Theoretical investigation of the infrared spectrum of small polyynes †

The full cubic and semidiagonal quartic force fields of acetylene (C 2 H 2 ), diacetylene (C 4 H 2 ), triacetylene (C 6 H 2 ), and tetraacetylene (C 8 H 2 ) are determined using CCSD(T) (coupled cluster theory with single and double excitations and augmented by a perturbative treatment of triple excitations) in combination with the atomic natural orbital (ANO) basis sets. Application of second-order vibrational perturbation theory (VPT2) results in vibrational frequencies that agree well with the known fundamental and combination band experimental frequencies of acetylene, diacetylene, and triacetylene (average discrepancies are less than 10 cm (cid:2) 1 ). Furthermore, the predicted ground state rotational constants ( B 0 ) and vibration–rotation interaction constants ( a i ) are shown to be consistent with known experimental values. New vibrational frequencies and rotational parameters from the presented theoretical predictions are given for triacetylene and tetraacetylene, which can be used to aid laboratory and astronomical spectroscopic searches for characteristic transitions of these molecules. 51–54 The combination of these studies shows that current quantum chemical theory, particularly coupled cluster theory with single and double excitations and augmented by a perturbative treatment of triple excitations (CCSD(T)), 55 is able to accurately reproduce equilibrium geometries, experimental vibrational frequencies, vibration–rotation interaction constants ( a i ), and ground state rotational constants ( B 0 ).

Extensive theoretical and experimental studies have been carried out for acetylene and diacetylene in the past few decades, including high-resolution spectroscopic studies of all the fundamental bands and a significant number of the combination bands, 29,[40][41][42][43][44][45][46][47][48][49][50] and high level ab initio calculations that take into account anharmonic effects. [51][52][53][54] The combination of these studies shows that current quantum chemical theory, particularly coupled cluster theory with single and double excitations and augmented by a perturbative treatment of triple excitations (CCSD(T)), 55 is able to accurately reproduce equilibrium geometries, experimental vibrational frequencies, vibration-rotation interaction constants (a i ), and ground state rotational constants (B 0 ).
Triacetylene and tetraacetylene are not as thoroughly studied, notably in terms of rotational information. While all of the fundamental vibrational modes of triacetylene have been measured, there is only rotational information for the IR active fundamental modes, 56 and the strongest IR combination band (n 8 + n 11 ). [57][58][59][60][61] However, theoretical studies of triacetylene do give rotational information for the remaining modes from CCSD(T) calculations of the vibration-rotation interaction constants 62 and the equilibrium geometry. 63 In addition, the harmonic frequencies of triacetylene were calculated using partial fourth-order many-body perturbation theory [SDQ-MBPT(4)]. 63 Conversely though, to the authors' knowledge, there is almost no rotational information for tetraacetylene. There has been only one low-resolution spectroscopic study of tetraacetylene, which measured three of the fundamentals (n 6 , n 8 , and n 14 at 3329.4, 2023.3, and 621.5 cm À1 , respectively), and one combination band (n 10 + n 14 at 1229.7 cm À1 ), and gives an estimate for the electronic ground state rotational constant, B 0 . 64 Unfortunately, the theoretical knowledge of tetraacetylene is equally limited, with only two studies of the equilibrium geometry (at the Hartree-Fock 65 and B3LYP 66 level of theory), and a calculation of the harmonic vibrational frequencies at the SVWN level of theory. 64 While the two modes that are most useful for astronomical identification (n 14 and n 10 + n 14 ) were measured, the uncertainty associated with the line positions is too large to allow for an unambiguous assignment. Moreover, some high-resolution IR searches have been attempted, 50,61,62,67 but so far no transitions have been assigned to tetraacetylene.
In this paper, we report the ab initio calculations for acetylene, diacetylene, triacetylene, and tetraacetylene. Due to the centrosymmetric nature of these molecules, observations in the laboratory and in space are most easily accomplished through their infrared spectra. As such, the properties computed and presented here are those related to that technique: fundamental vibrational frequencies, ground state rotational constants, and intramolecular interactions. The computational approach is calibrated using the well studied acetylene and diacetylene, and then extended to make predictions for triacetylene and tetraacetylene.

Computational methods
All calculations were carried out at the CCSD(T) level of theory, which with a sufficiently large basis set has been shown to accurately reproduce experimental values of semi-rigid molecules. [52][53][54][55][68][69][70][71][72] Equilibrium geometries were determined using the large core-valence correlation-consistent quadruple-z basis set (cc-pCVQZ), which features [8s7p5d3f1g] (nonhydrogen atoms) and [4s3p2d1f] (hydrogen) of (15s9p5d3f1g) and (6s3p2d1f) primitive basis sets, respectively. [73][74][75] All electron (AE)-CCSD(T)/cc-pCVQZ has been shown to give very accurate equilibrium geometries for unsaturated hydrocarbons. 54,[76][77][78] Optimizations were done using analytic energy derivatives, 79 and were considered converged when the root-mean-square (RMS) gradient fell below 10 À10 au. However, it is well known that correlation-consistent basis sets, such as cc-pCVQZ, tend to underestimate the vibrational frequencies of symmetric bending modes (p g ) of conjugated molecules, e.g., polyynes, due to their susceptibility to an intramolecular variant of basis set superposition error (BSSE). 54,80 It has been shown that one way to avoid this problem is to use basis sets with a large number of Gaussian primitives (particularly f-type), such as the atomic natural orbital (ANO) basis set (with the primitive basis set (13s8p6d4f2g) for non-hydrogen atoms and (8s6p4d2f) for hydrogen). 52,81,82 The basis set has two common truncations: [4s3p2d1f] for nonhydrogen atoms and [4s2p1d] for hydrogen (hereafter known as ANO1), and [5s4p3d2f1g] (non-hydrogen atoms) and [4s3p2d1f] (hydrogen) (hereafter known as ANO2). 74,75,81 In addition, only the valence electrons of carbon are considered in the correlation treatment, i.e., standard frozen-core (fc) calculations. (fc)-CCSD(T)/ ANO1 has been shown to accurately reproduce experimental frequencies and intensities for small molecules. 52,83,84 Using the (fc)-CCSD(T)/ANO1 optimized geometry, second-order vibrational perturbation (VPT2) theory calculations were determined from full cubic and the semidiagonal part of the quartic force fields obtained by numerical differentiation of analytic CCSD(T) second derivatives. 70,85 All calculations were performed with the development version of the CFOUR program. 86 3 Results and discussion

Equilibrium structure
The AE-CCSD(T)/cc-pCVQZ equilibrium geometries are shown in Fig. 1, with comparison to experimentally derived values (in italics) when known. 53,87,88 The theoretical equilibrium bond lengths for acetylene, diacetylene, and triacetylene all agree within 0.5% of the structures determined from experimentally measured rotational constants. As the length of the carbon chain increases, the C-H bond lengths stay essentially the same, B1.062 Å, consistent with a sp-H type C-H bond. However, the CRC bond lengths increase (particularly the internal CRC bonds), while the C-C bond lengths decrease, becoming closer to that typical of CC double bonds. This suggests that the p electrons become more delocalized over the internuclear axis, and the polyyne's configuration moves from a strict triple-single bond alternation to more of a consecutive double bond character of the CC bonds, making the overall structure more rigid as C 2 units are added, an effect that also qualitatively acts to increase the biradical character of the molecule as the size grows.
The equilibrium rotational constants, B e , obtained from the AE-CCSD(T)/cc-pCVQZ equilibrium geometries are summarized in Table 1, and agree well with experimental ground state rotational constants (B 0 ). As such, the equilibrium rotational constants suggest that the calculations predict the correct ground state geometry, because for linear molecules with more than three atoms the summation of vibration-rotation interaction constants (a i ) is expected to be close to zero, and from In addition, as seen for other carbon chains (e.g., HC n , HC 2n+1 N, and H 2 C n ) 89 the centrifugal distortion constant (D e ) decreases with increasing molecular size, with a theoretical D e = 1.6 Â 10 À6 cm À1 for acetylene, D e = 1.5 Â 10 À8 cm À1 for diacetylene, D e = 8.6 Â 10 À10 cm À1 for triacetylene, and D e = 1.2 Â 10 À10 cm À1 for tetraacetylene. These values are consistent with those found experimentally for the respective vibrational ground states ( Table 1). As noted by Thaddeus et al. 89 this behavior of increasing stiffness with chain length is a distinguishing characteristic associated with bona fide chains.

Spectroscopic properties of acetylene and diacetylene
The quality of the present calculations is checked by comparison to the experimentally well studied acetylene and diacetylene. The harmonic and VPT2 fundamental frequencies of the fundamental and combination bands are given in Table 2 and 3 for acetylene and diacetylene, respectively, and experimental values are included for comparison. The (fc)-CCSD(T)/ANO1 VPT2 fundamental frequencies show good agreement with experimental values, with most observed-calculated deviations (o-c) being less than 5 cm À1 and all being less than 15 cm À1 .
Based on previous studies of acetylene 52 and diacetylene, 53 the use of the ANO2 basis set was evaluated compared to the ANO1 basis set. For some of the vibrational modes, such as the n 4 mode of acetylene [612.88 cm À1 (observed)], 42 Martin et al. 52 showed that CCSD(T)/ANO2 can give a slightly better agreement (o-c value of B2 cm À1 ) compared to the ANO1 basis set (o-c value of B12 cm À1 ). However, the study by Thorwirth et al. 53 showed that, for diacetylene, the average o-c value with CCSD(T)/ANO2 is comparable to that for the ANO1 basis set (B6 cm À1 and B4 cm À1 , respectively). Moreover, the time cost of (fc)-CCSD(T)/ANO2 calculations compared to (fc)-CCSD(T)/ ANO1 far outweighs the minor frequency differences, and does not justify the higher computational cost of the ANO2 basis set in predicting the fundamental frequencies of longer polyynes.
The (fc)-CCSD(T)/ANO1 anharmonicity constants (x ij , ESI †) also accurately account for the known combination bands of acetylene and diacetylene (Tables 2 and 3, respectively). All the combination bands are within 5 cm À1 of their observed values. For both acetylene and diacetylene, the ANO1 basis set is able   to most accurately reproduce the C-H asymmetric stretch mode (n 3 and n 4 , respectively). Significant is the agreement between the experimental and our predicted frequencies of n 6 + n 8 [1241.060828 (38) cm À1 (observed) 46 and 1244.7 cm À1 (theoretical)], and 2n 6 + n 8 [1863.2512(5) cm À1 (observed) 44 and 1864.6 cm À1 (theoretical)] of diacetylene; both of which had only previously been calculated with CCSD(T)/cc-pCVQZ, and had o-c values greater than 20 cm À1 . 54 This suggests that the combination band VPT2 frequencies of polyynes determined using (fc)-CCSD(T)/ ANO1 are accurate to aid identification of molecules, such as in astronomical surveys.
The vibration-rotation interaction constants (Table 4) are also determined in the course of the VPT2 calculation, and are in good agreement with both previous theoretical studies 52,54 and experimentally determined values. 29,44,46,50,51,54 Based on the vibration-rotation interaction constants, the ground state rotational constants (B 0 ) were determined using the AE-CCSD(T)/ cc-pCVQZ determined B e values ( Table 1). For acetylene, B 0 = 1.175319 cm À1 , which is a 0.1% difference compared to the experimentally determined value of B 0 = 1.17664632 (18) cm À1 . 90 Diacetylene shows a similar 0.2% difference between the theoretical value of B 0 = 0.146167 cm À1 , and the experimentally determined value of B 0 = 0.1464123(17) cm À1 . 50 The consistent accuracy of these values suggests that the method presented is clearly good enough to be extrapolated to and aid high-resolution infrared spectroscopic searches for the larger polyynes.

Spectroscopic properties of triacetylene
The (fc)-CCSD(T)/ANO1 harmonic and VPT2 fundamental frequencies along with the experimental frequencies are given in Table 5. Comparison between theoretical VPT2 frequencies and experimental fundamentals measured with high-resolution techniques shows average o-c values that are smaller than those seen for acetylene or diacetylene (o-c B 2 cm À1 ). For the known combination band, the (fc)-CCSD(T)/ANO1 anharmonicity constants (x ij , ESI †) are able to reproduce the experimental value to within 5 cm À1 , suggesting other combination band frequencies are of equal accuracy.
For the modes observed in low-resolution studies (e.g., n 1 and n 12 ), the agreement is still good with o-c values less than 20 cm À1 . The notable exception is the internal CRC asymmetric stretch mode (n 7 ), which differs by 45 cm À1 . Since no rotationally resolved data can be found for this band, it is possible that the band observed at 1115.0 cm À1 59 was mis-assigned as the n 7 fundamental. A more likely assignment for this band is the n 9 + n 11 combination band, which has a predicted VPT2 frequency of 1107.1 cm À1 , a calculated intensity of 0.7 km mol À1 , and the same symmetry. Furthermore, the combination band is expected to be 3.5Â more intense than the n 7 fundamental at 0.2 km mol À1 , suggesting that n 9 + n 11 is more likely of the two to be observed. However, rotationally resolved measurements of this band are clearly needed to confirm this speculation.
We note that, a resonance between the n 5 fundamental and the n 2 + n 7 and 3n 7 combination bands must be addressed to achieve the very small (1 cm À1 ) o-c difference obtained for the C-H asymmetric stretch mode, n 5 . The vibrational frequencies as a result of resonant interactions are calculated by a deperturbation-diagonalization technique followed by transformation of the deperturbed transition moments, as discussed    85 This combination of Fermi and Darling-Dennison interactions shifts the n 5 predicted frequency from 3333.1 to 3329.5 cm À1 , which is able to reproduce the experimentally observed frequency [3329.0533(2) cm À1 61 ] with the same accuracy seen for diacetylene (o-c B 0.5 cm À1 ). The combination bands involved are similarly shifted: n 2 + n 7 from 3329.5 to 3362.2 cm À1 , and 3n 7 from 3526.7 to 3498.7 cm À1 . Since the shift is most pronounced for the two combination bands, future experimental work to observe either of these bands is required to confirm this prediction.

Spectroscopic properties of tetraacetylene
The (fc)-CCSD(T)/ANO1 harmonic and VPT2 frequencies of the fundamental and combination bands for tetraacetylene are given in Table 6, and the (fc)-CCSD(T)/ANO1 anharmonicity constants (x ij ) are given in the ESI. † For the four experimentally observed bands, agreement of the observed and calculated frequencies is good at 7 cm À1 , which is comparable to the uncertainty of the low resolution measurements. Furthermore, the ANO1 VPT2 frequencies are able to reproduce the experimental frequencies far better than the previous harmonic frequency calculations, which had o-c values of B20-100 cm À1 . 64 Of the predicted fundamental and combination bands, there are a number of bands that are found/predicted to have sufficient intensity and/or relatively unique frequency range that could offer viable target transitions to use to search for tetraacetylene in future laboratory or astronomical spectra. For example, in the IR the n 1 + n 6 at 6550.8 cm À1 or n 12 + n 15 at 871.9 cm À1 combination bands have both comparable predicted intensity to measured bands of di-and triacetylene, and have transitions in relatively clean regions of the spectrum. In terms of astronomical searches, the n 17 mode at 60.7 cm À1 , offers a unique target transition, since its low frequency makes it accessible by far-IR observations, similar to the n 2 bending mode of C 3 . 92 Based on the results discussed for the other small polyynes, the theoretical vibration-rotation interaction constants given in Table 4 are sufficient to assist in identification of ro-vibrational bands of tetraacetylene. The a i results in a theoretical ground state rotational constant of B 0 = 0.018844 cm À1 that agrees within errors with the experimentally determined value, B 0 = 0.020(3) cm À1 . 64 Overall, for polyynes the difference between the experimental and calculated rotational constants (DB 0 ) decreases from 0.001 to 0.00008 cm À1 as the chain length is increased, which is consistent with the trend seen for other carbon chain molecules (e.g., HC n N, HC n , C n O). 93 Therefore, if the trend continues as expected then the DB 0 for tetraacetylene is equal to or smaller than that seen for triacetylene, and the determined ground state rotational constant is a good approximation of the true value.

Conclusions
Accurate equilibrium geometries have been determined at the AE-CCSD(T)/cc-pCVQZ level of theory, and the full cubic and semidiagonal quartic force field have been determined at the (fc)-CCSD(T)/ANO1 level of theory for acetylene and the three smallest polyynes. No scaling or adjustments had to be included to match theoretical values with those determined by experiments. The resulting VPT2 fundamental vibrational frequencies and vibration-rotation interaction constants agree with known experimental values, showing about a 5 cm À1 deviation in frequencies for bands with high-resolution infrared information. For bands with only low-resolution data, the theoretical frequencies are able to confirm mode assignments or suggest a reassignment, as in the case of the observed band at 1115.0 cm À1 of triacetylene to the n 9 + n 11 combination band, which has previously been attributed to the n 7 fundamental. The provisional ab initio method used here is also able to accurately reproduce the observed frequencies of combination bands.
The calculated fundamental frequencies for triacetylene and tetraacetylene give insight as to why tetraacetylene has not yet been observed in space. Observation of centrosymmetric molecules in astronomical environments is mainly through infrared detection of the high intensity bending modes; e.g., n 8 [628.040776(36) cm À1 ] 29  64 and n 10 + n 14 [1229.7(5) cm À1 ], 64 and are predicted to be significantly weaker in intensity due to lower column densities. 34,35 Consequently, at these frequencies and resolutions of the previous infrared observations where polyynes were detected, [20][21][22]24,25 the transitions of tetraacetylene are blended with those of triacetylene. Other bands of tetraacetylene would be more suitable for identification, such as n 1 + n 6 , n 12 + n 15 , or n 17 that are expected to be equally strong as bands already used to identify di-and triacetylene.
Overall, the resulting computed geometries lead to equilibrium rotational constants (B e ), which when corrected for vibrational zero-point effects give ground state equilibrium constants (B 0 ) that agree with experimental values (0.2%).
Based on the small o-c values for acetylene, diacetylene, and triacetylene, we are confident that the fundamental frequencies and spectroscopic constants determined here offer an accurate guide for spectroscopic searches focused on detection of ro-vibrational bands of triacetylene and tetraacetylene. Such work is underway in our laboratory.

Conflicts of interest
There are no conflicts to declare.