Interaction potentials, spectroscopy and transport properties of C+(2PJ) and C+(4PJ) with helium

We calculate accurate interatomic potentials for the interaction of a singly charged carbon cation with a helium atom. We employ the RCCSD(T) method, and basis sets of quadruple-ζ and quintuple-ζ quality; each point is counterpoise corrected and extrapolated to the basis set limit. We consider the two lowest C+(2P) and C+(4P) electronic states of the carbon cation, and calculate the interatomic potentials for the terms that arise from these: 2Π and 2Σ+, and 4Π and 4Σ−, respectively. We additionally calculate the interatomic potentials for the respective spin–orbit levels, and examine the effect on the spectroscopic parameters. Finally, we employ each set of potentials to calculate transport coefficients, and compare these to the available data. Critical comments are made in the cases where there are discrepancies between the calculated values and measured data.


Introduction
Carbon ions, C + , appear in wide-ranging situations, including flames and plasmas [1], chemical vapour deposition (CVD) [2] and for use in radiotherapy [3]. They are also possibly present in small amounts in the ionospheres of the Earth [4] and those of other planets [5]. Additionally, they are thought to play a key role in the chemistry of the interstellar medium [6]. Chemistry mostly starts with interactions between species, and the interactions of atomic cations with rare gas atoms are generally agreed to be amongst the simplest interactions that can be probed, especially when the rare gas is the very non-polarisable helium atom. Indeed, the C + -He complex has been proposed as a possible species in the interstellar medium by Harrison et al. [7] and interactions between C + and He could be a cooling mechanism for C + ions in the interstellar medium and dense interstellar clouds [8].
Interatomic potentials are also important in the calculation of a range of quantities including collision cross sections -important in the calculation of ion transport dataand atomic collisional energy transfer. In turn, these underpin knowledge of transport time and loss mechanisms to walls in flow-tube experiments, transport of ions in plasmas, CVD and cooling of interstellar clouds.
Here we investigate the interatomic potentials that arise from the lowest atomic asymptotes of the open-shell C + -He complex, C + ( 2 P J ) + He( 1 S 0 ) and C + ( 4 P J ) + He( 1 S 0 ). From these we shall obtain accurate spectroscopic constants * Corresponding author. Email: Tim.Wright@nottingham.ac.uk and transport coefficients to determine whether the spinorbit (SO) interaction affects these significantly. The present work also serves as a precursor to the study of higher atomic number Group 14 atomic cation-rare gas species, where the inclusion of the SO interaction is certainly expected to be important in an accurate description of the interaction potentials.
When He interacts with C + , the degenerate atomic states are split. In the absence of the SO interaction, the 2 P ground electronic term of C + gives rise to lower 2 and higher 2 + diatomic terms, and the first excited 4 P atomic state of carbon leads to lower 4 − and higher 4 diatomic terms. These are the states that have been investigated by previous theoretical investigations; however, it is the SO split levels that are probed experimentally.
For the excited C + ( 4 P) ion, there are three SO levels: the lowest is 4 P 1/2 , the middle 4 P 3/2 is higher by 22.0 cm −1 and the upper 4 P 5/2 is 50.3 cm −1 higher [9]. The SO interaction leads to the 4 diatomic state splitting into 4 -1/2 , 4 1/2 , 4 3/2 and 4 5/2 levels, where it should be noted that C 2015 Taylor & Francis here = + rather than = | + | is employed to allow for the = −1/2 level to be differentiated from the = 1/2 one. This follows the convention used by Herzberg in his classic text [10]. The 4 − state gives rise to two levels, corresponding to = 1/2 and = 3/2.
Since levels of the same value can mix, an interaction between the 2 1/2 and 2 1/2 + levels is expected. The mixing of these is expected to be small, and so we will refer to the resulting = 1/2 levels with the original 2 1/2 and 2 1/2 + labels, rather than re-labelling the linear combinations that result; similar comments refer to the quartet states. Other smaller mixings can also occur with higher energy states, and indeed between the doublet and quartet states with the same value of ; however, since the separation between such manifolds of state exceed 30,000 cm −1 , these are neglected here. To simplify the presentation, we shall use the word 'state' to refer to atomic and molecular terms or levels, with the term symbol or context making it clear which is being referred to.
There has been significant work on the C + -He complex over the past 40 years, including calculating equilibrium internuclear separations, spectroscopic constants and binding energies of the systems, as well as measuring transport data for the carbon cation in He. The first such studies appear to be the almost contemporaneous papers by Toshima [8] and Harrison et al. [7]. The former constructed potentials from attractive and repulsive terms, yielding curves for 2 3/2 , 2 1/2 and 2 1/2 + ; however, no spectroscopic parameters were reported. Harrison et al. [7] used Hartree-Fock theory to calculate a potential energy curve for the 2 state; it was noted therein that this was only presumed to be the ground state, and that the calculations for the 2 + state did not converge. Both of these studies were prompted by interest in C + in the interstellar medium, with Harrison et al. suggesting it may be possible to detect the C + -He complex in interstellar space, while Toshima concluded that C + /He collisions could be important in cooling within dense interstellar clouds.
Cooper and Wilson [11] also studied C + -He by the Hartree-Fock method in 1981, where they were able to obtain curves for both the 2 and 2 + states, confirming that indeed the 2 state was lower in energy. In 1986, Young and Coggiola [12] made the first mass spectrometric observation of a stable ion involving helium, which they postulated to be either [CHe] + or [C 2 He 2 ] 2 + . A number of quantum chemical calculations then followed, using various split-valence Pople-style basis sets. The first report using correlated methods was by Koch and Frenking [13] in 1986 using the MP2 method; these were followed by a short communication using the MP4 method by Wong et al. [14]. Wider ranging MP2 studies across the first-row mono-and di-cations were also reported a few years later by Frenking et al. [15], with single-point MP4 calculations being employed for more reliable binding energies.
Most of these papers only reported equilibrium interatomic separations, rather than potential energy curves as such; even the reported curves [7,8] were quite limited. Similarly, limited curves at the MP4 (SDQ) level were reported by Grice et al. [16], who looked at potentials arising from each of the 2 P and 4 P states of C + interacting with He. Jemmis et al. [17] calculated the internuclear separation at both the MP2 and QCISD(T) levels, while Hughes and von Nagy-Felsobuki [18] calculated the same at the CCSD(T)/cc-pVQZ level, and also reported harmonic vibrational frequencies. Finally, Matoba et al. [19] calculated interaction potentials for the 2 , 2 + , 4 − and 4 states using the MRCI/d-aug-cc-pVQZ approach with transport data and spectroscopic constants also being reported.
Early measurements, in 1979, of C + ion mobilities at ∼300 K in He were made by Thomas et al. [20] (who also measured ND L , the longitudinal diffusion coefficient multiplied by the gas number density) and Dotan et al. [21], both concentrating on the C + ( 2 P) ground state. (A smoothed version of the Dotan et al. data has been presented in [22].) Data given in a 1981 thesis by Peska [23] are also available for C + ( 2 P) ion mobilities in He. A later study in 1986 by Twiddy et al. [24] reported measurements on both C + ( 2 P) and the metastable C + ( 4 P) ions; this is one of the few mobility studies to observe excited state ions. The potentials and calculated ion mobilities of Grice et al. [16] supported the observations by Twiddy et al. [24] of C + ( 4 P) in their flow-drift-tube experiment. More recently, Matoba et al. [19] studied the mobility of C + ( 2 P) and C + ( 4 P) at 77 and 4.3 K. They also used their calculated MRCI/d-augcc-pVQZ interaction potentials to calculate ion mobilities and compared these to their experimental data.
We also note that Rincon et al. [25] have studied the translational energy spectroscopy of ion beams containing C + ( 2 P) and C + ( 4 P). Collisions between these beams and neutral molecules, as well as He, were undertaken, which allowed the proportion of the different states to be determined. Further, it was found that spin-changing collisions only occurred with an open-shell collision partner, while closed-shell partners resulted in spin being conserved.

Quantum chemistry
Interaction potentials with and without the SO interaction have been computed for all diatomic states arising from the lowest two atomic asymptotes of C + -He in the following manner. Energies at more than 100 internuclear separations from 0.8 to 50Å were evaluated at the RCCSD(T) level of theory as implemented in MOLPRO [26,27]; all electrons were correlated. Standard aug-cc-pwCVXZ [28] and augcc-pVXZ (X = Q, 5) [29] basis sets were used for carbon and helium, respectively, and the interaction energies at each separation were counterpoise-corrected to account for basis set superposition error. Finally, the interaction potentials were point-by-point extrapolated to the basis set limit utilising the two-point (cubic) formula of Halkier et al. [30,31] at each separation; for simplicity, we denote the final potentials as RCCSD(T)/aV∞Z. These interaction energies were then used as the unperturbed eigenvalues of the Breit-Pauli SO matrix as implemented in MOLPRO to allow calculation of RCCSD(T) interaction energies inclusive of SO splitting at each separation [32]. From each interaction energy curve, the rovibrational energy levels were obtained using the LEVEL program [33]. The two lowest relevant levels were used in each case to obtain the spectroscopic constants from standard formulae.
We note that the potentials were calculated with a high precision in MOLPRO, with the integrals converging to within 10 −12 E h , energy converging to 10 −12 E h , orbitals in the SCF program to 10 −8 and the CCSD coefficients to 10 −7 . Our T 1 diagnostic values are ∼0.02 for the 2 P state and ∼0.04 for the 4 P state; these values are acceptable, but we also note that good agreement is seen with previously reported multireference configuration interaction (MRCI) values with similar basis sets, confirming that multi-reference behaviour is not unduly affecting our potentials.

Transport coefficients
We calculated the transport cross sections for C + in He from the ab initio interaction potential energy curves as functions of the ion-neutral collision energy using the classical-mechanical program PC [34] that is an improved version of the earlier program QVALUES [35,36]. The cross sections converged within 0.05%.
The cross sections as a function of collision energy were used in the program GC [35,37,38] to determine the standard mobility, K 0 , and the other gaseous ion transport coefficients as functions of E/n 0 (the ratio of the electric field to the gas number density) at gas temperatures, T, of 4.35, 77, 100, 200, 300, 400 and 500 K. We also used program VARY [39] to determine the zero-field values of K 0 as a function of T from 0.001 to 10,000 K. Calculations were performed for both 12 C + and 13 C + , while He was assumed to be the naturally occurring mixture of isotopes. The calculated mobilities are generally precise within the precision of the cross sections at E/n 0 values below 20 Td (1 Td = 10 −21 V m 2 ). The results are progressively less precise as E/n 0 increases to 1000 Td, but these details, as well as the mobilities and other transport properties, can be obtained from the tables placed in the database that is maintained from the University of Toulouse [40]. Various weightings of the cross sections were employed for each system, and these will be stated at the appropriate points below.
Additionally, it should be noted that each SO interaction potential was shifted such that the interaction energies computed at the longest internuclear separation, 100Å, were equal to the 1/R 4 ion-induced dipole interaction energy at this separation. This was done so that the asymptotes smoothly approached zero energy for the transport calculations; this shift has a negligible effect on the spectroscopic constants obtained from these potentials, but was essential for correctly calculating the zero-field mobility, especially at low T.

Spectroscopic constants
The RCCSD(T)/aV∞Z potentials are shown in Figure 1 for C + ( 2 P)-He and in Figure 2 for C + ( 4 P)-He, and the values are also included as Supplementary material. The spectroscopic constants obtained from the interaction potentials are presented in Table 1 for 12 C + and in Table 2 for 13 C + . We shall discuss the results for each C + state in turn, comparing with the previous work where available, concentrating on the 12 C + results, since previously reported results are   Table 1. Spectroscopic constants for 12 C + ( 2 P J )-He and 12 C + ( 4 P J )-He.
State  Tables 3 and 4 give the bound vibrational levels for the doublet and quartet states, respectively; only the levels from the SO states are given since these are the ones that would be observed experimentally.

C + ( 2 P) with helium
The ground state of C + has the electronic configuration 1s 2 2s 2 2p 1 , giving rise to a 2 P state. For C + ( 2 P), as shown previously [11], the 2 state is lower in energy than the 2 + one. This ordering is as expected since the unpaired electron can be located in the 2p z orbital, giving rise to the 2 + diatomic state, or in the 2p x,y orbitals, giving rise to the Table 3. Vibrational energy levels (cm −1 ) for the three spinorbit doublet states of 12 C + -He arising from the 12 C + ( 2 P J ) + He( 1 S 0 ) asymptote. Also given is the energy at the minimum, E min , for each case. Energies are given with respect to each dissociation asymptote. Values for 13  Our calculated 2 P 3/2 -2 P 1/2 SO splitting is 61.2 cm −1 , compared to the experimental value [9] of 63.4 cm −1 . The 2 1/2 state is lower in energy than the 2 3/2 state, in line with expectations for a less-than-half-filled orbital subshell.
We highlight that the spectroscopic constants for the 2 3/2 state are almost identical to those for the non-SO 2 state (see Table 1) as would be expected, since there is no other = 3/2 state with which it can interact in this three-state picture. On the other hand, the 2 1/2 state can interact with the 2 1/2 + one, and this is found to lead to notably different spectroscopic constants for the two resultant = 1/2 levels compared to the original (non-SO) terms, especially for the dissociation energies. (Recall that, despite this small interaction, we shall refer to the interacting states with the same designations as the non-interacting ones.) It can be seen that D e , ω e and k decrease upon inclusion of SO splitting from 2 to 2 1/2 , which correlates with the increase of these quantities for the 2 1/2 + state compared to the 2 + state. These changes are in line with the forms of the interaction potentials as shown in Figure 1. Except for D e , the spectroscopic parameters for the 2 1/2 state are very close to those for the 2 one, since the mixing with the 2 1/2 + state is small close to the minimum of the 2 1/2 curve owing to the large energetic separation. The curves are energetically much closer in the region of the minimum of the 2 1/2 + curve and so more mixing occurs, and hence we see the spectroscopic parameters for the 2 + state being perturbed more than for the 2 1/2 one. These changes represent changes in the shape of the 2 1/2 + potential relative to the 2 + one. We do not see such changes in the shape for the 2 3/2 level since, as mentioned above, there are no levels with which it can interact in the present three-state model and so it is unperturbed; any very small changes arise from numerical rounding in the SO diagonalisation procedure. Table 4. Vibrational energy levels (cm -1 ) for the six spin-orbit quartet states of 12 C + -He arising from the 12 C + ( 4 P J ) + He( 1 S 0 ) asymptote. Also given is the energy at the minimum, E min , for each case. Energies are given with respect to each dissociation asymptote. Values for 13 C + -He states are given as Supplementary material. In comparison to previous work (see Table 1, where we have only included results from correlated methods), for the 2 state, we see reasonably good agreement of the equilibrium internuclear separation with all of the MP2, MP4 (SDQ), QCISD(T) and CCSD(T)/cc-pVQZ calculations. The best agreement, however, is with the recent MRCI/daug-cc-pVQZ calculations of Matoba et al. [19], and this confirms that the single-reference RCCSD(T) method employed here is adequate. Similarly, good agreement is seen with the range of previous values for the dissociation energy. It is interesting to note that the agreement with the CCSD(T)/cc-pVQZ D 0 value from Hughes and von Nagy-Felsobuki [18] is not as good as the corresponding aug-cc-pVTZ value from that work, suggesting that diffuse functions are important. Indeed, if one uses the ratio of the cc-pVTZ and aug-cc-pVTZ D 0 values from [18] to scale the cc-pVQZ one, an 'aug-cc-pVQZ' value of 391 cm -1 is obtained -very close to the present value. The reason for this cc-pVQZ discrepancy appears to lie in the vibrational frequency calculated in [18], which is significantly higher than the present value (see Table 1).
For the 2 + state, only two previous reports of the equilibrium internuclear separation and dissociation energy have been reported. The MP4(SDQ) values of Grice et al. [16] are in reasonable agreement with the present ones, while the MRCI values from Matoba et al. [19] are in excellent agreement with the present values.

C + ( 4 P) with helium
The C + ( 4 P) state arises from a 2p ← 2s excitation, leading to a 1s 2 2s 1 2p 2 configuration. Upon interaction with helium, in the absence of SO coupling, the degeneracy is lifted and 4 and 4 − states result. In the former case, one of the two electrons in the 2p orbital is located in the 2p x,y orbitals, with the other in the 2p z ; in the 4 − state, both 2p electrons are located in the 2p x,y orbitals. As a consequence, we expect more electron repulsion with the He atom for the 4 term than for the 4 − term. Consistent with this, for C + ( 4 P) interacting with helium, the 4 − state was found to be lower in energy than the 4 state, as was shown previously [16]. Indeed, the magnitude of the binding energy of the helium atom in the 4 − state is quite remarkable. We note here that the 4 − state, as well as having both 2p electrons perpendicular to the internuclear axis, also has one fewer 2s electrons than the 2 state. As such, the binding energy of the helium atom would indeed be expected to be significantly greater, as shown in Table 1, as a result of its increased 'view' of the di-cationic core.
Since these states have quartet spin multiplicity, one more SO component arises here than was the case for C + ( 2 P J ). The SO splittings have been calculated to be 17.3 and 46.0 cm −1 , which compare well to the experimental values [9] of 22.0 and 50.3 cm −1 . The interaction potentials for all of these quartet SO states, as well as the non-SO states, are shown in Figure 2. For the = 1/2 and = 3/2 SO states arising from the 4 − term, we observe almost identical spectroscopic constants, since at the position of these minima, the curves are significantly energetically separated from the like-4 curves. As such, the constants are extremely close to each other and those for the non-SO 4 − state. The minima of the 4 1/2 − and 4 3/2 − states are found to be degenerate, both because near their minimum they are too far away to interact with like-states from the 4 manifold, and also since there is no SO coupling between them.
(They would be split a small amount by spin-spin coupling, in reality, but that is not included here.) For all four of the 4 states, the R e values are very similar, and very close to the 4 value. The other spectroscopic constants show more significant differences, however, with all of the dissociation energies, harmonic vibrational frequencies and force constants being greater than for the 4 state, with the exception of those for the 4 5/2 state which are indistinguishable from those of the non-SO term. The latter is expected, since there is no other = 5/2 state with which it can interact. The largest changes are observed for the 4 1/2 state where D e , ω e , k and ω e x e demonstrate significant changes. We note that the variations in the spectroscopic parameters for the 4 SO states are in contrast to those from 4 − where almost no variation is seen. We postulate that this is due to the significant depth of the 4 − potential compared to the much shallower 4 one, which leads to the minimum of the 4 curves being energetically close to the 4 − curves in this region. The ordering of the 4 states (energetically from the lowest to the highest, and using signed values for = 1/2) is = −1/2, 1/2, 3/2 and 5/2. Thus, using the non-crossing rule, we can see which SO states correlate to which atomic asymptote, with all involving He( 1 S 0 ). The 4 1/2 − state correlates to C + ( 4 P 1/2 ); 4 3/2 − and 4 -1/2 correlate to C + ( 4 P 3/2 ); and 4 5/2 , 4 3/2 and 4 1/2 correlate to C + ( 4 P 5/2 ). (Recall that states with the same value will be mixed to a small extent.) With regard to the previous work that did not include SO coupling, again there is very good agreement with the internuclear separation for the 4 − term; indeed, this is notably better than was observed for the 2 term. It has been noted by previous authors [18] that electron correlation is likely more important in the system with the larger equilibrium separation. The calculations of Grice et al. [16] show reasonably good agreement for the 4 − internuclear separation, but only fair agreement for the D e value; with regard to the 4 state, in both cases the agreement is good. Reasonable agreement is also seen with the results of Hughes and von Nagy-Felsobuki [18] and also with their ω e value, in contrast to the poorer agreement that was observed for the 2 state. In contrast, we note markedly poorer agreement with their cc-pVQZ D 0 value (11,501 cm −1 ) than the better agreement seen with their aug-cc-pVTZ value (9929 cm −1 ); this seems largely due to the diffuse functions as the same marked effect is seen between the cc-pVTZ and aug-cc-pVTZ results in [18]. Again, we can estimate the aug-cc-pVQZ value from the corresponding triple-ζ ratio, giving a value of 10,306 cm −1 , which is in excellent agreement with the present value.
As was the case for the C + ( 2 P)-He spectroscopic constants, we see very good agreement for both the 4 − and 4 states with the calculated internuclear separations and D e values from the MRCI calculations of Matoba et al. [19].

Transport coefficients 3.2.1. C + ( 2 P) with helium
In the present work, ion mobilities and diffusion coefficients have been calculated from both the non-SO and SO potentials arising from the lowest doublet and quartet states of C + interacting with helium. These are computed over a wide range of E/n 0 and at a variety of temperatures, including temperatures at which experimental data have been taken. Although ground, statistical and excited state weightings were used for zero-field and field-dependent calculations, only the values that can be compared to experiment are presented here; the full data-set is available in the Toulouse database [40].
Cross sections were calculated from each potential curve, and then various weightings of these were employed in generating the transport data to compare with experiment, since the actual ionic state populations are not definitively known owing to the ion production method, uncertainty regarding thermalisation, injection effects and the effect of the collisions during the ion drift region. For the doublet curves, the weightings chosen were those of the ground state (100% 2 or 100% 2 1/2 cross sections), the excited state (100% 2 + or a 1:1 weighting of the 2 3/2 and 2 1/2 + cross sections) and a statistical state (2:1 weighting of 2 and 2 + or a 1:1:1 weighting of 2 1/2 , 2 3/2, 2 1/2 + cross sections).
To avoid cluttered figures, we have plotted the different mobilities as follows. In Figure 3, we show three non-SO mobility curves: from the 2 and 2 + cross sections individually, as well as from a statistical (2:1) mix of these; only the latter makes physical sense for random orientation collisions, corresponding to the 2 P state. For comparison, we also show the mobilities obtained from a statistical (1:1:1) mix of the 2 1/2 , 2 3/2 and 2 1/2 + cross sections (corresponding to a statistical mix of 2 P 1/2 and 2 P 3/2 states) and also the smoothed data [22] from Dotan et al. [21]. It is clear that the calculated statistical results are in closest agreement with the experimental data; however, with the size of the cited error bars, it is not possible to decide between the SO and non-SO results, even though we would expect the former to be the more reliable. In Figure 4, we show all the available mobility data at room temperature, as well as the results from the individual SO curves and the statistical SO curve (also shown in Figure 3). It is clear that, particularly given the size of the various error bars, the experimental data are essentially in agreement across the whole range of E/n 0 . Also clear is that the best match through the whole range is with the statistical mix of SO states with just the odd experimental point falling off this curve, within the given error bars.
To compare the calculated and experimental mobilities numerically, we calculated the dimensionless statistical quantities, δ and χ , which are defined in [36]. The quantity, δ, is a measure of the relative difference between the exper-  imental and calculated values compared to the combined errors, and χ is a measure of the relative standard deviation compared to the sum of the squared error estimates. If |δ| is <1, there is substantial agreement between the two sets of values (and the converse is true). If χ is about the same value of |δ|, the agreement is about the same over the data-set and there is little scatter, while if it is substantially larger, then one of those two factors is not true.
Overall, we conclude that the mobilities indicate that statistical populations of the two 2 P J components were present in the experiments. C + was created in these experiments by charge transfer or electron impact ionisation, followed by thermalisation to ∼300 K. The Boltzmann population of the two SO states at 300 K may be calculated to be 1.00:0.37 for 2 P 1/2 : 2 P 3/2 ; therefore, one might expect different nascent populations. However, the ion mobility  [20], Dotan et al, [21], Peska [23] and Twiddy et al. [24]. measurements are carried out in a flow of helium, and as noted by Twiddy et al. [24], J-changing collisions (calculated [8] to have a rate of ∼10 −10 molecule −1 cm 3 s −1 at 150 K and expected to be significantly faster at 300 K) will scramble any nascent population differences. Coupled with the random orientation of the C + /He collisions, this is consistent with our observation that a statistical mix of the SO cross sections gives excellent agreement with the experiment. The fact that the non-SO statistical state also gives good agreement, while the other non-SO states do not a See text regarding the statistical quantities δ and χ . N is the number of experimental data points in each case. b These data correspond to C + ( 2 P) and arises from a 2:1 weighting of the 2 and 2 + results. c These data correspond to C + ( 2 P 3/2 ) and arise from a 1:1 weighting of the 2 3/2 and 2 1/2 + results. d These data correspond to C + ( 2 P 1/2 ) and arise from the 2 1/2 results. e These data correspond to a statistical mix of C + ( 2 P 1/2 ) and C + ( 2 P 3/2 ) and correspond to a 1:1:1 weighting of the 2 P 1/2 , 2 3/2 and 2 1/2 + results. f ND L represents the longitudinal diffusion coefficient multiplied by the gas number density. is again a result of the random orientations of the C + /He collisions. Below 100 Td, the data of Peska [23] are well represented by the calculated statistical data, but visually there is some deviation to higher E/n 0 values. Over the whole dataset, this is not a statistically significant deviation (Table 5), but it may point to more significant errors in the experiment at these larger values. We also note that the roomtemperature data reported by Thomas et al. [20] are gener-ally in good agreement with the present data and with previous results, but the data points at the lowest values of E/n 0 (and the apparent trend here) appear to be anomalous. Their diffusion data (see Table 5) are in marked disagreement with the present calculated results and we conclude they are unreliable.
We also note that our results indicate that the uncertainties attributed to the experimental data in [21,23,24] are perhaps somewhat pessimistic, particularly for E/n 0 values below 100 Td. We note that the magnitude of the experimental errors, coupled with the very small difference between the calculated values of statistical SO and non-SO, implies that we are unable to discriminate between these; we would, however, expect the SO results to be the more reliable. The conclusion that a statistical mix of states is more representative is (indirectly) in agreement with the conclusions of Grice et al. [16], who calculated mobilities from non-SO interaction potentials, obtained using the MP4(SDQ)/6-311 + G(3df,3pd) approach. They found that a 2:1 weighting of the 2 : 2 + cross sections gave reasonable agreement with experiment, while the individual results for the 2 and 2 + states were in poor agreement, again, a feature of the random C + /He collisional orientations.
We will now consider the more recent low-temperature mobility results of Matoba et al. [19]. Figure 5 and Table 5 show a comparison of their experimental 2 P data to our predicted mobilities using various weightings for the calculated mobilities at 77 K. These suggest that for E/n 0 in the range 10-100 Td, the statistical mix gives the best agreement, passing smoothly through the mobility maximum, while at E/n 0 < 10, the experimental results veer towards those of C + ( 2 P 1/2 ). Although this could point to an experimental issue, this would also be in line with the significantly greater Boltzmann population of the lower state at 77 K (population ratio 1.00: 0.15, assuming complete thermalisation), consistent with lower energy collisions at lower E/n 0 values which would not be expected to lead to efficient SO state change, and so maintaining the nascent Boltzmann populations.
When we compare our calculated mobilities with Matoba et al.'s 4.3 K experimental data [19] (Figure 6 and Table 5), little agreement is seen for any of the results, although there is perhaps an argument that the data are closest to that of the 2 P 1/2 state for moderate E/n 0 . We believe that the disagreement in the high E/n 0 regions is partly due to the small uncertainties cited, as evinced by the correct shape of the experimental mobility curve above about 50 Td, but there are other issues that are not so easy to ascribe.
First, the significant 'mobility minimum' seen in the experimental data at ∼10 Td is extremely unusual and is not present in the calculated values. Such results have often been found in the very low temperature work of Kaneko, Tanuma and colleagues [41,42,43] as well as the results in [19]. For molecular ions, it has been argued [44,45] that it occurs because of a Feshbach-type resonance; however, this explanation cannot account for the same behaviour being observed in atomic ion-atom systems. It has been argued [46] that calculations like the ones described here might show a mobility minimum if the cross sections were computed quantum-mechanically, rather than by using the classical-mechnical techniques inherent in program PC. However, a careful, exhaustive study [47] found this to not be the case for Ar + in He; therefore, it seems unlikely that it could be true here.
Second, it was found in 2000 [48] that there are no mobility minima for He + and Ar + ions in He when proper consideration is given to the thermal transpiration effects. These occur when a pipe is used to connect a very low-temperature drift tube with a room-temperature manometer. Unfortunately, it was not reported in [19] (E/n 0 ) / Td whether or how thermal transpiration effects were taken into account. Third, we note that the calculated mobilities at 77 and 4.3 K are very close to each other at high E/n 0 , and only deviate at low E/n 0 , and even then only to a small extent. Since there will be little difference between the regions of the interaction potentials sampled during such low-temperature collisions, this is in line with expectations; in contrast, the experimental data show significant differences between the 77 and 4.3 K data over all E/n 0 regions.
We have noted above that Matoba et al. [19] calculated MRCI interaction curves for C + ( 2 P) interacting with He( 1 S), and very good agreement is obtained for R e and D e with the present potentials. From their (non-SO) MRCI potentials, Matoba et al. [19] calculated mobilities using the first approximation of the two-temperature kinetic theory (which can be wrong by as much as 10%). They concluded that neither the 2 nor 2 + potentials gave good agreement with the experiment, but that the weighted average did (which would be consistent with random orientation collisions); however, they did not specify whether they used a 1:1 weighting of the 2 and 2 + cross sections, or the statistical weighting of 2:1, and apparently the mobilities were only calculated at a single (unspecified) temperature. We also note that their calculated mobilities are not in such good agreement with ours (compare our Figures 5 and 6 with their figure 5), particularly at high E/n 0 , but the simi-lar behaviour is observed at low E/n 0 , i.e. the experimental data drop below the calculated statistical mobilities and approach those of the 2 P 1/2 ( 2 for Matoba et al.) state. Finally, a small mobility minimum appears in Matoba et al.'s calculated data from the 2 + state, but this is not present in the weighted average they report. We thus conclude that the 4.3 K data of Matoba et al. [19] are unreliable. This could arise from incomplete thermalisation (at 4.3 K, the Boltzmann population of the 2 P 3/2 state is almost zero), but this would not explain the significant mobility minimum. One possible explanation is thermal transpiration, as mentioned above. Another is complexation of C + with He during the drift region followed by rapid collisional break-up. This would lead to a slight slowing of the passage of C + down the drift tube, and hence to lower mobilities. The significance of this process would depend on E/n 0 as there would need to be a balance of there being a high enough collisional frequency to form the complexes, but also enough energy for a significant number to dissociate (collisionally, or via unimolecular decay) rapidly, so that they arrived at close to the correct time of flight. Matoba et al. [19] present timeof-flight spectra in figure 1 of their work, and broadening of the spectral features does appear to occur at the E/n 0 values shown, although these are not as low as 10 Td. Although several pressures were used at both 4.3 and 77 K, with no significant changes detected in the mobilities, the pressures used, particularly at 4.3 K, do not appear to have differed Table 6. Statistical comparison of calculated ion transport quantities to experimental data for 12  a See text regarding the statistical quantities δ and χ . N is the number of experimental data points in each case. b These data correspond to C + ( 4 P) and arises from a 2:1 weighting of the 4 and 4 − results. c These data correspond to C + ( 4 P 5/2 ) and arise from a 1:1:1 weighting of the 4 5/2 , 4 3/2 and 4 1/2 results. d These data correspond to C + ( 4 P 3/2 ) and arise from a 1:1 weighting of the 4 3/2 − and 4 −1/2 results. e These data correspond to C + ( 4 P 1/2 ) and arise from the enough to rule out the formation of ion-neutral clusters as an explanation for the mobility minimum.

C + ( 4 P) with helium
With regard to the C + ( 4 P) state mobility data, there are only the studies of Twiddy et al. [24] at room temperature and Matoba et al. [19] at 77 and 4.3 K to which we can compare. A smoothed version of the Twiddy et al. [24] data has been compiled in the work of Viehland and Mason [49]. The room-temperature experimental data are compared to our calculated results in Figure 7 and Table 6, which show the results for each of the three 4 P J states, with the mobilities for 4 P 1/2 state coming from the 4 1/2 − curve, for the 4 P 3/2 state from a 1:1 mix of the 4 3/2 − and 4 (-1/2) curves and for the 4 P 5/2 from a 1:1:1 mix of the 4 1/2, 4 3/2 and 4 5/2 curves, in line with the curves that correlate to the different atomic asymptotes. Additionally, a statistical (equal) weighting of the six diatomic SO states is shown. It may be seen that only the statistical results are in good agreement with the experimental results of Twiddy et al. [24] indicating that the data are indeed for C + ( 4 P) ions, as also confirmed by Grice et al. [16]. As such, as was seen for C + ( 2 P), we conclude that a statistical mixture of the SO states is appropriate at room temperature, owing to the random nature of the collisions and, it is assumed, efficient SO-changing collisions; notwithstanding the thermalised nascent Boltzmann population ratios of 1.00: 0.45: 0.26 for 4 P 1/2 : 4 P 3/2 : 4 P 5/2 . We note that although the agreement between the Grice et al. [16]  Comparisons of our calculated mobilities to the 77 and 4.3 K C + ( 4 P) mobilities of Matoba et al. [19] are given in Figures 8 and 9, and Table 6. Owing to the small experimental uncertainties given, the agreement with the calculated values is poor, but the trend in the values is correct and perhaps points to the uncertainties being optimistic. If this were the case, then the statistical values seem most likely to describe the data: although the calculated data for the 4 P 3/2 state could be argued also to fit, there is no reason for this particular SO state to be preferentially populated. We note that, owing to the experimental difficulties in deconvoluting the 4 P mobility feature from the 2 P one, data at low E/n 0 were not obtained, and hence whether the movement of the mobility data towards the lowest SO state occurs or not is unestablished. At 4.3 K, again the experimental uncertainties seem too small, excluding the 4 P 3/2 state as being unlikely, it is difficult to say with any certainty what calculated curves fit the experimental data the best, although the statistical curve seems the most likely, albeit that the agreement is poor. Again, as noted above, it may be that complexation at these low temperatures occurs, leading to lower observed mobilities.
It is interesting to note that there is a small mobility minimum at 4.3 K in the calculated values from the statistical combination of non-SO states for quartet C + , but this is not present in the values from the statistical combination of SO states, although the data are close elsewhere. This minimum is much smaller than the experimental minimum at 4.3 K for doublet C + , and our minimum has been traced to small differences in the potentials at around 5Å. This shows the great sensitivity of mobility values to even very subtle changes in interatomic potentials.

Concluding remarks
We have calculated high-level interaction potentials for C + ( 2 P) and C + ( 4 P) interacting with He, both with and without the inclusion of SO coupling. Additionally, we present spectroscopic constants obtained using these potentials and show that these are mostly in agreement with the available published data. The present results are, however, more complete and also are expected to be the most reliable. Comparing the curves with and without SO coupling, we find that the perturbation to the spectroscopic constants obtained is small, but more significant for the upper states in each case, 2 + and 4 , owing both to their more weakly bound nature and the energetic proximity of the two non-SO curves at internuclear separations close to the minima of the respective upper states. We have also calculated transport data for both sets of ions, C + ( 2 P) and C + ( 4 P), with the focus on ion mobilities, for which there is the most experimental data. For C + ( 2 P), at ∼300 K we were able to show that the data of Thomas et al. [20] are unreliable for diffusion coefficients, while there is generally good agreement with the experimental mobility data of Thomas et al. [20], Dotan et al. [21], Peska [23] and Twiddy et al. [24], as well as with the calculated values of Grice et al. [16]. It was shown that a statistical mix of the ionic (SO) states gave the best agreement with experiment (a similar agreement was reached by ourselves in previous work on the open-shell O − and S − ions in He [50,51]), with the SO and non-SO results giving very similar values. This was notwithstanding the expected difference in nascent SO populations, and is a result of rapid J-changing collisions at this temperature as well as the random orientations of the collisions. At the lower temperature of 77 K, comparison between the experimental data of Matoba et al. [19] and the calculated values suggested that at high E/n 0 a statistical picture of the states matches the experimental data best, while at low E/n 0 there was significant deviation from this towards the results of the lower 2 P 1/2 state, in line with lower collision energies and the higher population of this state at this lower temperature. We found that there was stark disagreement between the calculated values and experimental data for high and low E/n 0 values at 4.3 K, and suggest that this could be the result of increased likelihood of short-lived complexation between C + and He, but may also be due to thermal transpiration effects or experimental difficulties in extracting data from overlapping mass peaks in the arrival spectra. There was some evidence at moderate E/n 0 values that the calculated 2 P 1/2 results fitted the data best here.
For C + ( 4 P), the data are more limited: good agreement was seen with the experimental ∼300 K data of Twiddy et al. [24] and the calculated values of Grice et al. [16]. For the lower 77 and 4.3 K temperatures, the data of Matoba et al. [19] were again limited, but the agreement with the calculated values was not good and seems partly attributable to optimistic cited experiment uncertainties; in particular, there were cited difficulties with overlapping features in the arrival spectra for the 4 P state. In addition, we note that there was the possibility of complexation of C + with He.
The present spectroscopic results could prove useful for trying to detect the C + -He complex in the interstellar medium, as suggested by Harrison et al. [7], although whether sufficient quantities would be present is not clear, with direct formation of the complex requiring three-body stabilisation by another atom/molecule or surface; also, ligand exchange is unlikely owing to the low binding energy. We note the importance of accounting for ion mobilities in determining accurate rate coefficients [24], particularly for slow reactions; the present calculated mobilities at a range of temperatures will clearly be useful in such investigations.
Finally, we highlight that our calculated mobilities suggest that, in principle, it could be possible to separate different atomic SO states on the basis of their mobilities. However, we have also shown that for experiments carried out at room temperature, SO-changing collision leads to complete scrambling of this, and mobilities calculated with statistical mixes of all contributing states need to be used to obtain agreement with the experiment. There is, however, some indication from the 77 K data from Matoba et al., notwithstanding the caveats noted above regarding the reliability of these data, that low temperatures and/or low collision energy may make such separation possible. For SO states with larger energetic separations than here, this may be more feasible and is something we shall consider in work on the heavier Group 14 cations.