.did not include diffuse (Rydberg) functions in their basis set, which start tobecome important for the higher states. Second, their CASPT2 perturbationmethod does not guarantee a lower bound to the energies. Our calculationsagree in the fact that none of the higher lying Π states have large oscillatorstrengths.The threshold for photodissociation, corresponding to dissociation of C H toC + H, lies around 4.7 eV (Table 11). Of the dipole-allowed transitions belowthe ionization potential at 9.6 eV, the 4 and 5 Σ + states at 7.7 and 8.711V, respectively, have orders of magnitude larger oscillator strengths thanother states and will thus dominate the photodissociation. Both states have1 π → nπ Rydberg character. Thus, our overall photodissociation rate will besignificantly larger than that using the data from Graf et al. (70) l − C H Like C H, l − C H is detected toward the carbon-rich evolved star IRC+10216(71) and in cold dark clouds (72). It also has linear symmetry but with a Π ground state. The equilibrium coordinates in our calculations were takenfrom Ref. (69). Table 10 summarizes our results. Comparison with Haubrichet al. (73) shows excellent agreement in both energies and oscillator strengths,except for the higher-lying Σ + states. This difference is likely due to the ex-plicit inclusion of Rydberg states in our work. For the 3 and 4 Π states, theoscillator strengths differ by a factor of 2 but these states show considerableinteraction between the 2 π → π and 3 π → π ( π → π ∗ ) excitations. Differ-ences in mixing ratios can lead to large changes in transition dipole momentsto individual states, but not in energies. The computed excitation energiesfor the lowest two excited states also agree well with those measured andcomputed by Ding et al. (74).The threshold for photodissociation lies around 3.6 eV (Table 11). Of thedipole-allowed transitions below the ionization potential at 7.4 eV, both the 4 Π (2 π → π ) state at 4.2 eV and the 6 Π (1 π → π ) state around 6.1 eV havelarge oscillator strengths. All other states have typical oscillator strengths ofa few × − and thus contribute at a lower level. In Table 12, the computed photodissociation rates in the unshielded interstel-lar radiation field cf. Draine (30) are presented, using the oscillator strengthsgiven in Tables 1 to 10 and asuming η u =1 for all states. Thus, these ratesshould be regarded as upper limits. Only states above the dissociation limitand below the ionization potential have been taken into account (see Table 11for adopted values). No corrections have been made for possible higher-lyingstates below the ionization limit not computed in this work. Since the oscilla-tor strengths for Rydberg states decrease roughly as 1 /n , it is assumed thatany such corrections would be small since the lowest Rydberg members arecalculated explicitly. For reference, inclusion of a hypothetical state at 9 eVwith an oscillator strength of 0.1 would increase the photodissociation ratesby only 3 . × − s − . 12or C H, our new rate is a factor of 3 larger than that given in van Dishoeck etal. (34), which was based on the energies and oscillator strengths of Shih et al.(24). The increase is mostly due to the higher Σ + and Π states which werenot computed in that work. For C , the new rate is only 30% larger, mostlybecause the photodissociation rate of this molecule is dominated by the verystrong absorption into the 1 Σ + u state around 8 eV, which was included inprevious estimates.It is seen that the rates for the various carbon-bearing molecules span a rangeof a factor of 8, with the rates for the bare carbon chains (C , C ) being largestand those for the odd-numbered C n H species lowest. Of the different C H iso-mers, l − C H has the largest photodissociation rate and c − C H the smallestby a factor of 3. However, c − C H differs from the other isomers in that ithas several states with large oscillator strengths above the ionization potentialof 9.15 eV. If those states were included, the c − C H photodissociation ratewould be increased by a factor of 2. For l − C H , the rate would drop from5 . × − to 4 . × − s − if the ˜ C state is not included.In spite of the range of values, all rates are above 10 − s − , coresponding to alifetime of less than 30 yr at the edge of an interstellar cloud. In regions suchas the Orion Bar and the Horsehead nebula, the radiation field is enhanced byfactors of 10 − compared with the standard field adopted here, decreasingthe lifetimes to less than 1 month. Thus, there must be rapid production routesof these molecules in order to explain their high abundances in UV-exposedregions. This conclusion is not changed if only 10%, say, of the absorptionswould lead to dissociation rather than the 100% assumed here. As argued in § We have presented vertical excitation energies, transition dipole moments andoscillator strengths for states up to the 9’th root of each symmetry for severalcarbonaceous molecules of astrophysical interest. For lower-lying states, goodagreement is generally found with previous studies. Several new, higher-lyingstates with large oscillator strengths, often of Rydberg character, are revealedin this work.The calculated photodissociation rates of the small carbon-bearing moleculesstudied here are substantial, leading to lifetimes at the edges of interstellarclouds of less than 30 yr. These high rates assume that all absorptions abovethe dissociation limit indeed lead to dissociation, so that the rates should be13iewed as upper limits. Further experimental work is needed to quantify thedissociation efficiencies for the strongest states found in this work.
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This Configurationwork (38) (37) (24) work1 Σ + b . . . ...4 σ σ π V2 Σ + c ...4 σ σ π π V3 Σ + σ σ π V4 Σ + σ σ π px R5 Σ + σ σ π px R6 Σ + σ σ π px R1 Π 0.68 0.60 0.54 0.96 1.7(-3) ...4 σ σ π V2 Π 7.63 7.29 7.07 8.11 1.0(-2) ...4 σ σ π π V3 Π 8.39 8.17 8.05 9.96 1.0(-1) ...4 σ σ π s R4 Π 9.00 8.68 8.34 8.48 3.0(-2) ...4 σ σ π pz R5 Π 9.47 8.70 3.0(-2) ...4 σ σ π s R6 Π 9.96 8.80 9.22 2.2(-2) ...4 σ σ π pz R7 Π 10.06 9.25 9.78 1.1(-2) ...4 σ σ π pz R8 Π 10.33 9.67 10.40 4.6(-2) ...4 σ σ σ π M1 Σ − σ σ π π V2 Σ − σ σ π π V3 Σ − σ σ π px R4 Σ − σ σ π px R5 Σ − σ σ π px R6 Σ − σ σ π px R1 ∆ 7.89 7.70 7.57 8.27 ...4 σ σ π π V2 ∆ 8.25 8.12 7.95 8.81 ...4 σ σ π π V3 ∆ 9.20 9.06 9.23 9.12 ...4 σ σ π px R a V=Valence; R=Rydberg; M=Mixed in this and subsequent tables b Ground state energy including Davidson correction: -76.432130 Hartree c Notation x(-y) in this and subsequent tables indicates x × − y able 2Vertical excitation energies, oscillator strengths and dominant configurations for C at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This Ref. This Configurationwork (46) work1 Σ + g a . . . ...4 σ g σ u π u V2 Σ + g σ u π u π g V3 Σ + g σ g σ u π u π g V1 Σ + u σ g σ u π u π g V1 Π u σ g σ u π u π g V2 Π u σ g σ u π u π g V3 Π u σ g σ u π u π g V4 Π u σ g σ u π u s R1 Π g σ g σ u π u π g V2 Π g σ g σ u π u π g V3 Π g σ g σ u π u π g V4 Π g σ g σ u π u π g V5 Π g σ g σ u π u π g V1 Σ − u σ g σ u π u π g V2 Σ − u σ g σ u π u π g V1 ∆ g σ g π u π g V2 ∆ g σ g σ u π u π g V3 ∆ g σ g σ u π u π u V4 ∆ g σ u π u π g V1 ∆ u σ g σ u π u π g V a Ground state energy including Davidson correction: -113.735304 Hartree able 3Vertical excitation energies, oscillator strengths and dominant configurations for l − C H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This Ref. This Configurationwork (51) work1 Π a . . . ...7 σ π π V2 Π 3.69 3.92 2.9(-4) ...7 σ π π V3 Π 4.99 5.33 1.4(-3) ...7 σ π π V4 Π 5.39 7.4(-3) ...7 σ π π V5 Π 6.51 6.6(-2) ...7 σ π p π R6 Π 7.56 2.9(-2) ...7 σ π π s R7 Π 7.82 3.5(-2) ...7 σ π p π R8 Π 7.87 1.3(-1) ...7 σ π p π R1 Σ + σ π π V2 Σ + σ π s R3 Σ + σ π p σ R4 Σ + σ π π V5 Σ + σ π p σ R1 Σ − σ π π V2 Σ − σ π π V3 Σ − σ π π V4 Σ − σ π π R5 Σ − σ π π p π R1 ∆ 2.79 2.96 7.6(-3) ...7 σ π π V2 ∆ 6.86 4.0(-3) ...7 σ π π V3 ∆ 7.63 2.6(-3) ...7 σ π π V4 ∆ 8.24 5.8(-3) ...7 σ π π R a Ground state energy including Davidson correction: -114.381395 Hartree able 4Vertical excitation energies, oscillator strengths and dominant configurations for c − C H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This Ref. This Configurationwork (51) work1 B a . . . ... a a b b b V2 B b → b V3 B a → a R4 B b → b R5 B a → a R6 B b → b V7 B a → a R8 B b → b R9 B a → a R1 A a → b V2 A a → b V3 A b → a R4 A b → a R5 A b → a R6 A b → a V7 A b → a R8 A a b → b b V9 A a ) → a b R1 A b → a V2 A a → b V3 A a → b V4 A a ) → b a R5 A a → b R6 A a → b R7 A b → a R8 A a → b R9 A a → b R1 B b → b V2 B b → b V3 B a → a V4 B a → a V5 B a ) → b b V6 B b → b R7 B b → b R8 B a → a R9 B a → a R a Ground state energy including Davidson correction: -114.393014 Hartree a footnote able 5Vertical excitation energies, oscillator strengths and dominant configurations forHC H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This This Configurationwork work1 B a . . . ...6 a a b b V2 B b → b V3 B a → a V4 B a → a R5 B a → a V6 B b → b V7 B a → a R8 B b → b R9 B a → a R1 A a → b V2 A b → a V3 A b → a V4 A a → b V5 A a → b R6 A b → a R7 A b → a R8 A b → a R9 A b → a R a Ground state energy including Davidson correction: -114.924751 Hartree able 6Vertical excitation energies, oscillator strengths and dominant configurations for c − C H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This This Configurationwork work1 A a . . . ... a a b b b V2 A a → a R3 A a → a R4 A a → a R5 A a → a R6 A a → a R7 A b → b V8 A a → a R9 A a → a R1 B a → b V2 B b → a V3 B a → b R4 B a → b R5 B b → a R6 B b → a R7 B b → a R8 B b → a R9 B a → b R1 B a → b R2 B a → b R3 B a → b V4 B b → a R5 B b → a R6 B b → a R7 B a → b V8 B b → a R9 B a → b R1 A a → a V2 A b → b V3 A a → a R4 A a → a V5 A b → b R6 A b → b R7 A b → b R8 A b → b R9 A b → b R a Ground state energy including Davidson correction: -115.037418 Hartree able 7Vertical excitation energies, oscillator strengths and dominant configurations for l − C H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This This Configurationwork work1 A a . . . ... a a b b V2 A b → b V3 A b ) → (2 b ) V4 A b → b R5 A b → b R6 A b → b R7 A b → b R8 A b → b R9 A b → b R1 B a → b V2 B a b → (2 b ) V3 B a → b R4 B a b → b b V5 B a → b R6 B b → a R7 B a → b R8 B a → b R9 B b → a R1 B a b → (2 b ) V2 B b → a V3 B b → a R4 B b → a R5 B b → a R6 B b → a R7 B b → a R8 B b → a R9 B < b → a R1 A b → b V2 A b b → (2 b ) V3 A b → b R4 A b → b V5 A b → b R6 A b → b V7 A b → b R8 A b → b R9 A b → b R a Ground state energy including Davidson correction: -114.965558 Hartree able 8Vertical excitation energies and oscillator strengths and dominant configurations for l − C at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This Ref. Exp. This Configurationwork (65) (65) work1 Σ − g a . . . ...4 σ u σ g π u π g V2 Σ − g σ u σ g π u π g V3 Σ − g σ u σ g π u π g π u V4 Σ − g σ u σ g π u π g π u V1 Σ + g σ u σ g π u π g π u V1 Σ − u σ u σ g π u π g V2 Σ − u σ u σ g π u π g π u V1 Σ + u σ u σ g π u π g V2 Σ + u σ u σ g π u π g V3 Σ + u σ u σ g π u π g π u V4 Σ + u σ u π u π g π u V5 Σ + u σ g π u π g π u V1 Π g σ u σ g π u π g V2 Π g σ u σ g π u π g V3 Π g σ u σ g π u π g π u V4 Π g σ u σ g π u π g π u V1 Π u σ u σ g π u π g V2 Π u σ u σ g π u π g V3 Π u σ u σ g π u π g π u V4 Π u σ u σ g π u π g π u V5 Π u σ u σ g π u π g π u V1 ∆ u σ u σ g π u π g V2 ∆ u σ u σ g π u π g π u V1 ∆ g σ u σ g π u π g π u V a Ground state energy including Davidson correction: -152.228684 Hartree able 9Vertical excitation energies,oscillator strengths and dominant configurations for l − C H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This Ref. This Ref. Configurationwork (70) work (70)1 Σ + a . . . . . . ...9 σ π π V2 Σ + < σ π π π V3 Σ + σ π π π V4 Σ + σ π π p π R5 Σ + σ π π p π R1 Π 0.37 0.44 7.8(-4) 8.4(-4) ...9 σ π π V2 Π 3.59 3.31 7.8(-4) 9.8(-4) ...9 σ π π V3 Π 5.31 4.71 3.4(-4) 7.2(-4) ...9 σ π π π V4 Π 7.15 5.92 2.2(-3) 9.2(-4) ...9 σ σ π π R5 Π 7.85 6.82 7.0(-3) 8.8(-5) ...9 σ σ π π R6 Π 7.90 7.84 1.4(-2) ...9 σ σ π π R7 Π 8.41 2.2(-3) ...9 σ σ π π R1 Σ − σ π π π V2 Σ − σ π π π V3 Σ − σ π π p π R4 Σ − σ π π p π R1 ∆ 5.76 5.20 ...9 σ π π π V2 ∆ 5.97 5.21 ...9 σ π π π V3 ∆ 7.85 ...9 σ π π p π R4 ∆ 8.10 ...9 σ π π p π R a Ground state energy including Davidson correction: -152.228684 Hartree able 10Vertical excitation energies, oscillator strengths and dominant configurations for l − C H at the ground state equilibrium geometryEnergy (eV) f el Dominant TypeState This Ref. This Ref. Configurationwork (73) b work (73) b Π a . . . . . . ...11 σ π π π V2 Π 3.05 3.21 5.1(-4) 1(-3) ...11 σ π π π V3 Π 3.91 3.99 1.8(-3) 4(-3) ...11 σ π π π V4 Π 4.18 4.19 5.7(-2) 3(-2) ...11 σ π π π V5 Π 5.20 5.10 3.5(-3) 5(-3) ...11 σ π π π V6 Π 6.09 6.35 5.2(-2) 1(-3) ...11 σ π π π V7 Π 6.13 6.11 3.0(-3) 1.4(-1) ...11 σ π π π V1 Σ + σ π π π V2 Σ + σ π π s R3 Σ + σ π π p σ R4 Σ + σ π π s R5 Σ + σ π π π V1 Σ − σ π π π V2 Σ − σ π π π V3 Σ − σ π π π V1 ∆ 2.71 2.62 4.8(-3) 3(-3) ...11 σ π π π V2 ∆ 6.06 5.92 2.6(-3) 1(-3) ...11 σ π π π V3 ∆ 6.63 6.05 1.5(-3) 2(-3) ...11 σ σ π π V a Ground state energy including Davidson correction: -189.983327 Hartree b Their cc-p-VTZ+SP triple ζ results if available, double ζ results otherwise. able 11Dissociation and ionization energies a with respect to ground-state equilibrium en-ergy (in eV) for various speciesSpecies Products ∆ E diss Ref. ∆ E ion Ref. l − C C + C( P ) 4.63 (75) 12.1 (76) l − C C + C( P ) 4.71 (66) 10.7 (77) l − C H C + H 4.90 TW b l − C H C + H 3.27 TW 8.6 TW c − C H C + H 4.29 TW 9.6 TW l − C H C + H 4.65 TW 9.6 TW l − C H C + H 3.56 TW 7.4 TWHC H C H + H 3.1 (59) 8.96 (58) c − C H C H + H 4.37 (59) 9.15 (78) l − C H C H + H 3.87 (59) 10.43 (78) a Experimental data refer to the adiabatic ionization potentials; computed valuesin this work to vertical ionization potentials b Computed in this work, using the same basis set and procedure for the products;dissociation energies were obtained at 20 Bohr able 12Photodissociation rates (in s − ) for various molecules in the unshielded interstellarradiation field Species Rate l − C l − C l − C H 1.6(-9) l − C H 1.8(-9) c − C H 1.1(-9) l − C H 3.7(-9) l − C H 1.3(-9)HC H 2.2(-9) c − C H l − C H5.1(-9)