Unimolecular Reaction of Methyl Isocyanide to Acetonitrile: A High

Apr 26, 2018 - Heats of formation calculated at 0 K using H (216.034 kJ mol–1), ... (44) Here, harmonic force fields were calculated using a ANO2(45...
1 downloads 0 Views 1MB Size
Download PDF
Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 2532−2538

pubs.acs.org/JPCL

Unimolecular Reaction of Methyl Isocyanide to Acetonitrile: A HighLevel Theoretical Study Thanh Lam Nguyen,† James H. Thorpe,† David H. Bross,‡ Branko Ruscic,‡,§ and John F. Stanton*,† †

Quantum Theory Project, Department of Chemistry and Physics, University of Florida, Gainesville, Florida 32611, United States Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States § Computation Institute, The University of Chicago, Chicago, Illinois 60637, United States ‡

S Supporting Information *

ABSTRACT: A combination of high-level coupled-cluster calculations and twodimensional master equation approaches based on semiclassical transition state theory is used to reinvestigate the classic prototype unimolecular isomerization of methyl isocyanide (CH3NC) to acetonitrile (CH3CN). The activation energy, reaction enthalpy, and fundamental vibrational frequencies calculated from first-principles agree well with experimental results. In addition, the calculated thermal rate constants adequately reproduce those of experiment over a large range of temperature and pressure in the falloff region, where experimental results are available, and are generally consistent with statistical chemical kinetics theory (such as Rice−Ramsperger−Kassel−Marcus (RRKM) and transition state theory (TST)).

T

reaction was reinvestigated with density functional theory (DFT).22,23 TST calculations were performed for thermal rate constants at the high-pressure limit (but, like ref 21, not in the falloff region). The rate constant calculated with DFT at 500 K was about 2.5 times smaller than the experimental value.23 Earlier, trajectory calculations done by Bunker and Hase24 using collisional energies of 293 to 837 kJ mol−1 showed nonRRKM unimolecular kinetics for CH3NC. This theoretical finding is potentially not relevant to thermal experiments8−15 since the collisional energies are so far above that at the reaction threshold. In this work, for the first time, thermal rate constants for the title reaction are computed from first-principles for a wide range of temperatures (400−1000 K) and pressures (10−2−105 mmHg) in order to construct falloff curves using state-of-theart calculations. The potential energy surface was first constructed using the HEAT-345Q protocol,25−27 followed by solving a two-dimensional master equation to obtain k(T,P),28−30 where microcanonical rate constants, k(E,J), were computed using the SCTST/VPT2 approach.31−35 Table 1 shows heats of formation for CH3CN and CH3NC calculated with HEAT-345Q: 81.3 ± 1.0 kJ mol−1 at 0 K (74.2 ± 1.0 at 298.15 K) and 184.6 ± 1.0 (178.0 ± 1.0) kJ mol−1, respectively, which are in excellent agreement with the benchmark ATcT values36−39 of 81.10 ± 0.25 kJ mol−1 at 0 K (74.04 ± 0.25 at 298.15 K) and 183.52 ± 0.60 (176.97 ±

he isomerization of methyl isocyanide (CH3NC) to acetonitrile (CH3CN) is a prototypical homogeneous unimolecular reaction1−7 that has long been studied with various experimental techniques.8−15 A half century ago, Rabinovitch and co-workers8−11 measured thermal rate constants over a wide range of temperatures and pressures in the falloff region. These authors reported an activation energy of 160.7 kJ mol−1 at the high-pressure limit and an Arrhenius pre-exponential factor of 1013.5 s−1,8 results that were verified by subsequent experimental studies.12−15 In addition, on the basis of models and statistical Rice−Ramsperger−Kassel−Marcus (RRKM) theory,16,17 Rabinovitch and co-workers were able to satisfactorily reproduce the measured rate constants in the falloff region (within a factor of 2).8 However, such kinetics calculations (using heuristic models for transition state structural and vibrational parameters) are ultimately not able to draw unambiguous conclusions about the nature or the kinetics. In the 1970s, the title reaction was theoretically investigated using both SCF/DZ+P18,19 and SDQ-MBPT(4)/ DZ+P20 levels of theory. However, vibrational analyses were not used in those works to obtain the zero-point vibrational energies (ZPE) and rovibrational parameters, which are needed to calculate thermal reaction rate constants. In 1980, Schaefer and co-workers21 used SCF/DZ+P theory to obtain ab initio rovibrational parameters for stationary points and an activation energy of 183 kJ mol−1, the latter lying quite far above the experimental value of 160.7 kJ mol−1. Using these SCF rovibrational parameters and the experimental activation energy, Hase21 computed thermal rate constants at the high-pressure limit with transition state theory (TST). The theoretical results were about 50% higher than experiment.21 Recently, this © 2018 American Chemical Society

Received: April 20, 2018 Accepted: April 26, 2018 Published: April 26, 2018 2532

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538

Letter

The Journal of Physical Chemistry Letters

reaction step is predicted to be exothermic by 103.3 kJ mol−1 at 0 K (103.8 kJ mol−1 at 298 K) and must overcome a barrier of 160.1 ± 1.0 kJ mol−1. The latter is in line with three (independent) experimental results: 160.7,8 159.8 ± 0.9,13 and 160.9 ± 1.5 kJ mol−1.15 As can be seen in Table 1, the SCF calculation overestimates the activation energy by 23 kJ mol−1 due to neglect of electron correlation. The CCSD(T)40,41 extrapolation gives the most important contribution: it recovers the lion’s share of the electron correlation and lowers the barrier by 14.9 kJ mol−1. The next most important contribution is the ZPE correction, which further decreases the barrier by 7.4 kJ mol−1. The remaining terms make smaller contributions (less than 1 kJ mol−1), but are nonetheless vital to achieve the desired accuracy. Next, we compare the theoretical fundamental vibrational frequencies of CH3CN and CH3NC (that were obtained using vibrational second-order perturbation theory42,43) with those of experiment (see Table 2).44 Here, harmonic force fields were

Table 1. Individual Contributions (kJ mol−1) of Various Terms to Total Atomization Energy (TAE), Heats of Formation (ΔfH°) for Various Species, and Reaction Barrier Calculated Using HEAT-345Q Protocol term δESCF δECCSD(T) δECCSDT δECCSDTQ δEScalar δEZPE δEDBOC δESpin−orbit TAE ΔfH° at 0 K

ΔfH° at 298 K

CH3CN (C3v,1A1)

CH3NC (C3v,1A1)

TS

1817.41 762.95 −4.37 3.54 −1.79 −117.18 0.34 −0.71 2460.19 81.29a (81.10 ± 0.25)c (81.11 ± 0.25)d 74.23b (74.04 ± 0.25)c (74.04 ± 0.25)d

1735.78 741.74 −3.88 2.75 −1.93 −117.11 0.29 −0.71 2356.93 184.55a (183.52 ± 0.60)c (183.84 ± 0.49)d 178.01b (176.97 ± 0.60)c (177.30 ± 0.49)d

1552.53 756.64 −4.19 3.48 −1.39 −109.76 0.21 −0.71 2196.81

reaction barrier 183.25 −14.90 0.31 −0.73 −0.54 −7.35 0.08 0.00 160.12

Table 2. Calculated Fundamental Vibrational Frequencies (cm−1) of CH3CN and CH3NC Using CCSD(T)/ANO2/ ANO1 Level of Theorya

Heats of formation calculated at 0 K using H (216.034 kJ mol−1), C(3P) (711.401 ± 0.05 kJ mol−1), and N (470.577 ± 0.024 kJ mol−1), which are taken from ATcT (refs 36−39). bHeats of formation calculated at 298 K. cIndependent ATcT results36−39 (TN ver. 1.122q; see the Supporting Information) dATcT results after inclusion of the current calculations (TN ver. 1.122q-bis) a

CH3CN index

symmetry

type of mode

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8

E A1 E A1 E A1 A1 E

CCN bending CC stretching CH3 rocking CH3 s-deformation CH3 d-deformation CN stretching CH3 s-stretching CH3 d-stretching CH3NC

index

symmetry

type of mode

ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8

E A1 E A1 E A1 A1 E

CNC bending CN stretching CH3 rocking CH3 s-deformation CH3 d-deformation NC stretching CH3 s-stretching CH3 d-stretching

0.60) kJ mol−1. As discussed in more detail in the Supporting Information, variance decomposition analysis39 indicates that the provenance of the ATcT heat of formation of acetonitrile is predominantly experimental, involving combustion calorimetry on liquid acetonitrile and its vaporization enthalpy. The excellent agreement between the experimental route and the current state-of-the-art computation is extremely gratifying and validates both the calculation and the experimental determination, particularly because analogous experimental routes can occasionally (and somewhat unpredictably) lead to a spurious enthalpy of formation for the gas phase species (see ref 39 for a recent example involving hydrazine). The isomerization of CH3NC to CH3CN as displayed in Figure 1 proceeds through a 1,2 methyl-shift. This unimolecular

Theoryb

Exptl.c

365 916 1041 1382 1447 2268 2953 3006

362 920 1041 1385 1448 2267 2954 3009

theory 268 943 1131 1426 1463 2158 2972 3009

exptl. 263 945 1129 1429 1467 2166 2966 3014

diff. 3 −4 0 −3 −1 1 −1 −3 diff. 5 −2 2 −3 −4 −8 6 −5

a

Experimental data are included for comparison. bHarmonic and anharmonic force fields were calculated using ANO2 and ANO1 basis sets, respectively. cFrom Shimanouchi.44

calculated using a ANO245,46 quadruple-ζ basis set while a smaller ANO145,46 triple-ζ basis set was used to obtain the anharmonic force fields. For small molecules, our previous calculations47 using the same approach were able to accurately reproduce observed fundamental frequencies, typically within a few wavenumbers. The same level of accuracy is also obtained here for both CH3CN and CH3NC. It should be mentioned that there is a weak Fermi resonance in CH3NC between ν7 and 2ν5, which was treated by deperturbation and diagonalization to obtain the value in the table.48 Such accuracy is more than sufficient to be applied in a master equation simulation where a typical energy grained bin size of 10 to 30 cm−1 is used (see below).28−30,49−51 It should be mentioned that there is no experimental data for a TS, so comparison between theory and experiment cannot be made. Given the nature of the electronic

Figure 1. Potential energy surface for the isomerization of CH3NC to CH3CN constructed using the HEAT-345Q method. 2533

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538

Letter

The Journal of Physical Chemistry Letters K =+J

structures of the minima and the TS, it is reasonable to expect that a similar level of accuracy is achieved for the latter. Furthermore, at the TS, there is a vibration with a rather low frequency of 195 cm−1, which corresponds to a hindered internal rotation of the methyl-group around an axis that is perpendicular to the CN bond during the methyl-migration from the N atom to the C atom. It is well established that VPT2 is strongly compromised by the presence of a largeamplitude motion (such as a hindered-internal rotation or an umbrella inversion). In this work, this low frequency vibration is assumed to be separable; it is projected out and treated as a separable one-dimensional hindered internal rotor. This assumption is only valid provided that couplings between this low frequency vibration and the remaining vibrations are weak, as is the case here. As shown in the Supporting Information, when the 1DHR treatment is included in the replacement of the low frequency vibration, the ZPE changes only negligibly (by 5 cm−1) and the calculated rate constants are also modified slightly (within 5%) under the conditions considered in this work. Therefore, VPT2 does not do that badly here, but the separable 1DHR approach is certainly to be preferred. With the potential energy surface, rovibrational parameters, and anharmonic constants at hand, we can now calculate thermal rate constants at the high-pressure limit from firstprinciples (i.e., without any empirical or heuristic parameters) using Miller’s semiclassical transition state theory,33−35 eq 1: k(T )P =∞ =

σ 1 × re × h Q CH NC 3

∫0

∞

Grv≠(E ,J )

3

∑ (2J + 1) ∫ J=0

0

Gv≠(E − Eo − Erot(J , K ))

∑

(3a)

K =−J K =+J

ρrv (E , J ) =

∑

ρv (E − Erot(J , K )) (3b)

K =−J

with Erot(J , K ) = J(J + 1)B̅ + (A − B̅ )K 2 , with

B̅ =

B·C

and

− J ≤ K ≤ +J

(4)

Instead of counting exactly ro-vibrational states with eq 3, one can do a convolution of rotational states with vibrational states through eq 5 (see below). It is shown in the Supporting Information that eq 5 gives numerical results that are essentially equivalent to those from eq 3. The advantages of the use of eq 5 is to speed up the calculations of ro-vibrational states. It is well established that the convolution technique is much faster than the direct count. Grv≠(E , J ) =

E − Eo

∑

Gv≠(E − Eo − Erot)ρr≠ (Erot)ΔE (5a)

Erot = 0 E

ρrv (E , J ) =

∞

∑ (2J + 1)

∑ Erot = 0

J=0

exp( −E /kBT ) dE

∞ re Q CH = NC

Grv≠(E , J ) =

ρv (E − Erot)ρr (Erot)ΔE (5b)

A ceiling energy of 40 000 cm−1 (relative to CH3NC) and a maximum total angular momentum (Jmax) of 200 are chosen to ensure that the numerical integration of eq 1 converges for all reaction temperatures considered in this work (up to 1000 K). An energy-grained bin as small as 10 cm−1 and a step size of ΔJ = 5 for total angular momentum are used in computing the integral numerically. The same bin size of 10 cm−1 is also applied to calculate the anharmonic vibrational density and sum of states for CH3NC and TS, respectively, using the BDENS and SCTST codes of the MULTIWELL software package.57 Note that CH3NC is a symmetric top molecule, but the TS is not. However, to a reasonable approximation the TS can be treated as a near prolate symmetric top because it has A = 1.357, B = 0.528, and C = 0.411 cm−1. For such species, the rotational energy levels are given by eq 4.58 SCTST/VPT2 rate constants calculated at the high-pressure limit over a temperature range of 400 to 1000 K are documented in Table S1 (see the Supporting Information) and increase significantly with temperature, as expected for a reaction with a high barrier. At three particular temperatures of 472.6, 503.6, and 533 K, where high-pressure limit experimental data are available,8 our ab initio rate constants of 8.9 × 10−5, 1.1 × 10−3, and 9.7 × 10−3 s−1 agree well (within 20%), respectively, with values of 7.5 × 10−5, 9.25 × 10−4, and 7.67 × 10−3 s−1,8 which we believe to be the most accurate results yet obtained for this reaction. These purely theoretical results are thus consistent with RRKM theory at the high-pressure limit (which is equivalent to TST theory59−61) for this reaction. As can be seen, the ab initio rate constants (calculated using the ab initio barrier Eo = 160.1 ± 1 kJ mol−1) slightly overestimate experimental results. When the experimental activation energy8 of Eo = 160.7 kJ mol−1 (that lies within a plausible error range of the present calculations) is used, the agreement between theory and experiment is better, within 5%.

(1)

∞

ρrv (E ,J ) exp( −E /kBT ) dE (2)

Here h is Planck’s constant, kB is Boltzmann’s constant, T is the reaction temperature, E is the internal energy relative to CH3NC, σ = 3 is the reaction path degeneracy, and QreCH3NC are the rovibrational partition functions for CH3NC (the superscripts “re” and ″ ≠ ″ designate reactants and transition state, respectively). Note that the translational and electronic partition functions of CH3NC and TS cancel and are excluded. While total angular momentum J is a good quantum number and always conserved, its projection (the K quantum number) is not. Therefore, including quantum states of three external rotations into vibrations depends sensitively on how the K quantum number is treated. As recommended by Hase and coworkers,52−56 there are four possible models: K active rotor for TS/K active rotor for CH3NC; K adiabatic rotor/K adiabatic rotor; K active rotor/K adiabatic rotor; and finally, K adiabatic rotor/K active rotor. Any of these can be used to compute microcanonical rate coefficients for the CH3NC →TS → CH3CN step. As given in the Supporting Information, of these four models, the K active model for both TS and CH3NC gives the best results as compared with experiment as well as with asymmetric top model. So, it is chosen for the basis of the discussion below. According to the K active/K active model,52 sums of rovibrational states for TS (G≠rv) and densities of ro-vibrational states (ρrv) for CH3NC are calculated as 2534

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538

Letter

The Journal of Physical Chemistry Letters

Figure 2. Thermal rate constants calculated at three different temperatures (199.4, 230.4, and 259.8 °C) in the falloff region using a collision efficiency8 βc = 0.6 (or ⟨ΔE⟩ = −1200 cm−1). Experimental data8 (unfilled, red symbols) are included for comparison.

Figure 3. Thermal rate constants calculated as functions of temperature and pressure in the falloff region for the unimolecular reaction of CH3NC to CH3CN: the MSC model uses a collision efficiency βc = 0.71, and the 2DME approach uses a downward energy transfer ⟨ΔE⟩down = 2500 cm−1. Experimental data8 (unfilled, red symbols) are also included for comparison.

So, Eo = 160.7 kJ mol−1 was chosen for use in the following master equation calculations. We now turn to the task of computing thermal rate constants as functions of temperature and pressure in the falloff region, and will then compare these theoretical results with Rabinovitch’s experiments.8 Two different approaches are used here to obtain solutions: one is to use a modification of the strong-collision energy transfer Lindemann model1,6 through a collision efficiency factor, which is similar to that used earlier by Rabinovitch and co-workers; and the other is to solve a master equation with a weak-collision energy transfer model.49 Full details are given in the Supporting Information, but briefly summarized here. Modification of Strong-Collision (MSC) Lindemann Model. The thermal rate constants in the falloff region are given by eq 6:1,6,62,63

∞

k(T , P)msc =

∑∫ J=0

∞

0

k(E , J ) 1+

k(E , J ) βc × ωLJ

× FB(E , J ) dE

≠ σ Grv (E , J ) · h ρrv (E , J )

k(E , J ) =

FB(E , J ) =

(6)

(7)

(2J + 1)ρrv (E , J ) exp(−E /kBT ) ∞ ∑ J = 0 (2J

+ 1) ∫

∞

0

ρrv (E , J ) exp( −E /kBT ) dE (8)

βc 1−

βc

≈

−⟨ΔE⟩ ∞

∫E FB(E , J ) dE o

× FB(Eo , J ) (9)

−1

where k(E,J) (in s ) is the microcanonical rate constant for the CH3NC → CH3CN step, FB(E,J) is the Boltzmann thermal 2535

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538

Letter

The Journal of Physical Chemistry Letters energy distribution function of CH3NC, ωLJ (in s−1) is the collision frequency, and βc is a collision efficiency between CH3NC and a bath gas, which can be estimated with eq 9.62,63 −⟨ΔE⟩ (in cm−1) is the average amount of energy transferred per collision. Eo = 160.7 kJ mol−1 is the activation energy. C2H6 is chosen as the bath gas here because it was used in Rabinovitch’s early experiment.8 Collisional parameters adopted from the literature9,10,57 are σ = 4.47 Å and ε/kB = 380 K for CH3NC and σ = 4.39 Å and ε/kB = 234 K for C2H6. With these parameters, a collision frequency of 1.16 × 1010 s−1 is obtained at T = 503.6 K and P = 1000 mmHg. To compute the integral in eq 6, one needs to know a collision efficiency, βc (or an average amount of energy transferred per collision, −⟨ΔE⟩; see eq 9). In previous studies, Rabinovitch and co-workers reported a collision efficiency βc = 0.6 for C2H6,8−10 from which we obtained −⟨ΔE⟩ ≈ 1200 cm−1 using eq 9. This ⟨ΔE⟩ value is (accidentally) equivalent to that obtained by Miller and Chandler64 in an independent study. Using βc = 0.6, we computed k(T,P) and display the results in Figure 2, where experimental data are also included for comparison. As can be seen in Figure 2, the present theoretical results agree well with experiment at 503.6 K, but slightly underestimates the experimental rate constants (by about 20%) at the other two temperatures, 472.6 and 533 K. It is obvious that fitting will be better when βc becomes larger than 0.6. We found that βc = 0.71 (or −⟨ΔE⟩ ≈ 2000 cm−1) gives a nearly perfect fit, as seen in Figure 3 above. It should be mentioned that the obtained collision efficiency of 0.71 is in very good agreement with 0.78 reported by Rabinovitch and co-workers in 1970.10 These results suggest that the thermally activated reaction system approaches the strong-collision limit, where βc becomes unity. It can be concluded that the MSC model can be applied to the title reaction. Two-Dimensional Master Equation (TDME) Approach. The TDME that describes a time-evolution for the thermally activated isomerization of CH3NC to CH3CN (as shown in Figure 1) in competition with the weak-collision energy transfer processes of CH3NC with the bath gas C2H6 is given by eq 10:49−51,65−71 ∂C(El , Jl ) ∂t

∞

=

∑∫ Jk = 0

⟨ΔE⟩down. To obtain a good fit, we varied ⟨ΔE⟩down from 500 to 3000 cm−1 based on the initial ⟨ΔE⟩ value from the MSC model (see above). The calculated rate constants as functions of pressure are plotted in Figure S1. These clearly depend on the value of ⟨ΔE⟩down. At very low pressure, P = 0.01 mmHg, k(T,P)ME increase by a factor of 2.5 as ⟨ΔE⟩down is raised from 500 to 3000 cm−1 (Figure S1, Supporting Information). With higher pressures, the calculated rate constants become less sensitive to the average down energy transfer. We observe good agreement (within 10%) with experiment when ⟨ΔE⟩down lies between 2000 to 3000 cm−1, with 2500 cm−1 yielding the best results. This value was used for subsequent calculations at other temperatures. It is worth nothing that while ⟨ΔE⟩down = 2500 cm−1 seems rather large, it is equivalent to a collision efficiency of 0.75 in the MSC model: fully consistent with the value of 0.78 obtained by Rabinovitch and co-workers in 1970.10 On the other hand, such a large value associated with collisions involving a molecule with “soft” (torsional) vibrations potentially acts as a “fitting parameter” that makes some nonstatistical behavior in the reaction. This latter possibility cannot be fully dismissed by the results of our work and appears to warrant further investigation. The calculated k(T,P)ME values are included in Figure 3 to compare with experiment and the MSC model. The excellent agreement (within 10%) between the two persists over an extensive range of temperatures and pressures in the falloff region. It is obvious that the two theoretical models used here are able to entirely replicate Rabinovitch’s experimental results,8 which can be viewed as a triumph for modern computational chemical kinetics.

■

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b01259. Optimized geometries of various stationary points, rovibrational parameters, one-dimensional hindered internal rotation, two-dimensional master equation approach, and the provenance analysis of the ATcT results are given (PDF)

∞

Ek = 0

■

P(El , Jl |Ek , Jk )ωLJC(Ek , Jk )dEk

− ωLJC(El , Jl ) − k(El , Jl )C(El , Jl )

⟨k(T , P)ME ⟩ =

*E-mail: johnstanton@ufl.edu. ORCID

David H. Bross: 0000-0002-8218-0249 Branko Ruscic: 0000-0002-4372-6990 John F. Stanton: 0000-0003-2345-9781 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The work at University of Florida by T.L.N., J.H.T., and J.F.S. was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Award DE-FG02-07ER15884 and the U.S. Air Force Office of Scientific Research (No. FA955016-1-0117). The work at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences, under Contract No. DE-AC0206CH11357, through the Gas-Phase Chemical Physics Program

∫

∫

AUTHOR INFORMATION

Corresponding Author

(10)

where C is the population of CH3NC, and P(El, Jl|Ek, Jk) is an energy/angular momentum transfer probability function from an initial state (Ek|Jk) to a final state (El|Jl). Solutions of eq 10 were previously reported elsewhere28−30 and are briefly given in the Supporting Information. Here, phenomenological rate constants, k(T,P)ME, in the falloff region can be computed from the smallest eigenvalues (|λ1|) weighted by a Boltzmann distribution that runs over all possible total angular momentum quantum numbers through eq 11: ∞ ∞ ∑J=0 |λ1(J )|FB(E , J ) dE 0 ∞ ∞ ∑J=0 FB(E , J ) dE 0

ASSOCIATED CONTENT

S Supporting Information *

(11)

In this work, we assume a single exponential function that is frequently used for P(E), energy transfer probability;49,50,65,66,68 and then we solve eq 10 for k(T,P)ME. To do this, one needs to find the (unknown) average amount of energy transferred per collision in a downward direction (i.e., deactivation path), 2536

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538

Letter

The Journal of Physical Chemistry Letters

(22) Jursic, B. S. Density functional theory and ab initio study of CH3NC and HNC isomerization. Chem. Phys. Lett. 1996, 256 (1−2), 213−219. (23) Halpern, A. M. Computational studies of chemical reactions: The HNC-HCN and CH3NC-CH3CN isomerizations. J. Chem. Educ. 2006, 83 (1), 69−76. (24) Bunker, D. L.; Hase, W. L. Non-Rrkm Unimolecular Kinetics Molecules in General, and Ch3nc in Particular. J. Chem. Phys. 1973, 59 (9), 4621−4632. (25) Tajti, A.; Szalay, P. G.; Csaszar, A. G.; Kallay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vazquez, J.; Stanton, J. F. HEAT: High accuracy extrapolated ab initio thermochemistry. J. Chem. Phys. 2004, 121 (23), 11599−11613. (26) Bomble, Y. J.; Vazquez, J.; Kallay, M.; Michauk, C.; Szalay, P. G.; Csaszar, A. G.; Gauss, J.; Stanton, J. F. High-accuracy extrapolated ab initio thermochemistry. II. Minor improvements to the protocol and a vital simplification. J. Chem. Phys. 2006, 125 (6), 064108. (27) Harding, M. E.; Vazquez, J.; Ruscic, B.; Wilson, A. K.; Gauss, J.; Stanton, J. F. High-accuracy extrapolated ab initio thermochemistry. III. Additional improvements and overview. J. Chem. Phys. 2008, 128 (11), 114111. (28) Nguyen, T. L.; Stanton, J. F. A Steady-State Approximation to the Two-Dimensional Master Equation for Chemical Kinetics Calculations. J. Phys. Chem. A 2015, 119 (28), 7627−7636. (29) Nguyen, T. L.; Lee, H.; Matthews, D. A.; McCarthy, M. C.; Stanton, J. F. Stabilization of the Simplest Criegee Intermediate from the Reaction between Ozone and Ethylene: A High-Level Quantum Chemical and Kinetic Analysis of Ozonolysis. J. Phys. Chem. A 2015, 119 (22), 5524−5533. (30) Nguyen, T. L.; McCaslin, L.; McCarthy, M. C.; Stanton, J. F. Communication: Thermal unimolecular decomposition of synCH3CHOO: A kinetic study. J. Chem. Phys. 2016, 145 (13), 131102. (31) Nguyen, T. L.; Stanton, J. F.; Barker, J. R. Ab Initio Reaction Rate Constants Computed Using Semiclassical Transition-State Theory: HO+H-2 -> H2O+H and Isotopologues. J. Phys. Chem. A 2011, 115 (20), 5118−5126. (32) Nguyen, T. L.; Stanton, J. F.; Barker, J. R. A practical implementation of semi-classical transition state theory for polyatomics. Chem. Phys. Lett. 2010, 499 (1−3), 9−15. (33) Miller, W. H. Semiclassical Theory for Non-Separable Systems Construction of Good Action-Angle Variables for Reaction-Rate Constants. Faraday Discuss. Chem. Soc. 1977, 62, 40−46. (34) Miller, W. H.; Hernandez, R.; Handy, N. C.; Jayatilaka, D.; Willetts, A. Abinitio Calculation of Anharmonic Constants for a Transition-State, with Application to Semiclassical Transition-State Tunneling Probabilities. Chem. Phys. Lett. 1990, 172 (1), 62−68. (35) Hernandez, R.; Miller, W. H. Semiclassical Transition-State Theory - a New Perspective. Chem. Phys. Lett. 1993, 214 (2), 129− 136. (36) Ruscic, B.; Pinzon, R. E.; Morton, M. L.; von Laszevski, G.; Bittner, S. J.; Nijsure, S. G.; Amin, K. A.; Minkoff, M.; Wagner, A. F. Introduction to Active Thermochemical Tables: Several ″Key″ Enthalpies of Formation Revisited. J. Phys. Chem. A 2004, 108, 9979−9997. Ruscic, B.; Pinzon, R. E.; von Laszewski, G.; Kodeboyina, D.; Burcat, A.; Leahy, D.; Montoy, D.; Wagner, A. F. Active Thermochemical Tables: thermochemistry for the 21st century. J. Phys.: Conf. Ser. 2005, 16, 561−570. (37) Ruscic, B.; Boggs, J. E.; Burcat, A.; Csaszar, A. G.; Demaison, J.; Janoschek, R.; Martin, J. M. L.; Morton, M. L.; Rossi, M. J.; Stanton, J. F.; Szalay, P. G.; Westmoreland, P. R.; Zabel, F.; Berces, T. IUPAC critical evaluation of thermochemical properties of selected radicals. Part I. J. Phys. Chem. Ref. Data 2005, 34 (2), 573−656. (38) Ruscic, B. Active Thermochemical Tables: Sequential Bond Dissociation Enthalpies of Methane, Ethane, and Methanol and the Related Thermochemistry. J. Phys. Chem. A 2015, 119 (28), 7810− 7837. (39) Ruscic, B.; Bross, D. H. Active Thermochemical Tables (ATcT) values based on ver. 1.122 of the Thermochemical Network (2016), available at ATcT.anl.gov. Ruscic, B.; Feller, D.; Peterson, K. A. Active

(B.R.) and the Computational Chemical Sciences Program (D.H.B.). We thank an anonymous reviewer for comments that helped to shape the final form of this manuscript.

■

REFERENCES

(1) Robinson, P. J.; Holbrook, K. A. Unimolecular reactions; WileyInterscience: London/New York, 1972; p xvii, 371 p. (2) Forst, W. Theory of unimolecular reactions; Academic Press: New York, 1973; p xv, 445 p. (3) Holbrook, K. A.; Pilling, M. J.; Robertson, S. H.; Robinson, P. J. Unimolecular reactions, 2nd ed.; Wiley: Chichester/New York, 1996; p xv, 417 p. (4) Forst, W. Unimolecular reactions: a concise introduction. Cambridge University Press: Cambridge, U.K./New York, 2003; p xi, 319 p. (5) Steinfeld, J. I.; Francisco, J. S.; Hase, W. L. Chemical Kinetics and Dynamics, 2nd ed; Prentice Hall International, Inc.: Upper Saddle River, NJ, 1999. (6) Eyring, H.; Lin, S. H.; Lin, S. M. Basic chemical kinetics; Wiley: New York, 1980; p vii, 493 p. (7) Gilbert, R. G.; Smith, S. C. Theory of unimolecular and recombination reactions; Blackwell Scientific Publications: Oxford/ Boston, 1990; p xii, 356 p. (8) Schneider, F. W.; Rabinovitch, B. S. Thermal Unimolecular Isomerization of Methyl Isocyanide - Fall-Off Behavior. J. Am. Chem. Soc. 1962, 84 (22), 4215−4230. (9) Fletcher, F. J.; Rabinovitch, B. S.; Watkins, K. W.; Locker, D. J. Energy Transfer in Thermal Methyl Isocyanide Isomerization. Experimental Survey. J. Phys. Chem. 1966, 70 (9), 2823−2833. (10) Chan, S. C.; Bryant, J. T.; Spicer, L. D.; Fujimoto, T.; Lin, Y. N.; Pavlou, S. P. Energy Transfer in Thermal Methyl Isocyanide Isomerization - a Comprehensive Investigation. J. Phys. Chem. 1970, 74 (17), 3160−3176. (11) Wang, F. M.; Rabinovitch, Bs Fall-Off Behavior of a Thermal Unimolecular System in Presence of a Weak Collider Inert Bath Gas. J. Phys. Chem. 1974, 78 (9), 863−867. (12) Lifshitz, A.; Carroll, H. F.; Bauer, S. H. Studies with a SinglePulse Shock Tube 0.2. Thermal Decomposition of Perfluorocyclobutane. J. Chem. Phys. 1963, 39 (7), 1661−1665. (13) Collister, J. L.; Pritchard, H. O. Thermal-Isomerization of Methyl Isocyanide in Temperature-Range 120−320degreesc. Can. J. Chem. 1976, 54 (15), 2380−2384. (14) Reddy, K. V.; Berry, M. J. Intracavity Cw Dye-Laser Photoactivation of Unimolecular Reactants - Isomerization of StateSelected Methyl Isocyanide. Chem. Phys. Lett. 1977, 52 (1), 111−116. (15) Wang, D. X.; Qian, X. M.; Peng, J. A novel method for the kinetic study of a chemical reaction. Determination of the kinetic parameters of the isomerisation reaction of methyl isocyanide using HeI photoelectron spectroscopy. Chem. Phys. Lett. 1996, 258 (1−2), 149−154. (16) Marcus, R. A.; Rice, O. K. The Kinetics of the Recombination of Methyl Radicals and Iodine Atoms. J. Phys. Chem. 1951, 55 (6), 894− 908. (17) Marcus, R. A. Unimolecular Dissociations and Free Radical Recombination Reactions. J. Chem. Phys. 1952, 20 (3), 359−364. (18) Liskow, D. H.; Schaefer, H. F.; Bender, C. F. Some Features of Ch3nc-]Ch3cn Potential Surface. J. Chem. Phys. 1972, 57 (11), 4509− 4511. (19) Liskow, D. H.; Schaefer, H. F.; Bender, C. F. Theoretical Reaction Coordinate for Methyl Isocyanide Isomerization. J. Am. Chem. Soc. 1972, 94 (15), 5178−5182. (20) Redmon, L. T.; Purvis, G. D.; Bartlett, R. J. Unimolecular Isomerization of Methyl Isocyanide to Methyl Cyanide (Acetonitrile). J. Chem. Phys. 1978, 69 (12), 5386−5392. (21) Saxe, P.; Yamaguchi, Y.; Pulay, P.; Schaefer, H. F. TransitionState Vibrational Analysis for the Methyl Isocyanide Rearrangement, Ch3nc-]Ch3cn. J. Am. Chem. Soc. 1980, 102 (11), 3718−3723. 2537

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538

Letter

The Journal of Physical Chemistry Letters Thermochemical Tables: Dissociation Energies of Several Homonuclear First-Row Diatomics and Related Thermochemical Values. Theor. Chem. Acc. 2014, 133, 1415. Feller, D.; Bross, D. H.; Ruscic, B. Enthalpy of Formation of N2H4 (Hydrazine) Revisited. J. Phys. Chem. A 2017, 121, 6187−6198. (40) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Headgordon, M. A 5th-Order Perturbation Comparison of Electron Correlation Theories. Chem. Phys. Lett. 1989, 157 (6), 479−483. (41) Bartlett, R. J.; Watts, J. D.; Kucharski, S. A.; Noga, J. Noniterative 5th-Order Triple and Quadruple Excitation-Energy Corrections in Correlated Methods. Chem. Phys. Lett. 1990, 165 (6), 513−522. (42) Mills, I. M., Vibration-Rotation Structure in Asymmetric- and Symmetric-Top Molecules. In Molecular Spectroscopy: Modern Research; Rao, K. N., Mathews, C. W., Ed.; Academic Press: New York. 1972, Vol. 1, p. 115. (43) Hoy, A. R.; Mills, I. M.; Strey, G. Anharmonic Force Constant Calculations. Mol. Phys. 1972, 24 (6), 1265−1290. (44) Shimanouchi, T. Tables of Molecular Vibrational Frequencies; Consolidated Vol. I, NSRDS-NBS 39; National Bureau of Standards: Washington, DC, 1972. (45) Almlof, J.; Taylor, P. R. General Contraction of Gaussian-Basis Sets 0.2. Atomic Natural Orbitals and the Calculation of Atomic and Molecular-Properties. J. Chem. Phys. 1990, 92 (1), 551−560. (46) Almlof, J.; Taylor, P. R. General Contraction of Gaussian-Basis Sets 0.1. Atomic Natural Orbitals for 1st-Row and 2nd-Row Atoms. J. Chem. Phys. 1987, 86 (7), 4070−4077. (47) McCaslin, L. M.; Stanton, J. F. Calculation of Fundamental Frequencies for Small Polyatomic Molecules: A Comparison of Correlation-Consistent and Atomic Natural Orbital Basis Sets. Mol. Phys. 2013, 111, 1492−1496. (48) Matthews, D. A.; Stanton, J. F. Quantitative analysis of Fermi resonances by harmonic derivatives of perturbation theory corrections. Mol. Phys. 2009, 107 (3), 213−222. (49) Glowacki, D. R.; Liang, C. H.; Morley, C.; Pilling, M. J.; Robertson, S. H. MESMER: an open-source master equation solver for multi-energy well reactions. J. Phys. Chem. A 2012, 116 (38), 9545−60. (50) Zhang, J.; Donahue, N. M. Constraining the mechanism and kinetics of OH + NO2 and HO2 + NO using the multiple-well master equation. J. Phys. Chem. A 2006, 110 (21), 6898−911. (51) Barker, J. R. Multiple-well, multiple-path unimolecular reaction systems. I. MultiWell computer program suite. Int. J. Chem. Kinet. 2001, 33 (4), 232−245. (52) Zhu, L.; Chen, W.; Hase, W. L.; Kaiser, E. W. Comparison of Models for Treating Angular-Momentum in Rrkm Calculations with Vibrator Transition-States - Pressure and Temperature-Dependence of Cl+C2h2 Association. J. Phys. Chem. 1993, 97 (2), 311−322. (53) Zhu, L.; Hase, W. L. Comparison of Models for Calculating the Rrkm Unimolecular Rate Constant-K(E,J). Chem. Phys. Lett. 1990, 175 (1−2), 117−124. (54) Song, K.; Hase, W. L. Role of state specificity in the temperature- and pressure-dependent unimolecular rate constants for HO2 -> H+O-2 dissociation. J. Phys. Chem. A 1998, 102 (8), 1292− 1296. (55) Hase, W. L. Some recent advances and remaining questions regarding unimolecular rate theory. Acc. Chem. Res. 1998, 31 (10), 659−665. (56) Song, K. H.; Sun, L. P.; Hase, W. L.; Grebenshchikov, S. Y.; Schinke, R. Relationship between mode specific and thermal unimolecular rate constants for HOCl -> OH+Cl dissociation. J. Phys. Chem. A 2002, 106 (36), 8339−8344. (57) Barker, J. R.; Ortiz, N. F.; Preses, J. M.; Lohr, L. L.; Maranzana, A.; Stimac, P. J.; Nguyen, T. L.; Kumar, T. J. D.; Li, C. G. L. MultiWell-2016 Software. http://clasp-research.engin.umich.edu/ multiwell/. (58) Baer, T.; Hase, W. L. Unimolecular reaction dynamics: theory and experiments; Oxford University Press: New York, 1996; p vi, 438 p. (59) Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 1935, 3 (2), 107−115.

(60) Evans, M. G.; Polanyi, M. Some applications of the transition state method to the calculation of reaction velocities, especially in solution. Trans. Faraday Soc. 1935, 31 (1), 875−893. (61) Truhlar, D. G.; Garrett, B. C.; Klippenstein, S. J. Current status of transition-state theory. J. Phys. Chem. 1996, 100 (31), 12771− 12800. (62) Troe, J. Theory of Thermal Unimolecular Reactions at LowPressures 0.1. Solutions of Master Equation. J. Chem. Phys. 1977, 66 (11), 4745−4757. (63) Troe, J. Theory of Thermal Unimolecular Reactions at LowPressures 0.2. Strong Collision Rate Constants - Applications. J. Chem. Phys. 1977, 66 (11), 4758−4775. (64) Miller, J. A.; Chandler, D. W. A Theoretical-Analysis of the Overtone-Induced Isomerization of Methyl Isocyanide. J. Chem. Phys. 1986, 85 (8), 4502−4508. (65) Miller, J. A.; Klippenstein, S. J.; Raffy, C. Solution of Some Oneand Two-Dimensional Master Equation Models for Thermal Dissociation: The Dissociation of Methane in the Low-Pressure Limit. J. Phys. Chem. A 2002, 106, 4904−4913. (66) Miller, J. A.; Klippenstein, S. J. Master equation methods in gas phase chemical kinetics. J. Phys. Chem. A 2006, 110 (36), 10528−44. (67) Jeffrey, S. J.; Gates, K. E.; Smith, S. C. Full Iterative Solution of the Two-Dimensional Master Equation for Thermal Unimolecular Reactions. J. Phys. Chem. 1996, 100, 7090−7096. (68) Frankcombe, T. J.; Smith, S. C. Time evolution in the unimolecular master equation at low temperatures: full spectral solution with scalable iterative methods and high precision. Comput. Phys. Commun. 2001, 141 (1), 39−54. (69) Barker, J. R. Monte-Carlo Calculations on Unimolecular Reactions, Energy-Transfer, and IR-Multiphoton Decomposition. Chem. Phys. 1983, 77 (2), 301−318. (70) Vereecken, L.; Huyberechts, G.; Peeters, J. Stochastic simulation of chemically activated unimolecular reactions. J. Chem. Phys. 1997, 106 (16), 6564−6573. (71) Vereecken, L.; Peeters, J. Detailed microvariational RRKM master equation analysis of the product distribution of the C2H2+CH(X-2 Pi) reaction over extended temperature and pressure ranges. J. Phys. Chem. A 1999, 103 (28), 5523−5533.

2538

DOI: 10.1021/acs.jpclett.8b01259 J. Phys. Chem. Lett. 2018, 9, 2532−2538