1 Introduction

The monohydrogenated linear butadiynyl radical C4H is an important intermediate and plays a significant role in planetary atmospheres and combustion reactions [1,2,3]. The C4H radical has been found in abundance in interstellar space than other small molecules [2,3,4]. It is an essential precursor for the formation of polycyclic aromatic hydrocarbons and fullerenes [1,2,3, 5]. The C4H was first synthesized in 1975 in low temperature (4 K) argon and neon noble gas matrices after the UV photolysis of diacetylene [6]. And it was identified again in the carbon-rich star IRC + 10216 [7] as well as in dense clouds in 1978 [8]. Many studies found that there were two low-lying electronic states (\( {\tilde{\text{X}}}^{2} \varSigma_{{}}^{ + } \) and \( {\tilde{\text{A}}}^{2} \varPi \)) for C4H molecule. The \( {\tilde{\text{X}}}^{2} \varSigma_{{}}^{ + } \) and \( {\tilde{\text{A}}}^{2} \varPi \) are the ground state and the lowest excited state, respectively [1, 6, 9,10,11]. The dipole moment of the ground state C4H (\( {\tilde{\text{X}}}^{2} \varSigma_{{}}^{ + } \)) is about 0.87 Debye, that is much smaller than the \( {\tilde{\text{A}}}^{2} \varPi \) one. In recent years, the gas phase kinetics of reactions of the linear butadiynyl radical C4H with a series of compounds have gained extensive attention due to its potential importance [1, 5, 12, 13]. Experimental investigations for C4H radical reactions with various hydrocarbons among the most abundant observed in Titan’s atmosphere have been reported [14,15,16]. The theoretical study of the C4H radical with a series of compounds, such as CH4, CH3OH, H2, C2H4, and C4H10, has been done by three study groups [17,18,19,20,21]. However, to the best of our knowledge, there is no available theoretical study of C4H + C2H6. In this paper, we have investigated the reaction of C4H with C2H6 using the density functional theory. Owing to different relative configurations in attacking process, four plausible reaction mechanisms are suggested. Depending on our calculated results, we obtained that M1a and M1b are the most effective reaction pathways.

2 Computational methods

All the electronic structural calculations were performed by Gaussian09 program [22]. The geometries involved in the title reaction were optimized at the ωB97X-D/6-311++G(3df,2p) level of theory [23]. Frequency calculations were carried out at the same level to characterize the nature of the computed stationary points. All the reactants, pre-reactive complexes and products were identified with zero imaginary frequency. All transition states presented in this work were marked with one and only one imaginary frequency. Intrinsic reaction coordinate (IRC) [24, 25] calculations were carried out using ωB97X-D/6-311++G(3df,2p) level of theory to verify that the transition states connect the designated local minima. The coupled-cluster (CC) theory [26] of triple excitations CCSD(T) method [27] with the aug-cc-pVTZ basis sets was used to obtain more accurate reaction energies for all species using the ωB97X-D-optimized geometries.

The Polyrate 9.7 program [28] was employed to calculate the thermal rate constants using the conventional transition state theory (TST) [29], canonical variational transition state theory (CVT) [30, 31], and canonical variational transition state including a small-curvature tunneling correction (CVT/SCT) method [29] over the wide temperature range of 145–1000 K. The pre-reactive complexes (Rc1a and Rc1b) were considered in the chemical kinetic calculations. TS1a and TS1b have two low vibrational frequencies, one is a torsional mode and the other is a bending vibration. In kinetic calculations, the torsional modes of TS1a (74 cm−1) and TS1b (72 cm−1) were treated by the hindered-internal-rotator approximation [VANHAR, TOR]; the low-frequencies bending modes of TS1a (51 cm−1) and TS1b (55 cm−1) were treated by semi-classical WKB with a quadratic–quartic fit to potential [VANHAR, QQSEMI], while all the other modes are treated by the default harmonic approximation.” Besides, frontier molecular orbital of selected points along the molecular electrostatic potential (MEP) was performed by ORCA 2.8 program package [32] and plotted using Chimera [33].

3 Results and discussion

3.1 Electronic structure calculations

The C4H + C2H6, two H-abstraction mechanisms (M1 and M2) are considered. M1 is defined that the hydrogen abstraction by C1 of C4H and M2 is the hydrogen abstraction by C4 of C4H. Owing to C4H and C2H6 attacking each other in a different direction, each mechanism M1 and M2 has two reaction channels M1a and M1b, M2a and M2b, respectively.

  • M1: hydrogen abstraction by C1 of C4H

    $$\cdot {\text{C}}^{1} \equiv {\text{C}}^{2} - {\text{C}}^{3} \equiv {\text{C}}^{4} {\text{H}} + {\text{C}}_{2} {\text{H}}_{6} \to {\text{H}}^{1} {\text{C}} \equiv {\text{C}}^{2} - {\text{C}}^{3} \equiv {\text{C}}^{4} {\text{H}} + \cdot {\text{C}}_{2} {\text{H}}_{5} \;(M1a,\,M1b)$$

  • M2: hydrogen abstraction by C4 of C4H

    $$\cdot {\text{C}}^{1} \equiv {\text{C}}^{2} - {\text{C}}^{3} \equiv {\text{C}}^{4} {\text{H}} + {\text{C}}_{2} {\text{H}}_{6} \to {\text{C}}^{1} \equiv {\text{C}}^{2} - {\text{C}}^{3} \equiv {\text{C}}^{4} {\text{H}}_{2} + \cdot {\text{C}}_{2} {\text{H}}_{5} \;(M2a,\,M2b)$$

The optimized geometries of reactants, pre-reactive complexes, transition states and products involved in above reaction mechanisms, with the selected bond lengths and bond angles at the ωB97X-D/6-311++G(3df,2p) level, are presented in Fig. 1. The coordinates of the reactants, pre-reactive complexes, transition states and products are provided in Supplementary material coord-xyz. The distances of forming and breaking C–H bonds in four transition states (TS1a, TS1b, TS2a and TS2b) are given in Table 1. From Table 1, one can see that the breaking C5–H1 bond in TS1a, TS1b, TS2a and TS2b is elongated by 4.7, 4.1, 31.5 and 30.3%, as compared to the C5–H1 equilibrium bond length in C2H6, respectively, and the forming C1/C4–H1 bonds in TS1a, TS1b, TS2a and TS2b are longer than the equilibrium bond length of C1/C4–H1 in HC4H/C4H2 by 55.4, 57.8, 16.7 and 17.3%, respectively. These structural studies reveal that TS1a and TS1b are more reactant-like, while TS2a and TS2b are more product-like. Transition states TS1a, TS1b, TS2a and TS2b possess one and only one imaginary frequency 255i, 208i, 1009i and 1113i cm−1, respectively, indicating that the TSs are real first-order saddle point (see Table S1). Structural characteristics and values of the imaginary frequency indicate that TS1a and TS1b are loose transition states, while TS2a and TS2b are tight transition states.

Fig. 1
figure 1

Geometric parameters of various species involved in the title reaction at the ωB97X-D/6-311++G(3df,2p) level. Bond length in unit of angstrom and angle in unit of degree

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Table 1 The distances of forming C1/C4–H1 bond and breaking C5–H1 bond in various transition states and the elongated ratio of compared with the corresponding values in the reactants and products
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Figure 2 shows the barrier heights of four reaction channels obtained at the CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-311++G(3df,2p) level. Electronic structure energies (Eelec), sum of electronic and zero-point energies (Eelec + ZPE), sum of electronic and thermal Enthalpies (Eelec + Hcorr) for various species at the ωB97X-D/6-311++G(3df,2p) level are listed in Table S2. Two isolated reactant molecules (C4H + C2H6) are used to define reference energy (0.0 kcal/mol). As shown in Figs. 1 and 2, the C1 atom of the liner C4H can attach to one of H atoms of C2H6, resulting in Rc1a and Rc1b. The shapes of Rc1a and Rc1b are cis-like structure and trans-like structure, respectively. The relaxed potential energy surface (PES) scans along the distance between the H atom in C2H6 and C1 atom of C4H show that this process is no barrier. The scan results indicate that the Rc1a and Rc1b are the initial adducts. Both Rc1a and Rc1b are also connected to the P1 (HC4H + ·C2H5) through the reactant-like transition states TS1a and TS1b, respectively. Intrinsic reaction path (IRC) calculations revealed that M1a and M1b involve reactant side complexes Rc1a and Rc1b with relative energy of − 1.2 and − 1.4 kcal/mol, before TS1a and TS1b, respectively. The C4 atom of the liner C4H can attach to one of H atoms of C2H6, resulting in Rc2a and Rc2b. The shapes of Rc2a and Rc2b are cis-like structure and trans-like structure, respectively. Both Rc2a and Rc2b are also connected to the P3 (C4H2 + ·C2H5) through the reactant-like transition state TS2a and TS2b, respectively. The relative free energies of four transition states TS1a, TS1b, TS2a and TS2b were calculated to be − 0.7, − 0.2, 20.4 and 21.7 kcal/mol, respectively, at the CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-311++G(3df,2p) level. Moreover, M1 is exothermic by 31.2 kcal/mol, but M2 is endothermic by 13.6 kcal/mol. The C1 atom in the C4H radical was demonstrated to be the most reactive site and M1a and M1b are mainly two competitive reaction pathways. M2 is kinetically less favorable owing to the much higher energy barriers compared to M1 and, thus, its contribution to the overall reaction is almost negligible and will not be discussed in the kinetic calculations.

Fig. 2
figure 2

Relative energy profiles for the title reaction at the CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-311++G(3df,2p) level, in units of kcal/mol

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3.2 Electron transfer behaviors

Direct electron transfer behaviors of M1a and M1b are investigated by quasi-restricted orbital. Figure 3 displays the schematic frontier molecular orbital diagrams for the reactants, transition states, and products involved in M1a and M1b. Figure 4 presents the changes in the spin density distribution of key atoms in M1a(a) and M1b(b). As shown in Fig. 3, on can see that at the starting point, there is a single unpaired electron in πC–C orbital of the C4H fragment. As the C4H and C2H6 approach each other gradually, the σC–H bond of C2H6 is going to attack the half-occupied πC–C orbital in C4H radical. The β electron in the C–H bond of C2H6 transitions to the single unpaired πC–C orbital of the C4H fragment and leading to the  ·C2H5 fragment with one α electron left. From Fig. 4, one can see that DFT computations of M1a or M1b, demonstrate that C4H is spin carriers; 40, 27 and 26% of spin density resides on the C2, C1 and C4 atom, respectively. Along the MEP of M1a or M1b, the spin density on the H1 and C3 has almost no change, with the density on C5 decrease and that on C1, C2 and C4 increase. Molecular orbital calculations suggest that M1a and M1b are typical hydrogen atom transfer (HAT) mechanism.

Fig. 3
figure 3

Schematic MO diagrams of reactants (C4H + C2H6), transition states (TS1a and TS1b) and products (HC4H + C2H5)

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Fig. 4
figure 4

The change of atomic spin densities on C1, C2, C3, C4 and H1 atom in M1a(a) and M2a(b) obtained from selected points on IRC pathways

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3.3 Dynamics calculations

The rate constants for the most favorable reaction pathways (M1a and M1b) are calculated using canonical variational transition state theory with small-curvature tunneling corrections (CVT/SCT). The symmetry numbers of C2H6, C4H, TS1a, C2H5 and HC4H are 6, 1, 1, 1 and 2, respectively (σC2H6 = 6, σC4H = 1, σTS1a = 1, σC2H5 = 1 and σHC4H = 2); therefore, the equivalent reaction channels of the forward and reverse reactions are 6 and 2, respectively [SIGMAF = 6, SIGMAR = 2]. The predicted rate constants in the temperature range 145–1000 K are plotted as functions of the reciprocal of temperature as shown in Figs. S2 and S3 for M1a and M1b, respectively. And the CVT/SCT rate constants k1 (M1a), k2 (M1b) and the overall rate constant k (k = k1+ k2) are plotted against 1000/T (k−1) as shown in Fig. 5. The overall CVT/SCT rate constants are also available by summing up the M1a (k1) and M2a (k2). These data and the existing experimental rate constants [14, 16] are listed in Table S3. In Fig. S2, the rate constants for TST and CVT are in good agreement with each other in the whole studied temperatures, while CVT and CVT/SCT curves of M1a have some differences. These results suggest that the variational effect in the whole temperature range is almost negligible, while the small-curvature tunneling correction plays a very important role.

Fig. 5
figure 5

Plot of the calculated individual CVT/SCT rate constants k1, k2 and the overall rate constant (k = k1+ k2), at the CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-311++G(3df,2p) level versus 1000/T between 145 and 1000 K

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Furthermore, from Fig. S2 one can see that the rate constants significantly increase as the temperature decrease, indicating negative temperature dependence in temperature range 145–500 K. In Fig. S3, the rate constants of TST and CVT are nearly same over the whole temperature range, which means that the variational effect for M1b is very small and almost negligible. The CVT rate constants are obviously greater than those of the CVT/SCT values in the 800–1000 K. For example, the kCVT/kCVT/SCT for M1b is 1.11 and 1.30 at 800 and 1000 K, respectively. Therefore, SCT correction plays an important role and should be considered in rate constant calculations in high-temperature range. It is clear that in this reaction there is negative temperature dependence at temperatures smaller than 298 K. From Table S3, we can see that the deviation between the theoretical and experimental values is 1.2, 4.8 and 4.8 times at 145, 298 and 300 K, respectively. The present calculated rate constants are less than the available experimental values. A possible explanation for this discrepancy may result from the basis set size and the frequency mode in the transition state calculation. Therefore, calculation at the CCSD(T) level with larger basis sets (extremely consuming CPU time and memory capacities) may tend to decrease the discrepancy, but it is not guaranteed.

The three-parameter (k3) rate-temperature expression fitting of the overall CVT/SCT rate constants are performed for convenience of future experimental measurements. (in units of cm3 molecule−1 s−1).

$$ k^{3} (145 - 200\,{\text{K}}) = 6.38 \times 10^{ - 16} (T)^{1.22} \exp \left( { - \frac{754.29}{T}} \right) $$
$$ k^{3} (200 - 300\,{\text{K}}) = 2.21 \times 10^{ - 17} (T)^{1.76} \exp \left( { - \frac{854.51}{T}} \right) $$
$$ k^{3} (300 - 1000\,{\text{K}}) = 2.77 \times 10^{ - 18} (T)^{2.07} \exp \left( { - \frac{942.04}{T}} \right) $$

4 Conclusion

Reaction mechanisms of linear butadiynyl radical with ethane are investigated at the CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-311++G(3df,2p). Four hydrogen abstraction channels are considered. Calculated results show that M1a and M1b are the main and competitive channels. Orbital analysis shows that M1a and M1b are the H atom abstraction mechanism. The conventional transition state theory (TST), canonical variational transition state theory (CVT) and canonical variational transition state including a small-curvature tunneling correction (CVT/SCT) method are used to calculate the rate constants for M1(M1a and M1b) at the CCSD(T)/aug-cc-pVTZ//ωB97X-D/6-311++G(3df,2p) levels of theory over a wide temperature range of 145–1000 K. The calculated results show that for M1a, the small-curvature tunneling correction is important and the variational effect is negligible. For M1b, the variational effect is insignificant in the whole temperature range, while the SCT is very important and should be taken into account in the rate constant calculations in high-temperature range. Three-parameter Arrhenius expressions are also provided within 145–1000 K.