Introduction

Tetrazoles are important high-energy self-organized combustion compounds, burning at the expense of their own components without any oxidizer [1]. Tetrazole derivatives are characterized by extreme properties among azoles, such as the highest N–H-acidity, the lowest basicity, and the presence of several “pyridine-like” nitrogen atoms [2]. These features along with biological activity of tetrazoles and their high nitrogen content and positive enthalpies of formation have made these substances and their complexes interesting for many different branches of science, pharmaceutical [3–5], and material chemistry including gas-generating and energetic materials [6–13], ferromagnets [14], and other materials for molecular electronics [15–17]. 1,5-Diaminotetrazole (DAT), as a fascinating component in the construction of high energy density materials because of its unique structure and remarkable properties, has attracted considerable attention [7, 18–25].

There has been considerable and continuing interest in the co-existence of tautomers of tetrazoles. A lot of experimental and theoretical studies have been devoted to the tautomerism and protonation of tetrazole [26, 27]. Results show that the temperature-dependent tautomeric amino-imino and prototropic transformations are important preliminary steps in reactions of 5-aminotetrazole (ATZ) [28]. We have studied the tautomerism of ATZ, and found that six possible tautomers could exist in theory, namely 1H,4H-5-imino-tetrazole, 1-H-5-amino-tetrazole, 2-H-5-amino-tetrazole, 1H,2H-5-imino-tetrazole, the mesoionic form, and 2H,4H-5-imino-tetrazole. In principle, DAT may coexist in amino and imino tautomeric forms in the solid state and probably also in the melted [29].

To the best of our knowledge, few studies have been concerned with 1,5-diaminotetrazole, and the present investigation was undertaken to study the mechanism and kinetics of the tautomerization of DAT in the same way as for ATZ. This paper focuses on the tautomers and intramolecular hydrogen shift of DAT (Fig. 1) and their kinetics in the gas phase. The aims of our study are to determine the stability of the tautomers and the mechanism of these tautomerization reactions, and obtain more kinetic information for the title reactions, including the rate constants over a wide temperature range. Specifically, we have focused on the importance of the tunneling effect. With a clearer knowledge of the existing forms, some new complexes may be obtained, and the results may prove useful in rationalizing certain phenomena and explaining some experimental data, such as the infrared spectrum, the various products of decomposition, and so on.

Fig. 1
figure 1

The tautomers and reactions of the tautomerism of DAT using the MP2/6-311G**

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Computational methods

All calculations were carried out with the Gaussian 03 program [30]. The optimized geometries and harmonic vibrational frequencies of all stationary points (the reactants, products, and transition states) were optimized with restricted second-order Møller−Plesset perturbation theory using the 6-311G** basis set. To yield more reliable enthalpies and barrier heights, we performed single-point energy calculations for the stationary points by the CCSD(T) method with the 6-311G** basis set. The intrinsic reaction coordinate (IRC)[31], and the minimum-energy path (MEP), were obtained at the MP2/6-311G** level to confirm that whether the transition states really connect with minima along the reaction path. The MEP was calculated in mass-weighted Cartesian coordinates with a gradient step size of 0.02 (amu)1/2 bohr. The energy derivatives, including gradients and Hessians at geometries of some selected points (17 points on each side) along the MEP, were calculated at the same level. The single-point energy calculations for the points along the MEP were then refined to establish the potential curve at the CCSD(T)/6-311G** level of theory.

All of the rate constants reported herein were evaluated over a wide temperature range from 200 to 2500 K using conventional, non-variational transition-state theory (TST) [32], TST rate constant calculations with the Eckart tunneling correction (TST/Eckart) [33], canonical variational transition-state theory (CVT)[34–36], canonical variational transition-state theory with small curvature tunneling correction (CVT/SCT) [37, 38], and canonical variational transition-state theory with zero-curvature tunneling correction (CVT/ZCT) [39, 40] by means of the Vklab program package [41] and the POLYRATE 8.2 program [42].

Results and discussion

The geometries and vibrational frequencies

We considered five tautomers of DAT in this study for comparison. The optimized structures for the reactants, products, and transition states consisting of the main bond parameters at the MP2/6-311G** level of theory, are given in Table 1. The experimental length of C = N bond is 1.273 Å in H2C = NH, and the bond length of C-N is 1.474 Å in CH3NH2 [43]. The distances of N-N in H-N = N-H and H2N-NH2 are 1.252 and 1.449 Å, respectively [44].

Table 1 The optimized geometries of tautomers of DAT and the transition states at MP2/6-311G**
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For tautomers a, c, d, and e, nearly all of the skeletal bonds are calculated to lie between isolated single and double bonds, except for the C4-N1 bond of a, the C4-N5 and N3-N2 bonds of c, and the C4-N1 and N5-N9 bonds of e, which are close to double bonds. The atoms of a and c are nearly in a plane with deviations of less than 0.5°, except for the hydrogen atoms of amino. The rings of d and e show a greater deviation from planarity and there is significant repulsion between the hydrogen atoms on N1 and N9 in e. For tautomer b, nearly all of the skeletal bonds are calculated to lie between isolated single and double bonds, and the atoms are not coplanar, showing greater deviations from planarity of 5°.

The skeletal bond lengths calculated for both b and c (Table 1) lie between those of isolated single and double bonds. The results clearly suggest that the lone pairs of electrons at N2 in b and at N5 in c are delocalized in planar five-center 6π electron aromatic systems, indicating that the molecules are aromatic, which may be the reason for the greater stability of these tautomers. 2H-5-amino-tetrazole has previously been suggested as a stable tautomer of ATZ, and our calculations show that 2,5-diamino-tetrazole (b) is also the most stable tautomer of DAT in theory.

Both the optimization and the frequency calculations were successfully completed for all of the reactants, products, and transition states of the above mentioned five reactions at the MP2/6-311G** level. Reactants and products were observed as local minima with positive vibrational frequencies. Only one imaginary frequency was found for each transition state, and the transition states of all five reactions were first-order saddle points in the potential barriers of the reaction paths. The infrared spectra of all five tautomers and the transition states of the tautomeric equilibria of the hydrogen shift are summarized in Table 2. The mechanisms of the tautomerization reactions are shown in Fig. 2. TSc-a, TSc-b, TSc-d, TSa-e, and TSd-e denote the transition states of reactions 1, 2, 3, 4, and 5, respectively, and normal-mode analysis revealed that they have only one imaginary frequency corresponding to exchange of the hydrogen atoms. The imaginary frequencies of TSc-a, TSc-b, TSc-d, TSa-e, and TSd-e are 1852i cm−1, 915i cm−1, 1584i cm−1, 1499i cm−1, and 1815i cm−1, respectively. Because a, b, c, d, and e were calculated to have similar IR spectra, it may not be straightforward to use IR spectroscopy to differentiate between these tautomers.

Table 2 Harmonic frequencies (cm−1) and intensities (km· mol−1), for the reactants, products, and transition states at MP2/6-311G**
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Fig. 2
figure 2

The mechanism of the tautomeric equilibrium and hydrogen shifts of DAT

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Energy barriers to tautomerism

The minimum-energy paths were obtained using the intrinsic reaction coordinate (IRC) theory at the MP2/6-311G** level of theory, and the potential energy profiles were further refined using CCSD(T)/6-311G**. Table 3 gives the reaction enthalpies (ΔH 0298K ), the reaction energies (ΔE), the classical potential barriers (VMEP), the ground-state adiabatic energy curve (V G a ), which is defined by the refined energies calculated using CCSD(T)/6-311G** with zero-point energies at the MP2/6-311G** level (VMEP + ZEP).

Table 3 The energy parameters (kcal · mol−1) of each tautomerism reaction of DAT
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These data show that, although less stable than tautomers b and c, tautomers a, d, and e are good candidates for experimental observation and worthy topics for future research. We can also conclude from Table 3 that all of these reactions are endothermic processes. These reactions may play an important role in both the atmosphere and in combustion systems of DAT. The order of V G a in the forward direction is reaction 4 > reaction 5 > reaction 2 > reaction 3 > reaction 1. The value of V G a in reaction 1 and reaction 3 are similar and that of V G a in reaction 2 is larger. Thus the result indicates that hydrogen shifts such as c → a and c → d are easier than c → b as amino shifts. A schematic representation of the tautomers and transition states for DAT is shown in Fig. 3.

Fig. 3
figure 3

Schematic representation of tautomers and transition states of DAT

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It can be seen that a, b, and c are relatively more stable than d and e. As a result, the value of V G a in reaction 4 and reaction 5 may be larger, and the low V G a of the reverse reactions in reactions 3, 4, and 5 also show that d and e are very unstable. For the reverse direction, the order of V G a is reaction 2 > reaction 1 > reaction 3 > reaction 4 > reaction 5. Finally, all of the results point to the fact that reactions 3, 4, and 5 are irreversible exothermic processes. All of these reactions need a suitable source of energy or an appropriate catalyst to proceed.

Figure 4 shows the adiabatic ground-state potential energy curves (V G a ) and the classical potential barriers (VMEP) of the reactions. As can be seen, the curve of reaction 2 is smoother, in accordance with the fact that the imaginary frequency of the transition state of this reaction is smaller, which indicates the tunneling effect of reaction 2 is smaller. Apart from reaction 2 as an amino-shift reaction, the other four reactions are quite similar because they are all hydrogen-shift reactions. In general, the transition states of hydrogen-shift reactions have higher imaginary frequencies and more cuspidaler curves with a greater tunneling effect.

Fig. 4
figure 4

The adiabatic ground-state potential energy curves (V a G) and the classical potential barriers (VMEP) of the reactions as a function of s (amu)1/2 bohr at the MP2/6-311G** level of theory

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In summary, the tautomers a and d are expected to be most easily derived from c experimentally. While b is the most stable, but is difficult to interchange with c. Again e is the least stable, while the probability of this reaction from c to e occurring is low because of the largest V G a of the transformation. All of the V MEP(s) and V G a (s) curves are similar in shape and their maxima are located at the same position (s = 0), which implies that the variational effect in calculating the rate constants of the reactions will be small to some extent.

Rate constants of tautomerization

Thermal rate constants are essential in modeling the tautomerization of DAT. Furthermore, accurate rate constants are increasingly needed for process simulations and are also necessary for quantifying the competing reactions. The interpolated variational transition-state theory, based on the minimum-energy path calculated at the MP2/6-311G** level of theory was used to calculate the reaction rates, and this approach was improved by interpolated corrections of barrier height at the CCSD(T)/6-311G** level. Overall reaction rate constants were calculated by TST and CVT theory, which corresponds to classical reaction-coordinate, and by TST with Eckart correction. Since the tunneling cannot be underestimated for reactions such as hydrogen exchange, dehydrogenation, and hydrogen migration, etc., a correction for tunneling was estimated using the ZCT and SCT approximations. The rate constants at several temperatures are depicted in Fig. 5 and detailed results are shown in Table 4.

Fig. 5
figure 5

Arrhenius plot of the rate constants calculated at the TST, TST/Eckart, CVT, CVT/SCT, and CVT/ZCT levels of theory. The rate constants are calculated based on the interpolated MEP at the MP2/6-311G** level of theory

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Table 4 Rate constants (in s−1) at several temperatures
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The variational effect, that is, the ratio of rate constants between the variational CVT and conventional TST, was considered as well. For these five reactions, throughout the studied temperature range, the CVT curves are nearly superimposable on the TST curves, and the data calculated using these two methods show only slight disparity, so there is little variational effect on the title reactions.

For reaction 1, the discrepancies in the rate constants calculated using the TST, TST/Eckart, CVT/SCT, and CVT/ZCT methods are significantly large only at lower temperatures (200 K < T < 500 K). With increasing temperature, the deviations become small. It can easily be concluded that tunneling has a distinct effect on this reaction at low temperatures but has little effect at high temperatures. At lower temperatures, the larger discrepancies occur due to the higher values of E a and the pre-exponential factor in the TST and CVT equations. At the higher temperatures, the smaller discrepancies appear due to the lack of a temperature in the pre-exponential factor. However, at 300 K, the rate constant calculated using the CVT/SCT method is 2.61 × 10−27, indicating that the thermal reaction is far too slow to play a role in atmospheric chemistry.

For reactions 3 and 5, the rate constants calculated using the TST/Eckart, CVT/SCT, and CVT/ZCT methods show only slight deviations compared to those calculated by the TST and CVT methods. It can be concluded that tunneling has smaller effects on these reactions at low temperatures than on reaction 1. For reaction 4, a smaller deviation compared to the values calculated by the TST method was found at low temperatures (200 K < T < 500 K), indicating that the effect of tunneling on this reaction is smallest among reactions 3, 4, and 5. In any case, our calculations indicate that the tunneling effect plays an important role in the hydrogen-shift tautomerization reactions of DAT than previously believed.

While for reaction 2 as an amino-shift reaction, the rate constants calculated by means of the TST, TST/Eckart, CVT, CVT/SCT, and CVT/ZCT methods are also almost identical at high temperatures (T > 500 K). At low temperatures (200 K < T < 500 K), the rate constants calculated by means of the TST/Eckart, CVT/SCT, and CVT/ZCT methods are almost identical, and there is little deviation compared to the values calculated by the TST and CVT methods. It can be concluded that tunneling has little effect on this reaction at low temperatures.

As can be seen in Table 4, the rate constant of reaction 4 is the highest, followed by those of reaction 5, reaction 2, reaction 3, and reaction 1. This order is in agreement with the adiabatic barriers (83.29 kcal · mol−1 (4), 73.78 kcal · mol−1 (5), 71.65 kcal · mol−1 (2), 63.71 kcal · mol−1 (3), and 61.53 kcal · mol−1 (1)). All of the rate constants increase rapidly with increasing temperature, especially in the lower temperature range.

Therefore, for the reactions under consideration, the variational effect is small over the whole temperature range. For the four hydrogen-shift reactions, V G a is larger, and the tunneling effect is smaller. In addition, the CVT/SCT rate constants in the range 200-2500 K were fitted by a three-parameter method (k = ATn exp(– Ea/RT)) for the five reactions as follows:

$$ \begin{array}{c}\hfill k(T)=2.262\times {10}^{-58}\times {T}^{20.42}\times {e}^{-\left(1.25\times {10}^4/T\right)}\kern0.5em {\mathrm{s}}^{-1}\hfill \\ {}\hfill k(T)=6.941\times {10}^3\times {T}^{3.07}\times {e}^{-\left(3.40\times {10}^4/T\right)}\kern0.5em {\mathrm{s}}^{-1}\hfill \\ {}\hfill k(T)=1.405\times {10}^{-13}\times {T}^{7.50}\times {e}^{-\left(2.544\times {10}^4/T\right)}\kern0.5em {\mathrm{s}}^{-1}\hfill \\ {}\hfill k(T)=2.500\times {10}^1\times {T}^{3.24}\times {e}^{-\left(3.88\times {10}^4/T\right)}\kern0.5em {\mathrm{s}}^{-1}\hfill \\ {}\hfill k(T)=8.418\times {10}^{-12}\times {T}^{6.87}\times {e}^{-\left(3.02\times {10}^4/T\right)}\kern0.5em {\mathrm{s}}^{-1}\hfill \end{array}. $$

Conclusions

The tautomerization reactions of DAT have been studied in detail. The geometries and frequencies of all stationary points, corresponding to the reactants, products and transition states, as well as the MEP, have been calculated at the MP2/6-311G** level of theory, and single-point energies have been calculated by CCSD(T)/6-311G**. Again the thermal rate constants for the reactions were evaluated using the TST, TST/Eckart, CVT, CVT/SCT, and CVT/ZCT methods in the temperature range 200-2500 K. For the tautomers of DAT, calculations predicted a reasonable cyclic structure. In the gas phase, 2,5-diaminotetrazole (b) is predicted to be the dominant form, followed by 1,5-diaminotetrazole (c). The stabilities of b and c can be ascribed to their aromatic character with delocalization of the lone pair on the saturated nitrogen atom into the five-membered ring to form a 6π electron aromatic system. As a result, the hydrogen-shift of c to a is the easiest transformation, while the tautomerization of a to e is the hardest one. Moreover, the transition-state theory calculations have indicated that tunneling plays a modest role in these hydrogen-transfer reactions, such as reactions 1, 3, and 5, while there is a smaller tunneling effect on reactions 2 and 4.

Finally, it is expected that the present theoretical study may be useful for estimating the kinetics of the reactions over a wide temperature range, for which no experimental data are available.