A global CHIPR potential energy surface for groundstate C_{3}H and exploratory dynamics studies of reaction C_{2} + CH → C_{3} + H†
Received
3rd September 2019
, Accepted 17th October 2019
First published on 18th October 2019
A fulldimensional global potentialenergy surface (PES) is first reported for groundstate doublet C_{3}H using the combinedhyperbolicinversepowerrepresentation (CHIPR) method and accurate ab initio energies extrapolated to the complete basis set limit. The PES is based on a manybody expansiontype development where the twobody and threebody energy terms are from our previously reported analytic potentials for C_{2}H(^{2}A′) and C_{3}(^{1}A′,^{3}A′), while the effective fourbody 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 builtin construction the appropriate permutational symmetry and describes in a physically reasonable manner all longrange features and the correct asymptotic behavior at dissociation. Exploratory quasiclassical trajectory calculations for the reaction C_{2} + CH → C_{3} + H are also performed, yielding thermalized rate coefficients for temperatures up to 4000 K.
1 Introduction
Small carbonbearing species C_{n} and C_{n}H (n = 1–3) are ubiquitous in the interstellar medium (ISM).^{1} They are particularly conspicuous and known to drive the Cchemistry^{2} in cold dense clouds^{3–5} and in circumstellar envelopes of evolved Crich stars.^{6–8} In these former environments, the radical^{9} C_{3}H – whose potential energy surface^{10} (PES) entails all the above smaller species as fragments – plays a prominent role, reaching relatively high fractional abundances (≈10^{−9}) when compared to H_{2}.^{2,11} In molecular cloud cores, both its cyclic^{12} (cC_{3}H; cyclopropynylidyne) and linear^{13} (C_{3}H; propynylidyne) isomers are thought to be formed via dissociative electron recombination of^{14}c,C_{3}H_{2}^{+}/C_{3}H_{3}^{+} or through the atomneutral pathway^{14–17} C + C_{2}H_{2}, the latter being an important prototypical reaction implied in the growth of Cchains in space.^{15–17} This has motivated further surmises^{18} on the role of cC_{3}H as a key intermediate (via cC_{3}H_{2} formation) in the synthesis of interstellar polycyclic aromatic hydrocarbon molecules – widely recognized as potential carriers of the socalled unidentified infrared bands.^{19}
The prevalence of C_{3}H in the ISM and its relevance to Cchain formation have stimulated considerable experimental^{12,20–28} and theoretical^{29–42} 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.
C_{3}H has a ^{2}Π electronic groundstate and and its two bending modes, ν_{4}(C–C–H) and ν_{5}(C–C–C), are perturbed by Renner–Teller (RT) and spin–orbit effects.^{13} The first laboratory detection of C_{3}H was reported by Gottlieb et al.^{20} who measured its microwave spectra in both ^{2}Π_{1/2} and ^{2}Π_{3/2} (ground) vibronic states. Subsequently, Yamamoto^{21} and Kanada et al.^{22} recorded pure rotational lines in ν_{4}(^{2}Σ^{μ}) and found that C_{3}H has an extremely low vibrationally excited state (≈27 cm^{−1} above ^{2}Π_{1/2}) which is caused by the strong RT effect in the ν_{4} 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 ^{2}Π_{r} and ν_{4}(^{2}Σ^{μ}) states^{23} and extending the range of rotational transitions within the ^{2}Σ vibrationally excited manifold;^{24} to our knowledge, no definitive assignment of the ν_{5}(^{2}Σ^{μ}) fundamental has yet been made. Infrared (IR) vibrational band centers for the stretching modes (ν_{1}, ν_{2} and ν_{3}) were provided by Jiang et al.^{25} in the Ar matrix, and bySheehan et al.^{26} in the gas phase.
cC_{3}H has a ^{2}B_{2}^{12} ground electronic state and was first detected in the laboratory by Yamamoto et al.^{12,27}via microwave spectroscopy. Based on the predicted rotational constants, the authors reported the molecular structure of cC_{3}H, confirming its C_{2v} nature.^{27} Moreover, using inertial defect considerations, they estimated the C–C asymmetric mode (ν_{4}; which lowers the symmetry from C_{2v} to C_{s}) to be fairly low (≈508 cm^{−1}) and attributed it to the vibronic interactions [i.e., pseudo Jahn–Teller (PJT) effects^{36}] between the ground ^{2}B_{2} and first ^{2}A_{1} excited states.^{27} In turn, Sheehan et al.^{26} reported gasphase estimates of the ν_{2} (ring stretching) and ν_{3} (scissoring) fundamentals of cC_{3}H; apart from such rough estimates, no accurate experimental IR band centers are available for this form^{39} thus far.
Early ab initio calculations^{12,22,25,29–34} on C_{3}H were primarily devoted to elucidate discrepancies between the predicted symmetries of the linear and cyclic forms and the ones actually inferred from microwave spectroscopy^{22,27} (energetically, the best estimate^{34} places cC_{3}H ≈ 14 kJ mol^{−1} more stable than C_{3}H). From these studies, a general trend can be stated: depending on the electronic structure method (singlereference versus truncatedspace multireference) and the utilized basis set (size), the cC_{3}H (C_{3}H) C_{2v} (C_{∞v}) structure might be in a transition state and that a slightly distorted C_{s} 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 equationofmotion coupled cluster method in the single and double approximations for ionized states (EOMIPCCSD), they first emphasized the role of the basis set effect to the accurate determination of its C_{2v} symmetry and to the increase of the X^{2}B_{2} → A^{2}A_{1} excitation energy. As noted by them, the PJT effect between such states – which had previously been considered^{27} as the cause of the possible C_{s} equilibrium structure – is weakened when the size of the oneelectron basis set is increased. Similar conclusions (regarding the Nelectron basis) were subsequently drawn by Halvick^{38} at the multireference configuration interaction level of theory [MRCI+Q] with ccpVXZ (X = D, T, Q) basis sets;^{43,44} he has shown that the PES along the C–C asymmetric w_{4} mode becomes increasingly stiff with correlation energy. Recently, Bassett and Fortenberry^{39} reported a quartic force field (QFF) for cC_{3}H from a composite scheme – based on accurate energies extrapolated to the completebasisset (CBS) limit at the CCSD(T)/augccpVXZ level of theory [briefly CCSD(T)/AVXZ] from basisset cardinal numbers X = T, Q, 5, then additively corrected for core correlation and scalar relativistic^{39} effects. From the QFF so obtained and using secondorder vibrational perturbation theory (VPT2),^{39} the authors reported rotational constants, structural parameters and anharmonic vibrational frequencies for the X^{2}B_{2} groundstate; to our knowledge, these are the best ab initio estimates so far. The spectroscopic characterization of the C_{3}H 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/ccpVTZ PESs (including relativistic effects) for both ^{2}A′ and ^{2}A′′ electronic states correlating with the ^{2}Π term. These PESs were subsequently used to compute the vibronic and spin–orbit structure of the C_{3}H spectrum using a variational approach.^{41} In their work, the authors highlight the extremely flat nature of the CCCH bending potential curve (^{2}A′) 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 selfconsistent field (CASSCF) and the CCSD(T) level of theory, the minimum structure of C_{3}H tends to change from a bent to a linear geometry with basis set size enhancement (from AVTZ to AVQZ). In turn, Mebel and Kaiser^{42} reported an ab initio study of the groundstate C_{3}H PES. Using CCSD(T)/6311+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 cC_{3}H) with a barrier of about 115 kJ mol^{−1}. Moreover, they remarked that, besides C(^{3}P_{j}) + C_{2}H_{2}(X^{1}Σ^{+}_{g}), the (barrierless) reactions of CH(X^{2}Π) + C_{2}(X^{1}Σ^{+}_{g}) and C(^{3}P) + C_{2}H(X^{2}Σ^{+}) represent facile neutral–neutral (exothermic) pathways yielding a carbon trimer in cold interstellar environments.
All the above features make C_{3}H 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 groundstate doublet C_{3}H that correlates with the ^{2}Π state at linear geometries, with ^{2}B_{2} 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 combinedhyperbolicinversepowerrepresentation (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 quasiclassical trajectory study of the C_{2} + CH reaction, while Section 6 gathers the conclusions.
2
Ab initio calculations
All calculations have been performed at the spinrestricted openshell RCCSD(T) [CC for brevity] level of theory^{48–50} using the ROHF determinant as reference.^{51} The AVXZ (X = D, T) basis sets of Dunning and coworkers^{43,44} 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) components^{53} 
E^{CBS}_{∞}(R) = E^{HF}_{∞}(R) + E^{cor}_{∞}(R),  (1) 
where E^{CBS}_{∞} is the total CBS energy and R a sixdimensional coordinate vector.
For the HF energy, a twopoint extrapolation protocol^{54} has been utilized,

E^{HF}_{X}(R) = E^{HF}_{∞}(R) + Ae^{−βx},  (2) 
where
x =
d(2.08),
t(2.96) are hierarchical numbers
^{55,56} associated with
X =
D,
T,
β = 1.62, and
A is a parameter to be calibrated from ROHF/AV
XZ energies.
^{54} In turn, the extrapolated core contributions are obtained
via the uniform singlet and tripletpair extrapolation (USTE) protocol
^{55,57–59} 
 (3) 
where
d(1.91) and
t(2.71) are CCtype
x numbers,
^{55} with
A_{3} calibrated from the raw CC core energies.
With the above CC/CBS(d,t) protocol, a total of 42538 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.†
3 CHIPR PES
Within the CHIPR framework,^{45,46} the groundstate doublet PES of C_{3}H assumes the following manybody 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 threebody interactions, and V^{(4)}(R) accounts for the fourbody ones. R = {R_{1}, R_{2}, R_{3}, R_{4}, R_{5}, R_{6}} is the set of interatomic separations (Fig. 1), with energy zero set to the infinitely separated groundstate C and H atoms.

 Fig. 1 Interparticle coordinate system employed in the construction of the CHIPR PES for groundstate C_{3}H; R_{1}–R_{3} define C–H bond distances, while R_{4}–R_{6} are C–C stretches.  
3.1 Twobody and threebody 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 pseudodiabatic matrix 
 (5) 
where 
 (6) 
and 
 (7) 
are diagonal terms constructed using previously reported two and threebody (groundstate) potentials for C_{2}H(^{2}A′),^{61} C_{3}(^{1}A′)^{62} and C_{3}(^{3}A′)^{60} (readers can refer to the original papers^{60–62} 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 _{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 C_{3}(^{1}Σ^{+}_{g}) + H(^{2}S), which can be attributed to the more attractive nature of the C_{3}(^{1}Σ^{+}_{g})'s threebody term. Note that this dissociation channel is the one with the largest experimental uncertainty, while an accurate estimate of the C_{3}'s atomization energy still needs to be determined. Indeed, fourbody 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 
Energy^{a} 
CC/CBS(d,t)^{b} 
V
^{(2+3)}

Exp. 
The units of energy are kJ mol^{−1}. Energies defined relative to the infinitely separated groundstate atoms.
CBS energies at CC/AVTZ optimized geometries.
Ref. 63.
Ref. 61.
Ref. 61 and 63.
Ref. 64.

C_{2}(^{1}Σ^{+}_{g}) + C(^{3}P) + H(^{2}S) 
−608.9 
−620.3 
−618.6 ± 2.1^{c} 
C_{2}(^{3}Π_{u}) + C(^{3}P) + H(^{2}S) 
−598.0 
−612.0 
−610.6 ± 2.1^{c} 
CH(^{2}Π) + 2C(^{3}P) 
−355.1 
−357.3 
−351.2 ± 0.4^{d} 
C_{2}(^{1}Σ^{+}_{g}) + CH(^{2}Π) 
−963.9 
−971.0 
−969.8 ± 2.5^{e} 
C_{2}(^{3}Π_{u}) + CH(^{2}Π) 
−953.1 
−963.3 
−961.2 ± 2.5^{e} 
C_{3}(^{1}Σ^{+}_{g}) + H(^{2}S) 
−1338.2 
−1373.9 
−1324.1 ± 13.0^{c} 
cC_{3}(^{3}A_{2}′) + H(^{2}S) 
−1255.5 
−1279.5 

C_{2}H(^{2}Σ^{+}) + C(^{3}P) 
−1108.5 
−1107.3 
−1110.7 ± 5.5^{f} 
3.2 Fourbody energy term
Once the V^{(2+3)}(R) PES is obtained, the (effective) fourbody 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 by^{45,46}

 (9) 
where
C_{i,j,k,l,m,n} are expansion coefficients of a
Lthdegree polynomial (
i +
j +
k +
l +
m +
n ≤
L) and
y_{p} (
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
i ≤
j ≤
k ≤
l ≤
m ≤
n 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 threebody contributions. In
eqn (9), the second summation runs over all permutation elements
g ∈
G, where
G is a subgroup of the
_{4} symmetric group.
^{65} Thus, for an AB
_{3}type molecule,
^{(i,j,k,l,m,n)}_{g} are the corresponding operators that reflect the action of the particle permutations [in cyclic notation
^{65}] (
), (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 y_{p} in eqn (9) is in CHIPR expanded as a distributedorigin contracted basis set^{45,46}

 (10) 
where
ϕ_{p,α} is expressed either as
^{45,46} 
ϕ^{[1]}_{p,α} = sech^{η}(γ_{p,α}ρ_{p,α}),  (11) 
or

 (12) 
In the above equations, ρ_{p,α} = R_{p} − R^{ref}_{p,α} defines the displacement coordinate R_{p} from the αth primitive's origin R^{ref}_{p,α}, γ_{p,α} are nonlinear 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 R^{ref}_{p,α} in eqn (11) and (12) are assumed related by^{45}

R^{ref}_{p,α} = ζ(R^{ref}_{p})^{α−1},  (13) 
where
ζ and
R^{ref}_{p} 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 cC_{3}H reference geometry; this is made up of a D_{3h} structure of C_{3} (R_{CC} = 2.596a_{0}) and the experimental^{27} CH bond distance (R_{CH} = 2.033a_{0}). With the above data, the parameters of eqn (10)–(13) have then been determined from a fit to the resulting fourbody energies. The final bases, referred to as CHb8 and CCb8, are depicted in Fig. 2. Note that their corresponding nonlinear parameters have been optimized only once and are kept fixed in all subsequent steps.

 Fig. 2 C–H (y_{1}–y_{3}) and C–C (y_{4}–y_{6}) contracted basis sets of eqn (10) as determined from a fit to CC/CBS(d,t) fourbody energies [ε^{(4)} in eqn (8)]; see the text. The calibrating cC_{3}H reference geometry is made up of a D_{3h} structure of C_{3} (R_{CC} = 2.596a_{0}) and the experimental^{27} CH bond distance (R_{CH} = 2.033a_{0}). Solid diamonds indicate the origins (R^{ref}_{p,α}) of each primitive basis ϕ_{p,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 c_{α}'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 c_{α}'s) were varied freely. The final optimized contracted bases are also shown in Fig. 2. For the weights (W), we employ the following function

 (14) 
where Δ
E =
E −
E_{min},
β = 1/2,
E_{min} = −0.645
E_{h} and
E_{0} = −0.62
E_{h}, with larger weights attributed to minima (min), transition states (ts) and minimum energy paths (MEPs).
With such an approach, all 42538 ab initio points could be leastsquares 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 χ^{2} [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 tests^{66–68} 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 results^{39}), i.e., T_{1} ≤ 0.02,^{66}D_{1} ≤ 0.05^{67} and ,^{68} the use of multireference electron correlation methods for C_{3}H 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 costeffective way.
Table 2 Stratified rootmeansquare deviations (in kJ mol^{−1}) of the final PES
Energy^{a} 
N
^{
}

rmsd 
_{1}
^{
}

_{1}
^{
}

_{1}/_{1}^{c} 
The units of energy are kJ mol^{−1}. Energy strata defined relative to the cC_{3}H global minimum with energy of −0.6427E_{h}.
Number of calculated points up to the indicated energy range.
Mean values of T_{1}, D_{1} 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 
14680 
7.1 
0.036 
0.098 
0.364 
200 
19699 
7.6 
0.035 
0.098 
0.363 
300 
24859 
7.6 
0.035 
0.097 
0.364 
400 
29052 
7.8 
0.035 
0.096 
0.368 
500 
30820 
8.0 
0.035 
0.096 
0.369 
1000 
42104 
8.0 
0.039 
0.103 
0.383 
2000 
42483 
8.0 
0.039 
0.103 
0.382 
3000 
42538 
8.0 
0.040 
0.104 
0.382 

 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 literature^{12,21,26,27,39–42,69} are also shown.

 Fig. 4 Contour plot for an H atom moving around partially relaxed cC_{3} with centerofmass fixed at the origin. Contours are equally spaced by 0.0135E_{h} starting at −0.65E_{h}. 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 massscaled a.u.) for the isomerization between symmetry equivalent cC_{3}H structures. In (b), the zero of energy corresponds to cC_{3}H. 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 (T_{e} in kJ mol^{−1}) for some lowlying excited states of c and C_{3}H as obtained from twostate CASSCF(13,13)MRCI+Q/CBS(D,T)^{60} calculations using the C_{2v} point group. The electronic states are labeled according to C_{2v} irreducible representations with the corresponding correlations with C_{∞v} and/or C_{s} also given in parenthesis. The energies are calculated at the corresponding experimental (groundstate) equilibrium geometries^{22,27} (see also Table 4)
cC_{3}H 
C_{3}H 
State 
Γ
_{el}
^{
}

T
_{e}
^{
}

State 
Γ
_{el}
^{
}

T
_{e}
^{
}

Electronic symmetry species.
MRCI+Q/CBS(D,T) values relative to the groundstate cC_{3}H minimum.
Experimental gasphase values reported in ref. 26.
Experimental gasphase ^{2}Π → ^{2}A′′(^{2}Δ) transition reported in ref. 28.

1 
^{2}B_{2}(^{2}A′) 
0.0 
1 
^{2}B_{2}(^{2}Π/^{2}A′), ^{2}B_{1}(^{2}Π/^{2}A′′) 
14.5 
2 
^{2}A_{1}(^{2}A′) 
113.2 
2 
^{2}A_{1}(^{2}Δ/^{2}A′), ^{2}A_{2}(^{2}Δ/^{2}A′′) 
281.0 

[96.8 ± 1.2]^{c} 

[268]^{d} 
3 
^{2}B_{1}(^{2}A′′) 
326.3 
3 
^{2}A_{2}(^{2}Σ^{−}/^{2}A′′) 
311.9 
4 
^{2}A_{2}(^{2}A′′) 
369.3 
4 
^{2}A_{1}(^{2}Σ^{+}/^{2}A′) 
346.1 
5 
^{2}A_{2}(^{2}A′′) 
437.5 
5 
^{2}B_{2}(^{2}Π/^{2}A′), ^{2}B_{1}(^{2}Π/^{2}A′′) 
370.7 
6 
^{2}B_{1}(^{2}A′′) 
439.3 



7 
^{2}A_{1}(^{2}A′) 
478.2 



8 
^{2}B_{2}(^{2}A′) 
678.2 



Accordingly, the predicted global mininum on the groundstate doublet of the PES corresponds to a cyclic C_{2v} geometry, cC_{3}H (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 cC_{3}H data and the recently reported CCF12/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 (w_{4}), all harmonic frequencies are well reproduced by CHIPR. Note that w_{4} corresponds to the cHC_{3}(^{2}B_{2}/1^{2}A′) symmetry lowering (from C_{2v} to C_{s}) which, according to previous work,^{27,36,38} is subject to vibronic mixing with the next ^{2}A_{1}/2^{2}A′ electronic state. Table 3 gives our best estimate of this excitation energy, namely ≈113.2 kJ mol^{−1} above ^{2}B_{2}. For the benefit of the reader, we also include exploratory estimates of stationary strutures for the lowest 7 excited states of c/HC_{3} (note that only geometries close to the one of equilibrium cHC_{3} or HC_{3} have been explored). Although the experimental ν_{4} value is itself largely questioned^{39} (it is derived from rotational distortions rather than vibrational observations;^{27} this is also true for the other experimental IR bands^{26}), difficulties in describing w_{4} seem to appear also in the case of CCF12 wave functions (see Table 4). Nevertheless, as noted in various occasions,^{38–40} extrapolations to the CBS limit tend to favor the highest symmetry C_{2v} structure over the (double minimum) C_{s} distorted ones. As shown in Fig. 4, the isomerization between the three symmetryequivalent cC_{3}H structures occurs via a (thus far unreported) C_{2v} transition state of ccC_{3}H. This form is predicted from CHIPR to be located 197.7 kJ mol^{−1} above cC_{3}H with an imaginary frequency of 1693.9 cm^{−1} along the H wagging motion (through cC_{3}'s centerofmass). 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 a_{0}, angles in deg) and harmonic (w_{i}) frequencies (in cm^{−1}) of minima on the C_{3}H groundstate PES. Energies (ΔE) in kJ mol^{−1} refer to global cC_{3}H minima
Structure 
Property^{a} 
CHIPR^{b} 
CC^{c} 
MRCI+Q^{d} 
CCF12^{e} 
QFF^{f} 
Exp. 
This work. spg stands for the symmetry point group.
This work.
This work, with the AVTZ basis set. ΔE values in parenthesis are CC/CBS(d,t) at CHIPR stationary points.
This work, with the AVTZ basis set. Geometries at the singlestate 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.
Ref. 40, with the AVTZ basis set.
Local PESs of ref. 39 (for cC_{3}H) and ref. 41 (for C_{3}H).
Ref. 27 and 12.
Ref. 26. Gasphase values.
Ref. 22. Gasphase values.
Ref. 69 and 25. Measured in the Ar matrix.
Ref. 21. Estimated fundamentals in the absence of Renner–Teller and spin–orbit effects.
Ref. 21. ν_{4}(^{2}Σ^{μ}) fundamental bands.

cC_{3}H 
R
_{HC1}

2.034 
2.038 
2.012 
2.039 
2.036 
2.033^{g} 
R
_{C1C2}

2.587 
2.526 
2.531 
2.600 
2.595 
2.596^{g} 
R
_{C2C3}

2.557 
2.622 
2.654 
2.608 
2.590 
2.602^{g} 
∠HC_{1}C_{2} 
150.4 
156.5 
157.0 
149.9 
150.1 
149.9^{g} 
∠C_{1}C_{2}C_{3} 
60.4 
64.0 
64.5 
59.9 
60.1 
59.9^{g} 
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 ± 25^{h} 
w
_{3} (CCH sym. bend) 
1754.7 
1606.0 
1566.8 
1214.0 
1219.5 
1161 ± 25^{h} 
w
_{4} (CC asym. stretch) 
1327.9 
919.4 
884.8 
2639.0 
619.5 
508 ± 25^{g}^{,}^{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 



C_{3}H 
R
_{HC1}

1.994 
2.017 
2.033 
2.013 
2.022 
1.922^{i} 
R
_{C1C2}

2.346 
2.366 
2.367 
2.351 
2.372 
2.370^{i} 
R
_{C2C3}

2.519 
2.539 
2.578 
2.538 
2.561 
2.506^{i} 
∠HC_{1}C_{2} 
180.0 
164.4 
175.2 
180.0 
180.0 

∠C_{1}C_{2}C_{3} 
180.0 
175.9 
178.2 
180.0 
180.0 

w
_{1} (CH stretch) 
3372.5 
3403.7 
3543.7 
3428.0 
3600.0 
3238^{j} 
w
_{2} (CCC asym. stretch) 
1856.5 
1844.4 
1807.1 
1870.0 
1830.0 
1839 ± 10,^{h} 1824.7^{j} 
w
_{3} (CCC sym. stretch) 
1094.5 
1137.7 
1137.7 
1136.0 
1160.0 
1167, 1159.8^{j} 
w
_{4} (CCH bend) 
247.7 
223.9, 236.8 
219.6, 297.6 
230.0 
202.0 
600,^{k} 20.3^{l} 
w
_{5} (CCC bend) 
302.6 
371.7 
355.8 
371.0 
374.0 
300^{k} 
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 C_{3}H groundstate PES. Properties and units as in Table 4
Property^{a} 
ccC_{3}H 
C_{3}H 
CHIPR^{b} 
CC^{c} 
MRCI+Q^{d} 
CHIPR^{b} 
CC^{c} 
MRCI+Q^{d} 
This work. spg stands for the symmetry point group.
This work.
This work, calculated with the AVTZ basis set. ΔE values in parenthesis are CC/CBS(d,t) at CHIPR stationary points.
This work, calculated using the AVTZ basis set. Geometries at the singlestate 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.
Ref. 42, calculated at the CC/6311+G(3df,2p)//B3LYP/6311G(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 
∠HC_{1}C_{2} 
103.5 
101.7 
100.8 
89.1 
96.8 
100.2 
∠C_{1}C_{2}C_{3} 
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) 
Property^{a} 
cC_{3}H 
CHIPR^{b} 
CC^{c} 
MRCI+Q^{d} 
CC^{e} 
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 
∠HC_{1}C_{2} 
173.6 
140.7 
140.2 
139.9 
∠C_{1}C_{2}C_{3} 
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 cC_{3}H, CHIPR predicts the linear isomer C_{3}H to be a local minimum in the groundstate 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), CCF12/AVTZ^{40} and MRCI+Q/CBS(D,T) levels are 11.8, 15.4 and 5.3 kJ mol^{−1}, respectively. Note that the CHIPR's C_{3}H structural parameters agree well with the experimental (vibrationally averaged) r_{0} values reported by Kanada et al.^{22} [at experimental geometries, our best estimate places C_{3}H about 14.5 kJ mol^{−1} above cC_{3}H (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 shown^{22,23} to couple with the ^{2}Π (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 ^{2}A′ PES very flat along w_{4} with the appearance of a quasilinear C_{3}H molecule, depending on the chosen ab initio method; see, e.g., Table S4 (ESI†) for the truncatedspace CASSCF/MRCI+Q estimates. Here, in accordance with the recent CCF12^{40} 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 C_{3}H are also in close agreement with the ones derived from the local ^{2}A′ PES studied by Perić et al.^{41} Note that, as for cC_{3}H, the experimental C_{3}H's IR bands but ν_{4}(^{2}Σ^{μ}) are largely uncertain.^{21,25,26,69}

 Fig. 5 Contour plot for an H atom moving around partially relaxed C_{3} which lies along the x axis with the central C atom at the origin. Contours are equally spaced by 0.0135E_{h} starting at −0.65E_{h}. 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 massscaled a.u.) for the isomerization between symmetry equivalent C_{3}H structures. In (b), the zero of energy corresponds to C_{3}H. 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).  

 Fig. 6 Contour plots for the CH and C_{2} collinear reactions yielding (a) C_{3} + H and (b) C_{2}H + C. Contours are equally spaced by 0.015E_{h}, starting at −0.65E_{h}. 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.  

 Fig. 7 Contour plot for a C atom moving around a partially relaxed C_{2}H which lies along the x axis with the C atom kept at the origin. Contours are equally spaced by 0.02E_{h} starting at −0.65E_{h}. 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 Tshaped (C_{2v}) transition state, hereafter denoted as C_{3}H, which is responsible for the degenerate isomerization between symmetryequivalent C_{3}H structures; its imaginary frequency amounts to 1111.7 cm^{−1} and points toward the H wagging motion through C_{3} centerofmass. Such a hitherto unreported form lies about 241.9 and 256.3 kJ mol^{−1} above C_{3}H and cC_{3}H, 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 cC_{3}H and C_{3}H occurs via cC_{3}H transition states^{14,42} 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 cC_{3}H; the corresponding CC/CBS(d,t) and MRCI+Q/CBS(D,T) values are 122.1 and 123.8 kJ mol^{−1}, respectively.

 Fig. 8 (a) Contour plot for CH moving around a C_{2} diatom with the centerofmass 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 0.00075E_{h} starting at −0.65 and −0.385E_{h}, respectively. The zero of energy is set relative to the infinitely separated atoms. (b) Minimum energy path (s is the reaction coordinate in massscaled a.u.) for isomerization between cC_{3}H and C_{3}H via the transition state of cC_{3}H. The zero of energy corresponds to cC_{3}H.  

 Fig. 9 Pathways connecting the stationary points on the CHIPR PES. All energies (in kJ mol^{−1}) refer to the global cC_{3}H 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 builtin construction, and also describes in a physically reasonable manner all longrange structures and the correct asymptotic behavior at dissociation. This is no doubt achieved by the use of the MBE/CHIPR methods^{10,45,46} with appropriate ^{(i,j,k,l,m,n)}_{g} operators in eqn (9). Indeed, as Fig. 8, 9 and Table 6 show, CHIPR also predicts longrange stationary points [(HC_{2}⋯C)_{min}, (HC_{2}⋯C)_{ts}, (C_{2}⋯HC)_{min} and (C_{2}⋯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 C_{2}H and CH to the C_{2} diatom, both toward cC_{3}H formation (see Fig. 9). In addition, Fig. 6 and 9 show that the formation of C_{3}H is barrierless for the collinear reactions of C_{2} + CH, C_{3} + H and C + C_{2}H, in accordance with ref. 42. Moreover, the cC_{3}H adduct is formed without any barrier by the perpendicular approach of H to both cC_{3}(^{3}A_{2}′) and C_{3}(^{1}Σ^{+}_{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 Longrange attributes of the C_{3}H groundstate PES. Properties and units as in Table 4
Property^{a} 
(HC_{2}⋯C)_{min} 
(HC_{2}⋯C)_{ts} 
CHIPR^{b} 
MRCI+Q^{c} 
CHIPR^{b} 
MRCI+Q^{c} 
This work. spg stands for the symmetry point group.
This work.
This work, calculated with the AVTZ basis set. Geometries at the singlestate 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 
∠HC_{1}C_{2} 
179.0 
180.0 
165.2 
150.0 
∠C_{1}C_{2}C_{3} 
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) 
Property^{a} 
(C_{2}⋯HC)_{min} 
(C_{2}⋯HC)_{ts} 
CHIPR^{b} 
MRCI+Q^{c} 
CHIPR^{b} 
MRCI+Q^{c} 
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 
∠HC_{1}C_{2} 
9.3 
15.3 
60.9 
70.3 
∠C_{1}C_{2}C_{3} 
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 C_{2}(v,j) + CH(v′,j′) → C_{3} + H by running quasiclassical trajectories (QCT)^{70} on the novel PES; a locally modified version of the VENUS96C code^{71} 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 b_{max} = 6 Å as optimized by trial and error. Average reaction cross sections were then obtained (for a given temperature T) as^{70} 〈σ_{R}(T)〉 = πb_{max}^{2}N_{R}/N, and the 68% associated errors by Δ〈σ_{R}(T)〉 = 〈σ_{R}(T)〉 [(N − N_{R})/(NN_{R})]^{1/2}, where N_{R} is the number of reactive trajectories out of a total of N (10^{4} per temperature) that were run. In turn, thermal rate constants for the formation of C_{3} + H were calculated as^{70} 
 (15) 
with the estimated 68% error being given by Δk(T) = k(T)[(N − N_{R})/(NN_{R})]^{1/2}; k_{B} is the Boltzmann constant, μ_{CH+C2} is the reduced mass of the reactants and g_{e} = 1/12 is the electronic degeneracy factor. The calculated results for temperatures up to 4000 K are shown in Fig. 10 and 11.

 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〈σ_{R}(T)〉] error bars.  

 Fig. 11 Rate constants and associated error bars for the C_{2} + CH reaction at temperatures up to 4000 K. Also shown are the experimental estimates of Dean and Hanson^{72} 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 〈σ_{R}〉 with respect to temperature, we have considered the following model function^{74}

〈σ_{R}(T)〉 = (a + bT + cT^{2})T^{n}exp(−mT),  (16) 
where
a,
b,
c,
n, and
m are parameters to be adjusted to the QCT calculated crosssections; they are numerically defined in
Table 7, with the final form of 〈
σ_{R}(
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 〈
σ_{R}(
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} cm
^{3} 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 cm
^{3} molecule
^{−1} s
^{−1})

k(T) = 6.98 × 10^{−11}(T/300)^{0.45}  (17) 
for temperatures up to ≈500 K. A similar dependence (
T^{0.6}) has also been found experimentally for the case of the barrierless reaction between C
_{2} and the (CHcongenere) N atom.
^{75} However, in high temperature regimes, it is predicted to be approximately constant,

k(T) ≅ 1.02 × 10^{−10} cm^{3} 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} cm
^{3} molecule
^{−1} s
^{−1} for the reaction C
_{2}(
v,
j) + CH(
v′,
j′) → C
_{3} + H. This is compared with the value of
^{15} 2.23 × 10
^{−10} cm
^{3} molecule
^{−1} s
^{−1} (at 10 K) for the C + C
_{2}H
_{2} → H
_{2} + C
_{3} reaction, a well established (neutral–neutral) source of carbon trimer in the ISM.
^{15,42}
Table 7 Parameters for 〈σ_{R}(T)〉 in eqn (16)
Parameter 
Value^{a} 
The power of 10 is in parenthesis.

a/Å^{2} K^{−n} 
7.52 (+1) 
b/Å^{2} K^{−(1+n)} 
−1.25 (−2) 
c/Å^{2} 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 groundstate doublet C_{3}H using accurate CBS extrapolated ab initio energies and the CHIPR method for modelling. The novel PES is based on a MBEtype development with the twobody and threebody energy terms from previously reported doubleMBE potentials for C_{2}H(^{2}A′)^{61} and C_{3}(^{1}A′,^{3}A′),^{60,62} while the effective fourbody energy term is modelled using the CHIPR formalism for AB_{3}type tetratomics. The final form reproduces accurately all known stationary structures of HC_{3} and their interconversion pathways, some unreported thus far to the best of our knowledge. Besides describing properly longrange interactions at all asymptotic channels and permutational symmetry by builtin construction, the PES reproduces reasonably well all exothermicities at dissociation regions – the best one can possibly afford with the above two and threebody potentials and our costeffective ab initio approach. By running exploratory quasiclassical trajectory calculations for the reaction C_{2} + CH → C_{3} + H, thermalized rate coefficients for temperatures up to 4000 K are also reported.
Conflicts of interest
There are no conflicts to declare.
Acknowledgements
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: CENTRO010145FEDER000014), cofinanced 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 0417/130). The support from A. J. C. V. from the China's Shandong Province “DoubleHundred Talent Plan” (2018) is also appreciated.
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Footnote 
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp04890a 

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