Highly Correlated Electronic Structure Calculations of the HeâC3 van

Article pubs.acs.org/JPCA

Highly Correlated Electronic Structure Calculations of the He−C3 van der Waals Complex and Collision-Induced Rotational Transitions of C3 Daniel G. A. Smith,† Konrad Patkowski,*,† Duy Trinh,‡ N. Balakrishnan,*,‡ Teck-Ghee Lee,§ Robert C. Forrey,*,∥ B. H. Yang,⊥ and P. C. Stancil*,⊥ †

Department of Chemistry and Biochemistry, Auburn University, 179 Chemistry Building, Auburn, Alabama 36849, United States Department of Chemistry, University of Nevada Las Vegas, 4505 South Maryland Parkway, Las Vegas, Nevada 89154, United States § Department of Physics, Auburn University, 206 Allison Laboratory, Auburn, Alabama 36849, United States ∥ Department of Physics, Penn State University, Berks Campus, 214 Luerssen Building, Reading, Pennsylvania 19610, United States ⊥ Department of Physics and Astronomy and Center for Simulational Physics, University of Georgia, Athens, Georgia 30602, United States ‡

ABSTRACT: An accurate 2D ab initio potential energy surface of the He−C3 collisional system is calculated using the supermolecular coupled-cluster method with up to perturbative quadruple excitations, CCSDT(Q). This interaction potential is then incorporated in full close-coupling calculations of rotational excitation/de-excitation cross sections in He + C3 collisions for rotational levels j = 0, 2, ..., 10 and collision energies up to 1000 cm−1. Corresponding rate coefficients are reported for temperature between 1 and 100 K. Results are found to be in excellent agreement with available theoretical data that were restricted to the temperature range of 5−15 K. Implications of the computed rate coefficients to astrophysical models of C3 and carbon clusters in interstellar and circumstellar environments are discussed. was recently computed by Ben Abdallah et al.12 They also performed preliminary scattering calculations for the rotational excitation for temperatures in the narrow range of 5 to 15 K for low-lying rotational levels, j ≤ 10. For astrophysical models to quantitatively predict the role and influence of C3 in stellar, interstellar, and cometary atmospheres, rate coefficients for collisional excitation and deexcitation of its rovibrational levels due to the dominant colliders must be available with sufficient accuracy. However, experimental data on this compound is limited and difficult to obtain, and a theoretical approach is warranted. In this work, helium is chosen as the collision partner with C3 for various reasons. H2 is the most abundant molecule in most interstellar environments, so its collisional interaction with C3 is the most important and requires stringent investigation. Collisional cross sections of He with C3 are expected to be qualitatively similar to those of H2. While details of the interaction and resulting cross sections are likely to be different, the reduced number of degrees of freedom for He−C3 allows for a more feasible theoretical treatment.13,14 Previous work of Ben Abdallah et al.12 on the He−C3 system used the size-consistent single- and double-excitation coupled

I. INTRODUCTION Tricarbon (C3) is a small carbon cluster. Being an astrophysically important molecule, it has been detected in stellar atmospheres, in the interstellar medium, and in cometary tails.1 It is also formed in electrical discharges and has been observed in flames,2 explosions,3 and in the vaporization of carbon.4 Soot formation around hot stars is attributed to C3 as a nucleation site.2 This carbon cluster has been detected around stellar objects including the circumstellar shell of the carbon star IRC +10216,5 as well as a few other carbon-rich sources in the direction of Sagittarius B2.1 C3 is also expected to be present in interstellar clouds and plays a crucial role in the formation of large carbon molecules and hydrocarbons.6 Besides observations of C2, it is the only pure carbon-chain molecule that has been detected with a recent observation made by Herschel Space Observatory in the sight-lines to W31C and W49N.7 Detailed analysis of C3 excitation in four lines of sight at visible wavelengths was performed by Roueff et al.,8 while Adamkovics et al.9 developed a novel method for the determination of column densities and analysis of excitation profiles involving the simulated observed rotationally resolved spectra of C3 for rotational levels j ≤ 14. Hence the excitation of C3 provides information on both the gas collisional temperature (from the low-lying rotational states) and the presence and nature of extrathermal excitation mechanisms (from higher j states). However, neither the Leiden Atomic and Molecular Database10 nor the BASECOL11 database contains collisional data on C3, but a 2D ab initio potential energy surface (PES) for He−C3 © 2014 American Chemical Society

Special Issue: Franco Gianturco Festschrift Received: December 9, 2013 Revised: January 28, 2014 Published: January 29, 2014 6351

dx.doi.org/10.1021/jp412048w | J. Phys. Chem. A 2014, 118, 6351−6360

The Journal of Physical Chemistry A

Article

Table 1. Frozen-Core Coupled-Cluster Interaction Energy Contributions (in cm−1), Computed in the XZ and aXZ Bases with and without Midbond (the letter M in the basis set symbol denotes the presence of the hydrogenic midbond set), for the NearMinimum Geometry of the He−C3 Complex (R = 6.75 bohr, θ = 90°)a X= method

D

T

Q

5

6

−15.499

−22.628 −25.598 −25.678 −27.951 −25.734 −25.457 −25.760 −27.051 −26.378 −25.686 −26.429 −27.264 0.379 0.448 0.432 0.495 0.370 0.495 0.404 0.485 −0.414 −0.472 −0.514 −0.536 −0.649 −0.679 −0.681 −0.717

−24.485 −25.926 −26.494 −27.091 −25.919 −26.066 −26.571 −27.149 −26.212 −26.103 −26.870 −27.176 0.426 0.460 0.472 0.500 0.432 0.500 0.463 0.507

−25.563 −26.686 −26.707 −26.932 −26.295 −26.710 −26.739 −26.911 −26.455 −26.730 −26.899 −26.926

−26.231 −27.120 −26.803 −26.933

CCSD(T)/aXZ ext. CCSD(T)/aXZM ext. CCSD(T)-F12b/aXZ

−26.853

CCSD(T)-F12b/aXZM

−23.151

CCSD(T**)-F12b/aXZ

−28.483

CCSD(T**)-F12b/aXZM

−24.904

ΔCCSDT/XZ ext. ΔCCSDT/XZM ext. ΔCCSDT/aXZ ext. ΔCCSDT/aXZM ext. ΔCCSDT(Q)/XZ ext. ΔCCSDT(Q)/XZM ext. ΔCCSDT(Q)/aXZ ext. ΔCCSDT(Q)/aXZM ext. ΔCCSDTQ/XZ ΔCCSDTQ/XZM

−21.054

0.217 0.284 0.189 0.213 −0.276 −0.460 −0.579 −0.595 0.020 0.041

The rows marked “ext.” display the CBS-extrapolated results where the values in the “X” column are obtained using the (X − 1,X) extrapolation. (T**) denotes that the triples were scaled using the dimer scale factor for both the dimer and monomer calculations. The results at the basis-set level employed for the entire surface are given in bold.

a

and compared with those of Ben Abdallah et al.12 Conclusions are presented in Section V.

cluster approach with perturbative triple excitations (CCSD(T)15) to calculate the PES. This potential was subsequently used by the same authors in scattering calculations for kinetic energies between 0.1 and 180 cm−1 and rotational levels j = 0−10.12 Corresponding rate coefficients were reported for temperatures T = 5−15 K. While the CCSD(T) method is considered to be the “gold standard” in computation of interaction potentials for weakly bound systems, higher-order coupled-cluster excitations as well as basis-set incompleteness effects may lead to inaccuracies in the medium- and long-range part of the potential. The current work presents a more stringent calculation of the PES, using the supermolecular coupled-cluster method with up to perturbative quadruple excitations, CCSDT(Q).16,17 An analytic fit of this potential energy surface is used in a quantum close-coupling method to calculate rotational excitation/deexcitation cross sections over collision energies of 10−6 to 1000 cm−1. This, in turn, permits calculations of rate coefficients for a wider range of temperatures, up to 100 K. The paper is organized as follows: In Section II, we discuss details of the electronic structure calculations of the PES and its analytical fit. In Sections III and IV, quantum-scattering calculations of cross sections and rate coefficients are presented

II. AB INITIO CALCULATION OF THE He−C3 POTENTIAL ENERGY SURFACE A 2D potential energy surface of helium interacting with C3 was calculated using the supermolecular coupled-cluster method with up to perturbative quadruple excitations, CCSDT(Q).16,17 The ground 1Σ+g state of the linear C3 molecule is dominated by the 1σ2g 2σ2g 1σ2u3σ2g 2σ2u1π4u4σ2g 3σ2u electronic configuration and is well separated (by about 24 000 and 17 000 cm−1, respectively18,19) from the lowest excited singlet and triplet states. Thus, the use of single-reference coupled-cluster approaches is appropriate. The C3 molecule was kept rigid at its linear geometry with a carbon−carbon bond distance at its vibrationally averaged value of 2.414 bohr12,20 so that the two internal coordinates are R (the distance between the helium atom and the central carbon in the C3 molecule) and θ (the angle between the molecular axis and the line connecting the helium atom and the middle carbon). The MOLPRO code21 was used to obtain all CCSD(T) and CCSD(T)-F12 energies as well as relativistic corrections. The CCSDT22 and diagonal Born−Oppenheimer corrections 6352



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geminal correlation factors (1.0 a−1 0 ); however, the aVXZ/ MP2FIT basis sets,39,40 the standard choice for the density fitting of the MP2-F12 pair functions, were also employed in the density fitting of the Fock matrix as well as for the complementary auxiliary basis set (CABS)41 in place of the standard VXZ/JKFIT and aVXZ/OPTRI bases. (The latter auxiliary sets would have led to inaccurate CABS singles corrections.) Because the triples contribution to CCSD(T)F12b does not include explicit correlation (the F12 triples correction has been derived42 but is significantly more computationally demanding), the basis-set incompleteness effects on ΔE(T) = ECCSD(T) − ECCSD were reduced by scaling the triples term by the ratio of the MP2-F12 and MP2 correlation energies:

(DBOCs) were calculated with the CFOUR program;23 the CCSDT(Q) and CCSDTQ24,25 corrections were obtained with the MRCC code26 interfaced to MOLPRO. All calculations utilized Dunning basis sets cc-pVXZ≡XZ and aug-cc-pVXZ≡aXZ.27,28 For basis sets involving midbond functions (indicated by adding “M” to the basis symbol), the additional functions were located halfway between the central carbon atom of the C3 molecule and the helium atom. These functions were chosen as the hydrogenic set from the same aug-cc-pVXZ orbital basis as the atom-centered functions. All calculations employed the counterpoise correction to eliminate basis-set superposition error.29 To establish the theory and basis-set level that provides the desired accuracy of the ab initio data points, we selected a near van der Waals minimum geometry (R = 6.75 bohr, θ = 90°) and examined the interaction energy contributions using an extensive test set of bases (X = D, T, Q, 5, 6) and methods (using the coupled-cluster theory up to CCSDTQ). The interaction energy was expressed as E int =

CCSD(T)/FC E int

+

rel ΔE int

+

+

T ΔE int

DBOC ΔE int

+

(Q) ΔE int

+

Q ΔE int

+

ΔE(T **) − F12 = ΔE(T)·

MP2 − F12 Ecorr MP2 Ecorr

(2)

To ensure size consistency, the scaling factor determined for the dimer was also used in the counterpoise-corrected calculations for monomers,36 as denoted by the double asterisk (T**).43 While explicitly correlated correlation energies formally exhibit a faster l−7 max convergence with respect to the maximum angular momentum present in the basis set,44,45 numerical tests indicate that the optimal extrapolation exponents for CCSD(T**)-F12b weak interaction energies are actually close to 3.0, especially when midbond functions are present.38 Thus, we have stuck to the same X−3 extrapolation as in the non-F12 case. Table 1 demonstrates the rapid convergence of the CCSD(T**)-F12b interaction energies compared with conventional CCSD(T). The best nonextrapolated CCSD(T) value, the a6ZM result, is surpassed in accuracy by CCSD(T**)-F12b at the aQZM level. Examining the best extrapolations at each level of theory, CCSD(T)/ (a5ZM,a6ZM), CCSD(T)-F12b/(aQZM,a5ZM), and CCSD(T**)-F12b/(aQZM,a5ZM), it is promising to see all three values agree to within 0.02 cm−1 with the most converged result apparently coming from CCSD(T**)-F12b/(aQZM,a5ZM). The CCSD(T**)-F12b method gives a CCSD(T)/CBS estimate of −26.93 ± 0.03 cm−1, consistent with conventional CCSD(T) but with a significant reduction in uncertainty. The higher-level coupled-cluster contributions shown in Table 1 indicate that, in order to achieve a similar accuracy for the total interaction energy, the effects of full triples and of (Q) perturbative quadruples have to be taken into account. Fortunately, similar to other small dimers studied at this level,46,47 the effects of full quadruples beyond CCSDT(Q) appear not to exceed a few hundredths of a wavenumber and can be safely neglected. As mentioned previously, the basis-set size constraints limit the accuracy of our estimates of the ΔETint (Q) and ΔEint contributions. Fortunately, these contributions largely cancel each other. Combining the best estimates of each higher-order correction gives a post-CCSD(T) correction of −0.21 ± 0.08 cm−1. The core, relativistic, and DBOC corrections were also core considered. The ΔEint correction was calculated as the difference between all-electron (AE) and frozen-core (FC) CCSD(T) interaction energies using the cc-pCVXZ≡CXZ and aug-cc-pCVXZ≡aCXZ basis sets.48 As Table 2 shows, the core correction is quite substantial, amounting to −0.10 cm−1 at the CBS limit. Relativistic effects were calculated using the secondorder Douglas−Kroll−Hess Hamiltonian 49,50 with the CCSD(T)/AE method and basis sets up to aC6ZM. As

core ΔE int

(1)

where Eint = EHe−C3 − EHe − EC3 at any level of theory and the first three ΔE corrections are defined with respect to the level of theory in the previous term, for example, ΔETint = (ECCSDT − int ECCSD(T) ). The last three corrections in eq 1 account for the int core−core and core−valence correlation, the relativistic effects, and DBOC and are discussed in detail later. Where applicable, standard X−3 extrapolations were carried out for the correlation energy.30 The shorthand notation method/(basis1,basis2) indicates the method and the two bases employed in the extrapolation. The SCF interaction energy was not extrapolated and was taken from the larger of the two bases used in the extrapolation. The N7 scaling and efficient implementation of the CCSD(T) method allows for calculations in bases up to a6ZM. However, the steeper scaling of methods past CCSD(T) (up to N10 for CCSDTQ) restricts the range of basis sets for which calculations are feasible. Table 1 shows the interaction energy contributions from different theory levels and basis sets at the near-minimum He− C3 geometry. As can be seen from this Table, the CCSD(T) interaction energy converges smoothly with respect to the basis-set size, and the addition of midbond functions as well as the complete basis set (CBS) extrapolation substantially enhance the basis-set convergence. The two largest extrapolations involving midbond, the (aQZM,a5ZM) and (a5ZM,a6ZM) results, are within 0.001 cm−1 of each other. This exceptional agreement is clearly accidental and cannot be used to determine the uncertainty of the CCSD(T)/CBS result. A more conservative uncertainty estimate for the (a5ZM,a6ZM) result, using the nonextrapolated a6ZM value, results in a CCSD(T)/CBS interaction energy of −26.93 ± 0.13 cm−1. To further explore the accuracy of the CCSD(T) interaction energy, we performed explicitly correlated CCSD(T)-F12 calculations. Among various approximations to the CCSD(T)F12 correlation energy,31,32 the CCSD(T)-F12b33,34 approximation with scaled triples has been shown to provide a very good agreement with large-basis CCSD(T) interaction energies, especially when midbond functions are employed.35−38 The CCSD(T)-F12b calculations employ the default MOLPRO explicitly correlated Ansätze (3C(FIX)) and 6353



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12.0, 14.0, 16.0, 20.0, and 25.0 bohr. As we are primarily interested in an accurate representation of the PES in the attractive region and in the low-energy part of the repulsive wall, the points with interaction energies above 1500 cm−1 were excluded from further consideration, leaving 226 ab initio data points for our analytic fit. As observed before for a similar system,52 the conventional CCSD(T) results at long-range (R ≥ 16 bohr) are better converged than the corresponding CCSD(T**)-F12b energies when comparing the CBSextrapolated results to the values calculated in the largest basis set. Therefore, the conventional frozen-core CCSD(T)/ (aQZM,a5ZM) interaction energies were taken for these points. A fitting function expressed in a mixture of (R,θ) and elliptic coordinates (R+,R−) was found to reproduce the data to a high accuracy using fewest variables. This function has the following form:

Table 2. Core, Relativistic, and DBOC Contributions to the Overall Interaction Energy (in cm−1) Computed in the XZ, aXZ, and aCXZ Bases with and without Midbond (the letter M in the basis set symbol denotes the presence of the hydrogenic midbond set) for the Near-Minimum Geometry of the He−C3 Complex (R = 6.75 bohr, θ = 90°)a X= method

D

T

Q

5

6

ΔEcore int /aCXZ ext. ΔEcore int /aCXZM ext. ΔErel int/aCXZ ext. ΔErel int/aCXZM ext. ΔEDBOC /XZ int ΔEDBOC /XZM int ΔEDBOC /aXZ int ΔEDBOC /aXZM int

−0.065

−0.106 −0.123 −0.096 −0.095 0.015 0.012 0.018 0.017 −0.002 0.073 −0.014 0.001

−0.111 −0.114 −0.101 −0.105 0.019 0.021 0.018 0.019 −0.005 −0.017 −0.015 −0.005

−0.107 −0.104 −0.101 −0.101 0.018 0.018 0.019 0.019

−0.105 −0.101 −0.101 −0.101 0.018 0.018 0.019 0.019

−0.098 0.021 0.024 0.004 0.206 −0.012 −0.034

k max

∑

V (R , θ ) =

e−DkR+Pk(R −)(Ak + Bk R+ + CkR+2)

k = 0,even

−

The rows marked “ext.” display the CBS-extrapolated results where the values in the “X” column are obtained using the (X − 1,X) extrapolation. The result at the basis-set level employed for the entire surface is given in bold.

a

∑

∑

fn (αR )

l = 0,2,4 n = 6,8,10

Cnl Pl(cos θ ) Rn

(3)

where R+ = Ra + Rb, R− = (Ra − Rb)/(2d), and Ra and Rb are the distances from the helium atom to the two points along the C3 molecular axis d bohr away from the center carbon atom. The value of d was optimized in the fit, and a typical value slightly exceeds the carbon−carbon bond length. In eq 3, f n(αR) represents the nth Tang−Toennies damping function53

expected, this correction is extremely small and has a wellconverged value of 0.02 cm −1 . The diagonal Born− Oppenheimer correction was estimated at the CCSD level51 with bases up to aQZM. While the value of this correction does not seem precisely converged at this level, it is clear that the magnitude of this correction does not exceed 0.02 cm−1. Combining the best CCSD(T)/FC/CBS estimate with the post-CCSD(T), core, relativistic, and diagonal Born−Oppenheimer corrections gives the best estimate for the nearminimum interaction energy at −27.22 ± 0.09 cm−1, where we have added the uncertainties of the ECCSD(T)/FC , ΔETint, ΔE(Q) int int , core DBOC ΔEint , and ΔEint terms quadratically and neglected the contributions to uncertainty from all remaining terms. As the results in Table 1 show, the CCSDT and CCSDT(Q) calculations for all data points cannot be avoided, and we performed them at the (DZM,TZM) and aDZM basis-set levels, respectively, for an optimal balance of accuracy and computational cost. At the near-minimum geometry of Tables 1 and 2, the theory and basis-set level employed for all data points, that is, the restriction of the CCSD(T**)-F12b, CCSDT, CCSDT(Q), and core-correction calculations to the (aQZM,a5ZM), (DZM,TZM), aDZM, and (aCTZM,aCQZM) bases, respectively, and the neglect of the CCSDTQ, relativistic, and diagonal Born−Oppenheimer contributions, leads to an interaction energy of −27.13 cm−1, an error of 0.09 cm−1 or 0.3% compared with our best result. For comparison, Ben Abdallah et al.12 calculated the near-minimum CCSD(T) interaction energy in an aug-cc-pVTZ+3s3p2d1f basis set and neglected all contributions beyond frozen-core CCSD(T), obtaining a result of −25.87 cm−1, with an error of 5.0% compared with our best value. The ab initio calculations at the level specified in the preceding paragraph were performed on a grid of points on the 2D energy surface. The angle θ, restricted by symmetry to the [0°, 90°] range, was varied in 10° increments, and the distance R took on 27 different values: 4.4, 4.7, 5.0, 5.3, 5.6, 5.9, 6.2, 6.4, 6.6, 6.75, 6.8, 6.85, 6.9, 7.1, 7.3, 7.5, 7.8, 8.2, 8.6, 9.0, 9.5, 10.0,

n

fn (αR ) = 1 − e−αR

∑ m=0

(αR )m m!

(4)

and Pn is the nth Legendre polynomial. One should note that −1 ≤ R− ≤ 1 and that only even Legendre polynomials are present because of symmetry. The expression for a 2D potential through four linearly dependent variables R, R+, R−, and θ might look redundant, but it accurately captures the different anisotropy of the short-range term and the asymptotic term with far fewer variables than for R and θ alone. On the contrary, Ben Abdallah et al.12 used the expansion in Legendre polynomials Pl(cos θ) for all terms and had difficulty converging the angular expansion for the short-range part of the potential. The parameters d, Ak, Bk, Ck, Dk, Cln, and α in eq 3 were optimized via a weighted least-squares fitting routine against the 226 ab initio interaction energies. The weights were taken as 1/ E2i , where Ei is the ab initio interaction energy. The nine linear variables Cln in the long-range part of the potential were fitted to points with R ≥ 14 bohr without the Tang−Toennies damping function and then kept frozen as the remaining linear and nonlinear parameters were fitted to all data points. Table 3 demonstrates the convergence of the fit as a function of kmax in eq 3. The table shows that the fitting function converges smoothly with respect to the total number of shortrange variables. By 31 total variables (the site location d, 5 nonlinear short-range, 15 linear short-range, one nonlinear Tang−Toennies damping parameter, and 9 linear long-range variables), the maximum error with respect to the ab initio data is lower than the estimated uncertainty of the ab initio results. Therefore, we select the 31-parameter fit, corresponding to kmax = 8, as the final analytic representation of the He−C3 potential. 6354



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with 0 ≤ λ ≤ 18 retained in the expansion. (Only even terms contribute.) This allows a direct comparison of our potential with that of Ben Abdallah et al.,12 who presented tabulated values of Vλ coefficients for λ = 0−18 as a function of R. In Table 4, we compare V0, V2, and V4 against those of Ben Abdallah et al.12 It can be seen that inclusion of higher-order electron correlation and larger basis sets in our calculations yields a slightly less repulsive short-range part and a more attractive long-range region. A graphical comparison of V0 and V2 is provided in Figure 2 to illustrate the differences. In most

Table 3. Performance of the Fit Benchmarked against the 226 Ab Initio Points as a Function of the Upper Limit kmax in the Short-Range Term in Equation 3 kmax

2

4

6

8

linear parameters nonlinear parameters mean absolute error maximum absolute error

15 4 11.86% 124.2%

18 5 0.36% 2.14%

21 6 0.19% 0.99%

24 7 0.07% 0.38%

A contour plot of the fitted potential energy surface, shown in Figure 1, illustrates a strong anisotropy of the potential due

Figure 2. Angular expansion coefficients of the potential Vλ(R). Solid curves denote results of present work and dashed curves denote that of Ben Abdallah et al.12 Black curves correspond to V0 and red curves correspond to V2.

Figure 1. Contour plot of the symmetry unique quadrant 0° ≤ θ ≤ 90° of the fitted He−C3 potential energy surface. Contours are in units of cm−1.

cases, the two results agree to within 5−10%, and for R > 10 bohr the agreement is near quantitative. However, as shown later, the small differences do not appear to have a significant effect on the computed cross sections and rate coefficients.

to the elongated character of the C3 monomer. Interestingly, the minimum predicted by the fit does not have a perfect Tshaped (θ = 90°) configuration. The lowest interaction energy of −27.20 cm−1 occurs for R = 6.80 bohr and θ = 81°. As far as the ab initio data points are concerned, the lowest interaction energy, −27.20 cm−1, was obtained at R = 6.80 bohr and θ = 80°; the lowest interaction energy for a θ = 90° structure, computed at R = 6.75 bohr, amounted to −27.13 cm−1. Thus, the T-shaped orientation corresponds to a saddle point with a very low energy barrier. Obviously, the linear θ = 0° configuration is associated with a saddle point and a much higher energy barrier, with the minimum interaction energy predicted by the fit amounting to −14.45 cm−1 at R = 9.53 bohr. The off-T-shape structure of the van der Waals minimum is obtained only for sufficiently large basis sets. Therefore, Ben Abdallah et al.12 found a perfectly T-shaped minimum. While the location of the minimum might seem quite surprising at first, it can be understood if one considers the dependence of the optimal approach angle θ on the intermolecular separation R. The linear θ = 0° approach is preferred asymptotically, while at short range the T-shaped θ = 90° structure allows for a closer approach to the C3 molecule. Thus, the optimal θ has to undergo a gradual switch from θ = 90 to 0° at some intermediate R, and it so happens that the onset of that switch occurs very close to the van der Waals minimum separation. In the scattering calculations, the interaction potential is expanded in Legendre polynomials according to V (R , θ ) =

III. SCATTERING CALCULATIONS The quantum close-coupling calculations were carried out using the MOLSCAT code54 with a rigid rotor approximation for the C3 molecule and the hybrid logarithmic derivative/Airy propagator for the radial integration of the coupled-channel equations.55 The rotational energy levels of the C3 molecule are computed using a value of B0 = 0.4305 cm−1 for the rotational constant,20,56 the same as adopted by Ben Abdallah et al.12 Because of the symmetry of the molecule and bosonic character of the 12C nucleus, only even rotational levels are present. Rotational levels up to j = 20 are included in the basis-set expansion to yield converged cross sections. For the computation of cross sections at low energies (from 10−6 to 1 cm−1), a logarithmic energy grid is used, such that every 10 data points bring the collision energy an order of magnitude higher. In this energy range, we use 1.0 ≤ R ≤ 100 bohr, with the Airy propagator being used from 80 to 100 bohr. For collision energies between 1 and 1000 cm−1, the energy range is carefully spanned to resolve resonance structures in the cross section. We use a 0.1 cm−1 grid from 1 to 10 cm−1, 1 cm−1 grid from 10 to 100 cm−1, and a 10 cm−1 grid from 100 to 1000 cm−1. At these upper energy ranges, 5.0 ≤ R ≤ 25.0 bohr is used. In Figure 3, we present Δj = 2 rotational excitation cross sections for initial rotational levels j = 2, 4, 6, 8, and 10. The results, which are in near-quantitative agreement with those

∑ Vλ(R)Pλ(cos θ) λ

(5) 6355



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higher partial waves. The centrifugal potentials for higher partial waves lead to effective barriers that dominate the van der Waals potential, preventing the formation of quasi-bound levels and accompanying resonances. De-excitation cross sections for j = 2 to j′ = 0 are shown in Figure 4 as a function of the collision energy from the ultracold

Table 4. Isotropic and Leading Anisotropic Coefficients of the He−C3 Interaction Potential as Functions of Ra R

V0

V2

V4

5.0 5.25 5.5 5.75 6.0 6.25 6.5 6.75 7.0 7.25 7.5 7.75 8.0 8.25 8.5 8.75 9.0 9.25 9.5 9.75 10.0 10.5 11.0 12.0 13.0 14.0 15.0

4331/4395 2896/2946 1897/1946 1219/1258 767/797 472/493 282/296 161.38/171.69 86.78/93.97 41.56/46.57 14.97/18.47 −9.28 × 10−4/2.44 −7.87/−6.15 −11.50/−10.28 −12.68/−11.82 −12.51/−11.89 −11.66/−11.21 −10.50/−10.18 −9.27/−9.04 −8.08/−7.92 −6.99/−6.88 −5.18/−5.12 −3.83/−3.81 −2.15/−2.14 −1.26/−1.26 −0.78/−0.77 −0.49/−0.49

13760/13944 9355/9469 6247/6379 4108/4216 2663/2744 1703/1761 1074/1115 667.26/695.24 407.10/426.41 242.88/256.15 140.61/149.71 77.89/84.14 40.14/44.44 17.96/20.92 5.36/7.41 −1.43/−0.02 −4.78/−3.82 −6.15/−5.49 −6.42/−5.96 −6.12/−5.80 −5.55/−5.32 −4.22/−4.11 −3.06/−3.01 −1.56/−1.54 −0.82/−0.80 −0.45/−0.44 −0.26/−0.25

11262/11670 7543/7643 4984/5092 3254/3337 2103/2159 1347/1384 854/880 537.33/554.31 334.81/346.23 206.54/214.22 125.96/131.13 75.78/79.26 44.81/47.16 25.90/27.47 14.49/15.54 7.72/8.41 3.77/4.22 1.53/1.83 0.32/0.50 −0.30/−0.18 −0.58/−0.50 −0.66/−0.63 −0.53/−0.52 −0.27/−0.26 −0.12/−0.11 −0.05/−0.05 −0.02/−0.02

Figure 4. De-excitation cross sections as a function of kinetic energy for the j = 2 rotational level (solid black curve). The PES was altered by ±0.5% to test the effect of the uncertainty of the PES on the cross sections. Dashed green curve: cross sections for j = 2 when PES multiplied by 1.005; dotted-dashed red curve: cross sections with PES multiplied by 0.995. The blue curve shows corresponding results for 3 He−C3 collisions on the unscaled PES.

Vλ are given in cm−1 and R is given in bohr. Each pair of Vλ coefficients for a given R is separated by a slash, with the datum on the left of the slash taken from our work, and the datum on the right taken from Ben Abdallah et al.12 a

regime to 1000 cm−1. To explore the sensitivity of results to any small errors resulting from the analytic fit or the residual contributions past the theory and basis set level employed for the ab initio points, we present results obtained by scaling the potential by 1.005 and 0.995, which amounts to modifying the potential by ±0.5%. This is comparable to the uncertainty of the potential energy surface. It is seen that calculations with the scaled PESs yield almost the same results as the unscaled PES, implying that any additional correlation and basis-set effects are not significant and that the PES is reliable for low and ultralow energy calculations. We also explored the sensitivity of results to the mass of the helium atom by performing limited calculations for the 3He−C3 system. As illustrated in Figure 4 for the j = 2 to j′ = 0 transition, the mass of the He atom has a more pronounced effect on the cross sections at low energies. A similar effect has been observed for 3He and 4He collisions with molecular hydrogen and its isotopomers.57 It is attributed to the displacement of the least bound or virtual state of the van der Waals complex relative to the initial channel threshold when the mass of the He atom is changed. Low-energy scattering cross sections are generally strongly influenced by bound or quasibound states that lie close to channel thresholds. The influence of these quasibound levels on the cross section fades as the collision energy is increased, and both isotopes yield nearly identical results for energies above 1.0 cm−1. At energies below 10−4 cm−1, the cross section varies inversely with the velocity. This is the Wigner threshold behavior of inelastic cross sections. Although cross sections in this regime are not directly relevant to astrophysics, they illustrate the correct threshold behavior and may be used to probe sensitivity of results to details of the interaction potential. This regime is currently of considerable interest to the molecule

Figure 3. Excitation cross sections for Δj = 2 transitions in C3 in collision with He as a function of the kinetic energy. Solid curve: j = 0, dashed curve: j = 2, dotted curve: j = 4, dashed-dotted curve: j = 6, dashed-double-dotted curve: j = 8.

reported by Ben Abdallah et al.,12 illustrate that rotational excitation cross sections decrease with increasing rotational level of the molecule due to the larger energy gap for transitions out of the higher rotational levels. The gradual suppression of the resonance structures with the increase in rotational excitation may be attributed to the increased contribution of 6356



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cooling and trapping community.58 The s-wave contribution dominates below 10−3 cm−1. The rapid oscillations in the cross sections between 1 and 10 cm−1 are attributed to quasibound states of the He···C3 van der Waals complexes formed in the collision. The energy dependence of the cross section is very similar to that of the He−CO2 system reported by Yang and Stancil.59 The van der Waals interaction potential of He− CO260 is considerably deeper (49.39 cm−1 at the global minimum corresponding to R = 5.79 bohr and θ = 90° compared with 27.20 cm−1 for the present system), and the minimum occurs at a shorter value of R. Consequently, He− CO2 supports resonances over a slightly broader range of energies compared with He−C3. Figure 5 shows the cross sections for transitions from j = 4 to j′ = 2 and j′ = 0 as a function of the kinetic energy. These cross Figure 6. Δj = −2 total quenching rate coefficients for C3 due to He collisions: solid curve: j = 2; dashed curve: j = 4; dotted curve: j = 6; dashed-double-dotted curve: j = 8; double-dashed-dotted curve: j = 10.

Figure 5. Same as in Figure 4 but for j = 4. Solid curve: j′ = 0; dashed curve: j′ = 2.

sections exhibit similar trends to the j = 2 → j′ = 0 transition shown in Figure 4 with the exception that there are fewer observable resonances. Also, the most probable final state corresponds to a Δj = −2 transition. This is due to the dominance of the λ = 2 term in the angular dependence of the interaction potential at long-range, which drives Δj = ±2 transitions. Similar results are found for j = 6, 8, and 10. The Δj = −2 and total quenching cross sections out of these levels are comparable to those of j = 4.

Figure 7. Δj = −4 total quenching rate constants for C3 due to He collisions: solid curve: j = 4; dashed curve: j = 6; dotted curve: j = 8; dashed-double-dotted curve: j = 10.

show a steeper dependence on temperature due to the larger energy gap involved. The de-excitation rate coefficients were used to compute rate coefficients of the reverse transitions, based on microscopic reversibility: (T ) = (2j + 1)/(2j′ + 1) exp[(kBT )−1(Ej ′ − Ej)] ′→j × k j → j (T ) (7) ′ We have verified that this expression yields the same results as that obtained using explicit excitation cross sections derived from the close-coupling calculations. Results are shown in Figures 8 and 9, respectively, for Δj = 2 transitions out of j = 0−8 and Δj = 4 transitions out of j = 0−6. Also included in these Figures are the corresponding results of Ben Abdallah et al.12 available in the 5−15 K range. Both results agree quantitatively. Figure 10 compares rate coefficients from our calculations with those of Ben Abdallah et al.12 for Δj = 2 and 4 transitions as functions of the initial rotational level at 10 K. The agreement is excellent. Comparison is also made to excitation rate coefficients of CO2 due to He from Yang and Stancil.59 The agreement between C3 and CO2 is remarkable, particularly for Δj = 2, given that the well depth for He−CO2 is 70% larger kj

IV. RATE COEFFICIENT CALCULATIONS The temperature-dependent rate coefficient for a given rotational transition is calculated by averaging the corresponding energy-dependent cross sections over the Maxwellian velocity distribution given by kj → j ′(T ) = (8kBT /πμ)1/2 (kBT )−2

∫ σj→j′(Ej)

× exp( −Ej /kBT )Ej dEj

(6)

where kB is the Boltzmann constant, σ is the cross section, μ is the reduced mass of the atom−molecule system, and Ej is the kinetic energy in the channel corresponding to the initial rotational state j. Quenching rate coefficients for Δj = −2 for j = 2 − 10 and Δj = −4 for j = 4 − 10 are shown in Figures 6 and 7, respectively, for temperatures from 1 to 100 K. For the Δj = −2 case, the rate coefficients increase rapidly until about 10−15 K and then plateau near 50 K. The Δj = −4 rate coefficients 6357



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Figure 8. Comparison of Δj = 2 excitation rate coefficients for C3 due to He collisions. Symbols (data from Ben Abdallah et al.): circles (j = 0), squares (j = 2), diamonds (j = 4), triangle up (j = 6), triangle down (j = 8); curves (this work): solid curve: j = 0, dashed curve: j = 2, dotted curve: j = 4, dashed-double-dotted curve: j = 6, double-dasheddotted curve: j = 8.

Figure 10. Variation of C3 and CO2 excitation rate coefficients at 10 K with initial j and Δj due to He collisions. Filled circle and square symbols: C3 from Ben Abdallah et al.;12 open circle and square symbols: C3 this work; triangles: CO2 from Yang and Stancil.59 Solid line: Δj = 2, dashed line: Δj = 4. Lines are only for visual aid.

found to be in excellent agreement with previous results of Ben Abdallah et al.12 computed using an interaction potential calculated at the CCSD(T) level of theory but restricted to temperatures between 5 and 15 K. We hope that the data provided here will improve astrophysical models of C3 in stellar atmospheres and interstellar gas. Further studies on this and related systems should go beyond the rigid-monomer approximation to explicitly include vibrational stretching and bending modes.

■

AUTHOR INFORMATION

Corresponding Authors

*K.P.: E-mail: [email protected]. *N.B.: E-mail: [email protected]. *R.C.F.: E-mail: [email protected]. *P.C.S.: E-mail: [email protected].

Figure 9. Comparison of Δj = 4 excitation rate coefficients for C3 due to He collisions. From this work: solid curve: j = 0, dashed curve: j = 2, dotted curve: j = 4, dashed-double-dotted curve: j = 6. Data from Ben Abdallah et al. are plotted as open symbols for comparison: circles: j = 0, squares: j = 2, diamonds: j = 4, triangles: j = 6.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS D.G.A.S. and K.P. were supported by startup funding from Auburn University. N.B. was supported by NSF grants PHY0855470 and PHY-1205838, R.C.F. was supported by NSF grant PHY-1203228, and B.H.Y. and P.C.S. were supported by NASA grant NNX12AF42G.

than that of He−C3, as previously discussed. Furthermore, Figure 3 shows that there is a considerable resonance structure between 10 to 20 cm−1, which would be expected to significantly affect the rate coefficients. The agreement is striking but may be due to a cancellation of numerical effects. All rate coefficients do show the expected decrease with j following an exponential-energy gap law behavior.

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REFERENCES

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V. CONCLUSIONS A new highly correlated electronically adiabatic ground-state potential energy surface for the interaction between He and C3 has been computed at the CCSDT(Q) level of theory. An analytical representation of this potential is adopted in quantum close-coupling calculations of excitation and de-excitation cross sections in He + C3 collisions within the rigid-monomer approximation. Rate coefficients for excitation and de-excitation collisions between rotational levels j = 0−10 are reported for temperatures ranging from 1 to 100 K. Rate coefficients are 6358



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