Vibrational Quenching of CN- in Collisions with He and Ar
Barry Mant, Ersin Yurtsever, Lola González-Sánchez, Roland Wester, Franco A. Gianturco
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Vibrational Quenching of CN − in Collisions with He and Ar Barry Mant, Ersin Yurtsever, Lola González-Sánchez, Roland Wester, and Franco A. Gianturco a) Institute for Ion Physics and Applied Physics, University of Innsbruck, Technikerstr. 25/3, 6020 Innsbruck,Austria Department of Chemistry, Koç University, Rumelifeneri yolu, Sariyer, TR-34450, Istanbul,Turkey Departamento de Química Física, University of Salamanca, Plaza de los Caídos sn, 37008 Salamanca,Spain (Dated: 21 January 2021)
The vibrational quenching cross sections and corresponding low-temperature rate constants for the ν = ν = − ( Σ + ) colliding with He and Ar atoms have been computed ab initio using new three dimensional potentialenergy surfaces. Little work has so far been carried out on low-energy vibrationally inelastic collisions for anions withneutral atoms. The cross sections and rates calculated at energies and temperatures relevant for both ion traps andastrochemical modelling, are found by the present calculations to be even smaller than those of the similar C − /He andC − /Ar systems which are in turn of the order of those existing for the collisions involving neutral diatom-atom systems.The implications of our finding in the present case rather small computed rate constants are discussed for their possiblerole in the dynamics of molecular cooling and in the evolution of astrochemical modelling networks. I. INTRODUCTION
Vibrationally inelastic collisions are fundamental processesin chemical physics and molecular dynamics. Gas phase col-lisions which can excite or quench a vibrational mode in amolecule have been studied both experimentally and theoreti-cally for decades and are generally well understood. Typi-cally the scattering cross sections and corresponding rates arerelatively small due to the generally large energy spacingbetween vibrational levels which require strong interactionforces between the colliding species to induce transitions. Onthe other hand, these processes still attract a great deal of at-tention and study as they have important applications in fieldssuch as cold molecules, where collisions are used to quenchinternal molecular motion, or astrochemistry, where accu-rate rate constants are necessary to model the evolution of gasclouds and atmospheres. There are also exceptional sys-tems such as the dramatic case of BaCl + + Ca where lasercooled calcium atoms can efficiently quench vibrational mo-tion with rates similar to rotational transitions. There continues to be many studies of diatom-atom vi-brationally inelastic collisions for both neutral andcationic species.
This is to be contrasted by with thecase for anions, where very little work has been carried outon vibrationally inelastic collision processes. Recently wehave tried to change this trend and have investigated vibra-tional quenching of the C − anion in collisions with noble gasatoms. This molecule is of direct interest as a possible can-didate for laser cooling mechanisms but a first step will re-quire the cooling of internal motion via collisions since spon-taneous dipole emission is forbidden for the rovibrational ex-cited states of this homonuclear species. The cross sectionsand rate constants for vibrational transitions were found byour calculations to be small, i.e. of the order of those for neu-tral species. a) Electronic mail: [email protected]
In this article we report the vibrational quenching of yetanother important anion, CN − in collisions with He and Aratoms. The cyanide anion is a well studied molecule, par-ticularly its spectroscopic properties have attracted a greatdeal of attention and investigations as well as the de-termination of its photodetachment energy . Recentwork in our group has further clarified important aspects ofits photodetachment behaviour at threshold from cold trapexperiments. This molecule has also been detected in theenvelope of a carbon star after its rotational constants werecarefully measured. Collisional processes of the anion withthe astrochemically relevant He and H species for ro-tational transitions have recently been studied and we havealso investigated the rotational cooling of this molecular an-ion with He, Ar and H as buffer gasses. The CN − anionis also thought to be an important participant as well in reac-tions in the interstellar medium and in the atmosphere ofTitan where it has been detected. We note in passing that the corresponding neutral speciesCN was one of the first molecules to be detected in space and cross sections and rates for this species have been investi-gated and obtained for various ro-vibrational processes in col-lisions with He and H . The cyanide cation is also sus-pected to be important to astrochemical processes but has yetto be detected. The cation’s vibrational energies have recentlybeen measured as well as a study has been carried out on itsrotational transitions induced by He collisions. Vibrationally inelastic collisions involving the CN − molec-ular anion with neutral atoms are a type of process rarely stud-ied for such systems. Although CN − can of course lose energythrough spontaneous emission, its wide relevance justifiesproviding an accurate assessment of the vibrational quench-ing processes involving He and Ar, typical buffer gases in iontraps.The paper is organised as follows: Section II presents theCN − potential energy and dipole moment curves along withthe anion’s vibrational energy levels and Einstein A coeffi-cients. The potential energy surfaces for the CN − /He andCN − /Ar systems are then discussed in Section III. The quan- −100−95−90−85−80−75 1 1.1 1.2 1.3 1.4 1.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 E ne r g y / c m − D i po l e / D eb y e r / Å Ab InitioLEVEL FitDipole Moment
FIG. 1.
Ab initio energies, PEC fit and DMC for CN − ( Σ + ). Thehorizontal lines show the first six vibrational energies. tum scattering methodology is described in Section IV andscattering cross sections and rates are discussed in Section V.Conclusions are given in Section VI. II. CN − POTENTIAL ENERGY CURVE AND DIPOLEMOMENT
Electronic energies for the ground Σ + state of the CN − anion were calculated at 19 internuclear distances r to obtainthe anion’s potential energy curve (PEC). Calculations werecarried out using the MOLPRO suite of quantum chemistrycodes at the CCSD(T) level of theory employing anaug-cc-pV5Z basis set. The expectation value of the non-relaxed CCSD dipole moment at each r distance was alsoobtained. The ab initio energies and dipole moment curve(DMC) for CN − are shown in Fig. 1.The LEVEL program was used to obtain the vibrationalenergies and wavefunctions, for the CN − molecule. The abinitio energies were used as input, interpolated using a cu-bic spline and extrapolated to r values below and above therange of calculated energies using functions implemented inLEVEL. The relative energies of the first three vibrational lev-els along with the rotational constants for each state are shownin Table I and compared with previously published calculatedtheoretical and experiment values. The agreement with pre-vious calculations and experimental values is quite good andcertainly sufficient to evaluate the cross sections and rates con-stants of inelastic collisions considered below.We have recently evaluated the dipole moment of CN − atits equilibrium bond length r eq using a variety of ab initio methods and basis sets and used it to evaluate the EinsteinA coefficients for pure rotational transitions. The best esti-mate of that work of 0.71 D is in quite good agreement withthe value of the DMC at r e of 0.65 D computed here. TheLEVEL program was also used to calculate the Einstein Acoefficients for ro-vibrational transitions of CN − using the abinitio calculated DMC. The values of A ν ′ j ′ , ν ′′ j ′′ for the first twovibrational states of the anion are shown in Table II and com- TABLE I. Comparison of vibrational energies and rotational con-stants with previous theoretical and experimental values. Literaturevalues calculated from Dunham parameters provided. Units of cm − .Relative energy B ν ν This work 0 1.864Calc. ν This work 2040 1.845Calc. ± ν This work 4055 1.831Calc. pared to those of neutral CN. The values for the anion andneutral molecule are broadly similar which is reasonable con-sidering they have very similar bond lengths and vibrationalenergies. The slightly larger values for neutral CN are a re-sult of the larger dipole moment for the neutral molecule. TABLE II. Einstein A coefficients A ν ′ , ν ′′ for selected CN − ( Σ + )vibrational transitions compared to those for neutral CN ( Σ + ) cal-culated by Brooke et al. For CN − the P(1) branch values were usedto compare to the Q-branch values for CN. Units of s − .Transition CN − CN ν → ν ν → ν ν → ν III. CN − /He AND CN − /Ar POTENTIAL ENERGYSURFACES AND VIBRATIONALLY AVERAGED MATRIXELEMENTS The interaction energies between CN − in its ground Σ + electronic state with He and Ar atoms were calculated us-ing ab initio methods implemented in the MOLPRO suite ofcodes. Geometries were defined on a Jacobi grid with R (the distance from the centre of mass of CN − to the atom)ranging from 2.5 to 20 Å and θ (the angle between R and theCN − internuclear axis r ) from 0 (C side) to 180 ◦ in 15 ◦ and10 ◦ intervals for He and Ar respectively. Seven values of theCN − bond length for each system between r = . r eq = . − and the noble gas atoms were determined by subtractingthe asymptotic energies for each bond length.For CN − /He, energies were calculated using the Multi-configurational self-consistent field (MCSCF) method with 8 occupied orbitals and 2 closed orbitals followed bya 1-state multi-reference configuration interaction (MRCI) calculation. An aug-cc-pV5Z basis was employed. In our R s i n ( q ) / Å −5 −5 −5 −5−10 −10 −10 −10 −10−20 −20 −20−30 −30 −30−40 −40−30−5 −5200−5 −50 −50 −50 −50−100 −100 −100 −100 −100−200 −200 −200−300 −300 −300−400 −400800−50 CN − −He −6 −4 −2 0 2 4 6 Rcos( q ) / Å R s i n ( q ) / Å CN − −Ar −6 −4 −2 0 2 4 6 Rcos( q ) / Å FIG. 2. Contour plots for CN − ( Σ + )/He (left) and CN − ( Σ + )/Ar (right) of vibrationally averaged matrix elements V , ( R , θ ) (top) and V , ( R , θ ) (bottom) projected onto Cartesian coordinates. Energies in cm − . See main text for further details earlier discussion of the CN-/He PES we discuss in detailthe reasons why we followed both methods for this systemand compared the CASSCF+MRCI results with the CCSD(T)with similar basis set expansions, finding them to be coin-cident in values. In particular, we corrected for the size-consistency possible shortcomings of the CASSCF+MRCI vsthe CCSD(T) methods by correcting the latter results usingthe Davidson ´ s correction as implemented in MOLPRO. Inour earlier work we showed that this correction brought thetwo sets of potential calculations to yield the same potentialvalues over a broad range of the employed grid. As an exam-ple, we note here that from our CBS (Complete Basis Set) ex-trapolated CCSD(T) calculations on CN − /He system we findthe minimum energy configuration as theta=40 deg., R=3.95Å with BSSE corrected energy at 49.522 cm − . CBS is calcu-lated by the default procedure in MOLPRO: it is the so-calledL3 extrapolation discussed in there. The results within theCAS(8,4) within the CASSCF+MRCI gave a theta=40 deg.,R=4.00 and an energy of 50.39 cm − for its minimum config-uration, showing the two methods to provide essentially thesame results.For the CN − /Ar system, energies were calculated using theCCSD(T) method with complete basis set (CBS) extrapola-tion using the aug-cc-pVTZ, aug-cc-pVQZ and aug-cc-pV5Zbasis sets. The basis-set-superposition-error (BSSE) wasalso accounted for all calculated points using the counterpoiseprocedure. The three-dimensional PESs were fit to an analytical formusing the method of Werner, Follmeg and Alexander where the interaction energy is given as V int ( R , r , θ ) = N r − ∑ n = N θ − ∑ l = P l ( cos θ ) A ln ( R )( r − r eq ) n , (1)where N r = 7 and N θ = 13 or 19 respectively are the numberof bond lengths r and angles θ in the ab initio grid, P l ( cos θ ) are the Legendre polynomials and r eq = . − . For each bond length r m and angle θ k , one-dimensional cuts of the PESs V int ( R , r m , θ k ) were fit to B km ( R ) = exp ( − a km R ) " i max ∑ i = b ( i ) km R i − [ + tanh ( R )] " j = j max ∑ j = j min c jkm R − j , (2)where the first terms account for the short range part of thepotential and the second part for the long range terms com-bined using the [ + tanh ( R )] switching function. For each r m and θ k Eq. 2 was least squares fit to the ab initio data(around 40 R points) using i max = j min = j max = − for CN − /He and0.27 cm − for CN − /Ar. From the 1D potential fits B km ( R ) ,the radial coefficients A ln ( R ) can be determined from the ma-trix product A ( R ) = P − B ( R ) S − where the matrix elementsof P and S are given as P kl = P l ( cos θ k ) and S nm = ( r m − r eq ) n respectively. The analytical representation of the PES, Eq.1, gives a reasonable representation of the ab initio interac-tion energies. An overall RMSE of 82 cm − for all pointsused in the fit was obtained for CN − /He but this drops to 0.26cm − for V <
500 cm − . For CN − /Ar an overall RMSE of 21cm − was obtained, a value which went down to 1.5 cm − for V <
500 cm − .The scattering calculations described in the next section re-quire the interaction potential to be averaged over the vibra-tional states of CN − χ ν ( r ) , which were obtained from LEVELas described in Section II, as V ν , ν ′ ( R , θ ) = h χ ν ( r ) | V int ( R , r , θ ) | χ ν ′ ( r ) i . (3)Fig. 2 shows the diagonal terms V , ( R , θ ) for both sys-tems. As expected for a molecule with a strong bond, so thatthe ground state vibrational wavefunction is strongly peakedaround r eq , the contour plots of the V , ( R , θ ) for each systemare very similar to our earlier rigid-rotor (RR) PESs whichwere obtained without the vibrational averaging. Bothsystem’s PES have a fairly similar appearance with the mostattractive part of the potential located on the nitrogen end ofCN − . The well depth is the main difference which increasesas expected from He to Ar due to the increasing number ofelectrons on the atoms and on the much larger dipole polar-izabiliy that dominates the long-range attractive terms with avalue of 1.383 a for He and 11.070 a for Ar. The off diagonal V , ( R , θ ) terms which directly drive vi-brationally inelastic ν = ν = − with Ar is more repulsive atclose range and for a broader range of geometries than is thecase for He. These findings suggest already that low-energycollisions with Ar will be likely to induce larger vibrationalcross sections than for the same collisions involving He atoms.Such expected behaviour will be in fact confirmed below byour actual calculations.The PESs for CN − /He and CN − /Ar can be compared tosimilar systems such as C − /He and C − /Ar which we have re-cently investigated. The location of the minimum interactionenergy for both anions interacting with He and Ar respectivelyare very similar with the main different being the perpendicu-lar angle of the well for C − . The off-diagonal matrix elementsfor these systems are also similar in magnitude and range butbeing slightly larger for the interaction of He and Ar with C − ,explaining the larger quenching rates for this anion (see be-low). The PES for the corresponding neutral systems CN/Heand CN/Ar which were reported by Saidani et al. can also becompared. In this case the well depth for He interacting withboth CN and CN − is similar but for Ar the interaction with theanion is somewhat weaker. As expected the interaction poten-tial for He and Ar interacting with the anion extends furtherthan the corresponding neutral systems. The off-diagonal ele-ments for the neutral and anionic systems are broadly similar.The close-coupling (CC) scattering calculations to be dis-cussed in the next section require to have the vibrationally av-eraged matrix elements in the form of the familiar multipole −40−30−20−10 0 10 20 30 3 4 5 6 7 8 9 10 11 12 CN − −He V l ( R ) / c m − R / Å V V V −50 0 50 100 150 200 3 4 5 6 7 8 9 10 11 12 CN − −ArR / Å V V V −400−300−200−100 3 4 5 6 FIG. 3. V λ , ( R ) expansion coefficients for λ = 0, 1 and 2 terms forCN − /He (left) and CN − /Ar (right). The rigid rotor (RR) values arealso plotted as dashed lines but essentially overlap the vibrationallyaveraged coefficients discussed in the present work. expansion given as V ν , ν ′ ( R , θ ) = λ max ∑ λ V λν , ν ′ ( R ) P λ ( cos θ ) . (4)Fig. 3 shows the multipole expansion coefficients for the firstthree V , ( R , θ ) terms for both systems. As anticipated fromthe broad spatial similarity of the contour plots, the multipoleexpansion for the vibrationally averaged matrix elements arevery close to those obtained from considering the anion as arigid rotor. This justifies our previous treatment of purely ro-tationally inelastic transitions where we considered the anionto behave as a rigid rotor (RR) and we refer the reader tothese works for a discussion of pure rotational transitions.It is also worthy of note about the diagonal coupling matrixelements reported in that Fig. 3 how the much more polar-izable Ar projectile gives the three lowest multipolar termsas attractive contributions to the interaction, thereby indicat-ing that their collective effects during the interaction wouldbe to draw the heavier partner closer to the anion. On theother hand, the same three coefficients for the lighter He part-ner (left-hand panel in Fig. 3) exhibit much shallower attrac-tive wells and only for two of the coefficients, with the λ =1 coefficient showing instead a slightly repulsive behaviour atintermediate distances.The off-diagonal expansion coefficients V λ , are shown inFig. 4. All terms quickly approach zero as R is increased.For both systems the V λ , ( R ) coefficients are mostly steeplyrepulsive as R decreases. As expected from the contour plots,the V λ , ( R ) terms are seen to be much more repulsive for theCN − /Ar interaction, with their turning points located at largerdistances than happens for the He partner. Such features ofthe interactions again suggest a larger dynamical vibrationalinelasticity for the case of Ar atoms than for the He collisionpartners. −20 0 20 40 60 80 100 2.5 3 3.5 4 4.5 5 5.5 6 CN − −He V l ( R ) / c m − R / Å V V V −20 0 20 40 60 80 100 2.5 3 3.5 4 4.5 5 5.5 6 CN − −ArR / Å V V V FIG. 4. V λ , ( R ) expansion coefficients for λ = 0, 1 and 2 terms forCN − /He (left) and CN − /Ar (right). IV. QUANTUM SCATTERING CALCULATIONS
Quantum scattering calculations were carried out using thecoupled channel (CC) method to solve the Schrödinger equa-tion for scattering of an atom with a diatomic molecule asimplemented in our in-house code, ASPIN. The method hasbeen described in detail many times before, from one of itsearliest, now classic formulations to one of its more recent,computation-oriented visitation from our own work . There-fore, only a brief summary of the method will be given herewith all equations given in atomic units. By starting withthe form employed for any given total angular momentum J = l + j the scattering wavefunction is expanded as Ψ JM ( R , r , Θ ) = R ∑ ν , j , l f J ν l j ( R ) χ ν , j ( r ) Y JMjl ( ˆ R , ˆ r ) , (5)where l and j are the orbital and rotational angular momen-tum respectively, Y JMjl ( ˆ R , ˆ r ) are coupled-spherical harmonicsfor l and j which are eigenfunctions of J . χ ν , j ( r ) are the ra-dial part of the ro-vibrational eigenfunctions of the molecule.The values of l and j are constrained, via Clebsch-Gordan co-efficients, such that their final summation is compatible withthe specific total angular momentum J one is considering. f J ν l j ( R ) are the radial expansion functions which need to bedetermined from the propagation of the radial coupled equa-tions.Substituting the expansion into the Schrödinger equationwith the Hamiltonian for atom-diatom scattering as defined indetail in , leads to the CC equations for each contributing J (cid:18) d dR + K − V − l R (cid:19) f J = . (6)Here each element of K = δ i , j µ ( E − ε i ) (where ε i is thechannel asymptotic energy), µ is the reduced mass of thesystem, V = µ U is the interaction potential matrix betweenchannels and l is the matrix of orbital angular momentum. For the ro-vibrational scattering calculations of interest in thepresent study,the matrix elements U are given explicitly as h ν jlJ | V | ν ′ j ′ l ′ J i = Z ∞ d r Z dˆ r Z d ˆ R χ ν , j ( r ) Y JMjl ( ˆ R , ˆ r ) ∗ | V ( R , r , θ ) | χ ν ′ , j ′ ( r ) Y JMj ′ l ′ ( ˆ R , ˆ r ) . (7)Since the intermolecular potential V ( R , r , θ ) is expressed as inEq. 4, then Eq. 7 can be written as h ν jlJ | V | ν ′ j ′ l ′ J i = ∞ ∑ λ = V λν , ν ′ ( R ) f J λ jl j ′ l ′ , (8)where the f J λ jl j ′ l ′ terms are the Percival-Seaton coefficients f J λ jl j ′ l ′ = Z dˆ r Z d ˆ R Y JMjl ( ˆ R , ˆ r ) ∗ P λ ( cos θ ) Y JMj ′ l ′ ( ˆ R , ˆ r ) , (9)for which analytical forms are known. Eq. 8 also makes useof the widely known approximation V λν , ν ′ ( R ) ≈ V λν j ν ′ j ′ ( R ) , (10)for all j such that the effect of rotation on the vibrational ma-trix elements is ignored for reasons that shall be further dis-cussed below.The CC equations are propagated outwards from the classi-cally forbidden region to a sufficient distance where the scat-tering matrix S can be obtained. The inelastic ro-vibrationalstate-changing cross sections are obtained as σ ν j → ν ′ j ′ = π ( j + ) k ν j ∑ J ( J + ) ∑ l , l ′ | δ ν l j , ν ′ l ′ j ′ − S J ν l j , ν ′ l ′ j ′ | . (11)To converge the CC equations, a rotational basis set wasalso used: for both systems it included up to j =
20 rotationalfunctions for each vibrational state. The CC equations werepropagated between 1.7 and 100.0 Å using the log-derivativepropagator up to 60 Å and the variable-phase method atlarger distances. The potential energy was interpolated be-tween calculated V λν , ν ′ ( R ) values using a cubic spline. For R < . V λν , ν ′ ( R ) were extrapolated as a λ R + b λ R whilefor R >
20 Å the λ = cR + dR . Asour ab initio calculated interaction energies were computed to R =
25 Å where the interaction energy is negligible for thetemperatures of interest here, the extrapolated form has also anegligible effect on cross sections. A number of parameters of the calculation were checked forconvergence. The scattering cross sections differed by around10-15% on going from 10 to 19 λ terms. This is less pre-cise than for rotationally inelastic cross sections where con-vergence to around 1% is typical and is due to the very smallcross sections for these processes which makes obtaining pre-cise and stable values more difficult to achieve. For produc-tion calculations, 10 λ terms were included for each V ν , ν ′ ( R ) as a compromise between accuracy and computational time.The effect of the vibrational basis set was also considered.It was found that for the ν = ν = −10 −9 −8 −7 −6 −5 −4 −3 −2 C r o ss S e c t i on / Å Collision Energy / cm −1 CN − −He n fi n n fi n n fi n CN − −Ar n fi n FIG. 5. Scattering cross sections for vibrationally inelastic collisionsof CN − with He and Ar. to only include these states. Including the ν = ν = ν = j =
20 basis gave convergence to betterthan 1% for the CN − /He while for CN − /Ar convergence toabout 10% was achieved.Scattering calculations were carried out for collision ener-gies between 1 and 1000 cm − using steps of 0.1 cm − for en-ergies up to 100 cm − , 0.2 cm − for 100-300 cm − , 1.0 cm − for 300-500 cm − and 10.0 cm − for 500-1000 cm − . Thisenergy grid was used to ensure that important features such asresonances appearing in the cross sections were accounted forand their contributions included when the corresponding rateswere calculated. At low collision energies, the positions andwidths of such resonances will be very sensitive to the detailsof the PES.For CN − /Ar the number of partial waves was increasedwith increasing energy as usual, requiring J =
120 for thehighest energies considered. For the CN − /He system how-ever, inverse behaviour was encountered: at low scatteringenergies below 100 cm − , more partial was were required (upto J =
80) to converge the vibrationally inelastic partial crosssections than at higher energies where only up to J =
35 wasrequired. We suspect this is due to the very small cross sec-tions so that at low energies it becomes difficult to convergethe calculations as all partial waves contribute uniformly verysmall values so that many more of them need inclusion for anacceptable convergent behaviour to occur.Vibrationally inelastic cross sections were computed for the ν = ν = − for collisions with He. Dueto time and memory constraints, only ν = − were considered for Ar collisions. We think, however, thatsuch calculations are already sufficient for our results to makeconvincingly our main points, as discussed further below. V. VIBRATIONALLY INELASTIC CROSS SECTIONS &RATE COEFFICIENTS
Fig. 5 compares vibrationally inelastic rotationally elastic(for j = j ′ =
0) cross sections for the de-excitation ν = → ν = ν = → ν = ν = → ν = − colliding with He and ν = → ν = − /Ar. At lowcollisions energies below 100 cm − the cross sections for Heare very small, orders of magnitude less than rotationally in-elastic collisions for this system. The cross sections showresonances at lower collision energies due to shape and/orFeshbach resonances. As expected due to the larger energydifference, the ν = → ν = ν = → ν = ν = → ν = − , the cross sections rapidly increasein value, a behaviour typically observed also in other systemsfor vibrationally inelastic cross sections. The CN − /Ar cross sections are found to be about four or-ders of magnitude larger than those we have obtained for Heat lower energies, also showing many distinct resonance fea-tures which are brought about by the presence of a strongerinteraction with the molecular anion. The detailed analysis ofsuch a forest of resonances would also be interesting and per-haps would be warranted in the case of existing experimentaldata on such processes, of which we are not aware till now,but would require a substantial extension of the present work.Thus, we do not intend to carry it out now, being somewhatoutside the main scope of the present study, and are leaving itfor future extension of this study in our own laboratory.The far larger cross sections we found for the Ar projec-tile are obviously a consequence of the deeper attractive wellfor the V ν , ν ( R , θ ) diagonal matrix elements and the larger off-diagonal V ν , ν ′ ( R , θ ) matrix elements (see Fig. 2), i.e. theystem from distinct differences in the strengths of the couplingpotential terms that drive the inelastic dynamics for the Arcollision partner.The general features of the vibrationally inelastic cross sec-tions shown in Fig. 5 are indeed similar to those which wehave obtained earlier for the C − anion colliding with He, Neand Ar set of systems that we have recently studied. Forboth of the anions, we have found that the vibrational quench-ing cross sections with He are uniformly very small, while wealso found that they increase by orders of magnitude when thelarger and more polarizable Ar atom becomes the collisionalpartner for either of these anionic molecules. Although suchgeneral behaviour could be reasonably expected from what weknow in these systems about their interaction forces, it is nev-ertheless reassuring to obtain quantitative confirmation on theextent of the size differences from detailed, and in principleexact, scattering calculations.The computed inelastic cross sections of the previous sec-tion can in turn be used to obtain the corresponding thermalrate constants over ranges of temperature of interest for plac-ing the present anion in cold environments. The correspond-ing k ν → ν ′ ( T ) can be evaluated, in fact, as the convolution ofthe computed inelastic cross sections over a Boltzmann dis-tribution of the relative collision energies of the interactingpartners as −20 −19 −18 −17 −16 −15 −14 −13 −12
10 20 30 40 50 60 70 80 90 100 n = 1 fi n = 0 k / c m m o l e c u l e − s − Temperature / K CN − −HeCN − −ArC −HeC −Ar 10 −20 −19 −18 −17 −16 −15 −14 −13 −12
10 20 30 40 50 60 70 80 90 100 n = 2 fi n = 1 Temperature / K −20 −19 −18 −17 −16 −15 −14 −13 −12
10 20 30 40 50 60 70 80 90 100 n = 2 fi n = 0 Temperature / K
FIG. 6. Rate constants k ν → ν ′ ( T ) for vibrationally inelastic transitions in CN − /He and Ar collisions. Also shown are the corresponding valuesfor C − /He and Ar. k ν → ν ′ ( T ) = (cid:18) πµ k B T (cid:19) / Z ∞ E c σ ν → ν ′ ( E c ) e − E c / k B T dE c (12)where E c = µ v / j = j ′ =
0) transitions correspond-ing to the cross sections in Fig. 5. The figure also showsrates for the corresponding transitions of the similar C − /Heand Ar systems. For CN − /He the rate constants for vibrationalquenching are very small, even lower than those for C − /Heand around nine orders of magnitude lower than those forCN − /He rotationally inelastic collisions. For CN − /Ar the ν = → ν = − /Ar. The ν = → ν = − /He is broadly similar tothose for ν = → ν = ν = → ν = − and C − and as shown in Fig. 6,vibrationally inelastic collisions for C − . This trend, whileseemingly in expectation with the stronger interaction poten-tial for the larger atom is not easy to predict a priori . Kato etal. and Ferguson measured vibrational quenching rates for N + in collisions with He, Ne, Ar, Xe and Kr and O + with He,Ne and Ar atoms respectively at 300 K. For both cations,quenching rates increased with the size of atom, suggestingthat the polarizability of the colliding atom plays an impor-tant role. In contrast Saidani et al calculated quenching ratesfor CN with He and Ar over a wide range of temperatures and found that cross sections and rates for Ar were ordersof magnitude lower than those for He. However, ionic in-teractions are driven by different forces than those acting be-tween neutrals, so it is not obvious how such a result relatesto the present findings for an anion. Analytical models canalso be used to gain insight into vibrational quenching such asthe work of Dashevskaya et al. where the quenching rate for ν = → ν = /He was calculated over a large range oftemperatures from 70-3000 K. The rates obtained were ingood agreement with experiment and similar to those foundhere for CN − /He at 100 K. It would be interesting to applythese models to the anion-neutral collisions of interest here.The work we have presented here, and the similar findingsfrom our previous study on a different diatomic anion likeC − , strongly suggests that the process of vibrational inelas-ticity in multiply bonded anionic molecules by low-T colli-sions with neutral noble gases is rather inefficient. The ratesfor quenching found here are even smaller than those we hadfound earlier for C − , also uniformly smaller than those knownfor many neutral diatomic molecules and cations .The quenching rates and Einstein A coefficients which wehave mentioned and shown earlier in this work, can be used toconsider the properties of the critical density n i crit ( T ) for CN − vibrations which is given as n i crit ( T ) = A i j ∑ j = i k i j ( T ) . (13)This quantity gives the gas density values which would berequired so that collisional state-changing processes matchin size those which lead to collision-less emission via spon-taneous decay. It is used in astronomical contexts to assesthe possible densities required for local thermal equilibrium(LTE) to be reached and are usually applied for rotational tran-sitions in molecules that can occur in the interstellar medium(ISM).In the present case of CN − /He, when we apply Eq. 13 tothe ν = ν = n i crit ( T ) ≈ − cm − at 100 K. Current kinetics mod-els which describe the density conditions in molecular cloudsindicate a wide variety of densities being present: from dif-fuse molecular clouds estimated at around 10 cm − to densemolecular clouds which are considered to be between 10 -10 cm − . The critical density obtained here for the vi-brational decay of CN − interacting with environmental Heatoms was found to be orders of magnitude larger than thoseexpected in the ISM regions where CN − has been detected,clearly suggesting that thermal equilibrium for these processwill likely never be attained and that the presently computedradiative transitions determine that CN − populates essentiallyonly the ground vibrational level in the ISM. VI. CONCLUSIONS
The cross sections and corresponding rate constants for vi-brationally inelastic transitions of CN − colliding with He andAr atoms have been calculated using new ab initio potentialenergy surfaces. As for atom-diatom vibrationally inelasticcollisions, the rate constants for both CN − /He and CN − /Arare very small, even smaller than those for corresponding val-ues of the similar C − /He and C − /Ar systems. Although morework is required before definitive conclusions can be drawn,it appears from the present calculations that vibrationally in-elastic collisions of molecular anions with neutral atoms (orat least noble gas atoms) are similar to neutral molecule-atomcollisions in that they generate similarly small transition prob-abilities and their collision mechanisms for transferring rel-ative energy, at sub-thermal and thermal conditions, to thevibrational internal motion of the anion is rather inefficient.This is in contrast with the generally more efficient collisionalenergy transfer probabilities which are found for molecularcation-atom systems in the current literature .For the anion of interest here this is not a crucial concernwhen wanting to find alternative paths which are more effi-cient in cooling its internal vibrational motion, since CN − candissipate energy through spontaneous dipole emission (Sec-tion II). On the other hand, in the case of homonuclear anionssuch as C − (of current interest for laser cooling cycles in coldtraps ) where this process is forbidden, collisions are likelyto be the primary means for quenching its vibrational motion.In such cases high gas pressures and the use of larger noblebuffer gases seem to be required.The present calculations confirm that collisional energytransfer paths which involve vibrational degrees of freedomfor a molecular anion under cold trap conditions are invariablyvery inefficient and are several orders of magnitude smallerthat the collisional energy-changing paths which involve theirrotational degrees of freedom. One can therefore safely esti-mate that these two paths to energy losses are markedly decou-pled with one another and can be treated on a separate footingwithin any kinetics modelling of their behaviour. ACKNOWLEDGMENTS
We acknowledge the financial support of the Austrian FWFagency through research grant n. P29558-N36. One of us(L.G-S) further thanks MINECO (Spain) for grant PGC2018-09644-B-100.
VII. DATA AVAILABILITY STATEMENT
Fortran programs and subroutines for the CN − /He andCN − /Ar PESs used are available in the Supplementary Ma-terial along with the vibrational coupling coefficients and vi-brational quenching rate constants. VIII. SUPPLEMENTARY MATERIAL
The multipolar coefficients for the Legendre expansion ofthe new vibrational PESs for CN − /He and CN − /Ar are pro-vided via Fortran program routines, as well as the couplingcoefficients for the vibrational dynamics. They are all areavailable as Supplementary Material to the present publica-tion. That Supplementary Material also contains subroutinesfor the inelastic and elastic rate coefficients for the two sys-tems studied in the present paper. K. Takayanagi, “Vibrational and rotational transitions in molecular colli-sions,” Prog. Theor. Phys. Supp. , 1–98 (1963). D. Secrest, “Theory of rotational and vibrational energy transfer inmolecules,” Annu. Rev. Phys. Chem. , 379–406 (1973). D. J. Krajnovich, C. S. Parmenter, and D. L. Catlett, “State-to-state vibrational transfer in atom-molecule collisions. beams vs. bulbs,”Chem. Rev. , 237–288 (1987). D. Secrest and B. Robert Johnson, “Exact quantum mechanical calcu-lation of a collinear collision of a particle with a harmonic oscillator,”J. Chem. Phys. , 4556 (1966). W. Eastes and D. Secrest, “Calculation of rotational and vibrational transi-tions for the collision of an atom with a rotating vibrating diatomic oscilla-tor,” J. Chem. Phys. , 640 (1972). W. C. Campbell, G. C. Groenenboom, H.-I. Lu, E. Tsikata, and J. M. Doyle,“Time-domain measurements of spontaneous vibrational decay of magnet-ically trapped NH,” Phys. Rev. Lett. , 083003 (2008). I. Kozyryev, L. Baum, K. Matsuda, P. Olson, B. Hemmerling, and J. M.Doyle, “Collisional relaxation of vibrational states of SrOH with He at 2K,” New J. Phys. , 045003 (2015). D. Caruso, M. Tacconi, F. A. Gianturco, and E. Yurtsever, “Quenchingvibrations by collisions in cold traps: A quantum study for MgH + ( X Σ + )with He ( S ),” J. Chem. Sci. , 93 (2012). W. Rellergent, S. S, S. Schowalter, S. Kotochigova, K. Chen, and E. R.Hudson, “Evidence for sympathetic vibrational cooling of translationallycold molecules,” Nature , 490 (2013). F. F. S. van der Tak, F. Lique, A. Faure, J. H. Black, and E. W. vanDishoeck, “The Leiden atomic and molecular database (LAMDA): Currentstatus, recent updates, and future plans,” Atoms , 2 (2020). C. Balança and F. Dayou, “Ro-vibrational excitation of SiO by collisionwith helium at high temperature,” MNRAS , 1673 (2017). R. Toboła, F. Lique, J. Kłos, and G. Chałasi´nski, “Ro-vibrational excitationof SiS by He,” J. Phys. B: At. Mol. Opt. Phys. , 155702 (2008). F. Lique and A. Spielfiedel, “Ro-vibrational excitation of CS by He,”Astron. Astrophys. , 1179 (2007). F. Lique, A. Spielfiedel, G. Dhont, and N. Feautrier, “Ro-vibrationalexcitation of the SO molecule by collision with the He atom,”Astron. Astrophys. , 331 (2006). T. Stoecklin, P. Halvick, M. A. Gannounim, M. Hochlaf, S. Kotochigova,and E. R. Hudson, “Explanation of efficient quenching of molecular ionvibrational motion by ultracold atoms,” Nat. Commun. , 11234 (2016). B. Yang, X. Wang, P. Stancil, J. Bowman, N. Balakrishnan, and R. Forrey,“Full-dimensional quantum dynamics of rovibrationally inelastic scatteringbetween CN and H ,” J.Chem.Phys. , 224307 (2016). Y. Kalugina, F. Lique, and S. Marinakis, “New ab initio potentialenergy surfaces for the ro-vibrational excitation of OH( X Π ) by He,”Phys. Chem. Chem. Phys. , 13500 (2014). I. Iskandarov, F. A. Gianturco, M. Hernández Vera, R. Wester,H. da Silva Jr., and O. Dulieu, “Shape and strength of dynamical cou-plings between vibrational levels of the H + , HD + and D + molecular ionsin collision with He as a buffer gas,” Eur. Phys. J. D , 141 (2017). T. Stoecklin and A. Voronin, “Vibrational and rotational cooling of NO + incollisions with He,” J. Chem. Phys. , 204312 (2011). T. Stoecklin and A. Voronin, “Vibrational and rotational energy transfer ofCH + in collisions with He and He,” Eur. Phys. J. D , 259 (2008). B. P. Mant, F. A. Gianturco, R. Wester, E. Yurtsever, and L. González-Sánchez, “Ro-vibrational quenching of C − anions in collisions with He,Ne and Ar atoms,” Phys. Rev. A , 062810 (2020). P. Yzombard, M. Hamamda, S. Gerber, M. Doser, and D. Comparat, “Lasercooling of molecular anions,” Phys. Rev. Lett. , 213001 (2015). S. E. Bradforth, E. H. Kim, D. W. Arnold, and D. M. Nue-mark, “Photoelectron spectroscopy of CN − , NCO − , and NCS − ,”J. Chem. Phys. , 800 (1993). D. Forney, W. E. Thomson, and E. Jacox, “The vibrational spectra ofmolecular ions isolated in solid neon. ix. HCN + , HNC + , and CN − ”J. Chem. Phys. , 1664 (1992). C. A. Gottlieb, S. Brunken, M. C. McCarthy, and P. Thaddeus, “The rota-tional spectrum of CN − ,” J. Chem. Phys. , 191101 (2007). P. Botschwina, “Spectroscopic properties of the cyanide ion calculated bySCEP CEPA,” Chem. Phys. Lett. , 58–62 (1985). K. A. Peterson and R. Claude Woods, “An ab initio investigation of thespectroscopic properties of BCl, CS, CCl + , BF, CO, CF + , N , CN − , andNO + ,” J. Chem. Phys. , 4409 (1987). L. T. J and C. E. Dateo, “Accurate spectroscopiccharacterization of C N − , C N − , C N − ,”Spectrochimica Acta Part A , 739 (1999). J. Berkowitz, W. A. Chupka, and T. A. Walter, “Photo ioniza-tion of HCN: The electron affinity and heat of formation of CN,”J. Chem. Phys. , 1497 (1969). R. Klein, R. P. McGinnis, and S. R. Leone, “Photodetach-ment threshold of CN − by laser optogalvank spectroscopy,”Chem. Phys. Lett. , 475 (1983). M. Simpson, M. Nötzold, A. Schmidt-May, T. Michaelsen, B. Bastian,J. Meyer, R. Wild, F. Gianturco, M. Milovanovi´c, V. Kokoouline, andR. Wester, “Threshold photodetachment spectroscopy of the astronmicalanion CN − ,” J.Chem.Phys. , 184309 (2020). Agúndez, M., Cernicharo, J., Guélin, M., Kahane, C., Roueff, E., Klos, J.,Aoiz, F. J., Lique, F., Marcelino, N., Goicoechea, J. R., González Gar-cía, M., Gottlieb, C. A., McCarthy, M. C., and Thaddeus, P., “Astro-nomical identification of CN − , the smallest observed molecular anion,”A&A , L2 (2010). L. González-Sánchez, B. P. Mant, R. Wester, and F. A. Gianturco, “Rota-tionally inelastic collisions of CN − with He: Computing cross sections andrates in the interstellar medium,” ApJ , 75 (2020). J. Kłos and F. Lique, “First rate coefficients for an interstellar anion: appli-cation to the CN − -H collisional system,” MNRAS , 271–275 (2011). L. González-Sánchez, E. Yurtsever, B. P. Mant, R. Wester, and F. A.Gianturco, “Collision-driven state-changing efficiency of different buffergases in cold traps: He( S ) Ar ( S ) and p-H ( Σ ) on trapped CN − ( Σ ),”Phys.Chem.Chem.Phys., Advance Article (2020), 10.1039/D0CP03440A. S. Petrie, “Novel pathways to CN − within interstellar clouds andcircumstellar envelopes: implications for is and cs chemistry,”Mon. Not. R. Astron. Soc. , 137–144 (1996). C. Romanzin, E. Louarn, J. Lemaire, J. Zabka, M. Polasek, J.-C. Guillemin,and C. Alcaraz, “An experimental study of the reactivity of CN − and C N − anions with cyanoacetylene (HC N),” Icarus , 242–252 (2016). S. Jerosimi´c, F. A. Gianturco, and R. Wester, “Associative detachment(AD) paths for H and CN − in the gas-phase: astrophysical implications,” Phys. Chem. Chem. Phys. , 5490 (2018). M. Satta, F. A. Gianturco, F. Carelli, and R. Wester, “A quantum study ofthe chemical formation of cyano anions in inner cores and diffuse regionsof interstellar molecular clouds,” ApJ , 228–235 (2015). L. Biennier, S. Carles, D. Cordier, J.-C. Guillemin, S. D. Le Pi-card, and A. Faure, “Low temperature reaction kinetics of CN − +HC N and implications for the growth of anions in titan’s atmosphere,”Icarus , 123–131 (2014). A. J. Coates, F. J. Crary, G. R. Lewis, D. T. Young, J. H. Waite Jr., andE. C. Sittler Jr., “Discovery of heavy negative ions in Titan’s ionosphere,”Geophys. Res. Lett. , L22103 (2007). V. Vuitton, P. Lavvasb, R. V. Yelle, M. Galand, A. Wellbrock, G. R. Lewis,A. J. Coates, and J. E. Wahlund, “Negative ion chemistry in Titan’s upperatmosphere,” Planet. Space Sci. , 1558–1572 (2009). A. McKellar, “Evidence for the molecular origin of some hitherto unidenti-fied interstellar lines,” PASP , 187 (1940). H. Burton, R. Mysliwiec, R. Forrey, B. Yang, P. Stancil, and N. Balakrish-nan, “Fine-structure resolved rotational transitions and database for CN+H collisions,” Mol. Astrophys. , 23–32 (2018). F. Lique, A. Spielfiedel, N. Feautrier, I. F. Schneider, J. Kłos, and M. H.Alexander, “Rotational excitation of CN( X Σ + ) by He: Theory and com-parison with experiments,” J. Chem. Phys. , 024303 (2010). F. Lique and J. Kłos, “Hyperfine excitation of CN by He,”MNRAS , L20–L23 (2011). Y. Kalugina, F. Lique, and J. Kłos, “Hyperfine collisional rate coefficientsof CN with H ( j = , 812 (2012). Y. Kalugina, J. Kłos, and F. Lique, “Collisional excitation ofCN( X Σ + ) by para- and ortho-H : Fine-structure resolved transitions,”J. Chem. Phys. , 074301 (2013). Y. Kalugina and F. Lique, “Hyperfine excitation of CN by para- and ortho-H ,” MNRAS , L21–L25 (2015). J. L. Doménech, O. Asvany, C. R. Markus, S. Schlemmer, and S. Thor-wirth, “High-resolution infrared action spectroscopy of the fundamental vi-brational band of CN + ,” J. Mol. Spec. , 111375 (2020). B. Anusuri, “Rotational excitation of cyanogen ion, CN + ( X Σ + ) by Hecollisions,” Comput. Theor. Chem. , 112748 (2020). H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, and M. Schütz,“MOLPRO: a general-purpose quantum chemistry program package,”WIREs Comput. Mol. Sci. , 242–253 (2012). H.-J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz, et al. C. Hampel, K. A. Peterson, and H.-J. Werner, “A comparison of the ef-ficiency and accuracy of the quadratic configuration interaction (QCISD),coupled cluster (CCSD), and brueckner coupled cluster (BCCD) methods,”Chem. Phys. Lett. , 1–12 (1992). M. J. O. Deega and P. J. Knowles, “Perturbative corrections to accountfor triple excitations in closed and open shell coupled cluster theories,”Chem. Phys. Lett. , 321–326 (1994). D. E. Woon and T. H. Dunning Jr, “Gaussian basis sets for use in cor-related molecular calculations. iii. the atoms aluminum through argon,”J. Chem. Phys. , 1358 (1993). D. E. Woon and T. H. Dunning Jr, “Gaussian basis sets for use in correlatedmolecular calculations. iv. calculation of static electrical response proper-ties,” J. Chem. Phys. , 2975 (1994). R. J. Le Roy, “LEVEL: A computer program for solving the ra-dial Schrödinger equation for bound and quasibound levels,”J. Quant. Spectrosc. Radiat. Transf. , 167 (2017). J. S. A. Brooke, R. S. Ram, C. M. Western, G. Li, D. W. Schwenke, andP. F. Bernath, “Einstein a coefficients and oscillator strengths for the A Π − X Σ + (red) and B Σ + − x Σ + (violet) systems and rovibrational transitionsin the X Σ + state of CN,” ApJS , 23 (2014). H. J. Werner and P. J. Knowles, “A second order multiconfiguration SCFprocedure with optimum convergence,” J. Chem. Phys. , 5053 (1985). P. J. Knowles and H. J. Werner, “An efficient second-order MC SCF methodfor long configuration expansions,” Chem. Phys. Lett. , 259 (1985). K. R. Shamasundar, G. Knizia, and H.-J. Werner, “A new in-ternally contracted multi-reference configuration interaction method,”J. Chem. Phys. , 053101 (2011). R. A. Kendall, T. H. Dunning Jr, and R. J. Harrison, “Electron affinitiesof the first-row atoms revisited. systematic basis sets and wave functions,”J. Chem. Phys. , 6796 (1992). A. K. Wilson and T. H. van Mourik, T amd Dunning, “Gaus-sian basis sets for use in correlated molecular calculations. vi. sex-tuple zeta correlation consistent basis sets for boron through neon,”Theochem , 339–349 (1996). S. F. Boys and F. Bernardi, “Calculation of small molecular interactionsby differences of separate total energies - some procedures with reducederrors,” Mol. Phys. , 553 (1970). H.-J. Werner, B. Follmeg, and M. Alexander, “Adiabatic and diabaticpotential energy surfaces for collisions of CN ( X Σ + , A Π / ) with He,”J. Chem. Phys. , 3139 (1988). C. Gaiser and B. Fellmuth, “Polarizability of helium, neon, and argon: Newperspectives for gas metrology,” Phys. Rev. Lett. , 123203 (2018). G. Saidani, Y. Kalugina, A. Gardez, L. Biennier, R. Georges, andF. Lique, “High temperature rection kinetics of CN( ν =
0) with C H and C H and vibrational relaxation of CN( ν =
1) with Ar and He,”J. Chem. Phys. , 124308 (2013). D. López-Duránn, E. Bodo, and F. A. Gianturco, “ASPIN: An all spinscattering code for atom-molecule rovibrationally inelastic cross sections,”Comput. Phys. Commun. , 821 (2008). A. M. Arthurs and A. Dalgarno, “The theory of scattering by a rigid rotator,”Proc. R. Soc. A , 540 (1960). D. E. Manolopoulos, “An improved log derivative method for inelastic scat-tering,” J. Chem. Phys. , 6425 (1986). R. Martinazzo, E. Bodo, and F. A. Gianturco, “A modified variable-phase algorithm for multichannel scattering with long-range potentials,”Comput. Phys. Commun. , 187 (2003). B. P. Mant, F. A. Gianturco, L. González-Sánchez, E. Yurt-sever, and R. Wester, “Rotationally inelastic processes of C − ( Σ + g ) colliding with He ( S ) at low-temperatures: Ab Initio in-teraction potential, state-changing rates and kinetic modelling,”J. Phys. B: At. Mol. Opt. Phys. , 025201 (2020). B. P. Mant, F. A. Gianturco, R. Wester, E. Yurtsever, and L. González-Sánchez, “Thermalization of C − with noble gases in cold ion traps,”J. Int. Mass Spectrom. , 116426 (2020). S. Kato, V. M. Bierbaum, and S. R. Leone, “Laser fluorescence and massspectroscopic measurements of vibrational relaxation of N + ( ν ) with He,Ne, Ar, Kr and Xe,” Int. J. Mass Spec. Ion Proc. , 469 (1995). E. E. Ferguson, “Vibrational quenching of small molecular ions in neutralcolliisons,” J. Phys. Chem. , 731 (1986). E. I. Dashevskaya, I. Litvin, E. E. Nikitin, and J. Troe, “Semiclassi-cal extension of the Landau-Teller theory of collisional energy transfer,”J. Chem. Phys. , 154315 (2006). T. P. Snow and M. B.J., “Diffuse atomic and molecular clouds,”Annu.Rev. Astronom. Astrophys. , 367–414 (2006). M. Agúndez and J. Cernicharo, “Oxygen chemistry in the CSE of thecarbon-rich star irc+10216,” ApJ , 374–393 (2006). r X i v : . [ phy s i c s . a t o m - ph ] J a n Rotational state-changing collisions of C H − and C N −−
1) with Ar and He,”J. Chem. Phys. , 124308 (2013). D. López-Duránn, E. Bodo, and F. A. Gianturco, “ASPIN: An all spinscattering code for atom-molecule rovibrationally inelastic cross sections,”Comput. Phys. Commun. , 821 (2008). A. M. Arthurs and A. Dalgarno, “The theory of scattering by a rigid rotator,”Proc. R. Soc. A , 540 (1960). D. E. Manolopoulos, “An improved log derivative method for inelastic scat-tering,” J. Chem. Phys. , 6425 (1986). R. Martinazzo, E. Bodo, and F. A. Gianturco, “A modified variable-phase algorithm for multichannel scattering with long-range potentials,”Comput. Phys. Commun. , 187 (2003). B. P. Mant, F. A. Gianturco, L. González-Sánchez, E. Yurt-sever, and R. Wester, “Rotationally inelastic processes of C − ( Σ + g ) colliding with He ( S ) at low-temperatures: Ab Initio in-teraction potential, state-changing rates and kinetic modelling,”J. Phys. B: At. Mol. Opt. Phys. , 025201 (2020). B. P. Mant, F. A. Gianturco, R. Wester, E. Yurtsever, and L. González-Sánchez, “Thermalization of C − with noble gases in cold ion traps,”J. Int. Mass Spectrom. , 116426 (2020). S. Kato, V. M. Bierbaum, and S. R. Leone, “Laser fluorescence and massspectroscopic measurements of vibrational relaxation of N + ( ν ) with He,Ne, Ar, Kr and Xe,” Int. J. Mass Spec. Ion Proc. , 469 (1995). E. E. Ferguson, “Vibrational quenching of small molecular ions in neutralcolliisons,” J. Phys. Chem. , 731 (1986). E. I. Dashevskaya, I. Litvin, E. E. Nikitin, and J. Troe, “Semiclassi-cal extension of the Landau-Teller theory of collisional energy transfer,”J. Chem. Phys. , 154315 (2006). T. P. Snow and M. B.J., “Diffuse atomic and molecular clouds,”Annu.Rev. Astronom. Astrophys. , 367–414 (2006). M. Agúndez and J. Cernicharo, “Oxygen chemistry in the CSE of thecarbon-rich star irc+10216,” ApJ , 374–393 (2006). r X i v : . [ phy s i c s . a t o m - ph ] J a n Rotational state-changing collisions of C H − and C N −− anions with He Rotational state-changing collisions of C H − and C N − anions with Heunder interstellar and cold ion trap conditions: a computationalcomparison. Jan Franz, Barry Mant, Lola González-Sánchez, Roland Wester, and Franco A. Gianturco a) Department of Theoretical Physics and Quantum Informatics, Faculty of Applied Physics and Mathematics,Gda´nsk University of Technology, ul. Narutowicza 11/12, 80-233 Gda´nsk, Poland Institute for Ion Physics and Applied Physics, University of Innsbruck, Technikerstr. 25/3, 6020 Innsbruck,Austria Departamento de Química Física, University of Salamanca, Plaza de los Caídos sn, 37008 Salamanca,Spain (Dated: 26 February 2020)
We present an extensive range of quantum calculations for the state-changing rotational dynamics involving two sim-ple molecular anions which are often included within evolutionary analysis of chemical networks in the Interstellarenvironments, C H − ( X Σ + ) and C N − ( X Σ − ). The same systems are also of direct interest in modelling selectivephoto-detachment (PD) experiments in cold ion traps where the He atoms function as the chief buffer gas at the lowtrap temperatures. This study employs accurate, ab initio calculations of the interaction potential energy surfaces (PESs)for these anions, treated as Rigid Rotors(RR) and the He atom to obtain a wide range of state-changing quantum crosssections and rates at temperatures up to about 100K.The results are analysed within both physical contexts to showdifferences and similarities between the state-changing dynamics of the two systems. I. INTRODUCTION
The discovery of carbon chain anions in interstellar and cir-cumstellar media has triggered and stimulated a large numberof theoretical and experimental studies on these species (e.g.see reference [1,2]). Their structures and spectral features, aswell as the clarification of their importance and of their role ininterstellar chemistry, and in gas phase ion-molecule reactionsin general, have therefore also attracted many specific studies[3-5] on their behaviour.The possible, and likely, existence of anions in astrophys-ical sources was first predicted theoretically and consideredin earlier chemical models [6,7], although the first negativehydrocarbon C H − was only detected in 2006 [8], therebyalso solving the problem of the unidentified lines discoveredby Kawaguchi et al. [9]. That identification was soon fol-lowed by the detection of other negatively charged species likeC H − [10], C H − [11,12], , C N − [13], C N − [4], and CN − [14]. The majority of these species were first detected ina well observed Circumstellar Envelope (CSE) IRC+10216,although these and other hydrocarbon anions were also dis-covered later on in other molecular clouds [15]. As it is tobe expected, the study and search for interstellar anions ofboth simple and increasingly more complex structural proper-ties is still current and relevant. More specifically, a simplespecies like C H − ( X Σ − ), has been expected to be amenableto observation with the new astrophysical instruments such asALMA, especially since it had been already observed as a sta-ble molecule in laboratory experiments [16,17]. Furthermore,its parent neutral form C H ( X Σ + ) has already been a well-known astrophysical molecule discovered by observation asearly as 1974 [18], thereby suggesting that the correspond- a) Electronic mail: [email protected] ing C H − anion should also be present, although perhaps asonly a very low-abundancy species, a fact justified in terms ofits high chemical reactivity and therefore expected rapid de-struction upon formation. Actual current numerical modelspredict, in fact, that the larger carbon chain anion formationsshould be in any case more probable than those for similarsmaller chains as the C H − [19], hence somehow supportingthe difficulties for its observation.In spite of the fact that the actual origin of many of thehydrocarbon anions has not yet been solved, it is generally ac-cepted that gas-phase processes are crucial for their formationand therefore the many observed hydrocarbon radicals C n Hmay also be the main precursors of the formation of C n H − anions through electron attachment or association processes,whereas associative detachment processes would contribute tothe generation of initial, neutral C n H species. As an example,a new mechanism for the formation of C H − from C H hasbeen proposed not long ago [1] from laboratory experiments.Along similar lines, another small C-bearing molecule, theneutral CCN radical ( X Π r ) has been also detected earlieron in the interstellar medium [20], where the molecule wasobserved at the 1 – 2 mK level toward the same circum-stellar envelope (CSE) of IRC + 10216 already mentionedearlier, using the facilities of the Arizona Radio Observa-tory (ARO). Lambda doublets of the J = . → . J = . → . Ω = / J = . → . N] abundance ratio was foundto be ∼
500 : 1 : 50, thus indicating again one could expecta rather low abundance for the neutral radical CCN. For thecorresponding anionic counterpart, however, no observationalevidence within the same CSE has been reported thus far, alsosuggesting a low abundance of the latter once formed withinotational state-changing collisions of C H − and C N −−
500 : 1 : 50, thus indicating again one could expecta rather low abundance for the neutral radical CCN. For thecorresponding anionic counterpart, however, no observationalevidence within the same CSE has been reported thus far, alsosuggesting a low abundance of the latter once formed withinotational state-changing collisions of C H − and C N −− anions with He 2that chemical network.In spite of this absence of direct detection, both the aboveanionic molecules, CCN − and CCH − , have been the objectof several laboratory studies which have analysed in some de-tail the photo-detachment (PD) mechanisms of these speciesand several other structural properties [21-26]. Furthermore,in general terms we should note here that possible astrophysi-cal abundances of either observed or not yet observed specieshave to be understood in terms of molecular stabilities, reac-tion probabilities and of both radiative and collisional excita-tions and relaxation of internal molecular modes: the accu-rate knowledge of all these facts can indeed help us to betterexplain the existence of a molecule and the probability of itbeing observed. The molecular stability and the spectroscopicproperties of both the C H − and C N − anions been studied invarious earlier investigations [22,23,26], while the modellingof molecular emission in the ISM environments where theycould be expected to exist requires collisional rate coefficientswith the most abundant interstellar species like He and H .Collisional data have been already presented for the C H − anion interacting with He [27], while no corresponding cal-culations, as far as we know, exist for the collisional rotation-state changes of C N − interacting with He. The actual abinitio PES is also not known for that system. The quantumdynamics of both these anions, besides not being extensivelystudied under CSE conditions, has also been only partially dis-cussed under the operating low temperatures of ion cold traps[28]. The present work is therefore directed to acquiring novelknowledge about the quantum dynamics of C N − in collisionwith He under both astrochemical and cold ion trap condi-tions, further implementing a comparison between its dynam-ical behaviour and that of the C H − polar anion under similarconditions.The following Section will only briefly remind readersabout the features of the anisotropic potential energy surfaceinvolving C H − and He atoms, since it has been presented andanalysed already in earlier work [27]. It will instead presentin greater details the new calculations of the ab initio pointsrelated to the Rigid Rotor (RR) PES associated to the interac-tion between C N − ( X Σ − ) and the neutral He atom. The twointeraction potentials will then be compared and their level ofspatial anisotropy will be analysed and discussed.The next Section 3 will present the state-changing rotation-ally inelastic cross sections for both systems and discuss theeffects from spin-spin and spin-rotation structural effects onthe C N − system via à vis the C H − ( X Σ + ) system. The cor-responding inelastic rates at the temperatures of interest willalso be presented, compared and discussed. The followingSection 4 shall examine the possible evolutionary dynamicsof C N − anions under the conditions of a cold trap when theyare undergoing laser photo-detachment processes. A compar-ison with the corresponding behaviour of the C H − systemwill also be presented and discussed. Our present conclusionswill be given in Section 5. II. FEATURES OF THE AB INITIO INTERACTIONS OFTHE PRESENT ANIONS WITH HE ATOMS
As mentioned in the previous Section, the electronic groundstate of C N − was taken to be ( X Σ − ) as indicated to be in ear-lier work [22]. The molecular geometry of this linear anion iswith r CC = .
344 Å and 1.207 Å and was given in the exper-iments from Garand et al. in ref. [22]. We have optimizedthe molecular geometry using the MRCI + Davidson correc-tion [29] and employing the aug-cc-pVQZ basis set. The finalvalues of the same bond distances have been the following: r CC = .
360 Å and r CN = .
212 Å . The ab initio calculationsof the 2D grid of points in the space (R, theta) were carried outwith the program package MOLPRO 2012 of ref. [29]. Forall the present calculations we have employed the internally-contracted multi-reference configuration interaction method(IC-MRCI) [30,31], using the aug-cc-pVQZ basis set [32] onall atoms. The reference space for the MRCI calculations con-sists of a complete - active space by distributing 14 electronsin 15 orbitals. All single and double excitations from the ref-erence configurations are included in the variational calcula-tion. The effect of quadruple excitations is estimated via theDavidson correction [29]. The angular grid involved calcu-lations of radial ` cut ` every 5°, from 0°to 180°. The radialgrid included a higher density of points around the variousminima regions at each selected angle. The total number ofradial points was: 44. The global minimum for the complexwas found at a distance of around 6.7 Bohr from the centre-of-mass, located at an angle of around 80°, with a well´ s depthof around 58 cm − . The data shown by the two panels ofFigure 1 provide a pictorial view of the new PES calculatedfor the C N − /He system (upper panel), while it also showsfor comparison the PES already calculated in earlier work forthe similar anion of C − interacting with He (from ref. [27]).One clearly sees from the two panels the expected similaritiesbetween the two interaction potentials since both exhibit thepresence of the minimum structure of their complex with Heto be slightly off the C geometry and at similar values of dis-tances from the c.-of-m. in each complex. Furthermore, giventhe larger number of electrons in the C N − case, the corre-sponding well appears to be deeper than that for the C H − /Hecomplex [27]. Both potentials will be asymptotically drivenby the polarizability term involving the spherical, dipole po-larizability of the He partner and therefore will behave verysimilarly in their long-range regions because of it. The lo-calization of the excess charge provided by the extra boundelectron of the anion is also an interesting item for the newCCN − /He PES discussed here. The Mulliken charges in theasymptotic situation (e.g. with the helium at a distance of 50bohr from the c.-o.-m. of the anionic target) turn out to beas follows (when computed with the MRCI // aug-cc-pVQZof the post-HF treatment): C = -0.59049; C = +0.22532 ;N = -0.63483. Here C is the Carbon atom in the middleof the molecule, so that the geometry of CCN- is given as:C -C -N , with no extra charge on the interacting He atom,as expected. The direction and value of the dipole momentare therefore given as: 2.1818 Debye, placed along the posi-tive direction of the molecular z-axis from the C -end of theotational state-changing collisions of C H − and C N −−
212 Å . The ab initio calculationsof the 2D grid of points in the space (R, theta) were carried outwith the program package MOLPRO 2012 of ref. [29]. Forall the present calculations we have employed the internally-contracted multi-reference configuration interaction method(IC-MRCI) [30,31], using the aug-cc-pVQZ basis set [32] onall atoms. The reference space for the MRCI calculations con-sists of a complete - active space by distributing 14 electronsin 15 orbitals. All single and double excitations from the ref-erence configurations are included in the variational calcula-tion. The effect of quadruple excitations is estimated via theDavidson correction [29]. The angular grid involved calcu-lations of radial ` cut ` every 5°, from 0°to 180°. The radialgrid included a higher density of points around the variousminima regions at each selected angle. The total number ofradial points was: 44. The global minimum for the complexwas found at a distance of around 6.7 Bohr from the centre-of-mass, located at an angle of around 80°, with a well´ s depthof around 58 cm − . The data shown by the two panels ofFigure 1 provide a pictorial view of the new PES calculatedfor the C N − /He system (upper panel), while it also showsfor comparison the PES already calculated in earlier work forthe similar anion of C − interacting with He (from ref. [27]).One clearly sees from the two panels the expected similaritiesbetween the two interaction potentials since both exhibit thepresence of the minimum structure of their complex with Heto be slightly off the C geometry and at similar values of dis-tances from the c.-of-m. in each complex. Furthermore, giventhe larger number of electrons in the C N − case, the corre-sponding well appears to be deeper than that for the C H − /Hecomplex [27]. Both potentials will be asymptotically drivenby the polarizability term involving the spherical, dipole po-larizability of the He partner and therefore will behave verysimilarly in their long-range regions because of it. The lo-calization of the excess charge provided by the extra boundelectron of the anion is also an interesting item for the newCCN − /He PES discussed here. The Mulliken charges in theasymptotic situation (e.g. with the helium at a distance of 50bohr from the c.-o.-m. of the anionic target) turn out to beas follows (when computed with the MRCI // aug-cc-pVQZof the post-HF treatment): C = -0.59049; C = +0.22532 ;N = -0.63483. Here C is the Carbon atom in the middleof the molecule, so that the geometry of CCN- is given as:C -C -N , with no extra charge on the interacting He atom,as expected. The direction and value of the dipole momentare therefore given as: 2.1818 Debye, placed along the posi-tive direction of the molecular z-axis from the C -end of theotational state-changing collisions of C H − and C N −− anions with He 3 FIG. 1. Computed PESs for the two molecular anions of the presentstudy. The data are presented as in-plane maps in 2D with r cos θ and r sin θ as coordinates. Energy levels in cm − . Upper panel: C N − from present calculations; lower panel: C H − from ref. [27]. molecule. Another type of presentation of the relevant PESfor both systems could be done by numerically generating theradial coefficients of the multipolar expansion of the RigidRotor (RR) 2D potential energy surfaces: V ( r = r e , R , θ ) = ∑ λ V λ ( R ) P λ ( cos θ ) (1)where r e is the geometry of the equilibrium structure of the an-ion, already discussed earlier, and the sum over the contribut-ing lambda values went up to 19, although only the dominant,stronger terms are shown in Figure 2. The panels of that figurealso compare the present findings for C N − with the earlierdata for C H − from ref. [27].The radial coefficients presented in that figure underlineonce more the similarities between the two systems, for whichthe overall spatial anisotropy is chiefly controlled by lambda= 1, 2 and 3 terms. This indicates that the ∆ N = ∆ N = N -level separations, without showing the spin- FIG. 2. Computed multipolar coefficients calculated from the initialPES data. Upper panel: lower values of the dominant radial coeffi-cients for the C2N- anion; lower panel: same data but for the C2H-anion. Energy values in cm-1 and distances in Å .FIG. 3. Schematic location of the rotational energy levels for themolecular anion of the present study. For the case of C N − ( X Σ − )only the pseudo-singlet levels without splitting effects are shown.See main text for further details. otational state-changing collisions of C H − and C N −−
212 Å . The ab initio calculationsof the 2D grid of points in the space (R, theta) were carried outwith the program package MOLPRO 2012 of ref. [29]. Forall the present calculations we have employed the internally-contracted multi-reference configuration interaction method(IC-MRCI) [30,31], using the aug-cc-pVQZ basis set [32] onall atoms. The reference space for the MRCI calculations con-sists of a complete - active space by distributing 14 electronsin 15 orbitals. All single and double excitations from the ref-erence configurations are included in the variational calcula-tion. The effect of quadruple excitations is estimated via theDavidson correction [29]. The angular grid involved calcu-lations of radial ` cut ` every 5°, from 0°to 180°. The radialgrid included a higher density of points around the variousminima regions at each selected angle. The total number ofradial points was: 44. The global minimum for the complexwas found at a distance of around 6.7 Bohr from the centre-of-mass, located at an angle of around 80°, with a well´ s depthof around 58 cm − . The data shown by the two panels ofFigure 1 provide a pictorial view of the new PES calculatedfor the C N − /He system (upper panel), while it also showsfor comparison the PES already calculated in earlier work forthe similar anion of C − interacting with He (from ref. [27]).One clearly sees from the two panels the expected similaritiesbetween the two interaction potentials since both exhibit thepresence of the minimum structure of their complex with Heto be slightly off the C geometry and at similar values of dis-tances from the c.-of-m. in each complex. Furthermore, giventhe larger number of electrons in the C N − case, the corre-sponding well appears to be deeper than that for the C H − /Hecomplex [27]. Both potentials will be asymptotically drivenby the polarizability term involving the spherical, dipole po-larizability of the He partner and therefore will behave verysimilarly in their long-range regions because of it. The lo-calization of the excess charge provided by the extra boundelectron of the anion is also an interesting item for the newCCN − /He PES discussed here. The Mulliken charges in theasymptotic situation (e.g. with the helium at a distance of 50bohr from the c.-o.-m. of the anionic target) turn out to beas follows (when computed with the MRCI // aug-cc-pVQZof the post-HF treatment): C = -0.59049; C = +0.22532 ;N = -0.63483. Here C is the Carbon atom in the middleof the molecule, so that the geometry of CCN- is given as:C -C -N , with no extra charge on the interacting He atom,as expected. The direction and value of the dipole momentare therefore given as: 2.1818 Debye, placed along the posi-tive direction of the molecular z-axis from the C -end of theotational state-changing collisions of C H − and C N −− anions with He 3 FIG. 1. Computed PESs for the two molecular anions of the presentstudy. The data are presented as in-plane maps in 2D with r cos θ and r sin θ as coordinates. Energy levels in cm − . Upper panel: C N − from present calculations; lower panel: C H − from ref. [27]. molecule. Another type of presentation of the relevant PESfor both systems could be done by numerically generating theradial coefficients of the multipolar expansion of the RigidRotor (RR) 2D potential energy surfaces: V ( r = r e , R , θ ) = ∑ λ V λ ( R ) P λ ( cos θ ) (1)where r e is the geometry of the equilibrium structure of the an-ion, already discussed earlier, and the sum over the contribut-ing lambda values went up to 19, although only the dominant,stronger terms are shown in Figure 2. The panels of that figurealso compare the present findings for C N − with the earlierdata for C H − from ref. [27].The radial coefficients presented in that figure underlineonce more the similarities between the two systems, for whichthe overall spatial anisotropy is chiefly controlled by lambda= 1, 2 and 3 terms. This indicates that the ∆ N = ∆ N = N -level separations, without showing the spin- FIG. 2. Computed multipolar coefficients calculated from the initialPES data. Upper panel: lower values of the dominant radial coeffi-cients for the C2N- anion; lower panel: same data but for the C2H-anion. Energy values in cm-1 and distances in Å .FIG. 3. Schematic location of the rotational energy levels for themolecular anion of the present study. For the case of C N − ( X Σ − )only the pseudo-singlet levels without splitting effects are shown.See main text for further details. otational state-changing collisions of C H − and C N −− anions with He 4 FIG. 4. Steady-state distributions of relative molecular populationsof rotational levels for temperatures up to 30K. Left panel: the C N − anion; right panel: the C H − anion. See main text for further discus-sion. spin and spin-rotation splitting for C N − . The sizes of thosesplitting constants are not known or available as yet and wewill be discussing further in the next Section how they will beincluded in the present study. Their energy splittings wouldin any event not be visible on the chosen energy scale in thatFigure. We clearly see, however, that the density of states overthe examined range of about 60 cm − , which should cover therelevant energy ranges of both the CSE environments and thecold ion traps preparation, is dramatically different betweenthe two systems, with the C N − anion showing a double num-ber of states being accessible within that energy range.The higher ` crowding ` of rotational states per unit en-ergy for the latter molecule translates into a higher numberof them being significantly populated at equilibrium tempera-tures in either environments. The comparison between steady-state rotational level populations in the two anions is reportedby the two panels in Figure 4, where we clearly see that attemperatures in a cold trap around 25K the anion in the leftpanel significantly populates levels up to N =
10, while theC H − anion on the right panel only has states up to N = III. QUANTUM CALCULATIONS OF STATE-CHANGINGROTATIONALLY INELASTIC CROSS SECTIONS ANDRATES
The inelastic cross sections involving collisional transitionsin both the title molecules were calculated using our in-housemultichannel quantum scattering code ASPIN, which we havealready described in many earlier publications of our group[33-35] and therefore will not be repeated here in detail. Theinterested readers are referred to the above publications for consultation.In the OH + ( Σ − ) state, we have three levels for each totalangular momentum ≥
1: the rotational levels are in fact splitby spin-spin and spin-rotation coupling effects. In the pureHund’s case (b) the electronic spin momentum S couples withthe nuclear rotational angular momentum N ( N = R for a Σ state) to form the total angular momentum j , given by j = N + S [36,37]. As a consequence of that coupling terms, therotational levels of the molecule may be labelled not only bythe above quantum numbers defined in the Hund’s case (b)but also by their parity index ε . The levels in molecules ofodd multiplicity with parity index equal to +1 are labelled e ,and those with parity index equal to -1 are labelled f [36].However, we shall omit this index in the following discussionsince we shall not be using it in our analysis.The extensive work we have done on collision of He witha Σ − molecule [36,37] have already shown that the ∆ N = S that isresponsible for the separations between j states. Thus, onecan say that rotational quenching transitions have much largerenergy gaps than those which simply cause spin-flipping pro-cesses within each N -labelled manifold. In a full close-coupling (CC) approach to the quantum dynamics the molec-ular Hamiltonian includes, in addition to the rotational con-tribution, a nuclear coupling contribution which induces hy-perfine energy splitting. These splittings are however lowerthan typically 10 − cm − , i.e. they are much lower than therotational spacing and collisional energies investigated in thiswork, as we shall further show later in this paper. In such sit-uation, the common approach (e.g. see: Alexander & Dagdi-gian [38]) is to neglect the hyperfine splitting and to decouplethe spin wave functions from the rotational wave functions,using a recoupling scheme. This simplifies considerably thedynamic problem which is then reduced to solving the usualspin-less CC equations associated to the more usual Σ case.This approximate but almost exact ` recoupling approach `will be considered as our reference approach in the followingcalculations, where our results will be compared with thosecarried out for the exact Σ case of the C H − molecular an-ion.Scattering calculations were carried out for collision en-ergies between 1 and 1000 cm − using steps of 0.1 cm forenergies up to 100 cm − , of 0.2 cm − for 100-200 cm − ,of 1.0 cm − for 200-500 cm − and of 2 cm − for 500-1000cm − . This fine energy grid was used to ensure that impor-tant features such as the many resonances appearing in thecross sections were accurately accounted for and their contri-butions correctly included when the corresponding rates werecalculated, as discussed below. The number of partial waveswas increased with increasing energy reaching J =
100 forthe highest energies considered. Inelastic cross sections werecomputed for all transitions between j = j =
12, whichwas deemed to be sufficient to model buffer gas dynamics in acold trap up to about 50 K (see below), while the same rangeof levels is expected to be the one most significantly popu-lated during low-energy collisional exchanges with He atomswithin ISM environments dynamical conditions.otational state-changing collisions of C H − and C N −−
12, whichwas deemed to be sufficient to model buffer gas dynamics in acold trap up to about 50 K (see below), while the same rangeof levels is expected to be the one most significantly popu-lated during low-energy collisional exchanges with He atomswithin ISM environments dynamical conditions.otational state-changing collisions of C H − and C N −− anions with He 5Another quantity that we shall compute from the cross sec-tions is the state-to-state inelastic rotational rates over a rangeof temperatures from thresholds up to about 100K, to coverthe range of T values expected to be significant for processesin the ISM environments we are discussing here. Hence, oncethe state-to-state inelastic integral cross sections are known,the rotationally inelastic rate constants k j → j ′ ( T ) can be evalu-ated as the convolution of the cross sections over a Boltzmanndistribution of the relative collision energies. In the equationbelow, all quantities are given in atomic units: k j → j ′ ( T ) = (cid:18) πµ k b T (cid:19) ( / ) Z ∞ E σ j → j ′ ( E ) e − E / k B T dE (2)Both the calculations of inelastic cross sections and of theircorresponding rates over the range of temperatures mentionedearlier have been carried out using the new interaction PESfor the C N − /He system and via the earlier PES already com-puted for the C H − /He system [27]. The comparison betweenour findings for these two systems will be further carried outin this Section by presenting our computed data below. IV. QUANTUM DYNAMICS OF LASERPHOTODETACHMENT PROCESSES IN COLD TRAPS
The additional process that can take place in cold ion traps[27, 39] where the present anions can be confined, and whichwe now want to describe specifically for the title moleculesof this study, involves the following dynamical steps: (i) col-lisional population re-distribution among the rotational lev-els of the anion by interaction with He atoms, as the preva-lent buffer gas, at the trap temperature; (ii) switching on thephoto-detaching laser after the previous rotational populationequilibration has been achieved. One could then be vary-ing its operational wavelength to selectively depopulate dif-ferent rotational states of the anion present in the trap afterthe step (i); (iii) further test other operating conditions suchas: buffer gas density, trap equilibration temperature, laserpower and laser wavelength in order to find the best choicesfor the state-selective photo-detachment (PD) processes tai-lored to the present system. We therefore have to know be-forehand the state-changing collisional rates for the molecularanion in the trap at different operating temperatures and the ki-netics of the evolution of the rotational state populations dur-ing collisional equilibration. We further require selecting thevalues of the PD rates (obtained from the PD cross sections asfunctions of the initial rotational state, as already discussed byus in [28,39]. As a comment about the calculated cross sec-tions and rate for state-to-state inelastic processes suffices itto say that the dominant inelastic processes we found for thepresent systems are those for which the state-changing val-ues of ∆ N ± , s −
1. Asa comparison, the corresponding Einstein spontaneous emis-sion coefficients for the C H − anion vary to be from about10 − to 10 − smaller than the collisional rates for this system[28] . In order to be modelling the rotational population evolutiondynamics, the master equations need to be solved using thecollisional thermal rates computed above at each chosen traptemperature and for the range of selected He density: dn i ( t ) dt = ∑ j = i n j ( t ) C ji ( T ) − n i ( t ) ∑ i = j P i j ( T ) (3)The quantities P i j ( T ) are the rates for the destruction of level i , while its formation rates are given by the C ji ( T ) terms.During the collisional step, i.e. before the laser is switchedon, the coefficients are given as: P i j ( T ) = η He k j → j ′ ( T ) and C ji ( T ) = η He k j → j ′ ( T ) . These relationships describes the `collision-driven ` time evolution process of thermalisation ofthe relative populations of the rotational levels of the anionwhich are populated at the selected buffer gas temperatureand for a given density η He in the trap. Once thermalisation isreached and the PD laser is turned on, one needs to modify theabove master equations by including the PD rates discussed inthe previous section and linked to the PD cross sections: dn i ( t ) dt = ∑ j = i n j ( t ) C ji ( T ) − n i ( t ) ∑ i = j P i j ( T ) + K PDi ! , (4)where K PDi is the additional destruction rate of the selectedlevel i caused by the PD laser. The set of rates K PDi is criti-cal in the experiments in general and for the present numericalsimulations because they drive the destruction of both the pop-ulation of one specific rotational level i and of all the molecu-lar ions which have been populating that specific state duringthe previous thermalisation step in the traps. In the experi-ments of this study these rates depend on the laser photon fluxand on the overlap between the laser beam and the ion cloudwithin the trap. Since these parameters, as well as the absolutevalues of the state-to-state PD cross sections, are presentlyunknown for the title molecular anions, we shall introduce ascaling parameter which we have already discussed in our ear-lier work [28,40] according to which the relation between therequired rate and the estimated cross section is given by: K PDJ ′′ = α ( ν ) σ PDJ ′′ ( ν ) (5)The above scaling parameter is helping us to include in themodelling the relative role of different features of the photo-detachment experiment which are effectively the laser-drivenfeatures of the PD dynamics. They include quantities likelaser-flux, spatial overlap, etc. which compete with the purecollisional rates that repopulate the anion’s rotational levelsvia its interaction with the buffer gas. This specific aspectof the modelling allows us to simulate either a ` collision-dominated ` situation or a ` PD-dominated ` situation when-ever the laser strength is markedly varied, as we will be furtherdiscussing below. The values of the σ PDJ ′′ ( ν ) can be obtainedthrough an analysis which we have presented before [28,40]and which will be not repeating here. Briefly, the PD crosssection is given as: σ PDJ ′′ ( ν ) ∝ J max ∑ J ′ = (cid:12)(cid:12)(cid:12) C J ′ J ′′ (cid:12)(cid:12)(cid:12) ( E − E th ) p Θ ( E − E th ) . (6)otational state-changing collisions of C H − and C N −−
1. Asa comparison, the corresponding Einstein spontaneous emis-sion coefficients for the C H − anion vary to be from about10 − to 10 − smaller than the collisional rates for this system[28] . In order to be modelling the rotational population evolutiondynamics, the master equations need to be solved using thecollisional thermal rates computed above at each chosen traptemperature and for the range of selected He density: dn i ( t ) dt = ∑ j = i n j ( t ) C ji ( T ) − n i ( t ) ∑ i = j P i j ( T ) (3)The quantities P i j ( T ) are the rates for the destruction of level i , while its formation rates are given by the C ji ( T ) terms.During the collisional step, i.e. before the laser is switchedon, the coefficients are given as: P i j ( T ) = η He k j → j ′ ( T ) and C ji ( T ) = η He k j → j ′ ( T ) . These relationships describes the `collision-driven ` time evolution process of thermalisation ofthe relative populations of the rotational levels of the anionwhich are populated at the selected buffer gas temperatureand for a given density η He in the trap. Once thermalisation isreached and the PD laser is turned on, one needs to modify theabove master equations by including the PD rates discussed inthe previous section and linked to the PD cross sections: dn i ( t ) dt = ∑ j = i n j ( t ) C ji ( T ) − n i ( t ) ∑ i = j P i j ( T ) + K PDi ! , (4)where K PDi is the additional destruction rate of the selectedlevel i caused by the PD laser. The set of rates K PDi is criti-cal in the experiments in general and for the present numericalsimulations because they drive the destruction of both the pop-ulation of one specific rotational level i and of all the molecu-lar ions which have been populating that specific state duringthe previous thermalisation step in the traps. In the experi-ments of this study these rates depend on the laser photon fluxand on the overlap between the laser beam and the ion cloudwithin the trap. Since these parameters, as well as the absolutevalues of the state-to-state PD cross sections, are presentlyunknown for the title molecular anions, we shall introduce ascaling parameter which we have already discussed in our ear-lier work [28,40] according to which the relation between therequired rate and the estimated cross section is given by: K PDJ ′′ = α ( ν ) σ PDJ ′′ ( ν ) (5)The above scaling parameter is helping us to include in themodelling the relative role of different features of the photo-detachment experiment which are effectively the laser-drivenfeatures of the PD dynamics. They include quantities likelaser-flux, spatial overlap, etc. which compete with the purecollisional rates that repopulate the anion’s rotational levelsvia its interaction with the buffer gas. This specific aspectof the modelling allows us to simulate either a ` collision-dominated ` situation or a ` PD-dominated ` situation when-ever the laser strength is markedly varied, as we will be furtherdiscussing below. The values of the σ PDJ ′′ ( ν ) can be obtainedthrough an analysis which we have presented before [28,40]and which will be not repeating here. Briefly, the PD crosssection is given as: σ PDJ ′′ ( ν ) ∝ J max ∑ J ′ = (cid:12)(cid:12)(cid:12) C J ′ J ′′ (cid:12)(cid:12)(cid:12) ( E − E th ) p Θ ( E − E th ) . (6)otational state-changing collisions of C H − and C N −− anions with He 6Here Θ ( E − E th ) is a step function so that only transitions with E th < E are considered, and the Clebsch-Gordan coefficientenforces the selection rule ∆ J ′′ = ± p exponential parameterand the actual wavelength ν of the laser source with respect tothe threshold of the specific electron-detachment process. V. PRESENT CONCLUSIONSACKNOWLEDGMENTS
We wish to acknowledge . . . .
VI. REFERENCES [1] M.A. Cordiner, T.J. Millar, C. Walsh, E. Herbst, D.C.Lis, T.A. Bell, E. Roueff, Organic Matter in Space, in: S.Kwok, S. Sandford (Eds.), Proceedings IAU Symposium No.251, 2008.[2] M.A. Cordiner, T.J. Millar, Astrophys. J. 697 (2009) 68.[3] E. Herbst, Y. Osamura, Astrophys. J. 679 (2008) 1670.[4] C. Walsh, N. Harada, E. Herbst, T.J. Millar, Astrophys.J. 700 (2009) 752.[5] B. Eichelberger, T.P. Snow, C. Barckholtz, V.M. Bier-baum, Astrophys. J. 667 (2007) 1283.[6] A. Dalgarno, R.A. Mc Cray, Astrophys. J. 181 (1973)95. [7] E. Herbst, Nature 289 (1981) 656. [8] M.C. Mc-Carthy, C.A. Gottlieb, H. Gupta, P. Thaddeus, Astrophys. J.652 (2006) L141. [9] K. Kawaguchi, Y. Kasai, S. Ishikawa,N. Kaifu, Publ. Astron. Soc. Japan, 47 (1995) 853. [10]J. Cernicharo, M. Guélin, M. Agúndez, K. Kawaguchi, M.C.McCarthy, P. Thaddeus, Astron. Astrophys. 467 (2007) L37.[11] A.J. Remijan, J.M. Hollis, F.J. Lovas, M.A. Cordiner,T.J. Millar, A.J. Markwick- Kemper, P.R. Jewell, Astrophys.J. 664 (2007) L47.[12] S. Brünken, H. Gupta, C.A. Gottlieb, M.C. McCarthy,P. Thaddeus, Astrophys. J. 664 (2007) L43.[13] P. Thaddeus et al., Astrophys. J. 677 (2008) 1132.[14] M. Agúndez et al., Astron. Astrophys. 517 (2010) L2.[15]N.Sakai,T.Shiino,T.Hirota,T.Sakai,S.Yamamoto,Astrophys.J.718(2010)L49.[16] S. Brünken, A. Gottlieb, H. Gupta, M.C. McCarthy, P.Thaddeus, Astron. Astrophys. 464 (2007) L33.[17] T. Amano, J.Chem.Phys. 129 (2008) 244305.[18] K.D. Tucker, M.L. Kutner, P. Thaddeus, Astrophys. J.193 (1974) 5425.[19] C. Barckholtz, T.P. Snow, V.M. Bierbaum, Astrophys.J. 547 (2001) L171.[20] J. K. Anderson and L. M. Ziurys, Astrophys. J. Lett.,795: L1 (2014). [21] K. Kawagouchi, T. Suzuki, S. Saito, E. Hirota, DyeLaser Excitation Spectroscopy of the CCN Radical, J. Molec.Spectr. 106, 320 (1984).[22] E. Garand, T. I. Yacovitch, D. M. Neumark, Slowphotoelectron velocity-map imaging spectroscopy of C N − ,C N − , and C N − Π ) with He: Propensity featuresin rotational relaxation at ultralow energies, Phys. Rev. A, 73,022703 (2006).[36] L. Gonzalez-Sanchez, E. Bodo, and F.A. Gianturco,Quenching of molecular ions by He buffer loading at ultralowenergies: rotational cooling of OH+( Σ − ) from quantum cal-culations, Eur. Phys. J. D 44, 65 - 72 (2007)[37] L. Gonzalez-Sanchez, R. Wester, F.A. Gianturco,Modeling Quantum Kinetics in Ion Traps: State-changingCollisions for OH+ ( Σ − ) ions with He as a Buffer Gas,ChemPhysChem, 19, 1866 (2018).[38] M. H. Alexander, P. J. Dagdigian, J. Chem. Phys. 79,302 (1983).[39] D. Hauser, S. Lee, F. Carelli, S. Spieler, O. Lakhman-skaya, E. S. Endres, S. S. Kumar, F. Gianturco, and R. Wester,Nat. Phys. 11, 467 (2015).[40] F. A. Gianturco, O. Y. Lakhmanskaya, M. Hernandez-Vera, E. Yurtsever, and R. Wester, Collisional relaxation ki-otational state-changing collisions of C H − and C N −−