Anharmonic Effect of the Decomposition Reaction in High-Temperature Combustion of Monomethylhydrazine Radicals

YU Hong-Jing XIA Wen-Wen SONG Li-Guo DING Yang HAO YuKANG Li-Qiang PAN Xin-Xiang YAO Li,*

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Anharmonic Effect of the Decomposition Reaction in High-Temperature Combustion of Monomethylhydrazine Radicals

YU Hong-Jing1XIA Wen-Wen1SONG Li-Guo2DING Yang2HAO Yu2KANG Li-Qiang2PAN Xin-Xiang2YAO Li1,*

(1;2)

In this work, the harmonic and anharmonic rate constants of the decomposition reaction of monomethylhydrazine (MMH) radicals have been calculated by using transition state (TS) and Rice-Ramsperger-Kassel-Marcus (RRKM) theories with either MP2 or B3LYP method at 6-311++G(,) basis set, respectively. The reaction mechanism and anharmonic effect of the MMH radicals are studied in detail and both of the harmonic and anharmonic rate constants increase sharply with increasing temperature in the canonical system. In the microcanonical system, these constants also show sharp increase with the energies.Overall, the anharmonic effect becomes more pronounced with the increasing temperature or energy in the canonical and microcanonical systems, respectively. These results indicate that the anharmonic effect of the decomposition reaction of MMH radicals is quite significant and cannot be ignored.

Anharmonic effect; Unimolecular reaction; RRKM theory; Rate constant

1 Introduction

The frequently applied liquid hypergolic propellants are hydrazine-based fuels with nitrogen tetroxide (NTO), liquid oxygen with kerosene and liquid hydrogen with liquid oxygen. Hypergolic propellants are fuel-oxidizer mixture that ignites spontaneously upon mixing at low temperature and pressure. They facilitate the design of rocket thrusters by simplifying the ignition, and are widely used in propulsion systems for applications in which variable and intermittent thrust capabilities are needed. Among the most commonly fielded bipropellant, the combinations are monomethylhydrazine (MMH) and NTO1. While toxic, these combinations offer higher performance, potentially increasing payload capacity, and beneficial long-term stability compared with other hypergolic propellants. Bipropellant systems of MMH and NTO are widely used in space propulsion. Before igniting, thermal decomposition reactions occured in MMH and NTO respectively. Different from the other decomposition reactions, the unmolecular decomposition reaction of MMH releases a lot of heat which plays an important role in the system of spontaneous combustion. For a long time, MMH has been researched as a significant derivative of hypergolic bipropellants and an extensively used hypergolic rocket fuel together with either nitrogen or red fuming nitric acid. And the related research has achieved a breakthrough1?4. Especially on the molecular reation dynamics, several methods have been utilized forcalculations of predominant channel for decomposition reaction of hydrazinium, and the reaction mechanism has been found5?7. The H-abstraction of MMH leads to the formations of four important MMH radicals:CH3NHNH,CH3NHNH, CH3NNH2, and CH2NHNH2radicals8. TheCH3NHNHradical can isomerize to the other radicals6. In 1962, Schlag and Sandsmark experimentally confirmed the existence and importance of anharmonic effect9. However, until now the anharmonic effect of MMH is rarely researched. Therefore,-CH3NHNH radical is the initial reactant, and the nucleus of all the channels isCH3NHNH in this paper. The structures of the reactants and transition states for four major MMHradicals are optimized with MP2 or B3LYP method and 6-311++G(3,2) basis set. The geometric parameters for TS1-4c in the reaction system are optimized at the MP2/6-311++G(3,2) level. Several structures of transition state fail to be located at thecalculations level. But the optimized geometric parameters for TS5-8b are found at the B3LYP/6-311++G(3,2) level of calculation. Then the vibrational harmonic and anharmonic frequencies are available, and the corresponding energy of the reaction is calculated at the same level. According to the RRKM theory, the total number of states of the transition state, the density of states of the reactants and the rate constant of the reaction are calculated. Therefore,the anharmonic effect of the unimolecular reaction can be researched in this paper10–15.

2 Calculation methodology and computational theory

2.1 Ab initio calculations

The equilibrium geometries of the reactants and transition states are calculated at B3LYP or MP2 function with the 6-311++G(3,2) basis set level. The parameters are identify, which are all of the stationary points as either minima or transition states at the same level, such as vibrational harmonic and anharmonic frequency, zero-point-energy and so on. Next, in order to ensure the transition state, intrinsic reation coordinate (IRC) is traced at the same level. Vibrational harmonic and anharmonic frequencies are calculated to identify all of the stationary points as either minima (zero imaginary frequency) or transition state (one imaginary frequency) on this condition. In order to obtain more exact and trustworthy analog data, the energy is corrected by the single-point energy and that is worked out with the coupled cluster QCISD(T) method and 6-311++G(3,2) basis set. In order to research its anharmonic effect, the rate constant is calculated by RRKM theory, which is based on the available statistic of this frequency and energy. All thecalculations of frequency and energy are appeared by Gaussian 09 program11.

2.2 Microcanonical and canonical ensemble

2.2.1 Canonical ensemble

In 1935, TS theory was proposed. According to its four basic assumptions, the unimolecular reaction rate constant for the canonical system expression with temperature is surfaced16?21:

where() andQ() are partition functions of reactant and transition state respectively. The rate constant for the canonical system can be described by vibrational partition function.

The expression of the partition function is:

And the logarithmic function is:

From (3), the value of the partition is decided on what is selected in the way of the analog potential. Here, to calculate the partition function. Morse oscillator (MO) potential is applied, and the anharmonic effect in the canonical case and microcanonical case are discussed, respectively.

For MO,

where,qis the partition function of-th vibration mode; andnindicates the-th energy level of the vibration mode. The energy of-th vibration mode is:

Hence, the canonical rate constant is worked out by the corresponding expressions.

2.2.2 Microcanonical ensemble

In 1951, RRKM theory was proposed. RRKM theory is suitable for the microcanonical case, which is a combination between statistical and transition state theory. It draws not only on the TS theory, but also on the intramolecular energy. At present, it is the most practical and successful theory for unimolecular reaction, which is analogous to the theoretical and experimental result. According to RRKM theory, the unimolecular reaction rate constant for the microcanonical system expression with energy is surfaced22:

Hence, the rate constant in the microcanonical system can be described by the total number of states for the transition state and the density of the state of the reactant, which are() and() respectively. In the calculation of the density of the state(), harmonic and anharmonic vibrational degrees of freedom of the reactant are 18 (3? 6,= 8,is the number of atoms included in a molecule monomethylhydrazine radical). To calculate the total number of states(), the harmonic and anharmonic imaginary frequencies are excluded and 17 vibrational degrees of freedom are for the transition state23–25.

The total number of states for the transition state is transmuted by Laplace transformation and inverse Laplace transformation of the distribution function:

Therefore, second-order approximation of the steepest descent method for() is emerged:

In like manner, second-order approximation of the steepest descent method for()d()dis obtained:

As indicated in Eqs.(9) and (10), the key is the partition function to solve() and().

Hence, the microcanonical rate constant is worked out by the corresponding expressions. The detail of derivation can be found in Refs.10 and 13.

3 Results and discussion

3.1 The MP2 calculation

As shown in Fig.1, there are four main channels of the reaction. The optimized geometries of reactants and transition states have been accomplished by MP2 method with 6-311++G(3,2) basis set in Fig.2. For the reaction process of channel 1, C―H bond is fissured. NH+transfers from N to C at the same time. For the reaction process of channel 2, N―H bond is fissured in the middle of the structure. For the reaction process of channel 3, two N―H bonds have rotations. So Intermediate 1 is formed, which is aisomerism ofCH3NHNH radical. Therefore, four different products are formed, which is corresponding with TS3a, TS3b, TS3c and TS3d, respectively. In the channel of TS3a, C―N bond is fissured. In the channel of TS3b, one of C―H bonds is fissured. The product is aisomerism of TS1. In the channel of TS3c, N―H bond is fissured in the middle of the structure and the dropped H transfers to lateral N. In the channel of TS3d, the process is contrary to TS3c, N―H bond is fissured on the edge, and the dropped H transfers to the N in the middle of the structure. For the reaction process of channel 4, N―H bond is fissured in the middle of the structure, and this dropped H transfers to N on the edge. So Intermediate 2 is formed. Therefore, three different products are formed, which is corresponding with TS4a, TS4b and TS4c, respectively. In the channel of TS4a, one of C―H bonds is fissured. In the channel of TS4b, one of N―H bonds is fissured on the edge. In the channel of TS4c, another N―H bond is fissured. The product is aisomerism of TS4b.

For these reactions, the energies of the reactants and transition states are calculated by MP2 method with 6-311++G(3,2) basis set. The barrier of the reaction is calculated at the same level. The results are shown in Table 1. In order to obtain more accurate data, the single-point energies are recalculated by employing the coupled cluster QCISD(T) method with 6-311++G(3,2) basis set. Thereby, the more reliable barrier is also shown in the Table 1, so that the line chart of process-barrier is given in the Fig.3. And they are used for calculation of harmonic and anharmonic rate constants from 300 to 4800 K. The interval of the temperature is 300 K. Harmonic and anharmonic rate constants in canonical system and microcanonical system are shown in Fig.4 to Fig.6. In both canonical system and microcanonical system, it’s clear that harmonic and anharmonic rate constants are increasing sharply with the temperature or energy from these charts.

Fig.1 Equation of four main channels of the reaction process by MP2 method with 6-311++G(3df,2p) basis set.

Fig.2 The optimized geometries of reactants and transition states.

3.1.1 Channel 1 by MP2 method

The junction of rate constant is between 900 and 1200 K for TS1 in the canonical system. When temperature is below 900 K, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than 1200 K, harmonic rate constant is less than anharmonic rate constant. According to the corresponding energy, minimum energy is 342.42 kJ·mol?1for the barrier. Corresponding temperature is 3300 K. When temperature is 300 K, the anharmonic rate constant is 0.67 times of the harmonic rate constant. When temperature is 3300 K, the anharmonic rate constant is 8.07 times of that harmonic. When temperature is 4800 K, the anharmonic rate constant is 27.75 times of the harmonic one. As shown in Fig.4(a), in the canonical system, harmonic and anharmonic rate constants are approximately similar. Thus, the anharmonic effect of the unimolecular reaction is not obvious. However, the anharmonic effect at those temperatures above 3300 K is more obvious, and the rate constant is more gradual than that at lower temperature range.

Harmonic rate constant is less than anharmonic rate constant for TS1 in the microcanonical system. When energy is 342.42 kJ·mol?1, the anharmonic rate constant is 3.01 times of the harmonic one. When energy is 557.52 kJ·mol?1, the anharmonic rate constant is 11.10 times of the harmonic one. As shown in Fig.4(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally.

3.1.2 Channel 2 by MP2 method

The junction of rate constant is between 300 and 600 K for TS2 in the canonical system. When temperature is below 300 K, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than 600 K, anharmonic rate constant is larger than harmonic rate constant. According to the corresponding energy, minimum energy is 218.32 kJ·mol?1above the barrier. Corresponding temperature is 2400 K. When temperature is 300 K, the anharmonic rate constant is 0.80 times of the harmonic one. When temperature is 2400 K, the anharmonic rate constant is 3.56 times of the harmonic one. When temperature is 4800 K, the anharmonic rate constant is 9.58 times of the harmonic one. As shown in Fig.4(a), in the canonical system, harmonic and anharmonic rate constants are similar integrally. Thus, the anharmonic effect of the unimolecular reaction is not obvious. However, the anharmonic effect at those temperatures above 2400 K is more obvious, and the rate constant is more gradual than that at lower temperature range.

Harmonic rate constant is less than anharmonic rate constant for TS2 in the microcanonical system. When energy is 218.32 kJ·mol?1, the anharmonic rate constant is 2.81 times of the harmonic one. When energy is 557.52 kJ·mol?1, the anharmonic rate constant is 11.93 times of the harmonic one. As shown in Fig.4(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally.

Table 1 Parameters calculated at MP2/6-311++G(3df,2p) level for TS1-4c pathway.

Fig.3 Barriers calculated at MP2/6-311++G(3df,2p) level.

3.1.3 Channel 3 by MP2 method

For TS3, the junction of rate constant is between 900 and 1200 K in the canonical system. For TS3a, the point is between 3300 and 3600 K. For TS3b, the point is between 600 and 900 K. For TS3c, the point is between 300 and 600 K. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than corresponding cross point, harmonic rate constant is less than anharmonic rate constant. For TS3d, there is not the point yet for these tempurature range, harmonic rate constant is larger than the anharmonic one. According to the corresponding energy, for TS3 to TS3d, minimum energies are 104.10, 178.49, 342.25, 218.15, 342.25 kJ·mol?1above the barrier respectively. Corresponding temperatures are 1500, 2100, 3300, 2400, 3300 K. When temperature is 300 K, the anharmonic rate constants are 0.69, 0.99, 0.78, 0.93, 1.01 times of these harmonic, separately. When temperature is close to the corresponding barrier, the proportions are 1.68, 0.91, 2.43, 1.03, 1.35 times, separately. When temperature is 4800 K, the ratios are 18.27, 1.18, 4.99, 1.52, 1.09 times, separately. In the canonical system, as shown in Fig.4(a) and Fig.5(a), harmonic and anharmonic rate constants are approximately similar. Thus, the anharmonic effect of the unimolecular reaction is not obvious. However, the anharmonic effect at higher barrier range is more obvious than that at lower corresponding temperature range, and the rate constant is more gradual than that at lower temperature range.

For TS3a, the intersection of rate constant is between 384.68 and 427.48 kJ·mol?1in the microcanonical system. When temperature is below 384.68 kJ·mol?1, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than 427.48 kJ·mol?1, harmonic rate constant is less than the anharmonic one. For TS3b, the intersection of rate constant is between 342.25 and 384.68 kJ·mol?1in the microcanonical system. When temperature is below 342.25 kJ·mol?1, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than 384.68 kJ·mol?1, harmonic rate constant is less than the anharmonic one. For TS3, TS3c, TS3d, there is not cross point, harmonic rate constant is less than anharmonic rate constant. When energy is minimum for the microcanonical system, which are 104.10, 178.49, 342.25, 218.15, 342.25 kJ·mol?1, the anharmonic rate constants are 1.10, 1.07, 0.95, 1.09, 1.54 times of these harmonic, separately. When energy is 557.52 kJ·mol?1for TS3, the proportion is 17.78 times. When energy is 557.35 kJ·mol?1for TS3a to TS3d, the proportions are 1.15, 1.99, 1.26, 1.72 times, separately. As shown in Fig.4(b) and Fig.5(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious for TS3 and the anharmonic effect is not obvious for TS3a-d integrally.

Fig.4 The canonical and microcanonical rate constants of TS1–4 at MP2/6-311++G(3df,2p) level.

Fig.5 The canonical and microcanonical rate constants of TS3a–d at MP2/6-311++G(3df,2p) level.

3.1.4 Channel 4 by MP2 method

For TS4 and TS4b, the junction of rate constant is between 600 and 900 K in the canonical system. For TS4a, the point is between 900 and 1200 K. For TS4c, the junction is between 300 and 600 K. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than corresponding cross point, harmonic rate constant is less than anharmonic rate constant. According to the corresponding energy, for TS4, minimum energy is 218.32 kJ·mol?1above the barrier. For TS4a to TS4c, it is 219.49 kJ·mol?1for everyone. But, corresponding temperature is 2400 K for anyone. When temperature is 300 K, the anharmonic rate constants are 0.78, 0.82, 0.94, 0.94 times of these harmonic, separately. When temperature is 2400 K, the proportions are 2.22, 2.11, 1.76, 1.66 times, separately. When temperature is 4800 K, the ratios are 7.80, 5.80, 2.64, 2.09 times, separately. In the canonical system, as shown in Fig.4(a) and Fig.6(a), harmonic and anharmonic rate constants are approximately similar. Thus, the anharmonic effect of the unimolecular reaction is not obvious. However, the anharmonic effect at higher barrier range is more obvious than that at lower corresponding temperature range, and the rate constant is more gradual than that at lower temperature range.

Fig.6 The canonical and microcanonical rate constants of TS4a–c at MP2/6-311++G(3df,2p) level.

For TS4c, the intersection is between 219.49 and 260.20 kJ·mol?1. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than the corresponding cross point, harmonic rate constant is less than anharmonic rate constant. For TS4, TS4a and TS4b, there is not cross point, harmonic rate constant is less than anharmonic rate constant. When energy is minimum for the microcanonical system, which are 218.32 kJ·mol?1for TS4 and 219.49 kJ·mol?1for TS4a-c, the anharmonic rate constants are 2.41, 1.38, 1.56, 0.88 times of these harmonic, separately. When energy is 557.52 kJ·mol?1for TS4, the proportion is 7.43 times. When energy is 558.90 kJ·mol?1for TS4a to TS4c, the proportions are 4.89, 2.99, 2.73 times, separately. As shown in Fig.4(b) and Fig.6(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally.

3.2 The B3LYP calculation

In this section, the optimized geometries of reactants and transition states have been accomplished by B3LYP method with 6-311++G(3,2) basis set in Fig.1. As shown in Fig.7, there are four main channels of the reaction. For the reaction process of channel 5, C―N bond is fissured. For the reaction process of channel 6, two N―H bonds have rotations. So Intermediate 1 is formed, which is aisomerism ofCH3NHNH radical. Therefore, three different products are formed, which is corresponding with TS6a, TS6b and TS6c, respectively. In the channel of TS6a, N―H bond is fissured in the middle of the structure. In the channel of TS6b, N―H bond is fissured in the middle of the structure, and the dropped H transfers to the N on the side. In the channel of TS6c, one of C―H bonds is fissured, and the dropped H transfers to the N on the side. For the reaction process of channel 7, N―H bond is fissured in the middle of the structure, and the dropped H transfers to the N on the side, then Intermediate 2 is formed. Therefore, three different products are formed, which is corresponding with TS7a, TS7b and TS7c, respectively. In the channel of TS7a, C―N bond is fissured. In the channel of TS7b, both one of C―H bonds and N―H bond on the side are fissured, then they form the H2. In the channel of TS7c, one of C―H bonds is fissured, and the dropped H transfers to the N in the middle of the structure, then Intermediate 3 is formed. For the reaction process of channel 8, one of C―H bonds is fissured, and the dropped H transfers to the Natom on the side, then Intermediate 3 is formed. Therefore, two different products are formed, which is corresponding with TS8a and TS8b, respectively. In the channel of TS8a, N―N bond is fissured. In the channel of TS8b, N―H bond is fissured in the middle of the structure.

Fig.7 The equation of four main channels of the reaction process by B3LYP method with 6-311++G(3df,2p) basis set.

For these reactions, the barrier of the reaction is calculated in the Table 2, and the line chart of process-barrier is given by QCISD(T) method with 6-311++G(3,2) basis set in Fig.8. According to these data, rate constants are calculated from 300 to 4800 K in Fig.9 to Fig.12. The interval of the temperature is also 300 K. In both canonical system and microcanonical system, it's clear that harmonic and anharmonic rate constants are increasing sharply with the temperature or energy from these charts.

Table 2 Parameters are calculated at B3LYP/6-311++G(3df,2p) level for TS5-8b pathway.

Fig.8 Barriers calculated at B3LYP/6-311++G(3df,2p) level.

Fig.9 The canonical and microcanonical rate constants of TS5–8 at B3LYP/6-311++G(3df,2p) level.

Fig.10 The canonical and microcanonical rate constants of TS6a–c at B3LYP/6-311++G(3df,2p) level.

3.2.1 Channel 5 by B3LYP method

There is not the intersection and harmonic rate constant is larger than anharmonic rate constant for TS5 in the canonical system. According to the corresponding energy, minimum energy is 180.75 kJ·mol?1above the barrier. Corresponding temperature is 2100 K. When temperature is 300 K, the anharmonic rate constant is 0.57 times of the harmonic one. When temperature is 2100 K, the anharmonic rate constant is 0.21 times of the harmonic one. When temperature is 4800 K, the anharmonic rate constant is 0.07 times of the harmonic one. According to Fig.9(a), in the canonical system, harmonic and anharmonic rate constants are approximately similar. Thus, the anharmonic effect of the unimolecular reaction is not obvious. However, the anharmonic effect at those temperatures above 2100 K is more obvious, and the rate constant is more gradual than that at lower temperature range.

The junction of rate constant is between 220.62 and 261.46kJ·mol?1for TS5 in the microcanonical system. When energy is below 220.62 kJ·mol?1, anharmonic rate constant is larger than harmonic rate constant. When energy is higher than 261.46 kJ·mol?1, anharmonic rate constant is less than harmonic rate constant. When energy is 180.75 kJ·mol?1, the anharmonic rate constant is 1.16 times of the harmonic one. When energy is 560.66 kJ·mol?1, the anharmonic rate constant is 0.14 times of the harmonic one. As shown in Fig.9(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally.

3.2.2 Channel 6 by B3LYP method

The junction of rate constant is between 900 and 1200 K for TS6 in the canonical system. For TS6a and TS6c, the junction is between 600 and 900 K. For TS6b, the junction is between 2100 and 2400 K. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than the corresponding cross point, harmonic rate constant is less than anharmonic rate constant. According to the corresponding energy, for TS6, minimum energy is 105.69 kJ·mol?1above the barrier, and, corresponding temperature is 1500 K. For TS6a to TS6c, it is 220.45 kJ·mol?1for everyone, and, corresponding temperature is 2400 K for anyone. When temperature is 300 K, the anharmonic rate constants are 0.69, 0.72, 0.77, 0.72 times of these harmonic, respectively. When temperature is close to the corresponding barrier, the proportions are 1.48, 5.28, 1.12, 6.87 times, respectively. When temperature is 4800 K, the ratios are 12.68, 20.22, 3.42, 27.48 times, separately. In the canonical system, As shown in Fig.9(a) and Fig.10(a), harmonic and anharmonic rate constants are approximately similar. Thus, the anharmonic effect of the unimolecular reaction is not obvious. But, for TS6a and TS6c, the anharmonic effect is a little obvious relatively. However, the anharmonic effect at higher barrier range is more obvious than that at lower corresponding temperature range, and the rate constant is more gradual than that at lower temperature range.

There is not the intersection and anharmonic rate constant is larger than harmonic rate constant for TS6 to TS6c in the microcanonical system. When energy is minimum for the microcanonical system, which is 105.69 kJ·mol?1for TS6 and 220.45 kJ·mol?1for TS6a-c, the anharmonic rate constants are 1.15, 2.57, 1.68, 2.40 times of these harmonic, respectively. When energy is 560.66 kJ·mol?1for TS6, the proportion is 13.58 times. When energy is 560.45 kJ·mol?1for TS6a to TS6c, the proportions are 16.58, 2.85, 22.70 times, respectively. As shown in Fig.9(b) and Fig.10(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally. But, for TS6b, the anharmonic effect is not obvious relatively.

3.2.3 Channel 7 by B3LYP method

For TS7, the junction of rate constant is between 900 and 1200 K in the canonical system. For TS7a, the junction is between 600 and 900 K. For TS7b and TS7c, the junction is between 300 and 600 K. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than the corresponding cross point, harmonic rate constant is less than anharmonic rate constant for TS7, TS7b and TS7c. But, on the contrary for TS7a, when temperature is below 600 K, anharmonic rate constant is larger than harmonic rate constant. When temperature is higher than 900 K, anharmonic rate constant is less than harmonic rate constant. According to the corresponding energy, for TS7 and TS7c, minimum energies are 220.62 and 221.46 kJ·mol?1above the barrier separately. And, corresponding temperature is 2400 K. For TS7a and TS7b, minimum energy is 262.34 kJ·mol?1above the barrier, and, corresponding temperature is 2700 K. When temperature is 300 K, the anharmonic rate constants are 0.67, 2.21, 0.85, 0.80 times of these harmonic, separately. When temperature is close to the barrier, the proportions are 2.55, 0.04, 7.11, 5.49 times, separately. When temperature is 4800K, the ratios are 10.79, 0.01, 16.44, 18.73 times, separately. In the canonical system, as shown in Fig.9(a) and Fig.11(a), harmonic and anharmonic rate constants are similar integrally. Overall, for TS7, the anharmonic effect of the unimolecular reaction is not obvious. But, for TS7a-c, the anharmonic effect of the unimolecular reaction is a little obvious. However, the anharmonic effect at higher barrier range is more obvious than that at lower corresponding temperature range, and the rate constant is more gradual than that at lower temperature range.

For TS7a, the intersection of rate constant is between 262.34 and 303.93 kJ·mol?1in the microcanonical system. When temperature is below 262.34 kJ·mol?1, anharmonic rate constant is larger than harmonic rate constant. When temperature is higher than 303.93 kJ·mol?1, anharmonic rate constant is less than harmonic rate constant. For TS7, TS7b and TS7c, there is not the intersection, harmonic rate constant is less than anharmonic rate constant. When energies are minimum for the microcanonical system, which are 220.62, 262.34, 262.34 and 221.46 kJ·mol?1for TS7 to TS7c, the anharmonic rate constants are 2.19, 4.84, 2.79, 1.64 times of these harmonic, separately. When energy is 560.66 kJ·mol?1for TS7, the proportion is 9.92 times. When energy is 561.66 kJ·mol?1for TS7a to TS7c, the proportions are 0.02, 11.86, 11.96 times, separately. As shown in Fig.9(b) and Fig.11(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally.

3.2.4 Channel 8 by B3LYP method

For TS8, the junction of rate constant is between 600 and 900 K in the canonical system. For TS8a, the junction is between 1800 and 2100 K. For TS8b, the junction is between 1200 and 1500 K. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than the corresponding cross point, harmonic rate constant is less than anharmonic rate constant. According to the corresponding energy, for TS8, minimum energy is 261.46 kJ·mol?1above the barrier, and, corresponding temperature is 2700 K. For TS8a, it is 74.43 kJ·mol?1, and corresponding temperature is 1200 K. For TS8b, it is 144.68 kJ·mol?1, and corresponding temperature is 1800 K. When temperature is 300 K, the anharmonic rate constants are 0.73, 0.66, 0.71 times of these harmonic, separately. When temperature is close to the corresponding barrier, the proportions are 3.01, 0.56, 1.86 times, separately. When temperature is 4800 K, the ratios are 8.04, 12.23, 11.92 times, separately. In the canonical system, as shown in Fig.9(a) and Fig.12(a), harmonic and anharmonic rate constants are similar integrally. Thus, the anharmonic effect of the unimolecular reaction is not obvious. But, for TS8b, the anharmonic effect is a little obvious. However, the anharmonic effect at higher barrier range is more obvious than that at lower corresponding temperature range, and the rate constant is more gradual than that at lower temperature range.

For TS8, there is not the intersection, harmonic rate constant is less than anharmonic rate constant in the microcanonical system. For TS8a and TS8b, the junction is between 183.05 and 222.76 kJ·mol?1. When temperature is below the corresponding cross point, harmonic rate constant is larger than anharmonic rate constant. When temperature is higher than the corresponding cross point, harmonic rate constant is less than anharmonic rate constant. When energies are minimum for the microcanonical system, which are 261.46, 74.43, 144.68 kJ·mol?1for TS8 to TS8b, the anharmonic rate constants are 3.40, 0.42, 0.51 times of these harmonic, respectively. When energy is 560.66 kJ·mol?1for TS8, the proportion is 8.62 times. When energy is 562.08 kJ·mol?1for TS8a and TS8b, the proportions are 12.39 and 16.30 times respectively. As shown in Fig.9(b) and Fig.12(b), in the microcanonical system, the anharmonic effect of the unimolecular reaction is obvious integrally.

4 Conclusions

In this paper, the geometries of reactants and the transition states for MMH radicals are optimized by using Gaussian 09 program at MP2/6-311++G(3,2) level for TS1-4c and at B3LYP/6-311++G(3,2) level for TS5-8b. The harmonic and anharmonic rate constants of the unimolecular decomposition for MMH radicals are calculated by the RRKM theory. Therefore, the anharmonic effect is found in the microcanonical and canonical case. In conclusion, harmonic and anharmonic rate constants are increasing sharply with the increasing of temperature in the canonical system, or increasing sharply with the increasing of energy in the microcanonical system. Overall, anharmonic effect becomes more and more obvious with the increasing of temperature in canonical system, and anharmonic effect is more and more obvious with the increasing of energy in microcanonical system. In both systems, for one channel, the growth trends of rate constants are approximately similar. Pathway has the lowest barrier by using MP2 or B3LYP method, which comes into being Intermediate 1. So, the pathway for Intermediate 1 is the most easy to generate. For all channels except TS5 and TS7a, the anharmonic rate constants are higher than harmonic rate constants at high temperature range, while the anharmonic rate constants are lower than harmonic rate constants at low temperature range in the canonical system, the anharmonic rate constants are higher than harmonic rate constants at high energy range, while the anharmonic rate constants are lower than harmonic rate constants at low energy range in the microcanonical system. Generally speaking, the junctions of rate constants differ from 300 to 4200 K. For all channels, the anharmonic effect of these unimolecular reactions at higher temperature range is more obvious than that at lower temperature range. In addition, at higher temperature range, the rate constants are more gradual than that at lower temperature range. So, it cannot be neglected. Furthermore, the anharmonic rate constant is calculated, it indicated that RRKM theory works for studying the rate constant of dissociation reaction. This result could make for the study that rocket fuel is MMH with nitrogen or red fuming nitric acid.

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甲基肼自由基在高溫燃燒分解反應下的非諧振效應

于洪晶1夏文文1宋立國2丁 楊2郝 宇2康利強2潘新祥2姚 麗1,*

(1大連海事大學物理系，遼寧 大連 116026；2大連海事大學輪機工程學院，遼寧 大連 116026)

基于過渡態(tài)理論(TST)和RRKM (Rice-Ramsperger-Kassel-Marcus)理論，使用MP2/6-311++G (3,2)//B3LYP/6-311++G(32)基組和方法，本文分別計算了甲基肼自由基在高溫燃燒下分解反應的諧振以及非諧振速率常數(shù)。因此，甲基肼自由基的詳細機理得以研究。在正則系綜中，諧振以及非諧振速率常數(shù)都隨著溫度的升高而增加；而在微正則系綜中，速率常數(shù)隨著能量的升高而增加。整體上看，在正則系綜和微正則系綜中，非諧振效應分別隨著溫度和能量的升高而變得越來越明顯。因此甲基肼自由基在高溫燃燒下分解反應的非諧振效應是不能被忽視的。

非諧振效應；單分子反應；RRKM理論；速率常數(shù)

O643

10.3866/PKU.WHXB201705227

April 24, 2017;

May 17, 2017;

May 22, 2017.

Corresponding author. Email: yaoli@dlmu.edu.cn; Tel: +86-13130432506.

The project was supported by the Major Research Plan of the National Natural Science Foundation of China (91441132) and Fundamental Research Funds for the Central Universities, China (3132016127, 3132016326, 3132016019).

國家自然科學基金(91441132)和中央高?；究蒲袠I(yè)務費專項資金(3132016127, 3132016326, 3132016019)資助項目