J. Phys. Chem. A 2010, 114, 5113–5118
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Whence the Energy Term of the Rate Constant? Andreas A. Zavitsas* Department of Chemistry and Biochemistry, Long Island UniVersity, UniVersity Plaza, Brooklyn, New York 11201 ReceiVed: February 6, 2010; ReVised Manuscript ReceiVed: March 12, 2010
Insights into the causes of energy barriers to reactions are obtained from our E* model for hydrogen abstractions. Results of the model are in good agreement with all eight experimentally studied symmetrical reactions of the type X-H + •X f X• + H-X. The E* energy is broken down into three components: energy needed to overcome triplet repulsion between the terminal Xs, energy needed to bring the H-X bonds to their transition state distances, and energy gain by the resonance stabilization of delocalization of the odd electron over three atoms. The strength of the X-H bond is a minor factor. The conclusion that triplet repulsion is a major factor is supported by the London equation and by results of recent high level theoretical calculations. The E* model requires inputs of bond dissociation energies, bond lengths, and infrared stretching frequencies of X-H and X-X. Some reported failures of E* are shown to have been caused by use of input values subsequently found to be incorrect. Intoduction Much is currently understood about rates of chemical reactions, regarding the pre-exponential term, A, of the Arrhenius expression of the rate constant for bimolecular reactions. The A factor can be decomposed into two terms, Z and p. Z denotes the collision frequency, obtained from gas collision frequency considerations, and p denotes the probability that the collision will have the proper geometry for reaction. In terms of Eyring’s formulation of the rate constant, the term not involving energy, the entropy of activation ∆S+, is understood as the change of the degree of order in going from reactants to the transition state (TS), calculated via partition functions, etc. The more ordered is the TS, the smaller is the rate constant, thus “tight” or “loose” transition states are postulated. The same level of intuitive understanding cannot be said to exist for the energy term of the rate constant. It is usually described as sufficient energy to break and form bonds or, more vaguely, as the energy needed for reaction to occur. Where does this energy go? Is bond making weaker than bond breaking at the TS? High level ab initio calculations or parametric empirical approaches for estimating the energy barrier generally simply provide a value for the energy but do not indicate where it goes. In general, all that is usually known is that, if any bond is being broken, an energy of activation is required, even for very exothermic reactions. Hydrogen atom transfer (HAT) radical reactions formally involve only three electrons, and they appear as the simplest reaction type to investigate causes of chemical reactivity. Unlike ion-molecule reactions, Arrhenius energies of activation, Ea, of HAT reactions are not affected greatly by solvent effects, and gas phase reactions often have about the same Ea as those in solution. Major effects that solvents can have on rates of ion-molecule reactions are eliminated. The simple HAT reactions provide insights into chemical reactivity as a function only of structure of the molecules involved, without significant solvent effects on the reacting neutral species. * To whom correspondence should be addressed. E-mail: zavitsas@ liu.edu.
A significant advance in obtaining an intuitive understanding of what causes the energy of activation was provided by the work of Isborn et al. for symmetrical reactions, which eliminate the effect or reaction enthalpy.1 For symmetrical hydrogen abstractions, X-H + •X f X• + H-X, the energy contributions to the barrier were broken down into five components: (1) Distorting X in X-H to its transition state (TS) geometry; (2) Breaking the X-H bond; (3) Distorting X• to its TS geometry; (4) Bringing the triplet-coupled X• radicals to their TS geometry; and (5) Adding the hydrogen atom with singlet coupling to both X• radicals to form the TS. Major conclusions were that the strength of the X-H bond is a small contributor to the energy barrier, so that other factors such ligand distortion and triplet repulsion can outweigh it. Depending on the nature of X, triplet repulsion was found to be is a significant factor, contributing between 4 and 14 kcal mol-1 to the barrier. Step (2) is somewhat counterbalanced by step (5), and the net effect of the two terms can contribute between +3 and -8 kcal mol-1 to the barrier. We have proposed a semiempirical model for estimating energies of activation in hydrogen atom transfer reactions.2 The model makes estimates, E*, of energies of activation based on the following information about reaction 1: For molecules X-H, H-Y and X-Y, bond dissociation energies (BDE); bond lengths (re), uncoupled stretching frequencies (ν); and masses of the atoms in groups X and Y that are bonded directly to the transferred H.
X - H + •Y f X• + H - Y
(1)
Potential energy Morse curves of reactant and product are used to calculate bonding as the X-H bond is being stretched and the H-Y bond is being made, with the equibonding assumption, that is, BDE[X · · · H] ) BDE[H · · · Y] at the TS. For simultaneous bonding of H to both X and Y at the TS, the canonical forms for placing the three electrons on three atoms must have the electron spins up-down-up, or down-up-down, so as to bring about simultaneous bonding of H to X and to Y. Repulsion arises from canonical form III of the TS, that is, triplet repulsion or antibonding between the terminal groups.
10.1021/jp101183w 2010 American Chemical Society Published on Web 03/31/2010
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Antibonding in X-Y is calculated by the Sato anti-Morse curve. The reliability of E* estimates of energies of activation has been demonstrated with over 120 experimentally studied reactions.2 The success of the E* model with so many experimental determinations lends credence to conclusions that can be drawn from the model: (a) The equibonding assumption constitutes proof of the Hammond postulate to the effect that the molecular geometries of the TS are similar to those of reactants for very exothermic reactions and vice versa. (b) In symmetrical reactions of the type X-H + •X f X• + H-X, the strength of the bonds being broken and made have only a minor effect on the energy of activation. (c) The energy of activation is needed primarily, but not exclusively, to overcome triplet repulsion between the terminal groups X and Y. Thus, reactions involving weak BDE[X-Y] (a molecule never formally present) have lower energies of activation and vice versa, because antibonding energies decrease with decreasing BDE[X-Y]. (d) The energy of activation is also affected significantly by the infrared stretching frequency of X-Y; the higher is the frequency, the smaller is the triplet repulsion and the energy of activation. (e) Increasing the energy barrier is the energy needed to bring the equally bonded X-H and H-Y to their transition state distances. (f) Significantly reducing the energy barrier is the resonance energy of delocalizing the odd electron over three atoms at the TS, reducing the barrier by 10.6-12 kcal mol-1, which is similar to the resonance stabilization energy of the allyl radical. In symmetrical reactions, the bond being broken must have the same strength as the bond being made.2c Despite the large number of successful applications of the E* method, its validity and/or accuracy has been questioned. Isborn et al. reported high level (UCCSD(T)/cc-pVTZ, MPW1K, and CBS-QB3) calculations of the activation enthalpies and potential energy barriers for six symmetrical hydrogen transfer reactions, X-H + •X f X• + H-X, with X ) CH3, NH2, OH, F, OOH, and ONH2.1 No particular reason was provided for selecting these reactions. The sum of the five steps of the theoretical calculations yield Etotal (the potential energy barrier), with step (4) being equivalent to the triplet repulsion term of E*. The theoretical results were compared to those obtained by three previously proposed semiempirical or empirical methods: the E* method,2c the empirical method of Roberts and Steel,3 and the “breathing orbital valence bond theory” (BOVB) of Shaik et al.4 All three methods were found to give poor agreement with theory for some of the reactions treated. Particularly bad performance of the E* model was found for HOO-H + •OOH, where the E* calculation was reported as giving the unreasonable value of -10 kcal mol-1 for the energy of activation. Experimental energies of activation, Ea, are available for only two of the six reactions chosen for this theoretical study, even though experimental values also exist for the following symmetrical reactions: H-H + •D, Cl-H + • Cl′, RCH2S-H + •SCH2R′, RO-H + •OR′, CH3CH2-H + • CH2CR′, and ArCH2-H + •CH2Ar′. Hence, a complete set of comparisons of the ab initio results and of E* was not made for all available experimental measurements of Ea for symmetrical reactions. Roberts and Steel3 questioned the strong dependence of the calculated E* on the antibonding energy of X-Y of eq 1, the molecule never present. The reason given was that an empirical parametric scheme they proposed for estimating energies of
Zavitsas activation did not require a term for antibonding (triplet repulsion) between X and Y. Shaik et al. also challenged the importance of such triplet repulsion on the basis of BOVB for estimating energies of activation for HAT reactions. Their concluding statement5 regarding the E* model (referred to as the Z-model) is “In conclusion, the VB model is a physically correct model. In contrast, the Z-model ignores the true origins of the barrier. Even if its quantitative performance appears good within a limited (albeit large) set of reactions, we prefer to use a correct model and continue to improve its quantitative aspects, which at the moment are still inaccurate.” It may not be exactly clear what is meant by “a limited (albeit large) set of reactions.” However, it is clear that the BOVB model is quantitatively inaccurate at the moment and can be improved in the future. The E* model which shows “good quantitative performance” obviously cannot be improved very much. The theoretical results of Isborn et al.1 demonstrated that triplet repulsion (antibonding) between the terminal groups is a significant factor in determining energies of activation, but also found that the triplet energy appears to be overestimated by the E* method. This is likely true, because the anti-Morse curve of Sato used by E* is qualitatively correct but not highly accurate. The same theoretical results support conclusions from the E* model to the effect that, in symmetrical reactions, the effect of the strength of the X-H bond is small enough so that other factors outweigh it.1 Thus, high level theoretical calculations, for the first time, support some of the major general conclusions of the E* model. Results and Discussion The issue of triplet repulsion is addressed first. In one of the earliest attempts to apply quantum mechanics to transition states of hydrogen abstraction (Scheme 1), London6 described the classical energy barrier by eq 2 in terms of Coulombic (A, B, C) and exchange (R, β, γ) energies, while specifying neglect of the “overlap integral” (resonance effects). SCHEME 1
Eclass ) A + B + C-{0.5[(β-R)2 + (γ-β)2 + (γ-R)2]}1/2
(2) In Scheme 1, (A + R), (B + β), and (C + γ), respectively, denote bonding energies at the TS between atoms 1 and 2, 2 and 3, and 1 and 3. (A - R), (B - β), and (C - γ), denote antibonding, respectively. For the TS of the symmetrical reaction XsHsX, A ) B and R ) β. Substitution of the equalities into eq 2 and algebraic simplification leads to Eclass ) (A + R) + (C - γ) + A. The term (C - γ) is the antibonding energy (triplet repulsion) between the terminal groups. Even though this was not understood when the E* model was first proposed,2a it is not surprising that London’s equation would have included, even though not explicitly, antibonding between the terminal groups. Claims against3-5 the importance of such repulsion must be weighed against London’s seminal and elegant eq 2, the success of the E* model with a large number of hydrogen abstractions, and the similar conclusion from the high level theoretical calculations of Isborn et al.1 In this work the focus is exclusively on symmetrical hydrogen abstractions, but application of the E* model is extended to all
Whence the Energy Term of the Rate Constant? TABLE 1: Symmetrical X-H + •X f X• + H-X: Calculated E* and Experimental Values of Energies of Activation (kcal mol-1) reaction H-H + D Cl-H + •Cl′ CH3-H + •CH3 CH3S-H + •SCH3 CH3O-H + •OCH3 HO-H + •OH CH3CH2-H + •CH2CH3 C6H5CH2-H + •CH2C6H5 •
E* 9.8 4.0 14.4 5.0 3.0 4.3 13.7 12.9
Ea (experimental) a
10.35, 9.90b 6.6,c 5.4,d 6.0e 13.5-14.5,c 14.65,f 14.51,g 13.0h 5.2i 2.6j 4.2 ( 0.5k 14.1,l 12.6,m 13.7n 13.4o
a Reference 7. b Reference 8. c Reference 9. Reference 11. f Reference 12. g Reference 13. i Reference 15. j Reference 16. k Reference 17. m Reference 19. n Reference 20. o Reference 21. e
d
Reference 10. Reference 14. l Reference 18.
h
symmetrical reactions for which experimental energies of activation are available, unlike the theoretical work.1 The symmetry removes any effects of reaction enthalpies on Ea and allows a clear evaluation to be made of other factors affecting it. In symmetrical reactions the strength of the bond being broken must be the same as that of the bond being made. This approach also provides a complete test of E* versus all experimentally available information. The results are shown in Table 1, using the latest available values of molecular properties required as input to the model. The input values used in this work for the E* calculation are provided in the Supporting Information (BDE at 298 K, re, and ν), with computational details and a copy of the program. Copies of the E* program are also available from the author on request. The Ea values for H-H + •D are the latest experimentally available7,8 and have been obtained at temperatures between 655 and 2100 K so that any possible tunneling effects should be minimal. For HO-H + H18O• f HO• + H-18OH at 300-420 K, the pre-exponential term is low, and the TS was characterized17 as “very tight,” consistent with our previous finding2c with the E* model that low triplet repulsions, such as caused by the weak X-X bond of HO-OH, allow close approach of the terminal groups to the H being transferred and lead to tight TS structures. Although ab initio calculations predict17 the existence of hydrogen-bonded complexes of the type [H2O · · · HO•], such complexes are not necessarily in the path of the reaction as often they are assumed to be. In the gas phase, this is particularly unlikely. It is more likely that the reality is [H2O · · · HO•] / HOH + •OH f HO• + H-OH and the theoretically calculated complex is not involved in the reaction, but may partially sequester •OH radicals from reacting. Consistent with this, the geometry of this complex has been found unfavorable for H atom abstraction.17 For all experimentally determined Ea of symmetrical reactions of Table 1, the E* model estimates energies of activation satisfactorily, keeping in mind that there are experimental uncertainties in both reported Ea and in the molecular values used in the E* model. No tunneling corrections are invoked. An example of support for conclusion (b) of the Introduction drawn from the E* model is available in Table 1 for the symmetrical reactions of methane, ethane, and toluene, which have similar energies of activation despite BDE values of the reacting bonds being 104.8, 100.5, and 89.6 kcal mol-1. An example of support of conclusion (c) is the reaction HO-H + · OH, where both experimental Ea and calculated E* are about 10 kcal mol-1 smaller than those for CH3-H + · CH3. This is due to the weak BDE[X-X]: BDE[HO-OH] ) 51.1 kcal mol-1
J. Phys. Chem. A, Vol. 114, No. 15, 2010 5115 versus BDE[CH3-CH3] ) 89.6. Conclusion (d) is supported in Table 1 by comparing H-H + •D to CH3-H + •CH3. The energy of activation of the former is about 4.6 kcal mol-1 smaller than the latter, even though the strengths of the X-H bonds involved are about the same, 104.2 and 104.8 kcal mol-1, respectively. The major difference between them is the stretching frequency of X-Y, ν[H-D] ) 3630 versus ν[CH3-CH3] ) 995 cm-1, leading to smaller triplet repulsion for the hydrogen reaction. Following the example of Isborn et al., this work also determines the components of the E* energy, a point that had not been addressed previously,2 by breaking it down into three parts. For H-H + •D, the energy needed to bring bonding to its TS distance is estimated as 3.9 kcal mol-1; the triplet repulsion, 16.6; and the resonance energy, -10.6. For 35Cl-H + 36Cl•, the corresponding values are: 3.1, 12.8, and -12.0. For CH3-H + •CH3: 5.1, 19.9, and -10.6. For CH3O-H + • OCH3: 1.8, 11.7, and -10.6. For CH3S-H + •SCH3: 3.5, 13.5, and -12.0. For HO-H + H18O•: 2.5, 12.4, and -10.6. For CH3CH2-H + •CH2CH3: 5.7, 18.7, and -10.6. For C6H5CH2-H + •CH2C6H5: 4.2, 19.3, and -10.6 kcal mol-1. Our very approximate correction for zero-point energies yields zero for perfectly symmetrical reactions and -0.3 kcal mol-1 for H-H + •D. The issue of the worst failure of E* ) -10 kcal mol-1 reported by Isborn et al.1 is addressed next for the case HOO-H + •OOH f HOO• + H-OOH. Their values of BDE, re, and ν for HOO-H used as input for their E* calculation conform reasonably to available experimental values. However, the values for the antibonding molecule, HOO-OOH, were all derived from theoretical calculations. BDE[HOO-OOH] ) 17.8 kcal mol-1 was taken from Mckay and Wright, obtained by G3(MP2) calculations at 0 K.22 We obtain 15.0 kcal mol-1 at 298 K by a full G3 calculation. For the bond length, Mckay and Wright report re[HOO-OOH] ) 1.430 Å, but re ) 1.374 Å was used instead,1 taken from previous CCSD(T) theoretical calculations of Fermann et al.23 The apparently unscaled harmonic ν[HOOOOH] ) 955 cm-1 was used for the central O-O stretching frequency as assigned by Fermann et al. in their Table 3, while a scaled harmonic value of ωe ) 880 cm-1 was reported in their Table 4. The above theoretical calculations of the polyoxides HOOH, HOOOH, and HOOOOH show an unusually peculiar, if not unique, pattern in that the weakest O-O bond has the shortest bond length and the highest stretching frequency. Isborn et al. used BDE[HOO-OOH] ) 17.8 kcal mol-1 and ν[HOO-OOH] ) 955 cm-1, whereas hydrogen peroxide has experimental BDE[HO-OH] ) 51.1 kcal mol-1 and ν ) 877 cm-1. The much weaker bond appears to have the higher force constant. It follows from the Morse equation that the equilibrium vibrational stretching frequency of a bond is proportional to the square root of the ratio of the force constant to the reduced mass: ωe ∝ (k/µ)1/2. Also, the force constant is proportional to the equilibrium bond dissociation energy: k ∝ De. Hence, ωe ∝ De via k, when the masses of the bonded atoms are the same. Hence, the observed pure stretching frequency is linearly related to the square root of the bond dissociation energy: ν ∝ (BDE)1/2. The stretching frequency of 955 cm-1 that was used1 had not been assigned on the basis of a normal-mode analysis. In the spectra of large molecules, it is often quite difficult to locate uncoupled stretching frequencies for use with the E* method, hence our frequent warnings2 about using coupled frequencies with it, that is, that are not fairly “pure” stretches. To ameliorate such difficulties we have introduced a procedure
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TABLE 2: BDE of O-O Bonds (kcal mol-1), Stretching Frequencies Estimated by ν ) 171(BDE)1/2 - 300, and Literature Values (cm-1) species and bond •
•
O-O (O2) HO-OH ClO-O• HO-O• CH3O-OH HO-OOH ClO-O• CH3O-OCH3 (CH3)3CO-OC(CH3)3 HOO-OOH
BDE (lit.) a
119.1 51.1a 59.67b 65.35d 45.2a 34.1g 60.45i 38.2a 38.9l 17.8n
ν (est.)
ν (lit.)
1566 921 1018 1082 850 699 1030 757 766 421
1556a 877a 1021c 1099e 863, 835f 799h 1031j 779k 771m (955?)n
a Reference 25. b Reference 26. c Reference 27. d Reference 28. Reference 29. f Reference 30. g Reference 31. h Reference 32. i Reference 33. j Reference 34. k Reference 35. l Reference 36. m Reference 37. n Reference 1. e
for estimating pure stretching frequencies by ν ) 171(BDE)1/2 - ci, where ci is characteristic of the directly bonded atoms.2c For O-O the value of ci is about 300 cm-1. The constant appears to be required because the Morse equation is accurate only at the equilibrium bond length, but it can fail quite badly and differently when the atoms involved are at the large distances of breaking the bond; also, the µ values are different for various atom combinations.24 The performance of this approximation with O-O bonds is shown in Table 2 for cases where BDE and uncoupled stretching frequencies ν are fairly well established experimentally. In no case does the approximation fail by more than 100 cm-1, except for the HOO-OOH case where it fails by 534 cm-1. As expected, the trend in Table 2 is for the weaker O-O bond to have a lower uncoupled ν, except for the case of HOO-OOH. It is conceivable that the higher stretching frequency of 955 cm-1 in HOO-OOH relative to the observed 877 cm-1 in HO-OH may be due to the shorter bond length of the former, 1.374 Å used by Isborn et al. versus 1.464 of the latter, despite the much weaker bond of HOO-OOH. Subsequent high level theoretical calculations (CCSD(T)/cc-pVTZ) of HOOH, HOOOH, and HOOOOH by Denis and Ornellas29 shed some light into these peculiar aspects. Relatively pure O-O stretching frequencies were calculated for HOOH as 912 cm-1 (BDE ) 51.1 kcal mol-1, re ) 1.4579 Å) and symmetrical and asymmetrical for HOOOH the calculated fundamentals are νa ) 784 and νs ) 879 for a properly weighted average32 of ν ) 833 cm-1 (BDE ) 34.1 kcal mol-1, re ) 1.4319 Å).29 The weaker O-O bond of HOOOH has a pure stretching frequency lower than that of HOOH, despite the calculated shorter bond length. For HOO-OOH the calculated29 bond length is again significantly longer than that used by Isborn et al., 1.4084 Å versus 1.374. Most importantly, the high level calculations of Denis and Ornellas find that vibrations involving the O-O bonds of the tetraoxide HOOOOH are highly coupled, and they carefully and explicitly refrain from making mode assignments for such bands at 479, 607, 629, 847, and 922 cm-1. The E* calculation is sensitive to the input molecular data of X-Y and, given the present uncertainties of the molecular properties of HOO-OOH for BDE, re and ν, an accurate answer cannot be expected of the E* model.38 Another example of similar, but less severe, problems is the Isborn et al. calculation of Etotal for H2N-H + •NH2,1 for which there are no measurements of Ea, but many of the molecular properties needed for the E* calculation are well established experimentally for the species involved in this reaction. The
following molecular properties were used1 as input to the E* model: BDE[H2N-H] ) 107.6 kcal mol-1 and BDE[H2N-NH2] ) 66.2, citing the NIST database.25 However, the current values in this database are 108.6 and 68.21 kcal mol-1, respectively. The infrared stretching frequency ν[H2N-NH2] ) 1077 cm-1 was used,1 citing infrared work by Gulaczyk et al.39 However, this value is questionable because it was derived from infrared studies where the symmetric N-N stretch would be either extremely weak or nonexistent (no change in dipole) and would fall in the region of NH2 wagging vibrations. Gigue`re and Liu have suggested ν[H2N-NH2] ) 960 cm-1, citing support from the known N-N stretching vibrations of methylhydrazine and of sym-dimethyhydrazine, 816 and 801 cm-1, respectively.40 Additional support for the value of 960 cm-1 was subsequently provided by the Raman spectrum of hydrazine-d4, where ν[D2N-ND2] ) 933 cm-1 is a clear, intense band in the vapor as reported by Durig et al.41 N,N-dimethylhydrazine also shows the N-N Raman band at 810 cm-1.42 With the more recent BDE values given above, ν[[H2N-NH2] ) 960 cm-1, and all other molecular properties as used by Isborn et al., E* is not 5.3, as was reported,1 but 7.8 kcal mol-1 versus the theoretical Etotal ) 11.4. An assumption seems to have been made by Isborn et al.1 that high level ab initio calculations yield the correct potential energy barrier (Etotal), to which E* may be approximately compared. However, for the only two symmetrical reactions treated for which experimental energies of activation are available, the reported theoretical results give values considerably higher: for CH3-H + •CH3, experimental Ea ≈ 14 kcal mol-1, calculated Etotal ) 18.1,1 and calculated E* ) 14.4; for HO-H + •OH, experimental Ea ) 4.2 ( 0.5 kcal mol-1, calculated Etotal ) 8.9,1 and calculated E* ) 4.3. The theoretical Etotal is greater than the experimental Ea by 4.1 and 4.7 kcal mol-1, respectively. The theoretical Etotal ) 11.4 reported for H2N-H + •NH2 is 3.6 kcal mol-1 higher than the E* results, which are in good agreement with all experimental measurements of Table 1. The usual rationalization for theoretical values being considerably higher than experimental Ea is to invoke extensive tunneling and this was also proposed to explain the high values of Etotal.1,43 Tunneling is likely when primary isotope effects, kH/kD, exceed about 8. Also, experimental evidence of tunneling generally is thought to be demonstrated when Arrhenius plots are curved. For the above two reactions, the experimental data show quite linear Arrhenius plots in the NIST chemical kinetics database.20 Hence, there is no clear experimental evidence for extensive tunneling. Even some extensively curved Arrhenius plots have been shown by the E* model not to be due to tunneling, but to changes with temperature of the BDEs involved.44 Success of high level calculations with stable molecules does not necessarily extend to transition states.45 Values of activation enthalpies, ∆H+, were also calculated by the CBS-QB3 method by Isborn et al.1 For CH3-H + •CH3 and HO-H + •OH, ∆H+ ) 15.8 and 7.7 kcal mol-1, respectively. Conversion to Ea requires 2RT to be added, where T is the mean temperature of the experimental range. Doing so leads to theoretical Ea ) 18.0 and 9.1 kcal mol-1, respectively.1 Again, the CBS-QB3 values of Ea are too high compared to experimental 14 and 4.2 kcal mol-1, respectively. Theoretical calculations of the HO-H + •OH reaction between 200 and 700 K have also been made by Masgrau et al.46 at various levels of theory, including CCD and CCSD using the 6-311G(3d, 2p) basis set. They also find high potential energy barriers of about 12 kcal mol-1 and postulate extensive tunneling to emulate the experimental results. Many types of
Whence the Energy Term of the Rate Constant? tunneling corrections have been proposed. Some examples are: the least action ground state method (LAG), the small-curvature semiclassical adiabatic ground state method (SCSAG), the minimum energy path SAG (MEPSAG), two large-curvature ground-state methods (LLG and LLG3), zero-curvature tunneling (ZVT), with and without corner-cutting, etc.47 Masgrau et al. used the canonical unified statistical theory (CUS) to obtain rate constants by direct multidimensional semiclassical dynamics. Rate constants and semiclassical transmission coefficients were obtained by canonical variational transition state theory (CVT). The CVT rate constants were then multiplied by a transmission coefficient obtained by the small-curvature tunneling (SCT) semiclassical adiabatic ground-state approximation, which is described as a multidimensional tunneling method. The corner-cutting aspect of the tunneling process was used, giving the greatest effect. Using such tunneling corrections, the barrier height of 11.7 kcal mol-1 was reduced to Ea ) 4.27 kcal mol-1 at 300-400 K, thus emulating experiment. The transmission coefficients used to do so increase the CVT rate constant by 2.5 × 106 at 200 K; they also increase the CVT rate constant for deuterium transfer at 200 K, DO-D + •OH f DO• + D-OH, by 3.4 × 104, an unprecedentedly large extent of deuterium tunneling. This remarkable result was explained in terms of the postulated importance of the theoretically calculated hydrogen bonded complex H2O · · · HO•. However, Hand et al.48 have pointed out that this theoretical global minimum complex would not lead to products, but another complex is relevant to the reaction, HOH · · · •OH. This complex is less stable than the global minimum by 3.1 kcal mol-1 and lies 2.1 kcal mol-1 below the separated HOH and •OH. The equilibrium constant for formation of complex HOH · · · •OH can be estimated in the gas phase at 300 K from ∆G ) ∆H - T∆S ) -2100 - (300 × -32) ) 7500 cal mol-1, where about -32 e.u. is the approximate entropy change for two particles joining. From ∆G ) -RT ln Ke, Ke is approximately 3.4 × 10-6 and hardly any of the reactants will be complexed. Hand et al. correct their QCISD(T)/6-311+G(3df,2p) barrier height of 10.1 kcal mol-1 by using the small-curvature semiclassical abiabatic ground state approach with corner-cutting for tunneling-corrected rate constants and obtain Ea ) 5.1 kcal mol-1 above the separated reactants at 300 K, thus emulating the experimental result fairly well.48 Hence, their complex will require an energy of activation of 5.1 + 2.1 ) 7.2 kcal mol-1 to lead to products, while the separated reactants require only 5.1, including a small zero point energy correction of 0.49 kcal mol-1. They suggest that the large-curvature method of tunneling correction to a QCISD(T)/ 6-311+G(3df,2p) potential energy surface might give a more accurate final Ea. It appears unlikely that the theoretical complex with a higher activation energy and at such low concentration will be a significant contributor to the gas phase reaction rate. The other complex proposed as important by Masgrau et al. at -6.6 kcal mol-1 relative to separated reactants will require 6.6 kcal mol-1 above their calculated Ea ) 4.3 to lead to products, which is unlikely with the equilibrium constant Ke ) 6.5 × 10-3 for its formation that can be similarly calculated from their value of ∆H for this complex. The simple model obtains E* ) 4.3 kcal mol-1, without invoking any complicating complexes between the reactants. The E* model assumes linear TS structures, whereas ab initio calculations find the [XsHsX]• angle to depend on the electronegativity of X.1 For example, the angle was found to decrease from 180° at the TS for [H3CsHsCH3]•, to 141° in [HOsHsOH]•, and to 133° in [FsHsF]•. Nevertheless, the E* model, with linear TS structures, estimates Ea accurately
J. Phys. Chem. A, Vol. 114, No. 15, 2010 5117 for the two reactions that have been studied experimentally (X ) H3C• and HO•; Table 1). For the HO-H + •OH reaction, with an OsHsO angle of 141° at the TS causing a closer approach of the terminal atoms, the E* model yields a triplet repulsion 4.8 kcal mol-1 greater than the linear TS. This increases the E* value to 4.3 + 4.8 ) 9.1 kcal mol-1 versus the reported Etotal of 8.9 of Isborn et al. with the same angle and Ea ) 9.1 by the CBS-QB3 calculation.1 The experimental Ea for F-H + •F is unlikely to ever be determined, because there are no isotopes of fluorine with sufficiently long halflives to perform the experiment; hence, this reaction will always be rife for speculation with various types of calculation. However, the reaction H2N-H + •ND2 is experimentally feasible, but not currently available, for testing results of calculations. The issue of hydrogen abstractions by the proton-coupled electron transfer (PCET) mechanism is not relevant to this work.49 The E* model was derived for HAT reactions. If a hydrogen atom is transferred by other mechanisms, the E* model is simply not applicable, but has been shown to predict accurately with over 120 simple hydrogen abstractions in the gas phase and in the nonpolar solvents commonly used for such radical reactions. The E* model is not without shortcomings. It can give slightly negative values for very exothermic reactions, and its zero-point energy correction is a gross approximation that does not reproduce primary deuterium isotope effects, kH/kD. The advantage of the E* model over many theoretical calculations is that it easily leads to conclusions (a-f) of the Introduction whereas ab initio calculations often do not lead to such insights. Also, if the molecular properties needed as input to the E* program are available, computational demands are minor for execution of the program relative to high level calculations; the E* program executes in a fraction of a second, and Occam’s Razor would seem to recommend it for such cases.50 Conclusions Estimates of energies of activation obtained by the E* model are quite accurate when compared to all available experimental values of Ea in symmetrical hydrogen abstractions. Major conclusions derived from the E* model regarding the importance of triplet repulsion between the terminal groups in HAT reactions have been confirmed by high level theoretical calculations, as has the conclusion that BDE[X-H] does not play an very important role in affecting the energy of activation of symmetrical reactions. The energy required for bringing to their TS geometries the bonds being broken and made is a significant, but not the major, factor. Acknowledgment. Support from the Intramural Research Program of Long Island University is gratefully acknowledged. Supporting Information Available: A complete set of molecular properties used to calculate the E* values of Table 1, with citations of their sources. Computational details and a copy of the source code of the E* program are also provided. This material is available free of charge via the Internet at http:// pubs.acs.org. References and Notes (1) Isborn, C.; Hrovat, D. A.; Borden, W. T.; Mayer, J. M.; Carpenter, B. K. J. Am. Chem. Soc. 2005, 127, 5794–5795. (2) (a) Zavitsas, A. A. J. Am. Chem. Soc. 1972, 94, 2779–2789. (b) Zavitsas, A. A.; Melikian, A. A. J. Am. Chem. Soc. 1975, 97, 2757–2763.
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(c) Zavitsas, A. A.; Chatgilialoglu, C. J. Am. Chem. Soc. 1995, 117, 10645– 10654 (E*, version 4). (3) Roberts, B. P.; Steel, A. J. J. Chem. Soc., Perkin 2 1994, 2157– 2162. (4) Shaik, S.; Wu, W.; Dong, K.; Song, L.; Hiberty, P. C. J. Phys. Chem. A 2001, 105, 8226–8235. (5) Shaik, S.; Wu, W.; Dong, K.; Song, L.; Hiberty, P. C. J. Phys. Chem. A 2002, 106, 5043–5045. (6) London, F. Z. Elektrochem. 1929, 35, 552. Heitler, W.; London, F. Z. Phys. Chem 1927, 44, 455. Wiberg, K. B. Physical Organic Chemistry; Wiley: New York, 1964. Laidler, K. J. Chemical Kinetics, 3rd ed.; Harper and Row: New York, 1987. (7) Michael, J. V.; Su, M. C.; Sutherland, J. W. J. Phys. Chem. A 2004, 108, 432–437, Between 1150 and 2100 K. (8) Michael, J. V.; Fisher, J. R. J. Phys. Chem. 1990, 94, 3318–3323, Between 655 and 1979 K. (9) Handbook of Bimolecular and Termolecular Gas Reactions, Kerr, J. A., Ed.; CRC Press: Boca Raton, FL, 1981. (10) Thommarson, R. L.; Berend, G. C. Int. J. Chem. Kin. 1973, 5, 629– 642. (11) Kneba, M.; Wolfrum, J. J. Phys. Chem. 1979, 83, 69–73. Based on the measurements with radioactive 36Cl of Klein, F. S.; Persky, A.; Weston, R. E. J. Chem. Phys. 1964, 41, 1799–1807. (12) Dainton, F. S.; Ivin, K. J.; Wilkinson, F. Trans. Faraday Soc. 1959, 55, 929–936, Using 14C tracer. (13) Arthur, N. L.; Bell, T. N. ReV. Chem. Intermed. 1978, 2, 37–74. (14) Chen, C.-J.; Back, M. H.; Back, R. H. Can. J. Chem. 1977, 55, 1624–1628, CH4 + •CD3 f •CH3 + HCD3. (15) Alnajjar, M. S.; Garrossian, M. S.; Autrey, S. T.; Ferris, K. F.; Franz, J. A. J. Phys. Chem. 1992, 96, 7037–7043. The experimental energy of activation pertains to hydrogen transfer in CH3(CH2)4CH2S-H + • SCH2(CH2)6CH3. (16) Griller, D.; Ingold, K. U. J. Am. Chem. Soc. 1974, 96, 630–631. The experimental energy of activation pertains to hydrogen transfer in RO-H + •OR′: (CH3)3CC(CH3)2CO-H + •OC(CH3)3. (17) Dubey, M. K.; Mohrschladt, R.; Donahue, N. M.; Anderson, J. G. J. Phys. Chem. A 1997, 101, 1494–1500. The tracer is H18O• and the experimental temperature range is 300-420 K. (18) Trotman-Dickenson, A. F.; Milne, G. S. Natl. Stand. Ref. Data Ser. (U.S. Natl. Bur. Stand.); 1967;. No. 9. Ea pertains to abstraction of 1° hydrogen of alkanes by ethyl radical. (19) Kerr, J. A. In Free Radicals; Kochi, J. K., Ed.; Wiley-Inrescience: New York, 1973; Vol. II. Ea pertains to abstraction of 1° hydrogen of alkanes by ethyl radical. (20) Review of Tsang, W. J. Phys. Chem. Ref. Data 1988, 17, 887. Ea cited in NIST Chemical Kinetics Database, Standard Reference Database 17, Version 7.0 (Web Version), Release 1.4.3. Data Version 2009.01 at http://kinetics.nist.gov/kinetics/. Ea pertains to CH3CH2CH2-H + •CH2CH3 f CH3CH2CH2• + H-CH2CH3. (21) Franz, J. A.; Alnajjar, M. S.; Barrows, R. D.; Kaisaki, D. L.; Camaioni, D. M.; Suleman, N. K. J. Org. Chem. 1986, 51, 1446–1456. The reported Ea pertains to ortho-CH2dCHCH2C6H4CH2• + para-xylene f ortho-CH2dCHCH2C6H4CH2-H + •CH2C6H4-p-CH3, i.e., benzyl hydrogen exchange. Reaction of meta-xylene has the same Ea. Unimolecular cyclization of ortho-CH2dCHCH2C6H4CH2• is the reference clock reaction used. (22) Mckay, D. J.; Wright, J. S. J. Am. Chem. Soc. 1998, 120, 1003– 1013. The total G2(MP2) energy at 0 K is obtained through the equation E(G2(MP2)) ) QCISD(T)/6-311G(d,p) + MP2/6-311+G(3df,2p)-MP2/ 6-311G(d,p) + ZPE + HLC, where ZPE ) zero-point energy and HLC ) high-level correction. (23) Fermann, J. Y.; Hoffman, B. C.; Tschumper, G. S.; Schaefer, H. F. J. Chem. Phys. 1997, 106, 5102–5108. re by TZ2P+CCSD(T)//TZ2p CISD calculation. (24) Zavitsas, A. A. J. Am. Chem. Soc. 1991, 113, 4755–4767.
Zavitsas (25) NIST Chemistry Webbook, NIST Standard Reference Database Number 69, Linstrom, P. J., Ed.; National Institute of Standards and Technology(http://webbook.nist.gov/chemistry/). (26) Karton, A.; Parthiban, S.; Martin, J. M. L. J. Phys. Chem. A 2009, 113, 4802–4816. (27) From ref 25. Obtained as the weighted average of reported symmetric and asymmetric vibrations, ns and na, by n ) [(ns2 + na2)/2]1/2. (28) From ref 25 and DfH°[HOO•] ) 3.5 kcal mol-1 from Matsunaga, N.; Rogers, D. W.; Zavitsas, A. A. J. Org. Chem. 2003, 68, 3158–3172. (29) Denis, P. A.; Ornellas, F. R. J. Phys. Chem. A 2009, 113, 499– 506. (30) From the NIST Standard Reference Database 101, http://cccbdb. nist.gov/. CCD/cc-pVDZ, 863 cm-1 (scaled); QCISD/cc-pVDZ, 835 cm-1 (scaled). (31) With DfH°[HOOOH] )-21.3 kcal mol-1 from ref 28, DfH°[HO•] ) 9.3 from ref 25, and DfH°[HOO•] ) 3.5 from ref 28. (32) Obtained as the weighted average of ns and na by n ) [(ns2 + na2)/2]1/2 of the experimental values of 776 and 821 cm-1 quoted in ref 29. (33) With DfH°[ClOO•] ) 23.3 kcal mol-1 from the CRC Handbook of Chemistry and Physics, 77th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1996-1997. DfH°[ClO•] ) 24.2 and DfH°[O] ) 59.55 kcal mol-1 from ref 25. (34) From ref 30. By CCD/cc-pVDZ. The animated vibrations entry shows the O-O stretch at 1077 cm-1 (unscaled) and 1031 cm-1 (scaled). (35) Della Vedova, C. O.; Mack, H. G. J. Mol. Struct. 1992, 274, 25– 32. (36) Reints, W.; Pratt, D. A.; Korth, H.-G.; Mulder, P. J. Phys. Chem. A 2000, 104, 10713–10720. (37) Bellamy, L. J. The Infrared Spectra of Complex Molecules, 3rd ed.; Chapman and Hall: London, 1975, p 137. (38) The E* model is not immune to the GIGO principle. (39) Gulaczyk, I.; Kreˆglewski, M.; Valentin, A. J. Mol. Spectrosc. 2003, 220, 132–136. (40) Gigue`re, P. A.; Liu, I. D. J. Chem. Phys. 1952, 20, 136–140. (41) Durig, J. R.; Bush, S. F.; Mercer, E. E. J. Chem. Phys. 1966, 44, 4238–4247. (42) National Institute of Advanced Industrial Science and Technology (AIST), Japan; at http://riodb01.ibase.aist.go.jp/sdbs/cgi-bin/cre_index. cgi?lang)eng. (43) Supporting information of ref 1 and other similar rationalizations of tunneling cited therein. (44) Zavitsas, A. A. J. Am. Chem. Soc. 1998, 120, 6578–6586. (45) Donahue, N. M. J. Phys. Chem. A 2001, 105, 1489–1497. (46) Masgrau, L.; Gonza´lez-Lafont, A.; Lluch, J. M. J. Phys. Chem. A 1999, 103, 1044–1053. (47) Steckler, R.; Dykema, K. J.; Brown, F. B.; Hancock, G. C.; Truhlar, D. G. J. Chem. Phys. 1987, 87, 7024–7035. Joseph, T.; Steckler, R.; Truhlar, D. G. J. Chem. Phys. 1987, 87, 7036–7049. Baldridge, K. K.; Gordon, M. S.; Steckler, R.; Truhlar, D. G. J. Phys. Chem. 1989, 93, 5107–5119. Lu, D. H.; Maurice, D.; Truhlar, D. G. J. Am. Chem. Soc. 1990, 112, 6206–6214. Truhlar, D. G.; Gordon, M. S. Science 1990, 249, 491–498. Zheng, J.; Truhlar, D. G. J. Phys. Chem. A 2009, 113, 11919–11925. Chuang, Y.-Y.; Truhlar, D. G. J. Phys. Chem. A 1997, 101, 3808–3814. Ferna´ndez-Ramos, A.; Truhlar, D. G. J. Chem. Theory Comput. 2005, 1, 1063–1078. (h) Tischenko, O.; Truhlar, D. G. J. Phys. Chem. A 2006, 110, 13530–13536. (48) Hand, M. R.; Rodriquez, C. F.; Williams, I. H.; Balint-Kurti, G. G. J. Phys. Chem. A 1998, 102, 5958–5966. (49) For some leading references see: Meyer, T. J.; Huynh, M. H. V.; Thorp, H. H. Angew. Chem., Int. Ed. 2007, 46, 5284–5304. Irebo, T.; Reece, S. Y.; Sjo¨din, M.; Nocera, D. G.; Hammarstro¨m, L. J. Am. Chem. Soc. 2007, 129, 15462–15464. Edwards, S. J.; Soudackov, A. V.; HammesSchiffer, S. J. Phys. Chem. B 2009, 113, 14545–14548. (50) Frustra fit per plura quod potest fieri per pauciora (It is futile to do with more what can be done with less). William of Occam (ca. 12861347). Quoted by Hoffmann, R.; Minkin, V. I.; Carpenter, B. K. Int. J. Phil. Chem. 1997, 3, 3–28.
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