A global CHIPR potential energy surface for ground-state C3H and exploratory dynamics studies of reaction C2 + CH → C3 + H

C. M. R. Rocha a and A. J. C. Varandas *ab
aDepartment of Chemistry and Coimbra Chemistry Centre, University of Coimbra, 3004-535 Coimbra, Portugal. E-mail: cmrocha@qui.uc.pt; varandas@uc.pt
bSchool of Physics and Physical Engineering, Qufu Normal University, 273165, P. R. China

Received 3rd September 2019 , Accepted 17th October 2019

First published on 18th October 2019

A full-dimensional global potential-energy surface (PES) is first reported for ground-state doublet C3H using the combined-hyperbolic-inverse-power-representation (CHIPR) method and accurate ab initio energies extrapolated to the complete basis set limit. The PES is based on a many-body expansion-type development where the two-body and three-body energy terms are from our previously reported analytic potentials for C2H(2A′) and C3(1A′,3A′), while the effective four-body one is calibrated using an extension of the CHIPR formalism for tetratomics. The final form is shown to accurately reproduce all known stationary structures of the PES, some of which are unreported thus far, and their interconversion pathways. Moreover, it warrants by built-in construction the appropriate permutational symmetry and describes in a physically reasonable manner all long-range features and the correct asymptotic behavior at dissociation. Exploratory quasi-classical trajectory calculations for the reaction C2 + CH → C3 + H are also performed, yielding thermalized rate coefficients for temperatures up to 4000 K.

1 Introduction

Small carbon-bearing species Cn and CnH (n = 1–3) are ubiquitous in the interstellar medium (ISM).1 They are particularly conspicuous and known to drive the C-chemistry2 in cold dense clouds3–5 and in circumstellar envelopes of evolved C-rich stars.6–8 In these former environments, the radical9 C3H – whose potential energy surface10 (PES) entails all the above smaller species as fragments – plays a prominent role, reaching relatively high fractional abundances (≈10−9) when compared to H2.2,11 In molecular cloud cores, both its cyclic12 (c-C3H; cyclopropynylidyne) and linear13 ([small script l]-C3H; propynylidyne) isomers are thought to be formed via dissociative electron recombination of14c,[small script l]-C3H2+/C3H3+ or through the atom-neutral pathway14–17 C + C2H2, the latter being an important prototypical reaction implied in the growth of C-chains in space.15–17 This has motivated further surmises18 on the role of c-C3H as a key intermediate (via c-C3H2 formation) in the synthesis of interstellar polycyclic aromatic hydrocarbon molecules – widely recognized as potential carriers of the so-called unidentified infrared bands.19

The prevalence of C3H in the ISM and its relevance to C-chain formation have stimulated considerable experimental12,20–28 and theoretical29–42 efforts toward understanding its intricate chemistry. Yet, most of these studies have thus far mainly concerned the determination of relative energetics, symmetry and spectra of its isomeric forms.

[small script l]-C3H has a 2Π electronic ground-state and and its two bending modes, ν4(C–C–H) and ν5(C–C–C), are perturbed by Renner–Teller (RT) and spin–orbit effects.13 The first laboratory detection of [small script l]-C3H was reported by Gottlieb et al.20 who measured its microwave spectra in both 2Π1/2 and 2Π3/2 (ground) vibronic states. Subsequently, Yamamoto21 and Kanada et al.22 recorded pure rotational lines in ν4(2Σμ) and found that [small script l]-C3H has an extremely low vibrationally excited state (≈27 cm−1 above 2Π1/2) which is caused by the strong RT effect in the ν4 bending mode;21 its molecular structure was first derived from the rotational spectral data.22 Subsequent studies were mainly devoted to improving previously reported spectroscopic constants for the 2Πr and ν4(2Σμ) states23 and extending the range of rotational transitions within the 2Σ vibrationally excited manifold;24 to our knowledge, no definitive assignment of the ν5(2Σμ) fundamental has yet been made. Infrared (IR) vibrational band centers for the stretching modes (ν1, ν2 and ν3) were provided by Jiang et al.25 in the Ar matrix, and bySheehan et al.26 in the gas phase.

c-C3H has a 2B212 ground electronic state and was first detected in the laboratory by Yamamoto et al.12,27via microwave spectroscopy. Based on the predicted rotational constants, the authors reported the molecular structure of c-C3H, confirming its C2v nature.27 Moreover, using inertial defect considerations, they estimated the C–C asymmetric mode (ν4; which lowers the symmetry from C2v to Cs) to be fairly low (≈508 cm−1) and attributed it to the vibronic interactions [i.e., pseudo Jahn–Teller (PJT) effects36] between the ground 2B2 and first 2A1 excited states.27 In turn, Sheehan et al.26 reported gas-phase estimates of the ν2 (ring stretching) and ν3 (scissoring) fundamentals of c-C3H; apart from such rough estimates, no accurate experimental IR band centers are available for this form39 thus far.

Early ab initio calculations12,22,25,29–34 on C3H were primarily devoted to elucidate discrepancies between the predicted symmetries of the linear and cyclic forms and the ones actually inferred from microwave spectroscopy22,27 (energetically, the best estimate34 places c-C3H ≈ 14 kJ mol−1 more stable than [small script l]-C3H). From these studies, a general trend can be stated: depending on the electronic structure method (single-reference versus truncated-space multireference) and the utilized basis set (size), the c-C3H ([small script l]-C3H) C2v (C∞v) structure might be in a transition state and that a slightly distorted Cs form might be preferred. For the cyclic isomer, such a symmetry breaking issue was first dealt with by Stanton and coworkers.36,37 Using the equation-of-motion coupled cluster method in the single and double approximations for ionized states (EOMIP-CCSD), they first emphasized the role of the basis set effect to the accurate determination of its C2v symmetry and to the increase of the X2B2 → A2A1 excitation energy. As noted by them, the PJT effect between such states – which had previously been considered27 as the cause of the possible Cs equilibrium structure – is weakened when the size of the one-electron basis set is increased. Similar conclusions (regarding the N-electron basis) were subsequently drawn by Halvick38 at the multireference configuration interaction level of theory [MRCI+Q] with cc-pVXZ (X = D, T, Q) basis sets;43,44 he has shown that the PES along the C–C asymmetric w4 mode becomes increasingly stiff with correlation energy. Recently, Bassett and Fortenberry39 reported a quartic force field (QFF) for c-C3H from a composite scheme – based on accurate energies extrapolated to the complete-basis-set (CBS) limit at the CCSD(T)/aug-cc-pVXZ level of theory [briefly CCSD(T)/AVXZ] from basis-set cardinal numbers X = T, Q, 5, then additively corrected for core correlation and scalar relativistic39 effects. From the QFF so obtained and using second-order vibrational perturbation theory (VPT2),39 the authors reported rotational constants, structural parameters and anharmonic vibrational frequencies for the X2B2 ground-state; to our knowledge, these are the best ab initio estimates so far. The spectroscopic characterization of the C3H isomers has also been done recently by Bennedjai et al.40 at the explicitly correlated CCSD(T) level of theory, CCSD(T)-F12/AVTZ. Note that, for the linear form, the most comprehensive theoretical study to date was carried out by Perić et al.41 who provided local MRCI+Q/cc-pVTZ PESs (including relativistic effects) for both 2A′ and 2A′′ electronic states correlating with the 2Π term. These PESs were subsequently used to compute the vibronic and spin–orbit structure of the [small script l]-C3H spectrum using a variational approach.41 In their work, the authors highlight the extremely flat nature of the CCC-H bending potential curve (2A′) and, like others,22,32 do not rule out the possibility of its quasilinearity based on the limited accuracy of their PESs. Despite this and the fact that their local forms assume C∞v equilibrium geometries, the values of the various spectroscopic parameters have shown excellent agreement with the experimental results. In fact, as shown by Ding et al.28 at both the complete active space self-consistent field (CASSCF) and the CCSD(T) level of theory, the minimum structure of [small script l]-C3H tends to change from a bent to a linear geometry with basis set size enhancement (from AVTZ to AVQZ). In turn, Mebel and Kaiser42 reported an ab initio study of the ground-state C3H PES. Using CCSD(T)/6-311+G(3df,2p), they first noted that the linear and cyclic forms rearrange to each other via a ring opening step through an asymmetric transition state (hereafter denoted as [small script l]c-C3H) with a barrier of about 115 kJ mol−1. Moreover, they remarked that, besides C(3Pj) + C2H2(X1Σ+g), the (barrierless) reactions of CH(X2Π) + C2(X1Σ+g) and C(3P) + C2H(X2Σ+) represent facile neutral–neutral (exothermic) pathways yielding a carbon trimer in cold interstellar environments.

All the above features make C3H a unique and challenging species from both chemical and astrophysical viewpoints. Certainly, the implications related to its spectroscopy and reaction dynamics in all those fields highlight the need for a global PES for the title system. The major goal of this work is therefore to provide such a form for the ground-state doublet C3H that correlates with the 2Π state at linear geometries, with 2B2 at cyclic ones and to the correct fragmentation channels. For this purpose, accurate CBS extrapolated ab initio energies are employed and modeled analytically using the combined-hyperbolic-inverse-power-representation (CHIPR) method.45–47

The paper is organized as follows. Section 2 provides an overview of the ab initio calculations and CBS extrapolations performed here. The details of the analytical modelling are described in Section 3, while the topographical features of the final PES are discussed in Section 4. Section 5 reports a preliminary quasi-classical trajectory study of the C2 + CH reaction, while Section 6 gathers the conclusions.

2 Ab initio calculations

All calculations have been performed at the spin-restricted open-shell RCCSD(T) [CC for brevity] level of theory48–50 using the ROHF determinant as reference.51 The AVXZ (X = D, T) basis sets of Dunning and co-workers43,44 were employed throughout, with the calculations done using MOLPRO.52 All ab initio energies have subsequently been CBS extrapolated. As usual,53 extrapolations were done separately for the HF and correlation (cor) components53
ECBS(R) = EHF(R) + Ecor(R),(1)
where ECBS is the total CBS energy and R a six-dimensional coordinate vector.

For the HF energy, a two-point extrapolation protocol54 has been utilized,

EHFX(R) = EHF(R) + Aeβx,(2)
where x = d(2.08), t(2.96) are hierarchical numbers55,56 associated with X = D, T, β = 1.62, and A is a parameter to be calibrated from ROHF/AVXZ energies.54 In turn, the extrapolated core contributions are obtained via the uniform singlet- and triplet-pair extrapolation (USTE) protocol55,57–59
image file: c9cp04890a-t1.tif(3)
where d(1.91) and t(2.71) are CC-type x numbers,55 with A3 calibrated from the raw CC core energies.

With the above CC/CBS(d,t) protocol, a total of 42[thin space (1/6-em)]538 symmetry unrelated grid points have been selected to map all relevant regions of the 6D nuclear configuration space while ensuring a balanced global representation of the PES; to assess all geometrical structures and coordinate systems considered here, see Fig. S1 of the ESI.


Within the CHIPR framework,45,46 the ground-state doublet PES of C3H assumes the following many-body expansion (MBE)10 form
V(R) = V(2+3)(R) + V(4)(R),(4)
where V(2+3)(R) accounts for the sum of two- and three-body interactions, and V(4)(R) accounts for the four-body ones. R = {R1, R2, R3, R4, R5, R6} is the set of interatomic separations (Fig. 1), with energy zero set to the infinitely separated ground-state C and H atoms.

image file: c9cp04890a-f1.tif
Fig. 1 Interparticle coordinate system employed in the construction of the CHIPR PES for ground-state C3H; R1R3 define C–H bond distances, while R4R6 are C–C stretches.

3.1 Two-body and three-body energy terms

Following ref. 60, V(2+3)(R) in eqn (4) is defined as the lowest PES arising from the diagonalization of the 2 × 2 pseudo-diabatic matrix
image file: c9cp04890a-t2.tif(5)
image file: c9cp04890a-t3.tif(6)
image file: c9cp04890a-t4.tif(7)
are diagonal terms constructed using previously reported two- and three-body (ground-state) potentials for C2H(2A′),61 C3(1A′)62 and C3(3A′)60 (readers can refer to the original papers60–62 for details on the corresponding functional forms). In eqn (5), ε(R) is a (small) coupling term chosen to warrant that the lowest eigenvalue of [script letter H]e [eqn (5)] is continuous everywhere. The energetics of the various dissociation channels predicted from V(2+3)(R) are given in Table 1. Also shown for comparison are the corresponding values obtained from the CC/CBS(d,t) protocol as well as experimental estimates.61,63,64 As seen, the predicted thermochemistry of the V(2+3)(R)'s fragments agrees quite well with the ones obtained from both theory and experiment. A somewhat larger discrepancy is observed for [small script l]-C3(1Σ+g) + H(2S), which can be attributed to the more attractive nature of the [small script l]-C3(1Σ+g)'s three-body term. Note that this dissociation channel is the one with the largest experimental uncertainty, while an accurate estimate of the [small script l]-C3's atomization energy still needs to be determined. Indeed, four-body energies vanish at these asymptotic channels, and hence play no role in the values reported in Table 1.
Table 1 Energetics (in kJ mol−1) of the various dissocition channels predicted from V(2+3)(R)
Channel Energya
CC/CBS(d,t)b V (2+3) Exp.
a The units of energy are kJ mol−1. Energies defined relative to the infinitely separated ground-state atoms. b CBS energies at CC/AVTZ optimized geometries. c Ref. 63. d Ref. 61. e Ref. 61 and 63. f Ref. 64.
C2(1Σ+g) + C(3P) + H(2S) −608.9 −620.3 −618.6 ± 2.1c
C2(3Πu) + C(3P) + H(2S) −598.0 −612.0 −610.6 ± 2.1c
CH(2Π) + 2C(3P) −355.1 −357.3 −351.2 ± 0.4d
C2(1Σ+g) + CH(2Π) −963.9 −971.0 −969.8 ± 2.5e
C2(3Πu) + CH(2Π) −953.1 −963.3 −961.2 ± 2.5e
[small script l]-C3(1Σ+g) + H(2S) −1338.2 −1373.9 −1324.1 ± 13.0c
c-C3(3A2′) + H(2S) −1255.5 −1279.5
[small script l]-C2H(2Σ+) + C(3P) −1108.5 −1107.3 −1110.7 ± 5.5f

3.2 Four-body energy term

Once the V(2+3)(R) PES is obtained, the (effective) four-body term ε(4)(R) can then be expressed as
ε(4)(R) = E(R) − V(2+3)(R),(8)
where E(R) is the CC/CBS(d,t) interaction energy (defined relative to the infinitely separated atoms) and V(2+3)(R) is the lowest characteristic value of eqn (5).

In CHIPR, ε(4)(R) [eqn (8)] can be conveniently modeled by45,46

image file: c9cp04890a-t5.tif(9)
where Ci,j,k,l,m,n are expansion coefficients of a Lth-degree polynomial (i + j + k + l + m + nL) and yp (p = 1, 2,…, 6) are transformed coordinates (see below). Note that, in the above equation, only those C's that excite at least four modes and satisfy ijklmn are included in the first summation. Following ref. 45 and 46, this former constraint is here specially devised so as to avoid any two- and three-body contributions. In eqn (9), the second summation runs over all permutation elements gG, where G is a subgroup of the [scr S, script letter S]4 symmetric group.65 Thus, for an AB3-type molecule, [scr P, script letter P](i,j,k,l,m,n)g are the corresponding operators that reflect the action of the particle permutations [in cyclic notation65] ([thin space (1/6-em)]), (3, 4), (2, 3), (2, 4), (2, 3, 4) and (2, 4, 3) onto the exponent set {i, j, k, l, m, n} brought by the first summation of eqn (9). These actions generate therefore the required symmetrized sums of monomials and make V(4)(R) [eqn (9)] invariant with respect to all permutations of like atoms.45,46

Like electronic structure theory,51 each yp in eqn (9) is in CHIPR expanded as a distributed-origin contracted basis set45,46

image file: c9cp04890a-t6.tif(10)
where ϕp,α is expressed either as45,46
ϕ[1]p,α = sechη(γp,αρp,α),(11)
image file: c9cp04890a-t7.tif(12)

In the above equations, ρp,α = RpRrefp,α defines the displacement coordinate Rp from the αth primitive's origin Rrefp,α, γp,α are non-linear parameters and η = 1, σ = 6 and β = 1/5 are constants.45,46 As usual,47 both ϕ[1]p,α and ϕ[2]p,α were employed in eqn (10), with the latter appearing only once as the last term in the summation. Note that all distributed origins Rrefp,α in eqn (11) and (12) are assumed related by45

Rrefp,α = ζ(Rrefp)α−1,(13)
where ζ and Rrefp are adjustable parameters.

The first step in CHIPR consists of calibrating the above contracted basis sets [eqn (10)]. For this, we have performed 1D CC/CBS(d,t) cuts along the C–C and C–H bond lengths of a previously selected c-C3H reference geometry; this is made up of a D3h structure of C3 (RCC = 2.596a0) and the experimental27 CH bond distance (RCH = 2.033a0). With the above data, the parameters of eqn (10)–(13) have then been determined from a fit to the resulting four-body energies. The final bases, referred to as CHb8 and CCb8, are depicted in Fig. 2. Note that their corresponding non-linear parameters have been optimized only once and are kept fixed in all subsequent steps.

image file: c9cp04890a-f2.tif
Fig. 2 C–H (y1y3) and C–C (y4y6) contracted basis sets of eqn (10) as determined from a fit to CC/CBS(d,t) four-body energies [ε(4) in eqn (8)]; see the text. The calibrating c-C3H reference geometry is made up of a D3h structure of C3 (RCC = 2.596a0) and the experimental27 CH bond distance (RCH = 2.033a0). Solid diamonds indicate the origins (Rrefp,α) of each primitive basis ϕp,α in eqn (10)–(13).

Having CHb8 and CCb8, the second step in CHIPR is the determination of the polynomial coefficients in eqn (9). To accomplish this, we have first set all cα's to their previously optimized values and the C coefficients [eqn (9)] to unity. The fitting solution so found was then employed as an initial guess for a subsequent optimization in which all parameters (including cα's) were varied freely. The final optimized contracted bases are also shown in Fig. 2. For the weights (W), we employ the following function

image file: c9cp04890a-t8.tif(14)
where ΔE = EEmin, β = 1/2, Emin = −0.645Eh and E0 = −0.62Eh, with larger weights attributed to minima (min), transition states (ts) and minimum energy paths (MEPs).

With such an approach, all 42[thin space (1/6-em)]538 ab initio points could be least-squares fitted to eqn (9) with a root mean square deviation (rmsd) of 8.0 kJ mol−1. This involves a total of 360 linear coefficients in the polynomial expansion (L = 10), and basis set contractions of the order M = 3 (8 parameters each), thence amounting to 113 fitted points per parameter; see the ESI to assess the numerical coefficients of all parameters outcoming from the fit. Shown in Fig. S2 (ESI) is the sensitivity of the energy fit with respect to the various parameters. In particular, this shows how the second derivative of χ2 [sum of the squared deviations of eqn (9) from the ab initio computed energies] varies for the first row that interrelates the first to all other polynomial coefficients. As one might expect, the sensitivity generally decreases with the total order of the polynomial coefficient. In turn, Table 2 displays the stratified rmsd and CC diagostic tests66–68 using the AVTZ basis set (see also Fig. S3–S5 of the ESI), while Fig. 3 shows the distribution of errors in the fitted data set. As noted, although the stratified diagnostics deviate slightly from their recommended values (in accordance with previous results39), i.e., T1 ≤ 0.02,66D1 ≤ 0.0567 and image file: c9cp04890a-t9.tif,68 the use of multireference electron correlation methods for C3H would make the task of calculating its global PES computationally unaffordable. Indeed, the relative cost between single point MRCI+Q/CBS(D, T) and CC/CBS(d,t) calculations amounts to ≈1200. This justifies our cost-effective way.

Table 2 Stratified root-mean-square deviations (in kJ mol−1) of the final PES
Energya N rmsd [T with combining macron] 1 [D with combining macron] 1 [T with combining macron] 1/[D with combining macron]1c
a The units of energy are kJ mol−1. Energy strata defined relative to the c-C3H global minimum with energy of −0.6427Eh. b Number of calculated points up to the indicated energy range. c Mean values of T1, D1 and their ratio within each energy stratum. Values taken from CC/AVTZ calculations.
50 1829 4.3 0.034 0.097 0.355
100 8100 5.6 0.036 0.098 0.363
150 14[thin space (1/6-em)]680 7.1 0.036 0.098 0.364
200 19[thin space (1/6-em)]699 7.6 0.035 0.098 0.363
300 24[thin space (1/6-em)]859 7.6 0.035 0.097 0.364
400 29[thin space (1/6-em)]052 7.8 0.035 0.096 0.368
500 30[thin space (1/6-em)]820 8.0 0.035 0.096 0.369
1000 42[thin space (1/6-em)]104 8.0 0.039 0.103 0.383
2000 42[thin space (1/6-em)]483 8.0 0.039 0.103 0.382
3000 42[thin space (1/6-em)]538 8.0 0.040 0.104 0.382

image file: c9cp04890a-f3.tif
Fig. 3 Distribution of deviations between calculated and fitted ab initio points. The curve indicates a normal distribution adjusted to the data where 80% of the errors fall within the rmsd.

4 Features of PESs

All major features of the CHIPR PES are depicted in Fig. 4–9; for the properties of its stationary points, see Tables 3–6, where the most recent/representative results available from the literature12,21,26,27,39–42,69 are also shown.
image file: c9cp04890a-f4.tif
Fig. 4 Contour plot for an H atom moving around partially relaxed c-C3 with center-of-mass fixed at the origin. Contours are equally spaced by 0.0135Eh starting at −0.65Eh. The zero of energy is set relative to the infinitely separated atoms. The insets show (a) optimized 1D cuts and (b) minimum energy paths (where s is the reaction coordinate in mass-scaled a.u.) for the isomerization between symmetry equivalent c-C3H structures. In (b), the zero of energy corresponds to c-C3H. Solid dots indicate ab initio CC/CBS(d,t) points while the dotted lines represent the dissociation energy limits predicted from V(2+3)(R).
Table 3 Vertical excitation energies (Te in kJ mol−1) for some low-lying excited states of c- and [small script l]-C3H as obtained from two-state CASSCF(13,13)-MRCI+Q/CBS(D,T)60 calculations using the C2v point group. The electronic states are labeled according to C2v irreducible representations with the corresponding correlations with C∞v and/or Cs also given in parenthesis. The energies are calculated at the corresponding experimental (ground-state) equilibrium geometries22,27 (see also Table 4)
c-C3H [small script l]-C3H
State Γ el T e State Γ el T e
a Electronic symmetry species. b MRCI+Q/CBS(D,T) values relative to the ground-state c-C3H minimum. c Experimental gas-phase values reported in ref. 26. d Experimental gas-phase 2Π → 2A′′(2Δ) transition reported in ref. 28.
1 2B2(2A′) 0.0 1 2B2(2Π/2A′), 2B1(2Π/2A′′) 14.5
2 2A1(2A′) 113.2 2 2A1(2Δ/2A′), 2A2(2Δ/2A′′) 281.0
[96.8 ± 1.2]c [268]d
3 2B1(2A′′) 326.3 3 2A2(2Σ/2A′′) 311.9
4 2A2(2A′′) 369.3 4 2A1(2Σ+/2A′) 346.1
5 2A2(2A′′) 437.5 5 2B2(2Π/2A′), 2B1(2Π/2A′′) 370.7
6 2B1(2A′′) 439.3
7 2A1(2A′) 478.2
8 2B2(2A′) 678.2

Accordingly, the predicted global mininum on the ground-state doublet of the PES corresponds to a cyclic C2v geometry, c-C3H (Fig. 4). As shown in Table 4, its predicted structural parameters are in excellent agreement with the ones observed via microwave spectroscopy.27 A close agreement is also found between the CHIPR's c-C3H data and the recently reported CC-F12/AVTZ results of Bennedjai et al.40 as well as the predicted QFF geometry of Bassett and Fortenberry.39 The latter includes, in addition to CBS energies, contributions from core–core/core–valence electron correlation and scalar relativistic effects.39 Moreover, with the exception of the C–C asymmetric stretching mode (w4), all harmonic frequencies are well reproduced by CHIPR. Note that w4 corresponds to the c-HC3([X with combining tilde]2B2/12A′) symmetry lowering (from C2v to Cs) which, according to previous work,27,36,38 is subject to vibronic mixing with the next 2A1/22A′ electronic state. Table 3 gives our best estimate of this excitation energy, namely ≈113.2 kJ mol−1 above [X with combining tilde]2B2. For the benefit of the reader, we also include exploratory estimates of stationary strutures for the lowest 7 excited states of c/[small script l]-HC3 (note that only geometries close to the one of equilibrium c-HC3 or [small script l]-HC3 have been explored). Although the experimental ν4 value is itself largely questioned39 (it is derived from rotational distortions rather than vibrational observations;27 this is also true for the other experimental IR bands26), difficulties in describing w4 seem to appear also in the case of CC-F12 wave functions (see Table 4). Nevertheless, as noted in various occasions,38–40 extrapolations to the CBS limit tend to favor the highest symmetry C2v structure over the (double minimum) Cs distorted ones. As shown in Fig. 4, the isomerization between the three symmetry-equivalent c-C3H structures occurs via a (thus far unreported) C2v transition state of cc-C3H. This form is predicted from CHIPR to be located 197.7 kJ mol−1 above c-C3H with an imaginary frequency of 1693.9 cm−1 along the H wagging motion (through c-C3's center-of-mass). The corresponding classical barrier heights calculated from CC/CBS(d,t) and MRCI+Q/CBS(D,T) protocols [for the description of the latter, see ref. 60] are estimated to be 194.2 and 188.8 kJ mol−1, respectively (Table 5). In fact, a close look at inset (b) of Fig. 4 shows that the CHIPR form reproduces accurately the MEP at the CC/CBS(d,t) level.

Table 4 Structural parameters (distances in a0, angles in deg) and harmonic (wi) frequencies (in cm−1) of minima on the C3H ground-state PES. Energies (ΔE) in kJ mol−1 refer to global c-C3H minima
Structure Propertya CHIPRb CCc MRCI+Qd CC-F12e QFFf Exp.
a This work. spg stands for the symmetry point group. b This work. c This work, with the AVTZ basis set. ΔE values in parenthesis are CC/CBS(d,t) at CHIPR stationary points. d This work, with the AVTZ basis set. Geometries at the single-state CASSCF(13,13)/AVTZ level. ΔE values in parenthesis are MRCI+Q/CBS(D,T) at CHIPR stationary points; see ref. 60 for the CBS protocol. e Ref. 40, with the AVTZ basis set. f Local PESs of ref. 39 (for c-C3H) and ref. 41 (for [small script l]-C3H). g Ref. 27 and 12. h Ref. 26. Gas-phase values. i Ref. 22. Gas-phase values. j Ref. 69 and 25. Measured in the Ar matrix. k Ref. 21. Estimated fundamentals in the absence of Renner–Teller and spin–orbit effects. l Ref. 21. ν4(2Σμ) fundamental bands.
c-C3H R HC1 2.034 2.038 2.012 2.039 2.036 2.033g
R C1C2 2.587 2.526 2.531 2.600 2.595 2.596g
R C2C3 2.557 2.622 2.654 2.608 2.590 2.602g
∠HC1C2 150.4 156.5 157.0 149.9 150.1 149.9g
∠C1C2C3 60.4 64.0 64.5 59.9 60.1 59.9g
w 1 (CH stretch) 3052.9 3269.9 3433.5 3246.0 3256.8
w 2 (CC sym. stretch) 1381.5 1159.4 1108.8 1578.0 1588.6 1613 ± 25h
w 3 (CCH sym. bend) 1754.7 1606.0 1566.8 1214.0 1219.5 1161 ± 25h
w 4 (CC asym. stretch) 1327.9 919.4 884.8 2639.0 619.5 508 ± 25g,h
w 5 (H wag) 951.0 894.9 840.6 900.0 1224.8
w 6 (H out of plane) 878.8 819.2 713.8 858.0 863.1
Spg C 2v C s C s C 2v C 2v C 2v
ΔE 0.0 0.0 (0.0) 0.0 (0.0) 0.0
[small script l]-C3H R HC1 1.994 2.017 2.033 2.013 2.022 1.922i
R C1C2 2.346 2.366 2.367 2.351 2.372 2.370i
R C2C3 2.519 2.539 2.578 2.538 2.561 2.506i
∠HC1C2 180.0 164.4 175.2 180.0 180.0
∠C1C2C3 180.0 175.9 178.2 180.0 180.0
w 1 (CH stretch) 3372.5 3403.7 3543.7 3428.0 3600.0 3238j
w 2 (CCC asym. stretch) 1856.5 1844.4 1807.1 1870.0 1830.0 1839 ± 10,h 1824.7j
w 3 (CCC sym. stretch) 1094.5 1137.7 1137.7 1136.0 1160.0 1167, 1159.8j
w 4 (CCH bend) 247.7 223.9, 236.8 219.6, 297.6 230.0 202.0 600,k 20.3l
w 5 (CCC bend) 302.6 371.7 355.8 371.0 374.0 300k
Spg C ∞v C s C s C ∞v C ∞v C ∞v
ΔE 14.4 7.3 (11.8) 0.8 (5.3) 15.4

Table 5 Attributes of transition states on the C3H ground-state PES. Properties and units as in Table 4
Propertya cc-C3H [small script l][small script l]-C3H
a This work. spg stands for the symmetry point group. b This work. c This work, calculated with the AVTZ basis set. ΔE values in parenthesis are CC/CBS(d,t) at CHIPR stationary points. d This work, calculated using the AVTZ basis set. Geometries at the single-state CASSCF(13,13)/AVTZ level. ΔE values in parenthesis are MRCI+Q/CBS(D,T) at CHIPR stationary points; see ref. 60 for the CBS protocol. e Ref. 42, calculated at the CC/6-311+G(3df,2p)//B3LYP/6-311G(d,p) level.
R HC1 2.361 2.443 2.461 2.226 2.142 2.150
R C1C2 2.540 2.537 2.555 2.533 2.559 2.589
R C2C3 2.540 2.537 2.555 5.065 5.081 5.096
∠HC1C2 103.5 101.7 100.8 89.1 96.8 100.2
∠C1C2C3 73.2 76.6 77.4 0.9 6.8 10.2
w 1 1693.9i 1676.2i 1616.9i 1111.7i 649.0i 1115.8i
w 2 751.1 637.7 616.5 164.0 91.2 307.2
w 3 879.9 797.3 840.1 350.1 302.6 429.9
w 4 1177.2 1348.1 1325.8 1094.5 1060.7 1023.3
w 5 2087.2 1630.1 1584.5 1512.8 1494.7 1279.0
w 6 2380.0 2242.0 2189.6 2581.7 2812.6 2754.3
Spg C 2v C 2v C 2v C 2v C 2v C 2v
ΔE 197.7 185.3 (194.2) 178.3 (188.8) 256.3 243.7 (258.1) 233.9 (251.1)

Propertya [small script l]c-C3H
R HC1 2.036 2.046 2.068 2.050
R C1C2 2.446 2.494 2.514 2.481
R C2C3 2.586 2.524 2.542 2.502
∠HC1C2 173.6 140.7 140.2 139.9
∠C1C2C3 94.0 90.2 89.1 89.4
d HC1C2C3 180.0 129.4 131.8 132.3
w 1 796.1i 1343.2i 1271.8i
w 2 160.7 618.0 586.5
w 3 443.6 690.1 704.2
w 4 1265.1 1394.2 1364.8
w 5 1523.9 1517.6 1479.7
w 6 3039.5 3216.5 3099.4
Spg C s C 1 C 1 C 1
ΔE 120.0 115.7 (122.1) 110.0 (123.8) 114.6

Besides c-C3H, CHIPR predicts the linear isomer [small script l]-C3H to be a local minimum in the ground-state of the PES lying 14.4 kJ mol−1 above the cyclic structure; see Fig. 5–7. As shown in Table 4, the corresponding energy differences at the CC/CBS(d,t), CC-F12/AVTZ40 and MRCI+Q/CBS(D,T) levels are 11.8, 15.4 and 5.3 kJ mol−1, respectively. Note that the CHIPR's [small script l]-C3H structural parameters agree well with the experimental (vibrationally averaged) r0 values reported by Kanada et al.22 [at experimental geometries, our best estimate places [small script l]-C3H about 14.5 kJ mol−1 above c-C3H (Table 3)]. Discrepancies, however, should a priori be expected in the description of the C–C–H and C–C–C degenerate bending modes (Table 4) whose vibrational angular momentum is shown22,23 to couple with the 2Π (total) electronic angular momentum. As first remarked by Kanada et al.22 and later by Perić et al.,41 such a strong RT effect makes the lowest 2A′ PES very flat along w4 with the appearance of a quasi-linear C3H molecule, depending on the chosen ab initio method; see, e.g., Table S4 (ESI) for the truncated-space CASSCF/MRCI+Q estimates. Here, in accordance with the recent CC-F1240 calculations and ref. 28, the extrapolations to the CBS limit tend to favor the highest symmetry C∞v species (Table 4). Suffice to add that the predicted structural parameters and harmonic frequencies for [small script l]-C3H are also in close agreement with the ones derived from the local 2A′ PES studied by Perić et al.41 Note that, as for c-C3H, the experimental [small script l]-C3H's IR bands but ν4(2Σμ) are largely uncertain.21,25,26,69

image file: c9cp04890a-f5.tif
Fig. 5 Contour plot for an H atom moving around partially relaxed [small script l]-C3 which lies along the x axis with the central C atom at the origin. Contours are equally spaced by 0.0135Eh starting at −0.65Eh. The zero of energy is set relative to the infinitely separated atoms. The insets show (a) optimized 1D cuts and (b) minimum energy paths (where s is the reaction coordinate in mass-scaled a.u.) for the isomerization between symmetry equivalent [small script l]-C3H structures. In (b), the zero of energy corresponds to [small script l]-C3H. Solid dots indicate ab initio CC/CBS(d,t) points while the dotted lines represent the dissociation energy limits predicted from V(2+3)(R).

image file: c9cp04890a-f6.tif
Fig. 6 Contour plots for the CH and C2 collinear reactions yielding (a) [small script l]-C3 + H and (b) [small script l]-C2H + C. Contours are equally spaced by 0.015Eh, starting at −0.65Eh. In the insets, the corresponding optimized 1D cuts are shown. Solid dots indicate ab initio CC/CBS(d,t) points while dotted lines represent the dissociation energy limits predicted from V(2+3)(R). The zero of energy is set relative to the infinitely separated atoms.

image file: c9cp04890a-f7.tif
Fig. 7 Contour plot for a C atom moving around a partially relaxed [small script l]-C2H which lies along the x axis with the C atom kept at the origin. Contours are equally spaced by 0.02Eh starting at −0.65Eh. The zero of energy is set relative to the infinitely separated atoms. Shown in the inset are the corresponding optimized 1D cuts. Solid dots indicate ab initio CC/CBS(d,t) points while the dotted line represents the dissociation energy limit predicted from V(2+3)(R).

Clearly visible from Fig. 5 is the presence of a T-shaped (C2v) transition state, hereafter denoted as [small script l][small script l]-C3H, which is responsible for the degenerate isomerization between symmetry-equivalent [small script l]-C3H structures; its imaginary frequency amounts to 1111.7 cm−1 and points toward the H wagging motion through [small script l]-C3 center-of-mass. Such a hitherto unreported form lies about 241.9 and 256.3 kJ mol−1 above [small script l]-C3H and c-C3H, respectively (Table 5). The corresponding isomerization barriers predicted from CC/CBS(d,t) and MRCI+Q/CBS(D,T) protocols are 246.3 and 245.8 kJ mol−1. Again, Fig. 5 [inset (b)] evinces the reliability of the CHIPR form in reproducing the CC/CBS(d,t) MEP for this process. In turn, as noted in Fig. 8(a) and (b), the isomerization between c-C3H and [small script l]-C3H occurs via [small script l]c-C3H transition states14,42 whose imaginary frequency (of 796.1 cm−1) points along the C–C bond breaking/forming process; see Fig. 9 to assess all geometries of the stationary structures discussed here. The classical barrier height is predicted to be 120.0 kJ mol−1 relative to c-C3H; the corresponding CC/CBS(d,t) and MRCI+Q/CBS(D,T) values are 122.1 and 123.8 kJ mol−1, respectively.

image file: c9cp04890a-f8.tif
Fig. 8 (a) Contour plot for CH moving around a C2 diatom with the center-of-mass fixed at the origin. All degrees of freedom but R and θ are partially relaxed as shown in the inset. Black solid and gray dotted contours are equally spaced by 0.0075 and 0.00075Eh starting at −0.65 and −0.385Eh, respectively. The zero of energy is set relative to the infinitely separated atoms. (b) Minimum energy path (s is the reaction coordinate in mass-scaled a.u.) for isomerization between c-C3H and [small script l]-C3H via the transition state of [small script l]c-C3H. The zero of energy corresponds to c-C3H.

image file: c9cp04890a-f9.tif
Fig. 9 Pathways connecting the stationary points on the CHIPR PES. All energies (in kJ mol−1) refer to the global c-C3H minimum. Dotted lines connect dissociative channels (in green), while solid ones connect isomeric structures (red for minima, blue for transition states).

Fig. 9 summarizes in a comprehensive way all major topographical attributes of the CHIPR form reported here. Fig. 4–8 accurately reproduce all known valence features of the PES, their isomerization pathways and permutational symmetries by built-in construction, and also describes in a physically reasonable manner all long-range structures and the correct asymptotic behavior at dissociation. This is no doubt achieved by the use of the MBE/CHIPR methods10,45,46 with appropriate [scr P, script letter P](i,j,k,l,m,n)g operators in eqn (9). Indeed, as Fig. 8, 9 and Table 6 show, CHIPR also predicts long-range stationary points [(HC2⋯C)min, (HC2⋯C)ts, (C2⋯HC)min and (C2⋯HC)ts] with structural parameters and energetics in close agreement with the ones obtained at CASSCF/AVTZ and MRCI+Q/CBS(D,T) levels. Such forms arise from the perpendicular attack of C to the central C atom of [small script l]-C2H and CH to the C2 diatom, both toward c-C3H formation (see Fig. 9). In addition, Fig. 6 and 9 show that the formation of [small script l]-C3H is barrierless for the collinear reactions of C2 + CH, [small script l]-C3 + H and C + [small script l]-C2H, in accordance with ref. 42. Moreover, the c-C3H adduct is formed without any barrier by the perpendicular approach of H to both c-C3(3A2′) and [small script l]-C3(1Σ+g) [in this case, the attack involves more than one reaction coordinate];42 the overall exotermicities of all such processes can be assessed from Fig. 9.

Table 6 Long-range attributes of the C3H ground-state PES. Properties and units as in Table 4
Propertya (HC2⋯C)min (HC2⋯C)ts
a This work. spg stands for the symmetry point group. b This work. c This work, calculated with the AVTZ basis set. Geometries at the single-state CASSCF(13,13)/AVTZ level. ΔE values in parenthesis are MRCI+Q/CBS(D,T) at CHIPR stationary points; see ref. 60 for the CBS protocol.
R HC1 2.008 1.938 2.004 1.940
R C1C2 2.281 2.081 2.317 2.195
R C2C3 6.960 6.459 6.873 5.980
∠HC1C2 179.0 180.0 165.2 150.0
∠C1C2C3 68.0 59.5 52.8 48.0
w 1 119.8 111.7 310.3i 149.0i
w 2 138.8 211.5 154.9 165.4
w 3 403.5 425.0 372.3 329.8
w 4 444.9 430.8 539.9 420.9
w 5 1817.0 2060.0 1384.2 1907.8
w 6 3430.1 3463.9 3284.3 3476.2
Spg C s C s C s C s
ΔE 567.2 552.0 (568.2) 569.5 555.0 (578.5)

Propertya (C2⋯HC)min (C2⋯HC)ts
R HC1 2.104 1.952 2.118 1.918
R C1C2 7.533 7.325 7.108 6.808
R C2C3 2.439 2.472 2.443 2.493
∠HC1C2 9.3 15.3 60.9 70.3
∠C1C2C3 80.7 70.5 80.1 67.5
d HC1C2C3 0.0 0.0 −80.4 −85.2
w 1 92.7 108.8 88.9i 193.5i
w 2 171.7 268.5 100.4 61.5
w 3 200.3 298.9 122.1 114.3
w 4 217.1 313.0 215.7 303.0
w 5 844.5 1790.0 1071.4 1615.4
w 6 3016.7 2988.2 2926.3 2859.3
Spg C 2v C 2v C 2v C 2v
ΔE 700.0 695.0 (715.6) 700.3 699.4 (716.1)

5 Preliminary QCT dynamics study

In this section, we test the reliability of the CHIPR form through preliminary calculations of rate constant for the reaction C2(v,j) + CH(v′,j′) → C3 + H by running quasi-classical trajectories (QCT)70 on the novel PES; a locally modified version of the VENUS96C code71 has been employed throughout. All calculations used a time step of 0.075 fs with the reactants initially separated by 8 Å, while the maximum value of the impact parameter is bmax = 6 Å as optimized by trial and error. Average reaction cross sections were then obtained (for a given temperature T) as70σR(T)〉 = πbmax2NR/N, and the 68% associated errors by Δ〈σR(T)〉 = 〈σR(T)〉 [(NNR)/(NNR)]1/2, where NR is the number of reactive trajectories out of a total of N (104 per temperature) that were run. In turn, thermal rate constants for the formation of C3 + H were calculated as70
image file: c9cp04890a-t10.tif(15)
with the estimated 68% error being given by Δk(T) = k(T)[(NNR)/(NNR)]1/2; kB is the Boltzmann constant, μCH+C2 is the reduced mass of the reactants and ge = 1/12 is the electronic degeneracy factor. The calculated results for temperatures up to 4000 K are shown in Fig. 10 and 11.

image file: c9cp04890a-f10.tif
Fig. 10 Cross sections and associated error bars as a function of temperature. The line shows the curve fitted to QCT data, eqn (16). For clarity, the results are shown with 99.6% [3σ ≡ 3Δ〈σR(T)〉] error bars.

image file: c9cp04890a-f11.tif
Fig. 11 Rate constants and associated error bars for the C2 + CH reaction at temperatures up to 4000 K. Also shown are the experimental estimates of Dean and Hanson72 and available data from KIDA kinetics database.73 The lines show the predicted QCT thermally averaged results from the excitation function in eqn (16)–(18). For clarity, the QCT results are shown with 99.6% [3σ ≡ 3Δk(T)] error bars.

To further explore the dependence of 〈σR〉 with respect to temperature, we have considered the following model function74

σR(T)〉 = (a + bT + cT2)Tn[thin space (1/6-em)]exp(−mT),(16)
where a, b, c, n, and m are parameters to be adjusted to the QCT calculated cross-sections; they are numerically defined in Table 7, with the final form of 〈σR(T)〉 plotted in Fig. 10. In turn, the predicted thermally averaged k(T)'s arising from substituting eqn (16) into (15) are shown in Fig. 11. Clearly, the model function of eqn (16) accurately fits the calculated 〈σR(T)〉's. As is visible from Fig. 10, the excitation function at T ≈ 100 K displays a maximum at very low collision energies, although no apparent barrier exists along the minimum energy path for the reaction. A possible implication is that other regions of the PES close to the minimum energy path are sampled by the trajectories, thus requiring a tiny barrier to be overcome. As seen, a good agreement is found between the calculated and predicted rate constants (see Fig. 11). In the high temperature range (2500 ≤ T/K ≤ 3800), the QCT calculated rate constants are predicted to be about 38% lower than the experimental estimates of Dean and Hanson.72 Conversely, at ≈280 K, our calculations agree remarkably well with the value of 6.30 × 10−11 cm3 molecule−1 s−1 (believed valid for the range of 10–280 K) recommended by the KIDA kinetics database;73 however, this clearly diverges for lower T. In fact, as seen from Fig. 11, the calculated rate constant is predicted to increase as a function of temperature, thus showing a positive slope. Specifically, it is well approximated by (in cm3 molecule−1 s−1)
k(T) = 6.98 × 10−11(T/300)0.45(17)
for temperatures up to ≈500 K. A similar dependence (T0.6) has also been found experimentally for the case of the barrierless reaction between C2 and the (CH-congenere) N atom.75 However, in high temperature regimes, it is predicted to be approximately constant,
k(T) ≅ 1.02 × 10−10 cm3 molecule−1 s−1(18)
over the 2000–4000 K interval; eqn (17) and (18) are also plotted in Fig. 11. A final remark to note is that at 10 K, a temperature of relevance in cold dense clouds, the model function of eqn (16) yields [using eqn (15)] a rate constant of 1.18 × 10−11 cm3 molecule−1 s−1 for the reaction C2(v,j) + CH(v′,j′) → C3 + H. This is compared with the value of15 2.23 × 10−10 cm3 molecule−1 s−1 (at 10 K) for the C + C2H2 → H2 + C3 reaction, a well established (neutral–neutral) source of carbon trimer in the ISM.15,42

Table 7 Parameters for 〈σR(T)〉 in eqn (16)
Parameter Valuea
a The power of 10 is in parenthesis.
a2 Kn 7.52 (+1)
b2 K−(1+n) −1.25 (−2)
c2 K−(2+n) 7.52 (−6)
n 7.51 (−2)
m/K−1 4.85 (−4)

6 Conclusions

We have reported the first global PES for the ground-state doublet C3H using accurate CBS extrapolated ab initio energies and the CHIPR method for modelling. The novel PES is based on a MBE-type development with the two-body and three-body energy terms from previously reported double-MBE potentials for C2H(2A′)61 and C3(1A′,3A′),60,62 while the effective four-body energy term is modelled using the CHIPR formalism for AB3-type tetratomics. The final form reproduces accurately all known stationary structures of HC3 and their interconversion pathways, some unreported thus far to the best of our knowledge. Besides describing properly long-range interactions at all asymptotic channels and permutational symmetry by built-in construction, the PES reproduces reasonably well all exothermicities at dissociation regions – the best one can possibly afford with the above two- and three-body potentials and our cost-effective ab initio approach. By running exploratory quasi-classical trajectory calculations for the reaction C2 + CH → C3 + H, thermalized rate coefficients for temperatures up to 4000 K are also reported.

Conflicts of interest

There are no conflicts to declare.


This work is supported by the Fundação para a Ciência e a Tecnologia and Coimbra Chemistry Centre, Portugal, in the framework of the project “MATIS – Materiais e Tecnologias Industriais Sustentáveis” (reference: CENTRO-01-0145-FEDER-000014), co-financed by the European Regional Development Fund (FEDER), through the “Programa Operacional Regional do Centro” (CENTRO2020) program. C. M. R. R. also thanks the CAPES Foundation (Ministry of Education of Brazil) for a scholarship (BEX 0417/13-0). The support from A. J. C. V. from the China's Shandong Province “Double-Hundred Talent Plan” (2018) is also appreciated.


  1. B. A. McGuire, ApJS, 2018, 239, 17 CrossRef CAS.
  2. R. I. Kaiser, Chem. Rev., 2002, 102, 1309–1358 CrossRef CAS PubMed.
  3. I. W. M. Smith, E. Herbst and Q. Chang, Mon. Notices Royal Astron. Soc., 2004, 350, 323–330 CrossRef CAS.
  4. I. W. M. Smith, A. M. Sage, N. M. Donahue, E. Herbst and D. Quan, Faraday Discuss., 2006, 133, 137–156 RSC.
  5. J.-C. Loison, V. Wakelam, K. M. Hickson, A. Bergeat and R. Mereau, Mon. Notices Royal Astron. Soc., 2014, 437, 930–945 CrossRef CAS.
  6. T. J. Millar and E. Herbst, A&A, 1994, 288, 561–571 Search PubMed.
  7. R. J. Hargreaves, K. Hinkle and P. F. Bernath, Mon. Notices Royal Astron. Soc., 2014, 444, 3721 CrossRef CAS.
  8. M. Agúndez, J. Cernicharo, G. Quintana-Lacaci, A. Castro-Carrizo, L. Velilla Prieto, N. Marcelino, M. Guélin, C. Joblin, J. A. Martín-Gago, C. A. Gottlieb, N. A. Patel and M. C. McCarthy, A&A, 2017, 601, A4 Search PubMed.
  9. NIST Standard Reference Database Number 101, http://cccbdb.nist.gov/, Release 16a edn, 2003.
  10. J. N. Murrell, S. Carter, S. C. Farantos, P. Huxley and A. J. C. Varandas, Molecular Potential Energy Functions, John Wiley & Sons, Chichester, 1984 Search PubMed.
  11. M. Agúndez and V. Wakelam, Chem. Rev., 2013, 113, 8710–8737 CrossRef PubMed.
  12. S. Yamamoto, S. Saito, M. Ohishi, H. Suzuki, S.-I. Ishikawa, N. Kaifu and A. Murakami, ApJ, 1987, 322, L55–L58 CrossRef CAS.
  13. P. Thaddeus, C. A. Gottlieb, A. Hjalmarson, L. E. B. Johansson, W. M. Irvine, P. Friberg and R. A. Linke, ApJ, 1985, 294, L49–L53 CrossRef CAS.
  14. J.-C. Loison, M. Agúndez, V. Wakelam, E. Roueff, P. Gratier, N. Marcelino, D. N. Reyes, J. Cernicharo and M. Gerin, Mon. Notices Royal Astron. Soc., 2017, 470, 4075–4088 CrossRef CAS PubMed.
  15. K. M. Hickson, J.-C. Loison and V. Wakelam, Chem. Phys. Lett., 2016, 659, 70–75 CrossRef CAS.
  16. D. C. Clary, E. Buonomo, I. R. Sims, I. W. M. Smith, W. D. Geppert, C. Naulin, M. Costes, L. Cartechini and P. Casavecchia, J. Phys. Chem. A, 2002, 106, 5541–5552 CrossRef CAS.
  17. R. I. Kaiser, C. Ochsenfeld, M. Head-Gordon, Y. T. Lee and A. G. Suits, Science, 1996, 274, 1508–1511 CrossRef CAS PubMed.
  18. T. Furtenbacher, I. Szabó, A. G. Császár, P. F. Bernath, S. N. Yurchenko and J. Tennyson, ApJS, 2016, 224, 44–58 CrossRef.
  19. A. G. G. M. Tielens, Annu. Rev. Astron. Astrophys., 2008, 46, 289–337 CrossRef CAS.
  20. C. A. Gottlieb, J. M. Vrtilek, E. W. Gottlieb, P. Thaddeus and A. Hjalmarson, ApJ, 1985, 294, L55–L58 CrossRef CAS.
  21. S. Yamamoto, S. Saito, H. Suzuki, S. Deguchi, N. Kaifu, S.-I. Ishikawa and M. Ohishi, ApJ, 1990, 348, 363–369 CrossRef CAS.
  22. M. Kanada, S. Yamamoto, S. Saito and Y. Osamura, J. Phys. Chem., 1996, 104, 2192–2201 CrossRef CAS.
  23. M. Caris, T. Giesen, C. Duan, H. Muller, S. Schlemmer and K. Yamada, J. Mol. Spectrosc., 2009, 253, 99–105 CrossRef CAS.
  24. M. C. McCarthy, K. N. Crabtree, M.-A. Martin-Drumel, O. Martinez, B. A. McGuire and C. A. Gottlieb, ApJS, 2015, 217, 10 CrossRef.
  25. Q. Jiang, C. M. L. Rittby and W. R. M. Graham, J. Chem. Phys., 1993, 99, 3194–3199 CrossRef CAS.
  26. S. M. Sheehan, B. F. Parsons, J. Zhou, E. Garand, T. A. Yen, D. T. Moore and D. M. Neumark, J. Chem. Phys., 2008, 128, 034301 CrossRef PubMed.
  27. S. Yamamoto and S. Saito, J. Chem. Phys., 1994, 101, 5484–5493 CrossRef CAS.
  28. H. Ding, T. Pino, F. Güthe and J. P. Maier, J. Chem. Phys., 2001, 115, 6913–6919 CrossRef CAS.
  29. H. Yamagishi, H. Taiko, S. Shimogawara, A. Murakami, T. Noro and K. Tanaka, Chem. Phys. Lett., 1996, 250, 165–170 CrossRef CAS.
  30. J. Takahashi and K. Yamashita, J. Chem. Phys., 1996, 104, 6613–6627 CrossRef CAS.
  31. K. Aoki, S. Ikuta and A. Murakami, J. Mol. Struct. THEOCHEM, 1996, 365, 103–110 CrossRef CAS.
  32. C. Ochsenfeld, R. I. Kaiser, Y. T. Lee, A. G. Suits and M. Head-Gordon, J. Chem. Phys., 1997, 106, 4141–4151 CrossRef CAS.
  33. R. K. Chaudhuri, S. Majumder and K. F. Freed, J. Chem. Phys., 2000, 112, 9301–9309 CrossRef CAS.
  34. J. C. Sancho-García and A. J. Pérez-Jiménez, J. Phys. B: At., Mol. Opt. Phys., 2002, 35, 3689–3699 CrossRef.
  35. Y. Wang, B. J. Braams and J. M. Bowman, J. Phys. Chem. A, 2007, 111, 4056–4061 CrossRef CAS PubMed.
  36. J. F. Stanton, Chem. Phys. Lett., 1995, 237, 20–26 CrossRef CAS.
  37. J. C. Saeh and J. F. Stanton, J. Chem. Phys., 1999, 111, 8275–8285 CrossRef CAS.
  38. P. Halvick, Chem. Phys., 2007, 340, 79–84 CrossRef CAS.
  39. M. K. Bassett and R. C. Fortenberry, J. Chem. Phys., 2017, 146, 224303 CrossRef PubMed.
  40. S. C. Bennedjai, D. Hammoutène and M. L. Senent, ApJ, 2019, 871, 255 CrossRef CAS.
  41. M. Perić, M. Mladenović, K. Tomić and C. M. Marian, J. Chem. Phys., 2003, 118, 4444–4451 CrossRef.
  42. A. M. Mebel and R. I. Kaiser, Chem. Phys. Lett., 2002, 360, 139–143 CrossRef CAS.
  43. T. H. Dunning, J. Chem. Phys., 1989, 90, 1007–1023 CrossRef CAS.
  44. R. A. Kendall, T. H. Dunning and R. J. Harrison, J. Chem. Phys., 1992, 96, 6796–6806 CrossRef CAS.
  45. A. J. C. Varandas, J. Chem. Phys., 2013, 138, 054120 CrossRef CAS PubMed.
  46. A. J. C. Varandas, J. Chem. Phys., 2013, 138, 134117 CrossRef CAS PubMed.
  47. A. J. C. Varandas, in Reaction Rate Constant Computations: Theories and Applications, ed. K. Han and T. Chu, The Royal Society of Chemistry, 2013, ch. 17, pp. 408–445 Search PubMed.
  48. P. J. Knowles, C. Hampel and H. Werner, J. Chem. Phys., 1993, 99, 5219–5227 CrossRef CAS.
  49. P. Piecuch, M. Wloch and A. J. C. Varandas, Topics in the Theory Of Chemical and Physical Systems, Dordrecht, 2007, pp. 63–121 Search PubMed.
  50. R. J. Bartlett and M. Musia&poll, Rev. Mod. Phys., 2007, 79, 291–352 CrossRef CAS.
  51. T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory, John Wiley & Sons, Chichester, 2000 Search PubMed.
  52. H. J. Werner, P. J. Knowles, G. Knizia, F. R. Manby and M. Schützet al., MOLPRO, a package of ab initio programs, version 2010.1, 2010, Cardiff, U.K., 2010, see: http://www.molpro.net Search PubMed.
  53. A. J. C. Varandas, Annu. Rev. Phys. Chem., 2018, 69, 177–203 CrossRef CAS PubMed.
  54. F. N. N. Pansini, A. C. Neto and A. J. C. Varandas, Theor. Chem. Acc., 2016, 135, 261–267 Search PubMed.
  55. A. J. C. Varandas and F. N. N. Pansini, J. Chem. Phys., 2014, 141, 224113 CrossRef CAS PubMed.
  56. F. N. N. Pansini, A. C. Neto and A. J. C. Varandas, Chem. Phys. Lett., 2015, 641, 90–96 CrossRef CAS.
  57. A. J. C. Varandas, J. Chem. Phys., 2007, 126, 244105 CrossRef CAS PubMed.
  58. A. J. C. Varandas, J. Phys. Chem. A, 2010, 114, 8505–8516 CrossRef CAS PubMed.
  59. A. J. C. Varandas, J. Phys. Chem. A, 2011, 115, 2668 CrossRef CAS.
  60. C. M. R. Rocha and A. J. C. Varandas, J. Phys. Chem. A, 2019, 123, 8154–8169 CrossRef CAS PubMed.
  61. S. Joseph and A. J. C. Varandas, J. Phys. Chem. A, 2010, 114, 2655–2664 CrossRef CAS PubMed.
  62. C. M. R. Rocha and A. J. C. Varandas, Chem. Phys. Lett., 2018, 700, 36–43 CrossRef CAS.
  63. K. A. Gingerich, H. C. Finkbeiner and R. W. Schmude, J. Am. Chem. Soc., 1994, 116, 3884–3888 CrossRef CAS.
  64. R. S. Urdahl, Y. Bao and W. M. Jackson, Chem. Phys. Lett., 1991, 178, 425–428 CrossRef CAS.
  65. B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer, New York, 2nd edn, 2001 Search PubMed.
  66. T. J. Lee and P. R. Taylor, Int. J. Quantum Chem., 1989, 36, 199–207 CrossRef.
  67. C. L. Janssen and I. Nielsen, Chem. Phys. Lett., 1998, 290, 423–430 CrossRef CAS.
  68. T. J. Lee, Chem. Phys. Lett., 2003, 372, 362–367 CrossRef CAS.
  69. J. W. Huang and W. R. M. Graham, J. Chem. Phys., 1990, 93, 1583–1596 CrossRef CAS.
  70. G. H. Peslherbe, H. Wang and W. L. Hase, Monte Carlo Sampling for Classical Trajectory Simulations, Wiley-Blackwell, 1999, ch. 6, pp. 171–201 Search PubMed.
  71. W. L. Hase, R. J. Duchovic, X. Hu, A. Komornik, K. F. Lim, D. H. Lu, G. H. Peslherbe, K. N. Swamy, S. R. V. Linde, A. J. C. Varandas, H. Wang and R. J. Wolf, QCPE Bull., 1996, 16, 43 Search PubMed.
  72. A. J. Dean and R. K. Hanson, Int. J. Chem. Kinet., 1992, 24, 517–532 CrossRef CAS.
  73. V. Wakelam, J.-C. Loison, E. Herbst, B. Pavone, A. Bergeat, K. Béroff, M. Chabot, A. Faure, D. Galli, W. D. Geppert, D. Gerlich, P. Gratier, N. Harada, K. M. Hickson, P. Honvault, S. J. Klippenstein, S. D. L. Picard, G. Nyman, M. Ruaud, S. Schlemmer, I. R. Sims, D. Talbi, J. Tennyson and R. Wester, ApJS, 2015, 217, 20 CrossRef.
  74. R. L. LeRoy, J. Phys. Chem., 1969, 73, 4338–4344 CrossRef CAS.
  75. J.-C. Loison, X. Hu, S. Han, K. M. Hickson, H. Guo and D. Xie, Phys. Chem. Chem. Phys., 2014, 16, 14212–14219 RSC.


Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp04890a

This journal is © the Owner Societies 2019