Benchmark Ab Initio Calculations of the Barrier Height and Transition

Sep 19, 2012 - Oksana Tishchenko* and Donald G. Truhlar* .... Evgeny Pliss , Viacheslav Machtin , Roman Pliss , Andrey Sirik , Denis Loshadkin , Alexa...
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Benchmark Ab Initio Calculations of the Barrier Height and Transition-State Geometry for Hydrogen Abstraction from a Phenolic Antioxidant by a Peroxy Radical and Its Use to Assess the Performance of Density Functionals Oksana Tishchenko* and Donald G. Truhlar* Department of Chemistry and Supercomputing Institute, University of Minnesota, 207 Pleasant Street Southeast, Minneapolis, Minnesota 55455-0431, United States S Supporting Information *

ABSTRACT: This Letter presents benchmark results for the transitionstate geometry and classical barrier height for the hydrogen atom abstraction from phenol by an organic peroxyl radical. We use multireference Møller−Plesset perturbation theory (MRMP2) based on a complete active space self-consistent field (CASSCF) wave function with a previously defined well-balanced nom-CPO+π active space and a triple-ζ one-electron basis set for the benchmark calculations, including full geometry optimization of the saddle point at the MRMP2 level. The classical barrier height for the abstraction reaction by the methylperoxyl radical is found to be 7.4 kcal/mol. A variety of density functionals are tested for their ability to reproduce the benchmark calculations for this reaction to provide guidance for selecting a reliable density functional in future calculations of larger systems involving phenolic antioxidants. The best-performing density functional is M05, and other functionals with above-average performance (for both transition-state geometry and transition-state barrier height) are B1LYP, B3LYP, B98, B971, and M06. SECTION: Molecular Structure, Quantum Chemistry, and General Theory

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phenolic antioxidants. Therefore, DFT reliability assessments for reactions of predominately single-reference character21 are not applicable to multireference reactions. Despite considerable interest, accurate barrier heights and saddle point structures for these, as for the majority of other multireference reactions, remain unknown because the optimization of transition-state structures for complex systems by multireference methods is in the category of grand challenges. The present study provides the most accurate to date transition-state geometry calculation of the barrier height of a reaction between phenol and a peroxyl radical. We use the results to assess the performance of a number of density functionals for this prototype reaction. Our benchmark calculations employ multireference Møller− Plesset perturbation theory (MRMP2)22 based on complete active space self-consistent field (CASSCF)23 reference wave functions. The CASSCF wave functions are constructed in the same way as those in a previous paper,24 in particular, nine “active” electrons are distributed in nine active orbitals. This active space is denoted nom-CPO+π;1,25 the nom-CPO active space for an arbitrary reactive system is defined in ref 25, and its

eactions in which a hydrogen atom is abstracted from the hydroxyl group attached to a π-electron system represent a challenge for conventional electronic structure methods due to delocalization of an unpaired electron over the π-electron system in the transition state; accurate treatment of such systems often requires using multireference electronic structure methods. A prototype system of this kind is an abstraction of the hydroxyl H-atom from ethenol by a free radical.1 Another interesting case is the hydroxyl H-atom abstraction from phenols. The phenol reactions have generated considerable interest over the last 2+ decades because they represent a viable mechanism of free radical scavenging in living organisms.2−5 Density functional theory (DFT)6 has been the de facto standard method for calculating reaction barrier heights and reaction rate constants in phenolic reactions,7−17 but the reaction barrier heights vary significantly depending on the choice of the density functional; for example, the reaction barrier for hydrogen abstraction from a model α-tocopherol by the hydroperoxy radical determined by two modern density functionals varied from 4.2 to 9.3 kcal/mol.10 A major concern is that, although DFT calculations provide an efficient way to calculate reaction barrier heights, the results are often inaccurate for systems with significant multireference character,18−20 such as in the case of hydrogen abstraction from © 2012 American Chemical Society

Received: August 13, 2012 Accepted: September 14, 2012 Published: September 19, 2012 2834

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extension to systems that involve π-bonding is defined in ref 1. The one-electron basis set is MG3S,26 which is the same as 6311+G(2df,2p)27,28 for the systems studied here. All molecular structures (separated reactants, the hydrogen-bonded complex, and the saddle point) are optimized at the MRMP2/nom-CPO +π/MG3S level using numerical gradients. Location of the reaction saddle point required evaluations of the Hessians of the energy at several geometries, and these were performed at the same level (lower-level Hessians appeared to be inadequate for this purpose because both CASSCF and DFT-based methods, each for its own reasons, yield distorted potential energy surfaces in the saddle point region). The benchmark reaction barrier height is calculated by MRMP2/nom-CPO+π/ aug-cc-pVTZ at MRMP2/nom-CPO+π/MG3S geometries, where aug-cc-pVTZ29 is a more heavily augmented valence triple-ζ basis set. All MRMP2 calculations were run with the GAMESS code,30 using up to 1024 processors. The mean (M) unsigned error (UE) for MRMP2/nom-CPO +π/aug-cc-pVTZ in the barrier heights for reactions in the DBH24 representative database31 is 1.4 kcal/mol,25 and the MUE in forming and breaking bond lengths and donor− acceptor distances for four reactions in the DBH24 database for the MRMP2/nom-CPO+π/MG3S model chemistry is only 0.007 Å;25 therefore, the present results for the transition-state geometry and reaction barrier height are expected to be accurate enough as benchmarks. All DFT calculations are performed with the MG3S basis set using a locally modified Gaussian code.32,33 In order to assess how well the DFT methods describe the molecular structure in the saddle point region, all molecular structures were optimized for each density functional, and the DFT barrier height for each functional was calculated using the geometry obtained with that functional. The functionals tested are as follows: B1LYP,34 B3LYP,35,41,42 B97-1,36 B97-2,37 B97-338 B98,39 BB1K,40 BHandH,32 BHandHLYP,32 BLYP,41,42 BMK,43 M05,44 M052X,45 M06,46,47 M06-2X,46,47 M06-L,48 M11-L,49 MPW1K,50 MPW1B95, 51 mPW1PW91, 52 MPW3LYP, 51 O3LYP, 53 PW6B95,54 SOGGA11,55 SOGGA11-X,56 τ-HCTHhyb,57 TPSS1KCIS,58 and TPSSh.59 In general, π-system radicals and transition states involving partial bonds at π centers are expected to have significant multireference character, which is defined as the inability of a system’s electronic wave function to be well represented by a single-configuration state function. It is convenient to evaluate multireference character quantitatively in terms of the occupation numbers of a CASSCF wave function at a given molecular geometry, and we defined the M diagnostic25 for this purpose. An M value less than 0.05 corresponds to small multireference character, 0.05 ≤ M ≤ 0.10 indicates moderate multireference character, and M > 0.10 denotes large multireference character.25 Table 1 shows the multireference character for the four stationary points considered here, and we see that the transition state does indeed have moderate-to-large multireference character. Complementary criteria are expectation values of S2 for the UHF wave function, where S is total electron spin; we find 0.76 and 1.29 for the reactant hydrogenbonded complex and the saddle point (for a pure doublet ground electronic state this value is 0.75); the difference between the values at the two key geometries indicates that potential energy surfaces calculated by UHF or by the singlereference Møller−Plesset (MP2) method based on the UHF wave function would be significantly distorted from the potential energy surface that is converged with respect to the

Table 1. M Diagnostics and Expectation Values of S2 for the UHF Wave Function M

S2

0.060 0.000a,c (0.061)a,d 0.098 0.102

0.76 0.76 1.29

species phenol • OOCH3 precursor complex transition state

a,b

a

Calculated for isolated reactant species. bUsing the (8/8) active space corresponding to the nom-CPO−π active space for H-abstraction from phenol. cUsing the “(1/1)” active space to reflect that only one orbital (SOMO) from •OOCH3 was included in actual calculations. dUsing the (9/7) active space that includes σOO, σ*OO, σCO, σ*CO, one lone pair orbital on each oxygen atom, and the π-type SOMO.

inclusion of electron correlation. This is confirmed by the finding that the unprojected UMP2/MG3S reaction barrier height with respect to the hydrogen-bonded complex is 35.0 kcal/mol, as compared to 12.3 kcal/mol calculated by MRMP2/MG3S; notice that this error is significantly higher than the MUE for a MP2/MG3S model chemistry of 5.0 kcal/ mol25 for single-reference reactions. The transition states for hydrogen atom abstractions from phenols are preceded by the formation of phenol···R• (where R• denotes a radical) hydrogen-bonded complexes. The transition state geometry is sketched in Figure 1. Geometrically,

Figure 1. Transition-state structure; see Table 1 for key geometrical parameters.

as in the smaller ethenol···R• hydrogen-transfer system,1 the reaction saddle point on a phenol···R• potential energy surface is characterized by a twist of the hydroxyl OH group and the hydrogen bond out of the molecular plane,7,8,24 as opposed to the corresponding equilibrium molecular structures with essentially planar hydrogen bonding. The dihedral angle ϕ between the OH bond and the molecular plane of the benzene ring, which is related to the location of the avoiding crossing between the ground and first excited electronic states of the system,8,24 is 70°. All reaction paths with the planar H-bond geometry (ϕ = 0) would inevitably pass through a conical intersection24 because interaction of the states correlating with the σ- and π-states of the phenoxyl radical at such geometries would not be allowed by symmetry. Table 2 shows that all density functionals that we have tested give a much smaller dihedral angle than the benchmark results, with the most accurate DFT values being 59° for M05-2X and 57° for M062X. 2835

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Table 2. Key Geometric Parameters of the Reaction Saddle Points for Hydrogen Atom Abstraction from Phenol by the • OOCH3 Radicala MRMP2 (best estimate) M06 CASSCF/6-31G(d,p)b M05 BMK B1LYP B3LYP MPW3LYP TPSS1KCIS B98 TPSSh SOGGA11-X B97-1 τ-HCTHhyb M05-2X MPW1K BB1K PW6B95 M11-L M06-2X mPW1PW91 B97-2 O3LYP PW1B95 MPW1B95 BHandHLYP SOGGA11 a

rOaH

rObH

rOO

ϕ

MUE (key bonds)

MUE (all bonds)

UE (ϕ)

1.119 1.125 1.158 1.143 1.161 1.170 1.177 1.174 1.192 1.174 1.197 1.162 1.178 1.184 1.174 1.153 1.161 1.176 1.164 1.177 1.176 1.180 1.203 1.177 1.177 1.152 1.252

1.299 1.249 1.230 1.233 1.207 1.205 1.203 1.202 1.197 1.196 1.196 1.195 1.195 1.195 1.198 1.188 1.189 1.189 1.187 1.189 1.186 1.184 1.183 1.184 1.184 1.187 1.139

2.397 2.350 2.382 2.362 2.351 2.361 2.367 2.362 2.376 2.355 2.380 2.342 2.357 2.364 2.349 2.327 2.335 2.350 2.338 2.346 2.346 2.350 2.373 2.345 2.345 2.313 2.376

70 46 80 36 48 39 32 36 25 39 24 49 38 33 59 52 47 42 27 57 42 39 0 43 43 56 9

0.034 0.041 0.041 0.060 0.060 0.061 0.062 0.065 0.066 0.066 0.067 0.067 0.067 0.068 0.071 0.071 0.071 0.072 0.073 0.073 0.074 0.074 0.075 0.075 0.076 0.104

0.013

24 10 34 22 31 38 35 45 31 46 21 32 37 11 18 23 29 43 13 28 31 70 27 27 14 61

0.014 0.017 0.017 0.018 0.018 0.019 0.018 0.020 0.016 0.019 0.020 0.017 0.020 0.021 0.020 0.026 0.017 0.019 0.019 0.022 0.020 0.020 0.021 0.029

MG3S basis set for all DFT calculations. bReference 24.

Table 2 also lists the key interatomic distances in the reaction saddle point; rOH(a) is between the phenolic oxygen and the transferred hydrogen atom, rOH(b) is between the terminal oxygen of •OOCH3 and the transferred hydrogen atom, and rOO is between the two heavy atoms involved in the hydrogen bond. The last three columns in that table list MUEs for the transition-state geometry by DFT methods. CASSCF results24 are also given for comparison. Previous studies10,17 indicated that the contribution of quantum mechanical tunneling to the overall rate constant of hydrogen abstraction from natural antioxidants α-tocopherol and coenzyme Q are 80 and 50%, respectively; therefore, it is essential to know accurate values for the distances rOO, rOH(a), and rOH(b). DFT calculations give consistently longer rOH(a) distances and consistently shorter rOH(b) and rOO than our best estimate. None of the density functionals are accurate to better than 0.03 Å for the key intermolecular distances for this reaction. The best performer among the density functionals for transition-state geometry is M06, which gives somewhat smaller errors for the key interatomic distances characterizing the H-transfer (MUEkey bonds) and for all interatomic distances (MUEall bonds) compared to other functionals. This is in line with a recommendation46,47 of this density functional as a good choice for systems with high multireference character. The density functionals are listed in Table 2 in order of increasing MUEkey bonds, which varies from 0.034 to 0.104 Å. The deviations of the transition-state geometries obtained by the DFT methods for the present reaction are, on average, more than twice larger than the corresponding MUEs25 for the same DFT methods for hydrogen-transfer reactions in the DBH2431

representative database; that database contains only singlereference systems. Table 3 presents the benchmark results and the assessment for barrier heights. In this table, we consider the reaction barrier height V‡ with respect to separated reactants and also the energy difference Ec between the transition-state structure and a local minimum for the hydrogen-bonded complex; Cartesian coordinates for this latter structure optimized at a benchmark level are given in the Supporting Information. Then, this same conformation of the complex was optimized with each density functional. (Note that this structure is slightly less stable compared to the structure with the coordination of OH to the primary oxygen atom of the peroxyl radical.8) Our best estimates for V‡ and Ec are 7.4 and 11.5 kcal/mol, respectively. The first column in Table 3 is the percentage X of Hartree− Fock exchange in the density functional. In general, functionals with X ≥ 30 have not been recommended47 for multireference problems, and functionals with X ≤ 10 have not been recommended21 for barrier height calculations, but both tables include some functionals with X out of the 10−30 range because some workers have used such functionals despite these recommendations, and including such functionals in the table can serve as a useful warning to the community. Table 3 lists the density functionals in order of MUEs for the two barrier heights. The 10 highest-ranked functionals all have 12 ≤ X ≤ 28. A rather interesting observation about Table 3 is that several of the better-performing functionals in the table, in particular, B98, B97-1, B97-2, B1LYP, B3LYP, and M05, were also among the best-performing functionals in our recent study60 of 1,3-dipolar addition reactions of ozone, which is a 2836

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Table 3. Barrier Heights (kcal/mol) with Respect to the Reactant Asymptote (V‡) and with Respect to a Minimum for the Hydrogen-Bonded Complex of Phenol + •OOCH3 System (Ec) Xa WFT 28 WFT 25 21 22 12 25 20 21 21 26 50 31 27 27 13 0 54 10 56 15 42 0 40 43 42 50

methodb MRMP2/aug-ccpVTZ//MRMP2/ MG3Sc M05 MRMP2/MG3S mPW1PW91 B97-2 B98 O3LYP B1LYP B3LYP B97-1 MPW3LYP PW6B95 BHandH MPW1B95 M06 B97-3 TPSS1KCIS M11-L M06-2X TPSSh M05-2X τ-HCTHhyb BMK SOGGA11 SOGGA11-X MPW1K BB1K BHandHLYP

V‡

Ec

7.4

11.5

7.0 8.2 8.6 8.5 6.6 7.1 8.9 6.4 6.0 5.7 9.3 7.7 10.3 5.0 10.6 4.8 4.7 10.5 4.1 11.2 3.6 12.5 2.3 13.3 15.0 15.4 17.6

11.5 12.3 12.1 12.4 10.4 9.7 12.4 9.8 10.1 10.1 13.2 15.2 13.2 9.2 13.8 8.4 8.1 15.1 7.6 15.8 7.2 15.7 5.9 17.0 18.7 18.6 21.7

UE (V‡)

UE (Ec)

MUE

0.4 0.8 1.2 1.1 0.9 0.3 1.5 1.0 1.4 1.7 1.8 0.3 2.9 2.4 3.2 2.6 2.7 3.1 3.3 3.8 3.8 5.1 5.1 5.9 7.6 8.0 10.2

0.0 0.8 0.5 0.8 1.1 1.9 0.8 1.7 1.4 1.5 1.7 3.6 1.7 2.3 2.2 3.1 3.4 3.6 4.0 4.2 4.4 4.2 5.7 5.5 7.2 7.0 10.2

0.2 0.8 0.9 1.0 1.0 1.1 1.2 1.4 1.4 1.6 1.7 1.9 2.3 2.3 2.7 2.9 3.1 3.3 3.6 4.0 4.1 4.6 5.4 5.7 7.4 7.5 10.2

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (O.T.); [email protected] (D.G.T.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Graham Fletcher for optimizing the performance of the highly parallel MCSCF code at the BG/P supercomputer and for his kind assistance with running GAMESS on the BG/P system at Argonne. This research was carried out using an INCITE grant using resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-06CH11357. Computer time at the Minnesota Supercomputer Institute is gratefully acknowledged. This work was supported in part by the National Science Foundation under Grant No. CHE09-56776.

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REFERENCES

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a

Rounded to the nearest integer for density functionals; WFT denotes wave function theory. bMG3S basis set for all DFT calculations. cBest estimate.

molecule with notoriously high61 multireference character. It is interesting to see consistent trends like this emerging, and we expect that benchmarks of the type reported here will eventually be available for enough multireference reactions for more general conclusions to be drawn. The eight best-performing density functionals in Table 3 agree with the best estimate within the 1.4 kcal/mol MUE (mentioned above) of the method used to obtain the best estimate. The best-performing functionals are those that appear in the higher rows of both Tables 2 and 3; these functionals are M05 (second for geometries and first for energetics), B1LYP (fourth for geometries and sixth for energetics), B3LYP (fifth for geometries and seventh for energetics), B98 (eighth for geometries and fourth for energetics), B97-1 (eleventh for geometries and eighth for energetics), and M06 (first for geometries and thirteenth for energetics).

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Letter

ASSOCIATED CONTENT

S Supporting Information *

Molecular geometries (Cartesian coordinates) and absolute energies for transition structure and reactants. This material is available free of charge via the Internet at http://pubs.acs.org. 2837

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