Theoretical Verification of Nonthermal Microwave Effects on

Article pubs.acs.org/JPCA

Theoretical Verification of Nonthermal Microwave Effects on Intramolecular Reactions Manabu Kanno,† Kosuke Nakamura,† Eri Kanai,† Kunihito Hoki,†,§ Hirohiko Kono,*,† and Motohiko Tanaka‡ †

Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan College of Engineering, Chubu University, 1200 Matsumoto-cho, Kasugai 487-8501, Japan

‡

ABSTRACT: There have been a growing number of articles that report dramatic improvements in the experimental performance of chemical reactions by microwave irradiation compared to that under conventional heating conditions. We theoretically examined whether nonthermal microwave effects on intramolecular reactions exist or not, in particular, on Newman−Kwart rearrangements and intramolecular Diels− Alder reactions. The reaction rates of the former calculated by the transition state theory, which consider only the thermal effects of microwaves, agree quantitatively with experimental data, and thus, the increases in reaction rates can be ascribed to dielectric heating of the solvent by microwaves. In contrast, for the latter, the temperature dependence of reaction rates can be explained qualitatively by thermal effects but the possibility of nonthermal effects still remains regardless of whether competitive processes are present or not. The effective intramolecular potential energy surface in the presence of a microwave field suggests that nonthermal effects arising from potential distortion are vanishingly small in intramolecular reactions. It is useful in the elucidation of the reaction mechanisms of microwave synthesis to apply the present theoretical approach with reference to the experiments where thermal and nonthermal effects are separated by screening microwave fields.

1. INTRODUCTION Industrial application of microwaves is widespread and makes a significant contribution to everyday life, for example, microwave ovens, satellite broadcasting, etc. In recent years, microwave irradiation has also been utilized to accelerate chemical reactions in the laboratory,1 and the number of such reports is increasing at a rapid rate.2 In those experiments, remarkable improvements in product yield and/or reduction of reaction time are observed over that obtained by classical processing techniques, and they are deemed to be caused by microwave effects, which can be classified into two components, namely, thermal and nonthermal effects. The thermal effects of microwaves are attributed to the rise in solvent temperature due to dielectric loss: the polar solvent molecules with a permanent electric dipole are forced to rotate with a phase lag from the alternating microwave field by molecular friction and dissipate energy in the form of heat.3 The nonthermal effects collectively mean those that cannot be included in the thermal effects, e.g., the direct interaction of the microwave field with the electric dipoles of solute molecules; in the case of intermolecular reactions, the solute alignment with a microwave field increases the probability of reactive collisions owing to the increase in activation entropy. There has been a long controversy as to whether the microwave effects on the kinetics of enhanced chemical reactions are in fact thermal or nonthermal. From an experimental aspect, separating thermal and nonthermal effects is generally a difficult task. Moreover, because of the difficulty in monitoring local © 2012 American Chemical Society

temperature over the reaction area, there may be a possibility that the reaction temperature was not precisely controlled in previously reported microwave-assisted experiments. However, very recently a breakthrough has been achieved by the experiments using a reaction vessel made out of silicon carbide (SiC), which enables the separation of thermal from nonthermal microwave effects.4 The material inside the reaction vessel is effectively shielded from a radiation field owing to the high microwave absorptivity of SiC but quickly heated by the ohmic resistance to the conduction current flowing in the semiconducting SiC under microwave irradiation. Also, the reaction temperatures inside and outside of the vessel are simultaneously monitored and well controlled with internal and external probes. Reference 4 compares the results of 21 inter- and intramolecular reactions performed in SiC and standard microwave-transparent Pyrex vials and shows that the product yields in both cases were almost identical with each other at the same reaction temperature and time. This clearly indicates that a microwave field has no direct (nonthermal) effect on the selected chemical reactions. As for theory, a real-time simulation to evaluate the accumulated effects of weak microwave−molecule interactions is time-consuming and computationally demanding. Here, we Received: December 25, 2011 Revised: February 8, 2012 Published: February 14, 2012 2177

dx.doi.org/10.1021/jp212460v | J. Phys. Chem. A 2012, 116, 2177−2183

The Journal of Physical Chemistry A

Article

where BiR represents the ith rotational constant of reactants. The symmetry number σR is the number of molecular orientations that are indistinguishable from each other. The expression for ZrTS is analogous to eq 3. For the vibrational partition functions, we adopt the harmonic-oscillator model

aim to theoretically investigate the existence of nonthermal microwave effects on chemical reactions without a real-time simulation of microwave−molecule interactions. In this context, we use the transition state theory,5 which provides the rate constants for elementary reactions at a given temperature, to verify whether the results in microwave organic synthesis can be explained only by thermal effects (temperature factors). A few theoretical papers have dealt with the behavior of rotationally hot molecules excited by microwaves in intermolecular reactions.6,7 In this article, in order to avoid a complicated analysis of microwave effects on intermolecular reactions to which various factors contribute, our particular attention is focused on those on intramolecular reactions, to be more specific, on the Newman− Kwart rearrangements (NKRs)8,9 and intramolecular Diels− Alder reactions (IDARs).10

Z vR =

j

ZrR

1 − exp( −hν Rj /kBT )

(4)

where is the harmonic frequency of the jth normal mode of reactants. ZvTS has a similar form to eq 4 except that the normal mode with imaginary frequency, which is regarded as the reaction coordinate, is excluded from the multiplication. The rate formula of eq 2 serves as the basis for evaluating the thermal effects (temperature effects) on microwave-assisted intramolecular reactions. In section 3.2, we extend the formula so that it can be applied to two-step intramolecular reactions in which an intermediate local minimum of the PES lies on the reaction path between reactants and transition states.

3. RESULTS AND DISCUSSION 3.1. Newman−Kwart Rearrangement (NKR). The NKR is a type of rearrangement reaction in which intramolecular aryl (Ar) migration of an O-aryl thiocarbamate, ArOC(S)NMe2, forms an S-aryl thiocarbamate, ArSC(O)NMe2. This methodology is one of the most efficient synthetic routes for the conversion of substituted phenols to thiophenols. The NKR, which proceeds via an O- to S-aryl migration, requires high activation energy and thus is conducted at temperatures in the range of 200−300 °C. Moseley et al. investigated the microwave effects on NKRs with a variety of aryl groups and concluded that the reaction rate is essentially unchanged from that obtained by conventional thermal heating.13,14 Furthermore, ref 4 presents the product yields of the NKR with a 4-cyanophenyl group (Scheme 1a) performed in SiC and Pyrex vials, which are identical to each other (Table 1). These experimental results

(1)

where kB, h, and T denote the Boltzmann constant, Planck constant, and absolute temperature, respectively. ZR and ZTS are the partition functions for reactants and transition states, respectively, while ER and ETS are their energies (excluding zeropoint vibrational energies). In the Born−Oppenheimer approximation,12 a molecular partition function can be decomposed into the product of translational, rotational, vibrational, and electronic parts. Among them, the translational partition function, which is associated with the center-of-mass motion of the molecule, is the same between reactants and transition states. The electronic partition function can be set to unity as long as the reaction progresses on the ground state potential energy surface (PES). Hence, one can rewrite eq 1 as ⎛ E − ER ⎞ k T Z rTSZ vTS k= B exp⎜ − TS ⎟ R R h Zr Z v kBT ⎠ ⎝

exp( −hν Rj /2kBT )

νjR

2. TRANSITION STATE THEORY This section outlines the rate formula for single-step intramolecular reactions in the standard transition state (or activated complex) theory.5 In this theory, the reaction rate is proportional to the concentration of activated complexes passing over the potential barrier to products per unit time. On the assumption that transition states are in quasi-equilibrium with reactants, the rate constant k for an intramolecular reaction can be expressed by11 ⎛ E TS − ER ⎞ k T ZTS k= B exp ⎜− ⎟ h ZR kBT ⎠ ⎝

∏

Scheme 1. Intramolecular Reactions Studied Using the Transition State Theory: (a,b) NKRs with (a) 4Cyanophenyl and (b) 4-Methoxyphenyl Groups Performed in a Microwave-Absorbing SiC or Microwave-Transparent Pyrex Reaction Vessel4 and (c,d) IDARs Conducted under Thermal Heating or Microwave Irradiation20a

(2)

ZvR

where and are the rotational and vibrational partition functions for the reactant, respectively, and ZrTS and ZvTS are their counterparts for transition states. The most important parameter in eq 2 is the activation energy ETS − ER in the exponential factor. Besides, the contribution of the rotational partition functions to the rate of intramolecular reactions should be even smaller than that of the vibrational ones. Therefore, we simply approximate the rotation of solute molecules in a solution by the free rotation of isolated molecules. Then, for the reactions studied in this article, the explicit forms of the rotational partition functions are given by that of a rigid asymmetric top 1/2 3 ⎛ kBT ⎞ π1/2 R Z r = R ∏ ⎜⎜ R ⎟⎟ σ i = 1 ⎝ Bi ⎠

a

NMP denotes N-methylpyrrolidone as the solvent. The reaction c produces no by-product; the reaction d competes with polymerization.

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Table 1. Experimentala and Theoreticalb Rate Constants, κ and k, for the NKR in Scheme 1a exptl condition

temp

time

microwave 224 °C 20 min (SiC or Pyrex)

product yield 82%

exptl rate constant κ

Table 2. Basis-Set Dependence of the B3LYP Activation Energy ETS − ER for the NKR in Scheme 1a basis set

theor rate constant k

activation energy (kcal/mol)

1.4 × 10−3 s−1 3.2 × 10−4 s−1

Experimental data are taken from ref 4. Experimental rate constant κ was determined by eq 7. bElectronic structure computations were done at the B3LYP/cc-pVTZ level of theory. Theoretical rate constant k was calculated from eq 2.

6-31G* 6-311G* 6-311+G* cc-pVDZ cc-pVTZ 36.7

35.8

35.0

34.9

34.7

a

examined the necessity of the long-range correction to the DFT exchange functional in the specific case of the NKR with a 2-nitrophenyl group for which activation energy has been experimentally determined to be about 28 kcal/mol.14 Using the cc-pVTZ basis set, the B3LYP activation energy for this reaction is 32.5 kcal/mol, whereas the single-point calculations with the long-range corrected functional LC-BOP17 at the B3LYP optimized geometries give ETS − ER = 38.7 kcal/mol, which is much higher than the experimental one. In this treatment, the long-range corrected DFT greatly overestimates the activation energy for NKRs. The computational results shown in the following are all at the B3LYP/cc-pVTZ level of theory. The optimized geometries for the NKR in Scheme 1a are depicted in Figure 1. From the frequency analysis, the normal mode of the transition state with imaginary frequency mainly consists of the cleavage of the Ar−O bond and the formation of the Ar−S bond. With ER ≡ 0, the energies of the transition state and product are 34.7 and −6.0 kcal/mol, respectively. The activation energy ETS − ER = 34.7 kcal/mol, symmetry numbers σR = σTS = 1, rotational constants {BiR} and {BiTS}, and harmonic frequencies {νjR} and {νjTS} for the NKR in Scheme 1a were inserted into eq 2 to figure out the theoretical rate constant k. The value of k at T = 497 K (224 °C) is 3.2 × 10−4 s−1, which is close to that of the experimental one κ. The transition state theory succeeded in reproducing the rate of this reaction semiquantitatively. Within the Onsager dielectric continuum model,18 electrostatic solvation lowers the energies of the reactant and transition state by 1.8 and 3.7 kcal/mol, respectively, in a polar solvent NMP, where we used the calculated electric dipole moments of the reactant and transition state (7.6 and 10.8 debye, respectively), a rough estimate of the cavity radius (6.0 Ǻ ), and the dielectric constant of NMP (32.2 at 25 °C). This implies that, with solvation included, the activation energy ETS − ER decreases by 1−2 kcal/mol and that the value of k becomes larger and closer to that of κ. However, in ref 4, the product yield Y(t) of the NKR in Scheme 1a was reported only at t = 20 min. If the reaction reached equilibrium at t < 20 min, the actual value of the experimental rate constant κ may be larger than that in Table 1. Therefore, we also evaluate the reaction rate of the NKR with a 4-methoxyphenyl group (Scheme 1b) of which the product yields Y(t) at two different times were reported in ref 4 (Table 3). The yields measured in SiC and Pyrex vials are virtually identical irrespective of the reaction time. As shown in Table 3, the value of the experimental constant κ for the NKR in Scheme 1b varies slightly depending on the reaction time but is almost regarded as a constant. The optimized geometries for the NKR in Scheme 1b are shown in Figure 2. Each of them resembles its counterpart in Figure 1 except for the methoxy group and the minor difference in the conformation of the dimethylamino group at the reactant geometry. The similarity is also found in the imaginary frequency mode of the transition state. The energies of the transition state and product are 40.4 and −6.9 kcal/mol, respectively. In this case, solvation hardly affects the activation energy because the magnitudes of the electric dipole moments of the reactant and transition state are

provide no evidence for nonthermal effects of microwaves and indicate that only reaction temperature is responsible for the enhancement of NKRs. Following the above-mentioned experimental studies, we theoretically quantify the thermal effects (temperature effects) on the NKR in Scheme 1a. First, the experimental rate constant κ was determined as follows. The rate law for a first-order reaction is [R] = C exp( −κt )

(5)

where t denotes the reaction time, [R] is the concentration of the reactant, and C is the initial concentration of the reactant. The product yield Y(t) is defined by Y (t ) =

[P] [R] =1− = 1 − exp( −κt ) C C

(6)

where [P] is the concentration of the product. The experimental rate constant κ is thus evaluated from the yield as

κ=−

ln[1 − Y (t )] t

(7)

By substituting the reaction time t and product yield Y(t) presented in Table 1 into eq 7 [Y(t = 20 min) = 0.82], we obtained κ = 1.4 × 10−3 s−1 for the NKR in Scheme 1a. Next, for comparison, we calculate the theoretical rate constant k from the transition state theory, i.e., eq 2. The polar solvent N-methylpyrrolidone (NMP) is considered not to be directly involved in the reaction process by forming a reaction intermediate with the reactant. Electronic structure computations for the reactant, transition state, and product were carried out using the quantum chemistry program Gaussian 09.15 For geometry optimization, we employed the second-order Møller− Plesset perturbation theory (MP2) and density functional theory (DFT) with the B3LYP hybrid functional,16 which require comparable computational efforts, using the 6-31G* basis set.16 The MP2 activation energy is smaller than the B3LYP one; in fact, for the IDARs featured in the next subsections, the difference in activation energy between the two methods reaches up to 10 kcal/mol. The more accurate coupled cluster method with single and double excitations (CCSD)16 was applied to single-point calculations at the MP2- and B3LYP-optimized geometries. In both of the two cases, the CCSD relative energy ETS − ER is fairly close to the B3LYP activation energy. We thus decided that for these systems the MP2 severely underestimates the activation energy and that the DFT is more suitable. Then, in order to check the basis-set dependence of the B3LYP activation energy, we again executed B3LYP geometry optimization using the 6-311G*, 6-311+G*, cc-pVDZ, and cc-pVTZ basis sets.16 The resultant activation energies are compared in Table 2. As the size of the basis set increases, the value of ETS − ER decreases and almost converges at the cc-pVTZ basis set. Furthermore, we 2179



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Figure 1. Optimized geometries of the reactant, transition state, and product for the NKR in Scheme 1a at the B3LYP/cc-pVTZ level of theory. The values in parentheses are their relative energies in the unit of kcal/mol. The white, green, blue, red, and yellow balls represent hydrogen, carbon, nitrogen, oxygen, and sulfur atoms, respectively. The arrows in the transition state indicate the directions of the normal-mode vector of imaginary frequency.

Table 3. Experimentala and Theoreticalb Rate Constants, κ and k, for the NKR in Scheme 1b exptl condition microwave (SiC or Pyrex)

temp

time

product yield

297 °C

15 min 25 min

52% 73%


thermal heating or microwave irradiation to produce bridged bicyclic compounds.20 One of the IDARs is illustrated in Scheme 1c. The values of the experimental rate constant κ for Scheme 1c in the two conditions, evaluated from eq 7, are listed in Table 4. The IDAR driven by microwave irradiation was


8.2 × 10−4 s−1 1.1 × 10−3 s−1 8.7 × 10−4 s−1

Table 4. Experimentala and Theoreticalb Rate Constants, κ and k, for the IDAR in Scheme 1c

Experimental data are taken from ref 4. Experimental rate constant κ was determined by eq 7. bElectronic structure computations were done at the B3LYP/cc-pVTZ level of theory. Theoretical rate constant k was calculated from eq 2.

a

small and almost the same (3.3 and 3.5 debye, respectively). The calculated value of the theoretical constant k at T = 570 K (297 °C) is 1.1 × 10−3 s−1, which is almost equal to that of κ. As in Scheme 1a, the reaction rate predicted by the transition state theory agrees well with the data from the precisely controlled experiment. These theoretical results support from a theoretical viewpoint that the enhancement of NKRs is due to thermal heating effects alone. This conclusion should be further enforced by comparing experimental and theoretical results at different temperatures. Experiments performed at different temperatures are desirable. 3.2. Intramolecular Diels−Alder Reaction (IDAR) without a Competitive Process. The IDAR is a [4 + 2] cycloaddition between a conjugated diene and a dienophile within a single molecule to form a cyclohexene-like structure. Microwave technology has been extensively applied to enhance or control IDARs.19 Starting from benzoic acid, Mihovilovic et al. prepared several kinds of precursors, each of which is a functionalized 1,3-cyclohexadiene derivative with a vinylterminated chain, and then conducted their IDARs under

exptl condition

temp

time

product yield

toluene reflux microwave

110 °C 135 °C

3 days 200 min

93% 94%



1.0 × 10−5 s−1 1.4 × 10−6 s−1 2.3 × 10−4 s−1 1.3 × 10−5 s−1

a

Experimental data are taken from ref 20. The temperature for reflux is the boiling point of toluene. Experimental rate constant κ was determined by eq 7. bElectronic structure computations were done at the B3LYP/cc-pVTZ level of theory. Theoretical rate constant k was calculated from eq 13.

about twenty times faster than that under toluene reflux. However, the temperatures for these two processing conditions were different. It is unclear whether the observed acceleration of this reaction stems from thermal or nonthermal microwave effects. Although the mechanisms of IDARs must be far more complex than in NKRs, we also apply the transition state theory to IDARs so as to gain at least qualitative insight into the microwave-acceleration mechanism of the reactions. In search of the theoretical rate constant k, we executed the geometry optimization and frequency analysis on the PES of this reaction in the same manner as in section 3.1. The resultant geometries are displayed in Figure 3. In addition to the reactant, transition state, and product, a local minimum corresponding to the intermediate structure between the reactant and transition state

Figure 2. Optimized geometries of the reactant, transition state, and product for the NKR in Scheme 1b at the B3LYP/cc-pVTZ level of theory. The values in parentheses are their relative energies in the unit of kcal/mol. The white, green, blue, red, and yellow balls represent hydrogen, carbon, nitrogen, oxygen, and sulfur atoms, respectively. The arrows in the transition state indicate the directions of the normal-mode vector of imaginary frequency. 2180



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Figure 3. Optimized geometries of the reactant, intermediate structure, transition state, and product for the IDAR in Scheme 1c at the B3LYP/ cc-pVTZ level of theory. The values in parentheses are their relative energies in the unit of kcal/mol. The white, green, and red balls represent hydrogen, carbon, and oxygen atoms, respectively. The arrows in the transition state indicate the directions of the normal-mode vector of imaginary frequency.

Finally, the rate constant k for a two-step intramolecular reaction is of the form

was found on the PES. The major constituent in the imaginary frequency mode of the transition state is the bond formation between the cyclohexadiene ring and vinyl group. The energies of the intermediate structure, transition state, and product are 1.5, 27.5, and −24.5 kcal/mol, respectively. The presence of the intermediate local minimum makes it inappropriate to use eq 2 for the IDAR in Scheme 1c, which is not a single-step reaction. Instead of eq 2, we derive the rate constant that describes the kinetics of two-step consecutive intramolecular reactions. The chemical equation for such a reaction can be described by the two-step model k1

⎛ E − ER ⎞ k T Z rTSZ vTS k= B exp⎜ − TS ⎟ R R h Zr Z v kBT ⎠ ⎝

In contrast to eq 11, the partition functions for the reactant, not intermediate structure, arise in eq 13, and the effective activation energy for the whole reaction process is the energy of the transition state relative to that of the reactant, ETS − ER. It should be noted that although eq 13 appears to be identical to eq 2 for single-step intramolecular reactions, the reactant and transition state in this case are not directly connected: the transition state is not of the lower-barrier equilibrium R ⇄ I but of the ratedetermining step I → P. Now, let us calculate the theoretical rate constant k for the IDAR in Scheme 1c from the extended rate formula of eq 13. The effective activation energy ETS − ER = 27.5 kcal/mol, symmetry numbers σR = σTS = 1, rotational constants {BiR} and {BiTS}, and harmonic frequencies {νjR} and {νjTS} were inserted into eq 13. The calculated values of k for the IDAR in a reflux or microwave condition are shown in Table 4. They are smaller than the respective values of the experimental constant κ by about 1 order of magnitude. The transition state theory failed to give the absolute rates of this reaction that coincide with the experimentally observed ones. In the meantime, as the temperature rises from 110 to 135 °C, the theoretical constant k increases about 9-fold, while the experimental one κ becomes 23-fold. What is the possible reason for this discrepancy? In a long-time experiment, the reaction may be saturated before the product yield is measured. In that case, as noted above, the actual value of κ should be larger than that in Table 4; this holds true especially for the experiment under toluene reflux that lasted for three days. Hence, in reality, the increasing ratio of κ is expected to be less than 23 and closer to that of k, which takes into account only temperature factors. In this sense, the transition state theory accounts for the increase in the reaction rate of the IDAR in Scheme 1c by microwave irradiation, but there remains room for the existence of nonthermal contribution to this reaction and further thorough investigations are needed. 3.3. Intramolecular Diels−Alder Reaction (IDAR) with a Competitive Process. We turn our attention toward another interesting finding on IDARs reported in ref 20. The reactant in Scheme 1d appears similar to that in Scheme 1c but possesses an acryloyl group instead of an allyl one. The behavior of this molecule under thermal heating or microwave irradiation is totally different from that of the reactant in Scheme 1c. As shown in Table 5, the IDAR in Scheme 1d competes with polymerization. In a traditional reflux condition, polymerization completely overwhelms the IDAR. For the purpose of efficiently constructing a polymer out of its monomer units, a

k2

R XooooY I → P k−1

(8)

where the symbols R, I, and P stand for the reactant, intermediate structure, and product, respectively, and k1, k−1, and k2 are the rate constants for individual elementary reaction processes in eq 8. Here, we resort to the steady-state approximation: since the barrier between R and I is only a few kcal/mol, which is much lower than that between I and P, i.e., k−1 ≫ k2, the forward and backward reactions R ⇄ I can be considered to be in equilibrium above room temperature. The equilibrium constant K is defined from the law of mass action by K=

k1 [I] = k−1 [R]

(9)

where [I] denotes the concentration of the intermediate structure. From eq 9, the reaction rate of the rate-determining step I → P is d[P] = k2[I] = k2K[R] dt

(10)

Accordingly, the apparent rate constant is given by k = k2K. In analogy to eq 2, the analytical expression for k2 is ⎛ E − EI ⎞ k T Z rTSZ vTS k2 = B exp⎜ − TS ⎟ I I h kBT ⎠ ⎝ Zr Z v

(11)

where ZrI and ZvI are the rotational and vibrational partition functions for the intermediate structure, respectively, and EI is its energy. As seen in eq 11, the activation energy for the ratedetermining step is the energy of the transition state relative to that of the intermediate structure, ETS − EI. The equilibrium constant K is equivalent to the ratio of partition functions multiplied by the Boltzmann factor, that is, K=

⎛ EI − ER ⎞ exp ⎜− ⎟ kBT ⎠ ⎝ Z rR Z vR

(13)

Z rIZ vI

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Table 5. Experimentala and Theoreticalb Rate Constants, κ and k, for the IDAR in Scheme 1d exptl condition

temp

time

product yield



toluene reflux microwave

110 °C 210 °C

7 days 500 min

polymerization 32%

not determined 1.3 × 10−5 s−1

1.2 × 10−10 s−1 1.6 × 10−6 s−1

a Experimental data are taken from ref 20. The temperature for reflux is the boiling point of toluene. Experimental rate constant κ was determined by eq 7. bElectronic structure computations were done at the B3LYP/cc-pVTZ level of theory. Theoretical rate constant k was calculated from eq 13.

Figure 4. Optimized geometries of the reactant, intermediate structure, transition state, and product for the IDAR in Scheme 1d at the B3LYP/ cc-pVTZ level of theory. The values in parentheses are their relative energies in the unit of kcal/mol. The white, green, and red balls represent hydrogen, carbon, and oxygen atoms, respectively. The arrows in the transition state indicate the directions of the normal-mode vector of imaginary frequency.

polymerization initiator that decomposes into free radicals is usually added to the system. Yet, no initiator was used in the experiments in ref 20, which were not intended to synthesize a polymer. We conceive that spontaneous generation of free radicals, which is quite rare, triggered polymerization and that this may account for such a long reaction time (seven days). On the contrary, microwaves promoted the IDAR: the desired product was isolated in a moderate yield, and the reaction time was drastically reduced (500 min). Is the transition state theory capable of clarifying why the IDAR in Scheme 1d is facilitated by microwaves over a competitive polymerization process? Although the experimental rate constant κ cannot be determined for the IDAR under toluene reflux, it is still possible to work out the theoretical one k from eq 13. The optimized geometries of the reactant, intermediate structure, transition state, and product for the IDAR are exhibited in Figure 4. As expected, except for the acryloyl group, they are analogous to those in Figure 3 as well as the imaginary frequency mode of the transition state. The energies of the intermediate structure, transition state, and product are 1.0, 35.2, and −15.2 kcal/mol, respectively. The relative energies ETS − ER and EP − ER are much higher than the respective energies in Scheme 1c, which is mainly due to the deconjugation between the vinyl and carbonyl double bonds in the regions around the transition state or product. The rather high effective activation energy led to an extremely small value of k at the temperature for reflux, T = 383 K (110 °C), as shown in Table 5. The obtained value of k = 1.2 × 10−10 s−1 gives only a small product yield of 0.007% for seven days. However, the theoretical constant k increases more than 104-fold as the temperature reaches T = 483 K (210 °C) by microwave irradiation. The experimental product yield of 32% corresponds to κ = 1.3 × 10−5 s−1, which is more or less consistent with the theoretical value of k = 1.6 × 10−6 s−1. The temperature elevation of as much as 100 °C dramatically speeds up the IDAR. In contrast, the temperature dependence of the polymerization rate is presumed to be small since the ratedetermining step of polymerization is a radical addition to the vinyl double bond. Hence, we speculate that in the lower temperature range polymerization is much faster than the IDAR, while at higher temperature this relationship is reversed. The transition state theory offers a reasonable qualitative explanation for the marked promotion of this reaction over polymerization,

although a conclusive decision cannot be made without detailed experimental data such as polymerization rate. 3.4. Nonthermal Microwave Effects on Intramolecular Reactions. As demonstrated above, the observations in microwave-assisted experiments of NKRs can be interpreted by bulk temperature effects. Nevertheless, we cannot yet rule out the possibility of nonthermal microwave effects on other intramolecular reactions such as IDARs. It is meaningful to develop a general discussion concerning which molecular structure is most likely to be sensitive to microwave fields. The electric dipole interaction between a molecule and the electric field of microwaves, ε(t), is represented by −μ(x)ε(t), where x denotes a set of nuclear coordinates of the molecule and μ(x) is the molecular electric dipole moment as a function of x. The effective PES for the internal nuclear degrees of freedom of the molecule varies in time by this interaction. The deviation of this PES from the field-free one in the electronic ground state depends on the geometry of the molecule. In the vicinity of the reactant or product, the field-free PES can be approximated by a quadratic function of x that opens upward (vertex down). In this region, the molecule is stable, and its motion is well confined even in the presence of a microwave field ε(t); the trajectory is trapped in the potential well without heating. On the contrary, the field-free PES around the transition state opens downward (vertex up), which means that a subtle change in the PES can readily push down the molecule from its vertex. The deformation of the PES may temporarily lower the potential barrier; what is more, a low microwave frequency (several GHz for use in chemistry) allows the PES to be deformed for a long time. These may help the molecule approaching the transition state to cross over the potential barrier. Therefore, if microwave fields were to have a nonthermal effect on intramolecular reactions by distorting the PES, it should be most pronounced when a molecule is near the transition state. In the first place, a molecule needs to be thermally excited to reach the transition state. However, the magnitude of the electric dipole interaction is on the order of 10−6 kcal/mol for a typical microwave power (∼10−1 W/cm2). After all, out of the possible nonthermal microwave effects, that arising from the deformation of the PES is minor in intramolecular reactions. 2182



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4. CONCLUSIONS We have theoretically examined whether the origin of the enhancement/acceleration of intramolecular reactions observed in microwave-assisted experiments is thermal or nonthermal microwave effects. Our approach to accomplish this objective is to quantify the thermal effects using the transition state theory. In this article, calculations have been carried out for the cases of Newman−Kwart rearrangements (NKRs) and intramolecular Diels−Alder reactions (IDARs). The rate constants numerically calculated for NKRs show good agreement with those determined from previously reported experimental data, and thus, no substantial nonthermal effect of microwaves can be identified on NKRs. The theoretical estimation of the rate constants for IDARs is more difficult than in the cases of NKRs: the reaction paths in IDARs from the reactant to the transition state go via intermediate geometrical isomers. The rate constants theoretically obtained for the twostep model are smaller than the experimental values, but the temperature dependence of the rate constant can be reproduced in consideration of the saturation of the reaction. The overwhelming acceleration of IDARs over polymerization under microwave irradiation can also be understood qualitatively in terms of temperature factors: The rate of IDARs exceeds that of polymerization as the solvent temperature is raised by microwave heating. At this point, thermal effects are likely to be dominant in IDARs; however, the existence of nonthermal effects cannot be completely excluded without quantitative experimental data such as relevant activation energies. Microwave fields may in general exert nonthermal effects on molecules around a transition state by distorting the PES, but such effects are negligible in the intramolecular reactions we investigated. In conclusion, we proposed a theoretical approach to quantify nonthermal microwave effects: the transition state theory, which is feasible in practice (at least for single-step intramolecular reactions), can quantify the thermal effects on the reaction rate and the deviation from this theory is attributed to nonthermal effects. The application of the transition state theory is not only limited to organic systems but also suitable to nonthermal microwave phenomena in other research fields21 such as inorganic chemistry and material science. This approach should be combined with the results obtained by the experimental technique recently developed by Kappe et al.4 in which nonthermal effects can be separated from thermal ones by controlling microwave penetration through a reaction vessel. The next step is to improve the quantitative performance of the present approach by using QM/MM methods22 and extend our survey to intermolecular reactions.23 The combination of the experimental and theoretical procedures is promising to reveal the role of microwaves in chemical reactions more in detail.

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Article

ACKNOWLEDGMENTS We appreciate valuable discussions with Professors T. Komeda, H. Oikawa, K. Maruyama, Y. Zempo, N. Yoshikawa, and S. Ohuchi. This work was supported by a Grant-in-Aid for Scientific Research on Priority Area No. 465 (Grant No. 18070005 “Theoretical and Molecular Dynamics Studies of Microwave Heating Mechanisms of Magnetic and Dielectric Materials”) from the Japan Ministry of Education, Science, and Culture.

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*E-mail: [email protected]. Present Address §

Center for Frontier Science and Engineering, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182−8585, Japan. Notes

The authors declare no competing financial interest. 2183


Theoretical Verification of Nonthermal Microwave Effects on

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