Precise equilibrium structures of 1H- and 2H-1,2,3-triazoles (C₂H₃N₃) by millimeter-wave spectroscopy

Abstract

The 1H- and 2H-1,2,3-triazoles are isomeric five-membered ring, aromatic heterocycles that may undergo chemical equilibration by virtue of intramolecular hydrogen migration (tautomerization). Using millimeter-wave spectroscopy in the 130–375 GHz frequency range, we measured the spectroscopic constants for thirteen 1H-1,2,3-triazole and sixteen 2H-1,2,3-triazole isotopologues. Herein, we provide highly accurate and highly precise semi-experimental equilibrium (r_e^SE) structures for the two tautomers based on the spectroscopic constants of each set of isotopologues, together with vibration–rotation interaction and electron-mass distribution corrections calculated using coupled-cluster singles, doubles, and perturbative triples calculations [CCSD(T)/cc-pCVTZ]. The resultant structures are compared with a “best theoretical estimate” (BTE), which has recently been shown to be in exceptional agreement with the semi-experimental equilibrium structures of other aromatic molecules. Bond distances of the 1H tautomer are determined to <0.0008 Å and bond angles to <0.2°. For the 2H tautomer, bond angles are also determined to <0.2°, but bond distances are less precise (2σ ≤ 0.0015). Agreement between BTE and r_e^SE values is discussed.

INTRODUCTION

1H- and 2H-1,2,3-Triazole (C₂H₃N₃, Fig. 1) are near-oblate, asymmetric rotors (κ = 0.94, C_s and 0.82, C_2v, respectively). The tautomers, as well as a few of their isotopologues, have been studied using ultraviolet,¹ infrared (IR),^1–4 Raman,³ and microwave^5–8 spectroscopy. The 1H/2H tautomerism and equilibrium have received a fair amount of interest.^5,8–15 Both of the tautomers are present in the solid,¹¹ liquid,^4,10,12,13 and gas^{5,9,10,13–15} phases. Although 1H-1,2,3-triazole is preferentially stabilized in polar solvents,¹² the 2H-1,2,3-triazole tautomer is estimated to be more stable by ∼3.5–4.5 kcal/mol in the gas phase.^5,9,10,13 Despite the greater abundance of the 2H tautomer in the gas phase, its smaller dipole moment (μ_b = 0.10 D) relative to that of the 1H tautomer (μ_a = 4.2 D, μ_b = 0.77 D) results in both molecules being observable in the gas phase by millimeter-wave spectroscopy at room temperature.

FIG. 1.

(a) 1H-1,2,3-triazole and (b) 2H-1,2,3-triazole structures with principal inertial axes and atom numbering.

While the structures of the triazoles have been computed at a variety of levels of theory,^3,16,17 and there has been a crystal-structure determination,¹¹ there has been only a single attempt to determine a gas-phase, equilibrium structure of 1,2,3-triazole. Begtrup et al.⁵ used electron diffraction and microwave spectroscopy data to approximate an equilibrium structure for 2H-1,2,3-triazole. That structure determination, however, used microwave data from only three isotopologues. To date, there have been no attempts to determine the gas-phase structure of 1H-1,2,3-triazole. Following the pioneering work of Pulay, Meyer, and Boggs,¹⁸ semi-experimental structure (r_e^SE) determination has evolved into a powerful method for establishing gas-phase molecular structure.^19,20 Semi-experimental structures for a variety of small heterocyclic compounds have been reported using a density functional theory version of the method.²¹ Recently, we used coupled-cluster computations to achieve high-accuracy and high-precision semi-experimental equilibrium structure (r_e^SE) determinations of pyrimidine,²⁰ pyridazine,²² thiophene,²³ thiazole,²⁴ and benzene.²⁵ We, therefore, sought to determine semi-experimental equilibrium structures for the 1,2,3-triazoles.

In our recent studies of structures of the six-membered ring molecules (which include only first- and second-row elements), the r_e^SE structures displayed exceptional agreement with the purely computational, “best theoretical estimate” (BTE) structures. Structures of two sulfur-containing, five-membered ring molecules, however, had BTE parameters that did not uniformly fall within the r_e^SE parameter uncertainties. Those discrepancies were plausibly ascribed to the proximity of certain atoms to principal axes,^23,24,26 combined with the inability to shift those atoms from the principal axes upon isotopic substitution (because the heavy sulfur atom essentially set the axes’ orientation). 2H-1,2,3-Triazole offers the possibility to examine the above assertion—one nitrogen atom and one hydrogen atom lie directly on the b-axis, but the principal axes are easily rotated upon isotopic substitution. The ability to obtain well-determined parameters involving the on-axis atoms would support the idea that the near-axis sulfur atom was the culprit behind difficulties in previously examined five-membered ring structure determinations.

The precise determination of spectroscopic constants of the 1,2,3-triazoles achieved in the current study—particularly the determination of the centrifugal distortion constants—enables accurate prediction of the high-intensity transitions across a large range of frequencies for possible search in the interstellar medium (ISM) or in other harsh environments, such as Saturn’s moon, Titan. There is considerable current interest in identifying aromatic molecules²⁷ and nitrogen heterocycles^28,29 in these environments, and such studies may be particularly relevant to the nitrogen-rich atmosphere of Titan, where the structurally related molecule, 1H-1,2,4-triazole, has been suggested as a likely tholin component.^30,31 Although these molecules are not believed to be naturally occurring on Earth, they have been found to act as effective mimics in a variety of biological applications^32,33 with beneficial properties, such that 1,2,3-triazole-containing species are now widely targeted pharmaceuticals.³⁴ In terms of organic chemistry, 1H-1,2,3-triazole is the stable product of the Huisgen cycloaddition (a 1,3-dipolar cycloaddition) between hydrazoic acid and acetylene.³⁵ This is a quintessential example of “Click chemistry” where two small molecules generate an aromatic product in a highly exothermic reaction.^36,37 Acetylene has been detected in the ISM^38,39 and on Titan.⁴⁰ Amberger et al. assigned the rotational spectrum of hydrazoic acid,⁴¹ allowing for its search and potential identification. The detection (or not) of these molecules could be informative regarding the chemistry in various environments.

In this study, we observed transitions between 130 and 375 GHz for 1H- and 2H-1,2,3-triazoles, as well as several of their heavy-atom and deuterium-substituted isotopologues. Spectroscopic constants for many of the non-standard isotopologues are reported for the first time, and the spectroscopic constants for the standard isotopologues determined here include a full set of quartic, along with several sextic, centrifugal distortion constants. Presently, these and the spectroscopic constants of the isotopologues are used to obtain semi-experimental equilibrium structures for 1H- and 2H-1,2,3-triazoles, using a total of 13 isotopologues for the former and 16 for the latter.

EXPERIMENTAL METHODS

1H-1,2,3-triazole and 2H-1,2,3-triazole were obtained from Sigma-Aldrich (97% purity) and Oakwood Chemical (98% purity). The initial sample obtained (from Sigma-Aldrich) was pink and that from Oakwood was colorless. Initial rotational spectra were collected using the Sigma-Aldrich sample without further purification. In order to observe low-abundance isotopologues, a sample was later purified by Kugelrohr distillation. The sample from Sigma-Aldrich contained pyridazine and pyrazole as major impurities, identified by the previously collected rotational spectrum,⁴² and measured rotational constants.^43–45 Deuteriated samples were prepared as described later.

Continuous broadband spectra in the 130–230 and 250–375 GHz ranges were collected using 5–30 mTorr sample pressures at room temperature. The instrument has been described previously.^41,42,46 The low- and high-frequency spectra were combined into a single spectral file using Assignment and Analysis of Broadband Spectra (AABS) software.^47,48 Additional short-frequency-range scans were collected with increased numbers of scans for low-abundance isotopologues. Data were analyzed using ASFIT and ASROT,⁴⁹ along with the AC program.^50,51 In our least-squares fits, we assume a uniform 50 kHz frequency measurement uncertainty for all transitions, including those from Stiefvater et al.⁸ All least-squares output files are provided in the supplementary material.

COMPUTATIONAL METHODS

Calculations were carried out using a development version of CFOUR.⁵² The 1H- and 2H-1,2,3-triazole structures were first optimized at the CCSD(T)/cc-pCVTZ level of theory. The optimized geometries and the same level of theory were subsequently used for anharmonic, second-order vibrational perturbation theory (VPT2) calculations, wherein cubic force constants are evaluated using analytical second derivatives at displaced points,^53–55 for each 1,2,3-triazole tautomer and magnetic properties calculations for each isotopologue. The “best theoretical estimate” (BTE) for each tautomer was evaluated as described previously,^20,22–24 and details are provided in the supplementary material.

Using the experimental, averaged determinable rotational constants and computational electron-mass distribution and vibration–rotation interaction corrections as input, the xrefit module of CFOUR calculates the moments of inertia and semi-experimental equilibrium structure. Since (1) the Z-matrix whose parameters are determined in this routine must comprise the minimal parameters necessary to geometrically define the entire molecular structure and (2) standard error propagation assumes independent uncertainties, multiple Z-matrices had to be employed for each 1,2,3-triazole tautomer in order to obtain all parameters of interest and their proper uncertainty values. The xrefiteration routine,²² which begins by determining a structure using a single isotopic substitution at each position and then sequentially adds the most uncertainty-minimizing isotopologue to the structural least-squares fit until all available isotopologues are incorporated, was used to determine the suitability of the triazole isotopologues for the structure determination.

Computational output files are provided in the supplementary material.

SYNTHESIS OF DEUTERIO TRIAZOLES

The preparation of deuterium-containing isotopologues of 1,2,3-triazole was not as straightforward as for pyridazine⁴²and pyrimidine.²⁰H/D exchange using a strong base (t-BuOLi/t-BuOD), adapted from Esselman et al.⁴²and depicted in Scheme 1(a), was attempted, but produced a detectable quantity of only the N-deuteriated species. Alternate synthetic routes involving Grignard or organolithium reagents were considered, and the route depicted in Scheme 1(c) resulted in the successful generation of 1,2,3-triazole isotopologues with deuterium at the carbon positions. During the course of attempting to generate singly carbon-deuteriated 1,2,3-triazole using a set of steps similar to those in Scheme 1(c), however, it was recognized that weak bases, such as carbonate, result in the exchange of both bromine and hydrogen/deuterium at the carbon atoms. This observation led us to develop the reaction in Scheme 1(b). Detailed synthetic procedures for the reactions depicted in Scheme 1 are provided in the supplementary material.

SCHEME 1.

Deuterium incorporation in 1H- and 2H-1,2,3-triazoles: (a) N-deuteriation by strong base, (b) C-deuteriation by weak base, and (c) three-step sequence to achieve a high level of deuterium content at C-4 and C-5.

ANALYSIS OF ROTATIONAL SPECTRA

Transitions of 2H-1,2,3-triazole are more intense than those of the 1H tautomer, despite the larger dipole moment of the latter, confirming the greater population of the 2H than 1H tautomer in the gas phase. The spectrum of 2H-1,2,3-triazole (normal isotopologue) features b-type, R-branch transitions (^bR_1,1, and ^bR_−1,1). The spectrum of 1H-1,2,3-triazole (normal isotopologue) exhibits both a- and b-type, R-branch transitions (^aR_0,1, ^bR_1,1, and ^bR_−1,1), of which the a-type transitions are more intense. Certain isotopic substitutions in both triazole tautomers result in substantial rotation of the principal axes. For the 1H tautomer, such rotations can alter which of the dipole components (and corresponding transitions) are larger, and for the 2H tautomer, some isotopologues have a-type or both a- and b-type transitions. Figure 2 shows a small portion of an experimental spectrum in the 291.0–296.5 GHz range, which includes four isotopologues of each tautomer. Despite the presence of multiple triazole species, the spectrum is quite sparse as a result of each tautomer’s bands being separated by ∼10 GHz (∼2C). Such a separation fortuitously made it possible to analyze multiple isotopologues in a single spectrum without substantial loss of useable transition data. Even so, several of the lowest-intensity species could not be identified in the experimental spectra using their purely computationally predicted spectroscopic constants. These species were identified only after an r_e^SE structure of each tautomer (obtained using most, but not all, of the isotopologues in the final isotopologue set) was used to reverse-predict the rotational constants. These constants, combined with the computational centrifugal distortion constants, enabled the identification and fitting of transitions for the very low-intensity isotopologues.

FIG. 2.

Predicted stick spectra from 291.0 to 296.5 GHz (top) and experimental spectrum (bottom) of the normal and several deuteriated isotopologues of 1,2,3-triazoles. Transitions belonging to vibrationally excited states are also discernible.

Transitions of the ground vibrational states of observed isotopologues have typical oblate-type band structures; transitions progress to higher frequency away from the K_a = 0 transition with increasing K_a and concurrently decreasing J in our frequency region. Bandheads of 1H-1,2,3-triazole appear as four degenerate transitions: two ^aR_0,1 (J = K_a + K_c and J + 1 = K_a + K_c) transitions and two b-type (^bR_1,1 and ^bR-_1,1), while those of the 2H tautomer are made up of the two degenerate b-type transitions. For both tautomers, these transitions lose degeneracy near K_a = 12 in our frequency region. Q-branch transitions were also observed for both tautomers. Data distribution plots for the all isotopologues, showing the breadth of quantum numbers observed, are provided in the supplementary material.

Table I provides the 1H and 2H-1,2,3-triazole spectroscopic constants determined in this work alongside those determined by Begtrup et al.⁵ and those predicted computationally in the S reduction, III^r representation. The predicted rotational constants for both triazoles are within 0.1% of the experimental values. All of the predicted centrifugal distortion constants are within 10% of the experimental values for the 1H tautomer, with the exceptions of d₂ and H_J (whose predicted values are in error by 50% and 80%, respectively). All of the predicted centrifugal distortion constants of the 2H tautomer are within 22% of their experimental values. Despite the previous determination of the spectroscopic constants by Begtrup et al.⁵ using only partial quartic Hamiltonians (using τ terms), the rotational constants are in reasonable agreement with those determined in the current work. The spectroscopic constants of the triazoles determined in the A reduction (provided in the supplementary material) are also in reasonable agreement with the computed values, validating the use of the computed values when centrifugal distortion constants could not be determined experimentally. (Due to the similar values of A₀ and B₀ for each of the triazoles, the A reduction, III^r representation does not effectively determine the spectroscopic constants, so only the A reduction, I^r representation constants were determined.) Using both reductions enabled us to calculate the determinable constants from each reduction, compare the values, and use their average in the structure determination to eliminate the effects of reduction on the structure. A comparison of the determinable constants calculated from each reduction is provided in the supplementary material and gives confidence that the parameters used for the r_e^SE were determined accurately.

TABLE I.

Experimental and computational spectroscopic constants for 1H- and 2H-1,2,3-triazole, normal isotopologue, ground vibrational state (S-reduced Hamiltonian, III^r representation).

	1H-1,2,3-triazole			2H-1,2,3-triazole
	CCSD(T)a	Current work	Begtrup et al. (Ref. 5)b	CCSD(T)a	Current work	Begtrup et al. (Ref. 5)b
A0 (MHz)	10 036	10 030.800 90 (26)	10 030.785 (8)	10 257	10 252.030 69 (14)	10 255.006 (13)
B0 (MHz)	9 877	9 870.690 16 (27)	9 870.675 (8)	9 783	9 776.988 76 (14)	9 776.969 (13)
C0 (MHz)	4 976	4 972.941 28 (21)	4 972.933 (8)	5 005	5 002.457 56 (15)	5 002.448 (10)
DJ (kHz)	3.61	3.683 91 (31)		3.58	3.653 40 (22)
DJK (kHz)	−5.65	−5.772 36 (19)		−5.59	−5.716 73 (11)
DK (kHz)	2.43	2.490 22 (25)		2.41	2.465 545 (99)
d1 (MHz)	0.036 3	0.039 02 (35)		0.093 5	0.113 422 (57)
d2 (kHz)	0.008 04	0.016 033 (77)		0.018 5	0.015 176 (85)
HJ (Hz)	0.001 38	0.000 77 (15)		0.001 43	0.001 18 (11)
HJK (Hz)	−0.006 00	−0.006 040 (54)		−0.006 05	−0.006 138 (35)
HKJ (Hz)	0.007 85	0.007 94 (13)		0.007 80	0.007 836 (58)
HK (Hz)	−0.003 23	−0.002 95 (15)		−0.003 18	[−0.003 18]
h1 (Hz)	0.000 154	[0.000 154]		0.000 013 8	[0.000 013 8]
h2 (Hz)	−0.000 248	[−0.000 248]		−0.000 145	[−0.000 145]
h3 (Hz)	0.000 125	[0.000 125]		0.000 015 7	[0.000 015 7]
N lines c		838d			915
σ (MHz)		0.033			0.023

^a B_e values evaluated using the cc-pCV5Z basis set and corrected for vibration–rotation interaction and electron-mass distribution evaluated using the cc-pCVTZ basis.

^b Constants as reported in Ref. 5.

^c Number of independent transitions.

^d Transitions reported by Stiefvater et al.⁸ are included in the least-squares fit.

The heavy-atom isotopologues for both triazoles were observed at natural abundance in the same spectrum as the normal isotopologues. Using the previously described deuteriation methods, isotopologues substituting each H atom, and even some of their heavy-atom variants, were also analyzed. In total, 13 isotopologues were used for the 1H-1,2,3-triazole structure determination, and 16 isotopologues contributed to that of 2H-1,2,3-triazole. Although the 1H tautomer benefitted from the presence of both a- and b-type transitions, its smaller transition intensity and its lower symmetry (C_s) relative to that of the 2H tautomer (C_2v) made analyzing heavy-atom isotopologues more challenging. Where isotopologues substituting two of the nitrogen atoms and both of the carbon atoms of the 2H tautomer had their transition intensities doubled due to symmetry in any species that retained C_2v symmetry prior to heavy-atom substitution, none of the 1H isotopologues had such intensity enhancement. Most of the isotopologues were identified in the spectra based on their predicted spectroscopic constants, although some of the very low-intensity species required a more accurate prediction of their rotational constants using xrefit after the other isotopologues had been analyzed and included in the structural least-squares fit. For species where some spectroscopic constants could not be determined, these parameters were held constant at their predicted values. Tables of spectroscopic parameters for all measured isotopologues of 1H-1,2,3-triazole and 2H-1,2,3-triazole (S reduction, III^r representation; A reduction, I^r representation; comparison of determinable constants), along with data distribution plots, are provided in the supplementary material.

SEMI-EXPERIMENTAL EQUILIBRIUM STRUCTURE (r_e^SE)

Spectroscopic constants determined in both A and S reductions (B₀) were converted to determinable constants (B″) using Eqs. S(1)–S(6) in the supplementary material. The agreement between these two sets of parameters (within 0.04 MHz) confirmed that the rotational and quartic centrifugal distortion constants were determined with sufficient accuracy for use in the structure determination. The two sets of determinable constants were averaged and input into xrefit, along with vibration–rotation interaction and electron-mass distribution corrections predicted at the CCSD(T)/cc-pCVTZ level of theory. The xrefit module uses the values to calculate the equilibrium rotational constants (B_e) using Eq. (1), where the second term is the vibration–rotation interaction correction, $\frac{m_e}{M_p}$ is the electron-to-proton mass ratio, B″ is the determinable constant, and g^ββ is the corresponding magnetic g-tensor component.

$$\begin{array}{c}\hfill B_e^{\beta }=B^{″\beta }+\frac{1}{2}\sum _i{\mathrm{\alpha }}_i^{\beta }-\frac{m_e}{M_P}{g}^{\beta \beta }{B}^{\beta }.\hfill \end{array}$$ (1)

These rotational constants are then used to calculate the inertial defect (Δ_i), which is precisely zero for a rigid, planar molecule. Absent either of the computational corrections, it is evident from Table II that the observable inertial defect (Δ_i0) is ∼0.04 uŲ for both triazole tautomers. As observed for other heterocyclic molecules,^20–24 inclusion of the vibration–rotation interaction correction decreases the inertial defect (Δ_ie) to a small negative value, and subsequent application of the electron-mass correction brings the magnitude of the inertial defect (Δ_ie) quite close to zero. In close similarity to several other heterocyclic aromatic molecules, all of the fully corrected inertial defect values for 1H-1,2,3-triazole are positive, with an average value of 0.0013 ± 0.0001 uŲ. Strikingly, the fully corrected inertial defect values for the 2H tautomer are both positive and negative. Their average value is 0.000 09 ± 0.0003 uŲ, a value that is approximately two orders of magnitude smaller than that observed in previous structure determinations. For the nitrogen-substituted benzene analog, pyrimidine,²⁰ and pyridazine,²² the fully CCSD(T)-corrected inertial defect values were consistently negative with an average value of −0.0011 uŲ. For thiophene²³ and thiazole,²⁴ these inertial defect values were consistently positive with average values also close to 0.001 uŲ. It is, therefore, interesting, but not obvious as to why, the 2H-1,2,3-triazole isotopologues exhibit corrected inertial defect values over an order of magnitude smaller than those observed previously. The numbers of transitions observable for the 2H-1,2,3-triazole species are smaller than those observable for other previously examined molecules, many of whose transitions numbered in the thousands. It is, therefore, unlikely that the spectroscopic constants for 2H-triazole are somehow better determined than those for other molecules. It is alternately conceivable that the spectroscopic constants used for the structure determination were heavily affected by having too few transitions measured to determine many parameters, resulting in many parameters held constant at their predicted values, generating an artificially close agreement between the experimentally determined and computed constants. If the experimental constants were to agree perfectly with computed values, then the corrections at the same level of theory would result in ideal inertial defect values. This, however, is not the case, as some isotopologues with a small number of transitions (21 for [4-²H, 5-¹³C] and 41 for [4,5-²H, 4-¹³C]) and with the fewest spectroscopic parameters able to be determined have the largest magnitude of fully corrected inertial defect values. Meanwhile, those species with inertial defect values particularly close to zero vary from 60 to 632 transitions analyzed and have between three and all five quartic centrifugal distortion constants determined. Importantly, this discrepancy between 2H-1,2,3-triazole and other molecules does not appear to indicate a problem. The small values of the fully corrected inertial defects for both 1H and 2H tautomers indicate again that the spectroscopic constants are sufficiently well determined to be used in the structure determination.

TABLE II.

Inertial defects (Δ_i) of triazole isotopologues from various determinations of the moments of inertia.

1H-1,2,3-triazole				2H-1,2,3-triazole
Isotopologue	Δi0 (uÅ2)	Δie (uÅ2)a	Δie (uÅ2)b	Isotopologue	Δi0 (uÅ2)	Δie (uÅ2)a	Δie (uÅ2)b
C2H3N3	0.043 0	−0.008 86	0.001 35	C2H3N3	0.0399	−0.009 60	0.000 239
[4-13C]-	0.043 6	−0.008 94	0.001 27	[4-13C]-	0.0404	−0.009 68	0.000 162
[5-13C]-	0.043 6	−0.008 85	0.001 35	[2-15N]-	0.0401	−0.009 76	0.000 087
[1-15N]-	0.043 3	−0.008 98	0.001 23	[1-15N]-	0.0405	−0.009 58	0.000 258
[2-15N]-	0.043 4	−0.009 02	0.001 18	[4-2H]-	0.0401	−0.009 47	0.000 380
[4-2H]-	0.043 1	−0.008 97	0.001 23	[2-2H]-	0.0343	−0.009 97	−0.000 123
[5-2H]-	0.042 7	−0.008 86	0.001 34	[2-2H, 4-13C]-	0.0349	−0.009 82	0.000 030
[1-2H]-	0.038 5	−0.008 93	0.001 27	[2-2H, 1-15N]-	0.0347	−0.009 88	−0.000 028
[1,4-2H]-	0.038 5	−0.009 04	0.001 17	[2,4-2H]-	0.0343	−0.009 86	−0.000 010
[1,5-2H]-	0.038 2	−0.008 95	0.001 26	[2,4,5-2H]-	0.0346	−0.010 05	−0.000 201
[1,4,5-2H]-	0.038 3	−0.009 01	0.001 20	[1,2,3–15N]-	0.0413	−0.009 55	0.000 288
[1,2,3-15N]-	0.044 6	−0.008 78	0.001 43	[2-2H, 1,2,3-15N]-	0.0356	−0.009 83	0.000 015
[4,5-2H]-	0.042 93	−0.008 88	0.001 33	[4,5-2H]-	0.0405	−0.009 69	0.000 153
Average (®)	0.041 8	−0.008 9	0.001 3	[4,5-2H, 4-13C]-	0.0411	−0.009 32	0.000 528
Std. Dev. (s)	0.002 4	0.000 1	0.000 1	[4-2H, 4-13C]-	0.0404	−0.009 65	0.000 199
				[4-2H, 5–13C]-	0.0398	−0.010 35	−0.000 503
				Average (®)	0.0383	−0.009 75	0.000 092
				Std. Dev. (s)	0.0029	0.000 25	0.000 245

^a Vibration–rotation interaction corrections only.

^b Vibration–rotation interaction and electron-mass corrections.

The semi-experimental equilibrium structure parameters of 1H and 2H-1,2,3-triazole resulting from 13 to 16 isotopologues, respectively, are depicted in Fig. 3 and enumerated in Tables III and IV. The 2σ statistical uncertainties of the 1H-1,2,3-triazole bond distances are all <0.0008 Å. Corresponding uncertainties in the internal bond angles are <0.06°, while those of external angles are <0.2°. Overall, the parameters are determined to a similar degree of precision as those of thiazole.²⁴ Surprisingly, despite the larger number of isotopologues used to determine fewer structural parameters, the 2σ bond distance uncertainties for 2H-1,2,3-triazole are approximately a factor of two greater than for 1H-1,2,3-triazole. The only bond distance in the symmetrical isomer determined to better than 0.001 Å is that of the two equivalent C–H bond distances. All three angles of 2H-1,2,3-triazole have uncertainties just under 0.1°. The observation that the 2H-1,2,3-triazole structure is less well-determined than the 1H tautomer would seem to suggest that perhaps some of the isotopologues in the structure determination were providing conflicting information, resulting in larger parameter uncertainties. While the effect of isotopologue inclusion on structure indeed supports this idea, it is unclear whether the inclusion of the isotopologues in question is detrimental to the accuracy of the resultant parameters (vide infra).

FIG. 3.

Semi-experimental equilibrium structures (r_e^SE): (a) 1H-1,2,3-triazole and (b) 2H-1,2,3,-triazole with 2σ statistical uncertainties from least-squares fitting the isotopologue moments of inertia.

TABLE III.

Structural parameters of 1H-1,2,3-triazole.

	reSE CCSD(T)/cc-pCVTZ	CCSD(T) BTE	CCSD(T)/cc-pCV5Z
RC4–H (Å)	1.073 5 (3)	1.073 69	1.073 58
RC4–C5 (Å)	1.369 2 (5)	1.369 49	1.369 39
RC5–H (Å)	1.073 1 (3)	1.073 30	1.073 2
RN3–C4 (Å)	1.362 8 (7)	1.362 72	1.362 22
RN1–C5 (Å)	1.350 2 (6)	1.350 35	1.349 90
RN2–N3 (Å)	1.300 8 (7)	1.301 14	1.300 07
RN1–H (Å)	1.003 8 (4)	1.003 90	1.003 70
RN1–N2 (Å)	1.341 1 (7)a	1.342 21b	1.340 26b
θC5–C4–H (°)	129.75 (12)	129.662	129.636
θC4–C5–H (°)	133.28 (12)	133.421	133.360
θN3–C4–C5 (°)	108.715 (38)	108.714	108.693
θN1–C5–C4 (°)	103.625 (33)	103.654	103.603
θN2–N3–C4 (°)	108.981 (55)	109.003	109.000
θC5–N1–H (°)	129.10 (11)	128.900	129.152
θN2–N1–C5 (°)	111.620 (43)a	111.59b	111.63b
θN1–N2–N3 (°)	107.059 (43)a	107.04b	107.07b
N isotopologues	13

^a Redundant parameters and uncertainties determined by using alternate Z-matrices.

^b Redundant parameters not determined directly; calculated using geometry.

TABLE IV.

Structural parameters of 2H-1,2,3-triazole.

	Begtrup et al. (Ref. 5)	reSE CCSD(T)/cc-pCVTZ	CCSD(T) BTE	CCSD(T)/cc-pCV5Z
RN2–H (Å)	0.997	1.003 (1)	1.003 48	1.003 3
RN1–N2 (Å)	1.317	1.323 (1)	1.322 37	1.321 0
RN1–C5 (Å)	1.339	1.333 (1)	1.333 54	1.332 9
RC4–H (Å)	1.069	1.0744 (6)	1.074 16	1.074 06
RC4–C5 (Å)	1.398	1.399 (2)a	1.399 35b	1.399 41b
θN1–N2–H (°)	121.5c	121.601 (81)	121.569	121.560
θN2–N3–C4 (°)	103.0c	102.912 (95)	102.894	102.92
θN1–C5–H (°)	138.5c	120.869 (86)	120.887	120.928
θN1–N2–N3 (°)	117.1d	116.80 (16)a	116.862b	116.881b
θN1–C5–C4 (°)	108.5c	108.689 (43)a	108.675b	108.636b
N isotopologues	3e	16

^a Parameters and uncertainties determined by using alternate Z-matrices.

^b Parameters not determined directly; calculated using geometry.

^c Parameters not determined directly in original work; calculated here for comparison.

^d Angle held constant at ab initio value.

^e Microwave work examined standard isotopologue, as well as [1,2,3-¹⁵N]- and [2-²H, 1,2,3-¹⁵N]-2H-1,2,3-triazoles.

BEST THEORETICAL ESTIMATE

As described previously,^20,22–25 the best theoretical estimate (BTE) is based on a CCSD(T)/cc-pCV5Z optimized structure with four additional corrections that address:

• 1.residual basis set effects beyond cc-pCV5Z,
• 2.residual electron correlation effects beyond the CCSD(T) treatment,
• 3.effects of scalar (mass-velocity and Darwin) relativistic effects,
• 4.the fixed-nucleus approximation via the diagonal Born–Oppenheimer correction.

Equations used to calculate these corrections and the values of each of these corrections for both 1,2,3-triazole tautomers are provided in the supplementary material (Tables S-VIII and S-IX). One of the most important factors of the algorithm used to determine the BTE is the estimation of residual basis set effects, specifically estimated as the difference between the (directly computed) cc-pCV5Z geometry at the CCSD(T) level of theory and the estimate of the CCSD(T) geometry at the basis set limit. Following others,¹⁹ the latter is obtained by assuming an exponential convergence pattern with respect to the highest angular momentum basis functions present in the basis. For 1H-1,2,3-triazole, there are decidedly non-exponential patterns exhibited by a few of the geometrical parameters, as revealed by careful consideration of the computational data that is provided in the supplementary material. For example, the N–H bond distance computed with the cc-pCVQZ basis is actually shorter (but minimally so) than that with cc-pCV5Z, a direct violation of the standard pattern of decreasing bond lengths with respect to basis expansion. In this case—where the cc-pCV5Z distance is shorter than cc-pCVTZ and longer than cc-pCVQZ—the behavior is not even qualitatively exponential, and extrapolation is meaningless. Similarly, while the angles θ_C4–C5–H and θ_C5–N1–H exhibit a monotonic pattern of change between cc-pCVTZ and cc-pCV5Z, the cc-pCVTZ/QZ difference is smaller than the cc-pCVQZ/5Z difference. The triple- to quintuple-zeta N-H bond distance and θ_N1–C5–H of the 2H tautomer also exhibit non-exponential behavior. This leads to quite unreliable extrapolated values for all of these parameters.

It is notable that two of the three problematic parameters of 1H-1,2,3-triazole correspond to those which are conspicuous in the comparison of BTE and semi-experimental parameters. For the θ_C4–C5–H angle, the BTE (133.42°) falls just above the range of the semi-experimental determination (133.28 (12)°), while the respective values for θ_C5–N1–H are 128.90° and 129.10 (11)° (Table III). If the cc-pCV5Z angles are used instead of the dubious extrapolated values, the so-modified BTE angles become 133.36° and 129.16°, respectively, both of which lie within the corresponding semi-experimental uncertainty ranges. The source of the non-exponential behavior is unclear, although challenges posed by the electronic structure effects associated with the corresponding geometric displacements might be at work here. To further investigate this issue, calculations were carried out with the core-weighted (cc-pwCVXZ) basis sets (Table S-VIII), but the three problematic parameters still exhibit the same qualitative behavior. The dubious nature of the corresponding extrapolations is further evidenced by the fact that all extrapolated distances and angles with the cc-pwCVXZ extrapolation are within 0.0002 Å and 0.02°, respectively, of the corresponding cc-pCVXZ results for the non-problematic results, but differences in the problematic parameters are at least an order of magnitude larger (Table S-VIII). This results from the fact that the underlying data are not well-described by the approximation of exponential convergence, and it might well be better here to substitute the cc-pCV5Z results for those obtained by the extrapolation procedure.

DISCUSSION

As has become standard in our recent structure determinations, the effect of including the available isotopologues in the r_e^SE structure is examined using xrefiteration. This script has been described in detail previously.²² Briefly, the analysis provides the r_e^SE parameters based on a set of isotopologues substituting each unique atom once. The program then iteratively adds one isotopologue at a time to sequential r_e^SE determinations until all isotopologues are incorporated. The order in which the isotopologues are incorporated is based on the extent to which they decrease the uncertainty in the parameters; those that most decrease the uncertainty are added first. Figures showing a composite structural uncertainty as a function of a number of incorporated isotopologues are provided in the supplementary material. Figure 4 shows the structural parameter values and their uncertainties as a function of number of isotopologues for 1H-1,2-3-triazole. As depicted in the left-side plots, bond distances appeared to be fairly well converged even using single isotopic substitutions at each atom, though the incorporation of even a single additional isotopologue results in a noticeable change that brings the value into closer agreement with the BTE value (dashed, colored line) for several parameters (e.g., R_N1–H, R_N2–C4). The inclusion of further isotopologues has only a minor effect on the uncertainties in the bond distances, but it is not possible to know a priori how many isotopologues are necessary to converge a parameter. Examination of the angles in the right-hand column, similarly, reveals that θ_C5–C4–H does not converge until the ninth isotopologue is incorporated, and the additional isotopologues appear important for the determination of the external, H-involving angles of 1H-1,2,3-triazole (the top three plots in the right-hand column of Fig. 4). As mentioned previously, the BTE values for two angles (θ_C4–C5–H and θ_C5–N1–H) fall outside the 2σ uncertainties of the semi-experimental values. If, however, the correction for residual basis set effects beyond cc-pCV5Z (which are manifestly unreliable for these parameters) is omitted (black, dotted lines), these two parameters fall into agreement with the computed values. In fact, the value of θ_C5–N1–H moves farther from the BTE value and closer to that excluding the basis set correction with the incorporation of more isotopologues. One parameter whose basis set extrapolation is reasonable (vide supra), θ_C5–C4–H, falls into disagreement, though this may be an indication that all corrections are important for some parameters, as observed previously.^20,22

FIG. 4.

Plots of the structural parameters of 1H-1,2,3-triazole as a function of the number of isotopologues (N_iso) and their 2σ uncertainties with consistent scales for each distance and for each angle. The table at the bottom indicates the xth isotopologue added to the r_e^SE. Colored dashed lines indicate the BTE value; black dotted lines indicate the value obtained by adding all corrections except the basis set extrapolation to the CCSD(T)/cc-pCV5Z value.

Figure 5 shows the analogous structural parameter plots for the 2H-1,2,3-tautomer. In this case, the θ_N1–N2–H and θ_N2–N3–C4 values are reasonably stable through the 14th isotopologue. The value of θ_N1–C5–H, however, changes with the incorporation of even a few isotopologues beyond the minimal set, and its uncertainty decreases noticeably. The bond distance values and some of their uncertainties are refined by the inclusion of numerous isotopologues. It is visually striking that, with the last two isotopologues, [2-²H, 1-¹⁵N] and [4-²H, 5-¹³C], several of the parameters change in a discontinuous way. Furthermore, inclusion of the last four … increases the uncertainty of several parameters (see also Fig. S27). The inclusion of these four species is responsible for the aforementioned surprisingly lower precision of multiple parameters in the 2H tautomer relative to the 1H tautomer. Such effects may be due to errors in the data (rotational constants or corrections) used for the corresponding isotopologues or due to the fact that these isotopologues provide new or different information than the prior isotopologues. The last four isotopologues are indeed those whose spectroscopic constants are determined using the fewest transitions: 11 transitions for [4-²H, 4-¹³C], 41 transitions for [4,5-²H, 4-¹³C], 60 transitions for [2-²H, 1-¹⁵N], and 21 transitions for [4-²H, 5-¹³C]. Potential support for the hypothesis that the relatively small number of transitions identified for these species results in inaccurate rotational constants could be that the fully corrected inertial defects (Table II) of two of the carbon-deuteriated species ([4,5-²H, 4-¹³C] and [4-²H, 5-¹³C]) have the largest magnitudes of the 2H isotopologues, ranging from 0.000 50 to 0.000 53 uŲ. The N-deuterio isotopologue, [2-²H, 1-¹⁵N], however, has a quite small inertial defect magnitude (0.000 03 uŲ), and the other carbon-deuteriated species, [4-²H, 4-¹³C], also has among the lowest magnitudes (0.000 20 uŲ).

FIG. 5.

Plots of the structural parameters of 2H-1,2,3-triazole as a function of the number of isotopologues (N_iso) and their 2σ uncertainties with consistent scales for each distance and for each angle. The table at the right indicates the xth isotopologue added to the r_e^SE. Colored dashed lines indicate the BTE value; black dotted lines indicate the value obtained by adding all corrections except the basis set extrapolation to the CCSD(T)/cc-pCV5Z value.

Moreover, all of these inertial defect values would have been considered not only acceptable but also among the smallest values of previous r_e^SE structures examined in this way. If these four isotopologues are excluded (x = 12 in Fig. 5) from the r_e^SE of 2H-1,2,3-triazole, the 2σ uncertainties of all bond distances fall to <0.001 Å, but two of the bond distances (R_N1–N2 and R_C4–H) fall out of agreement with the corresponding BTE values. If only the last two isotopologues are excluded, the BTE and r_e^SE remain in complete agreement, but the desired precision is still not achieved. Parameters for four additional sets of r_e^SE structures for 2H-1,2,3-triazole, excluding the isotopologues in question here, are provided in the supplementary material. It is typical to consider the agreement of computational results in comparison to experimental (or semi-experimental) results, the latter being generally accepted to be more accurate in reality. Based on the exceptional agreement of several other BTEs and their respective r_e^SE structures (especially those containing only C, N, and H atoms),^20,22,25 however, it would appear surprising for the BTE presented here—calculated using the same levels of theory in all components as those of the cited works—to suddenly fail to predict the structural parameters of the triazoles to the same degree of accuracy. Since two bond distances that fall out of agreement upon removal of the last four isotopologues are expected to be properly estimated by the BTE (i.e., their basis set extrapolation correction is appropriate), we consider the r_e^SE parameters of 2H-1,2,3-triazole presented here to be the most accurate. The inclusion of all 16 isotopologues in the determination of the parameters is necessary, albeit at the cost of the parameters' precision.

Figures 6 and 7 provide, respectively, for 1H- and 2H-1,2,3-triazole, a means to visualize the effects of the sum corrections on the respective CCSD(T)/cc-pCVXZ parameters and the agreement between r_e^SE and computational structures. The patterns that emerge upon examination of these plots are similar for both tautomers. In many cases, the triple–zeta parameter value is substantially outside of the r_e^SE 2σ uncertainty. The quadruple-zeta values fall within the 2σ uncertainty for approximately half of the 1H parameters and nearly all of the 2H parameters. The quintuple-zeta values are within the 2σ uncertainty of the r_e^SE value for all of the 2H parameters and all but a single 1H parameter. The 1H parameter whose quintuple-zeta value falls outside the r_e^SE value (θ_C5–C4–H) shows a relatively large spread in the triple-, quadruple-, and quintuple-zeta values (Fig. 6, right-hand panel, top graph). The corresponding BTE value is in agreement with the r_e^SE. Conversely, the three direct computational values are closely clustered and all within the r_e^SE uncertainty for the θ_C4–C5–H and θ_C5–N1–H parameters of 1H-1,2,3-triazole (Table S-VIII)—it is the corrections applied to obtain the BTE (in particular, the aforementioned problematic extrapolated basis set correction) that put the BTE values out of agreement with the semi-experimental structure.

FIG. 6.

Graphical comparison of the 1H-1,2,3-triazole structural parameters with bond distances in angstroms (Å) and angles in degrees (°). Bond distances are set to the same scale, and bond angles are set to the same scale. Uncertainties shown for current, CCSD(T) r_e^SE are 2σ. Cyan star indicates the BTE value; red star indicates the value obtained by adding all corrections except the basis set extrapolation to the CCSD(T)/cc-pCV5Z value.

FIG. 7.

Graphical comparison of the 2H-1,2,3-triazole structural parameters with bond distances in angstroms (Å) and angles in degrees (°). Bond distances are set to the same scale, and bond angles are set to the same scale. Uncertainties shown for current, CCSD(T) r_e^SE are 2σ. Where symbols of predicted values do not appear, they are sufficiently far from the CCSD(T) r_e^SE as to not fit within the respective displayed ranges. Cyan star indicates the BTE value; red star indicates the value obtained by adding all corrections except the basis set extrapolation to the CCSD(T)/cc-pCV5Z value.

CONCLUSION

This work represents a continuation of our recent research efforts to determine highly accurate equilibrium structures and, in particular, extends the number of heterocyclic organic molecules that have been studied.^20,22–24 Unlike pyrimidine²⁰ and pyridazine,²² however, the 1,2,3-triazole structures presented in this work are exceptional in the sense that 1H-1,2,3-triazole is the first molecule containing only carbon, nitrogen, and hydrogen for which the BTE and r_e^SE structures are not in complete agreement. These differences—two of the BTE parameters fall outside the 2σ uncertainties of the parameters of the r_e^SE structure—have been traced to issues associated with the extrapolation of structural parameters computed with large core-correlating basis sets (cc-pCVXZ, with X = T, Q, and 5) to the complete basis set limit. The atypical behavior observed in this study might be ameliorated by using alternative extrapolation strategies, but it is by no means obvious how this might be accomplished. It is only certain from our analysis that the use of the raw, uncorrected cc-pCV5Z result is a superior strategy to a nonsensical extrapolation correction.

In other respects, however, the results obtained for the 1,2,3-triazole isomers are among the best that we have been able to obtain in studies of this sort. For example, in recent studies on thiophene²³ and thiazole,²⁴ the process of determining accurate semi-experimental structures has been compromised by the presence of atoms that lie near, but not along, inertial axes. Although the N3 and C4 atoms of 1H-1,2,3-triazole are also fairly close to inertial axes, they are easily rotated away from principal axes by isotopic substitution and geometric parameters involving these atoms are readily determined with high accuracy. The positions of N2 and the bonded H atom of the 2H tautomer, which lie along an inertial axis in the main isotopologue, appear no less poorly determined than other positions in the molecule. The inertial defects defects obtained from the semi-experimental equilibrium rotational constants in this work are the smallest (by more than an order of magnitude) yet found in our recent semi-experimental structural determinations. Furthermore, the high quality of data in this work facilitated the detection of several low-abundance isotopologues in the experimental spectrum. The latter underscores the virtue of using such highly accurate r_e^SE structures to predict rotational constants that can subsequently be used to seek isotopologues of interest in complex spectra, including observational astronomical spectra. Finally, this work supplies the scientific community with spectroscopic constants including quartic and sextic centrifugal distortion constants necessary to identify these species in the ISM or other harsh environments. The detection (or lack thereof) of these triazoles in various locations would be interesting and potentially shed light on the types of chemistry in these environments.

SUPPLEMENTARY MATERIAL

Computational output files, least-squares fitting for all isotopologues, data distribution plots for all isotopologues, mass spectra, xrefiteration outputs, equations used for calculating determinable constants and BTE corrections, and tables of S reduction, A reduction, and determinable constants, as well as 2H-1,2,3-triazole r_e^SE parameters determined using 12–15 isotopologues, are provided in the supplementary material.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Maria A. Zdanovskaia: Formal analysis (equal); Writing – original draft (lead). Brian J. Esselman: Formal analysis (equal); Writing – review and editing (equal). Samuel M. Kougias: Methodology (equal). Brent K. Amberger: Investigation (supporting). John F. Stanton: Conceptualization (equal); Formal analysis (equal); Writing – review and editing (equal). R. Claude Woods: Conceptualization (equal); Formal analysis (equal); Supervision (equal); Writing – review and editing (equal). Robert J. McMahon: Funding acquisition (lead); Project administration (lead); Supervision (equal); Writing – review and editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.