Comparison of Relative Activation Energies Obtained by Density

Article pubs.acs.org/JPCB

Comparison of Relative Activation Energies Obtained by Density Functional Theory and the Random Phase Approximation for Several Claisen Rearrangements Madeline K. Hartley, Seanna Vine, Elizabeth Walsh, Sara Avrantinis, G. William Daub,* and Robert J. Cave* Department of Chemistry, Harvey Mudd College, 241 Platt Blvd., Claremont, California 91711, United States ABSTRACT: We investigate several representative density functional theory approaches for the calculation of relative activation energies and free energies of a set of model pericyclic reactions, some of which have been studied experimentally. In particular, we use a standard hybrid functional (B3LYP), the same hybrid functional augmented with a basis set superposition error and dispersion correction, a meta-hybrid functional developed to treat transition states and weak interactions (M06-2X), and the recently implemented random phase approximation (RPA) based on Kohn−Sham orbitals from conventional density functional theory by Furche and co-workers. We apply these methods to calculate relative activation energies and estimated free energies for the amide acetal Claisen rearrangement. We focus on relative activation energies to assess the effects of steric and weak interactions in the various methods and compare with experiment where possible. We also discuss the advantages of using this set of reactions as a test bed for the comparison of treatments of weak interactions. We conclude that all methods yield similar trends in relative reactivity, but the RPA yields results in best agreement with the experimental values.

1. INTRODUCTION The accurate calculation of reaction energetics has been a goal for quantum chemistry since its inception1−3 and has grown increasingly more realistic for systems of chemical interest. This is due, in part, to increased computational power, but to a greater extent this growth in accuracy has been realized by development of extrapolation methods such as the GN,4−11 CBS,12,13 and related approaches coupled with the increased accuracy of density functional theory (DFT) methods.14−28 In particular, it is the previously unimagined accuracy of density functional theory for large systems that has led to an explosion in its application across all subdisciplines of chemistry. The growth in use of DFT has helped fuel continued functional development, progressing from commonly used generalized gradient approximation (GGA)14 and hybrid GGA 17 , 21 functionals to various families of metaGGAs15,23−26,29 so that newer methods offer considerable accuracy for a wider range of systems and properties. Continued improvement stems both from the increased flexibility of new functionals and from expanded “training sets” through which the functionals are parameterized.26 This latter point is a benefit and a curse in that any set of parameters is a compromise in achieving accuracy among various properties of chemical interest. The alternative, optimizing “best” functionals for individual properties, might work, but is intellectually unsatisfying and still does not guarantee the “right answer for the right reasons”30 if a method is unable to treat a © XXXX American Chemical Society

wide variety of chemical behaviors. Thus, it is important to continually push methodology forward to seek higher accuracy in ever-widening chemical arenas. Accurate treatment of steric interactions, both weak and strong, is an important goal for DFT methods because subtle differences in nonbonded interactions can often play a significant role in controlling kinetics. It is an important challenge because DFT methods are relatively insensitive to small density changes in areas of low density, but this is exactly where steric interactions reside energetically. A number of DFT approaches have attempted to include improved treatment of steric interactions, such as the M06 and M08 functionals of Zhao and Truhlar,23−28 the TPSS functional,15 the more recent B97M-V functional,31 the combined basis set superposition and dispersion corrections developed by Grimme and co-workers32 and Becke and Johnson,22,33 and more recent work by both the groups of Becke34 and of Johnson.35−37 A different approach has been recently taken by the Furche, Scuseria, Toulouse and Hesselmann groups.38−48 Reasoning that while DFT methods often provide improved densities relative to Hartree−Fock wave functions, most do not include Special Issue: Bruce C. Garrett Festschrift Received: July 10, 2015 Revised: August 26, 2015

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DOI: 10.1021/acs.jpcb.5b06646 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

has received considerable study experimentally and theoretically;60−67 thus, we have some sense of how well DFT performs for these reactions. It has two stereochemically distinct transition states (chair and boat, Scheme 2) with similar bonding in each so one expects similar errors in activation energies for both pathways. In this regard, the transition-state energy dif ference can be viewed as a type of isodesmic transformation.1 There are several positions at which we can vary substituents, giving rise to distinctly different steric interactions for the boat and chair transition states, and the changes in relative activation energies should be indicative of, to a large extent, the weak steric interactions present in the transition state. The CR can be performed with several ring structures (Schemes 3 and 4; here we use the so-called amide acetyl

the explicit physics necessary to treat dispersion and weak interactions; these groups have developed the random phase approximation (RPA) based on Kohn−Sham DFT orbitals. The RPA includes explicit treatment of the double excitations necessary to model dispersion correctly, and recent work has shown38 it is a powerful tool for describing weak interactions and reaction chemistry.38−41 This suggests that the RPA based on DFT orbitals might be a useful tool for treating reactions in which steric interactions play an important role in determining kinetic behavior. RPA implementations have been based on full range functionals39−41,49 as well as range-separated functionals,42,43,45−47,50 and their application to chemical processes is steadily growing based on their demonstrated accuracy and speed. In this contribution, we use the RPA to predict relative activation energies for several variants of the Claisen rearrangement (CR, see Schemes 1 and 2). Previous work has examined

Scheme 4. Chair and Boat Transition States Lead to syn and anti Products

Scheme 1. Sample Claisen Reactants and Products

Scheme 2. Sample Claisen Transition States

Claisen rearrangement, AACR) allowing subtle modifications of nonbonded interactions leading to a family of parallel reactions that can be used to assess theoretical results. Finally, experimental results exist for a number of these systems against which comparisons of theoretical results can be made. These features led us to pursue the present work. In the experimental work referenced below, the syn:anti product ratios were determined by performing the amide acetal Claisen rearrangements at 80 and 125 °C and subjecting the resulting product mixtures to both capillary gas chromatography and 1H NMR spectroscopic analysis. The syn and anti products were characterized by 1H and 13C NMR spectroscopy

application of the RPA to model activation energies45,46,48 (see the extensive set of comparisons in ref 48, for example), but to the best of our knowledge the method has not been applied to the activation energies of pericyclic reactions. The CR has been studied experimentally by one of us51−54 and has a number of features that make it a natural test case for methods that include dispersion explicitly. It is a member of the family of pericyclic reactions (an analogue of the Cope rearrangement55−59) that

Scheme 3. 5- and 6-Membered-Ring Claisen Rearrangements Illustrating Distinct Products Arising from Chair and Boat Transition States

B


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3 with sufficient accuracy to obtain quantitative activation energies. Relative activation energies allow us to, hopefully, factor out absolute errors common to the two similar transition states. Examining changes in relative activation energies further reduces issues with errors due to subtle differences between the two types of transition states and minimizes effects associated with the “chameleonic” nature of Cope-like transition states.55,56 While we do extensive basis set and geometry testing for a sample system (for which we do not have experimental data, thus minimizing bias in our examination of the remaining systems), we focus largely on one basis set and method for obtaining transition states throughout the study in order to reduce the number of “moving pieces.” The remainder of the article is organized as follows: in section 2 we outline the theoretical methods used and discuss the computational details of our study. We then present our results in section 3 and discuss our results in section 4. Section 5 presents the conclusions.

as well as infrared spectroscopy, mass spectrometry, and elemental analysis. The nature of the reaction did not lead to a convenient method for determining reaction kinetics, but the fact that the reaction is not reversible allowed for the determination of the activation free energy difference between the chair and boat processes directly from the syn and anti product ratios (Scheme 5); it has been shown that the syn and Scheme 5. Relative Free Energy Difference between the Chair and Boat Transition States Determines the Product Distribution for the Claisen Rearrangement

2. THEORETICAL METHODS AND COMPUTATIONAL DETAILS A panoply of density functional approaches exist, and there is no doubt that a semiexhaustive search using our target reactions could turn up a “best” method for mimicking experiment if we cast our net widely. However, we believe it is more instructive to examine a few representative approaches and ask whether qualitative differences or trends are observed that might argue for greater reliability for one or more approaches in the present case. With that goal in mind we apply the following methods: (1) DFT with the B3LYP functional.17−19,68 This functional still sees heavy use in studies of reaction chemistry, and it has been previously shown to be a useful tool for pericyclic reactions.63 Because it was not explicitly developed to treat dispersion it is a useful point of comparison for the remaining methods used here. (2) DFT with the M06-2X functional.23−27 The meta-hybrid M06-2X functional is one of a family of functionals developed by Zhao and Truhlar that has seen increasing use in recent years. The training set for these functionals highlights the importance of activation energies and dispersion interactions. (3) DFT with the B3LYP functional and corrections for BSSE and dispersion (GD3BJ correction).32,69,70 The addition of the semiempirical correction is an attempt to account for finite basis effects and long-range interactions, two factors that could have differential impacts on the relative activation energies of Claisen rearrangements. Note that while we occasionally highlight the fact that this correction approximately treats dispersion and BSSE, the BSSE correction is not based on a traditional counterpoise correction,71 or a method of this type. (4) CBS-QB3.12,13 One of a range of methods for extrapolating energies to the highly correlated large basis set limit, CBS-QB3 begins with a B3LYP optimization. We limit its use to cases in which we compare relative transition-state energies at separately optimized geometries. (5) Random phase approximation.38−41 We use the implementation of the Furche group and have tested several choices of functional (PBE,14 BLYP,19,68 and B3LYP17−19,68) to generate orbitals for the RPA treatment. (6) HF. This uncorrelated method is included for completeness. (7) MP2. This is the simplest ab initio method that contains terms that treat dispersion. Core electrons were not correlated.

anti products do not interconvert under the reaction conditions. As a result, if theory is able to accurately evaluate relative transition-state free energies, there would be an independent way to probe the energetics of the chair and boat transition-state topologies and shed light on the chemistry of these reactions. To give context to the results and also assess whether the nonbonded interactions are an insurmountable challenge for commonly used DFT methods, we apply the RPA and several conventional DFT approaches to study the CR and AACR. In particular, we use a well-studied GGA-hybrid (B3LYP17−19), a more recent and heavily used meta-GGA (M06-2X23−25) that has been parametrized to treat activation energies and dispersion, B3LYP augmented with basis set superposition error (BSSE)- and dispersion-corrections,22,32 and the RPA.38−41 This selection of methods is far from exhaustive but includes several common approaches to dealing with dispersion’s impacts on reactivity (including no explicit treatment, i.e., B3LYP). In selected examples we also use the CBS-QB312,13 method, MP2, and CCSD(T) results as points of comparison with our explicitly DFT-based results. The questions we address include the following: (1) How important is it to include dispersion treatment for the relative activation energies of this series of reactions? (2) Are there trends among the results of these methods for these reactions? (3) Do activation energies tell the same story as activation free energies for these reactions? (4) How do the results of the various methods compare with experimental results for these systems? Note that we focus here on relative activation energies for the pair of reaction pathways rather than activation energies themselves. As with many other pericyclic reactions, there are detailed questions about whether they are concerted or stepwise, and at present, no multireference methods are available that can treat systems of the size shown in Scheme C


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The Journal of Physical Chemistry B (8) CCSD and CCSD(T).3 Various ab initio approaches suggest themselves as representative high-level correlation treatments, but the robustness and accuracy of coupled cluster methods make them a natural point of comparison. Core electrons were not correlated. (9) CASSCF and CASMP2.72 We utilize the method of McDouall and co-workers72 to account for correlation effects on top of a 6-electron/6-orbital complete active space SCF (6/ 6 CASSCF). This is the minimal MCSCF capable of characterizing the 3,3 sigmatropic shifts of which the Cope and Claisen rearrangements are examples. The perturbation theory method is based on localized CASSCF orbitals that approximate an orthogonalized valence bond treatment. We use the CASMP2 approach largely to test whether any multireference effects are present that would invalidate the DFT and/or CCSD(T) results. Because the RPA method is a relatively new approach, we offer a brief summary here, though the reader is encouraged to explore the method in greater detail in refs 38−41. In a density functional theory context, the RPA energy can be written as41

reactions and there is no natural separation into separate reactants here. A number of past studies of the RPA for activation energies have been done without explicit treatment of BSSE,45,46,48 and previous work by the Furche group on systems having weak interactions40 indicates that the extrapolation we perform leads to similar interaction energies with and without the counterpoise correction. However, it should also be kept in mind that the basis sets used there are somewhat larger that those we use for our biggest systems. On the other hand, by calculating relative transition-state energies, we appear to benefit from cancellation of BSSE effects (vida infra). All geometry optimizations; vibrational analyses; and singlepoint B3LYP, M06-2X, B3LYP+GD3BJ (denoted simply as GD3BJ here), CASSCF, CASMP2, MP2, and CCSD(T) were performed using Gaussian 09.81 All RPA and associated DFT calculations were performed using Turbomole.82

3. RESULTS We begin our analysis using two simple Claisen rearrangements, those of compounds 1a and 1b in Scheme 1. Compound 1a (“parent”) can undergo the Claisen rearrangement via both chair and boat transition states (see Scheme 2), but it possesses no substituents beyond hydrogen. Compound 1b can react through similar chair and boat transition states, but methylation at positions 1 and 4 leads (vide infra) to increased steric interactions in the boat relative to the chair transition state, which we assess using the methods discussed previously. Tables 1 and 2 present results from geometry optimizations of the chair and boat transition states for compounds 1a and 1b

ERPA = EKS + E CRPA = T S + V ext + EH + E X + E CRPA

where, in the second equality, the first four terms are evaluated using the Kohn−Sham orbitals from an initial DFT calculation as an expectation value of the nonrelativistic electronic Hamiltonian. The term ECRPA is the correlation energy relative to the Kohn−Sham reference energy. The RPA correlation energy can be evaluated as a difference in zero-point energies (one-electron excitations) between time-dependent DFT (TDDFT) electronic excitation energies for the system evaluated in the full RPA and in the Tamm−Dancoff approximation to RPA.41 The computational breakthrough for RPA applications to large systems was the combination of this expression with resolution of the identity and imaginary frequency integration.38,39,41,49 In a wave function perspective, RPA begins with the KS determinant defining the zeroth-order energy of the system. ECRPA is defined using double-excitation coefficients obtained from TDDFT “collective excitation” coefficients (off-diagonal terms in TDDFT eigenvectors).41 Alternatively, Scuseria, Henderson, and Sorenson44 showed that ECRPA can be viewed as arising from doubles coefficients obtained from a truncated CCD formalism. In either perspective, the RPA includes correlation effects that allow it to capture dispersion effects. In addition, because it is not perturbative, the method is bettersuited for treating small gap systems, of which transition-state calculations are a natural example.38−40,49 The basis sets used were taken from the family of correlation consistent (CC) basis sets of Dunning and co-workers.73−76 While Pople basis sets77−80 tend to be more common in applications of electronic structure theory to organic reactions, we chose the CC basis sets because they afford a systematic extrapolation to the large-basis limit. These are large systems, and we use several traditional correlated methods as well as RPA; all of these approaches are susceptible to BSSE, likely more so than standard density functional theory methods. A common approach to treating BSSE is the counterpoise correction71 in which separate reactant calculations are performed with the partner basis and these reactant energies are used to calculate overall energetics and/or activation energies. Such an approach is not possible for our systems because we are performing intramolecular

Table 1. Relative Transition-State Energies for Chair and Boat Transition States of the Unsubstituted and 1,4Substituted Claisen Speciesa method

ΔETS parent

ΔETS 1,4 substituted

ΔΔE

B3LYP M06-2X B3LYP-GD3BJ CBS-QB3

−3.83 −5.63 −3.96 −4.50

−4.43 −6.39 −4.55 −5.20

−0.60 −0.76 −0.59 −0.70

a

Results are in kilocalories per mole based on optimizations for each transition state in the given method and the cc-pVDZ basis, except for CBS-QB3 where a modified 6-31(d) basis is used for optimization and frequency analysis. Each stationary point was confirmed by vibrational analysis and was shown to possess a single imaginary frequency corresponding to the 3,3 sigmatropic shift characteristic of Claisen rearrangements. Energy differences (ΔETS) are calculated as ETS(chair) − ETS(boat); thus, a negative value indicates the chair transition state is lower in energy than that of the boat form. The value of ΔΔE is the change in relative transition-state energy upon substitution and is (ΔETS)1,4 − (ΔETS)unsubstituted. A negative value implies the chair form is stabilized relative to the boat form upon substitution.

using various methods and some form of double-ζ basis. Careful perusal of the results suggests the following: (1) All methods predict a larger transition-state energy difference (ΔETS = activation energy difference) for the 1,4 methyl substituted species (compound 1b). (2) The O−C and C−C bond lengths are somewhat shorter for the M06-2X transition states than for the other methods. (3) The transition-state energy differences (ΔETS) and ΔΔE are largest for M06-2X. Because the transition-state geometries are somewhat different (though reassuringly close in many respects), it is D


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(3) The MP2 ΔETS and ΔΔE values show the greatest disagreement with the CCSD(T) results of all the methods we tested. (4) CCSD(T) in the cc-pVDZ basis yields results in reasonable agreement with CASMP2, M06-2X, and both RPA methods. These methods tend to give ΔETS values larger than those provided by BLYP, B3LYP, with GD3BJ. (5) The RPA results are relatively insensitive to the KS orbitals on which they are based. Furche and co-workers suggest use of nonhybrid orbitals as input to the RPA, but we examined B3LYP orbitals (results not shown) in order to compare results based on the same zeroth-order KS density for DFT (B3LYP) and RPA results (RPA/B3LYP). The B3LYPbased RPA results are quite similar to those of PBE or BLYP RPA. (6) Of the DFT methods used here, M06-2X is most accurate for ΔETS (in comparison with CCSD(T) results) in this small basis. Both points (4) and (5) are borne out as the basis set is expanded, as shown in Table 4. In Table 4 (see also Figure 1) we treat the chair and boat transition states of compounds 1a and 1b using extended basis sets (up to aug-cc-pVQZ) and examine ΔETS and ΔΔE for a subset of the methods included in Table 3. Not surprisingly the non-RPA DFT results show little change upon basis set extension. While the results of correlated methods change somewhat more with basis set, the variations in ΔETS and ΔΔE are only on the order of a few tenths of a kilocalorie per mole. (The extrapolation method used here is detailed in ref 40.) Given the geometries of the boat and chair transition states, one would expect that if lessening BSSE were the dominant effect of basis set expansion, ΔETS and ΔΔE would increase with basis set expansion (the chair would be preferentially stabilized as the basis set improved). On the other hand, if improved correlation were the dominant effect of basis set expansion, ΔETS and ΔΔE should decrease with basis set expansion (expected greater correlation error for the more congested boat transition state). The modest decrease in ΔETS and ΔΔE observed here does not imply BSSE effects are irrelevant, but it does suggest they are not the dominant effect associated with basis set expansion. Examining ΔETS and ΔΔE separately, one arrives at different conclusions about the relative accuracy of the methods. The ΔETS results (and associated “error” values in comparison with the extrapolated CCSD(T) results) indicate that M06-2X is the most accurate of the DFT methods, yielding differences in chair and boat transition states for each compound in significantly better agreement with CCSD(T). The error values of M06-2X are essentially half those of B3LYP and slightly more than half those of B3LYP with GD3BJ corrections. The errors in M062X are comparable to either of the RPA methods, but M06-2X tends to overestimate ΔETS while the RPA methods tend to underestimate it. If one instead focuses on ΔΔE (the change in chair versus boat transition states upon substitution), then M06-2X is marginally worse than the other DFT methods and RPA. This stems from somewhat larger M06-2X errors for the 1,4substituted species compared to the unsubstituted species. The errors for the other DFT methods for both species are larger than those for M06-2X, but they are more balanced. The RPA methods also tend to be more balanced for ΔΔE than M06-2X, and as noted above, the errors are similarly small.

Table 2. Bond Distances (Angstroms) between Terminal Atoms in the Claisen Rearrangement as a Function of Method for Optimized Transition Statesa system

bond

B3LYP

M06-2X

B3LYP-GD3BJ

CBS-QB3

parent, chair

O−C C−C O−C C−C O−C C−C O−C C−C

1.92 2.32 1.99 2.42 1.91 2.30 1.98 2.43

1.82 2.16 1.88 2.24 1.81 2.14 1.85 2.23

1.91 2.34 1.98 2.41 1.91 2.33 1.97 2.43

1.94 2.36 2.01 2.46 1.94 2.35 2.00 2.46

parent, boat 1,4 chair 1,4 boat a

Results are based on optimizations for each transition state in the given method and the cc-pVDZ basis. Each stationary point was confirmed by vibrational analysis and was shown to possess a single imaginary frequency corresponding to the 3,3 sigmatropic shift characteristic of Claisen rearrangements.

difficult to draw unequivocal conclusions about relative accuracy based on the results of Table 1. Thus, in the remaining calculations we base all calculations on B3LYP ccpVDZ geometries. Nevertheless, the results of Tables 1 and 2 are important because they indicate that the various methods do not yield radically different ΔETS and ΔΔE when separate optimizations are performed. Table 3 compares results from an array of methods for the energy difference between B3LYP/cc-pVDZ chair and boat Table 3. Relative Energies of the B3LYP/cc-pVDZ Transition States Using Various Methods in the cc-pVDZ Basisa method

ΔETS parent


ΔΔE

B3LYP M06-2X BLYP B3LYP-GD3BJ MP2 CCSD(T) CASSCF(6/6) CASMP2(6/6) RPA/BLYP RPA/PBE

−3.83 −5.75 −3.01 −3.98 −6.44 −4.85 −3.52 −4.72 −4.50 −4.55

−4.43 −6.58 −3.59 −4.57 −7.59 −5.74 −4.07 −5.32 −5.30 −5.36

−0.60 −0.83 −0.58 −0.59 −1.04 −0.89 −0.55 −0.60 −0.80 −0.81

a

Results are in kilocalories per mole based on the chair and boat B3LYP/cc-pVDZ optimized geometries. Energy differences are for the given method and the cc-pVDZ basis. Energy differences are as defined in Table 1.

transition states (values less than zero indicate that the chair transition state is lower in energy than the boat). Without making too much out of relatively small energy differences, these results indicate the following: (1) The relative energy differences between methods in Table 1 are likely due more to the methods themselves than small differences in geometries. (2) The similarity of the ΔETS and ΔΔE for CASSCF/ CASMP2 and the other methods indicates that large dif ferential multireference effects are not important here. Note that multireference effects could be important for the actual activation energies (rather than activation energy differences), but the results of Table 3 would not reveal this. E


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In somewhat of a surprise, Table 4 also indicates that HF yields quite good results for ΔΔE and the individual ΔETS. This is likely a function of the near-isodesmic nature of the quantities we are calculating (chair−boat activation energies for ΔETS and substitution differences giving rise to ΔΔE) rather than an inherent high accuracy for HF in these systems. Indeed, were we to calculate activation energies for these systems, HF would likely have errors significantly larger than those of many of the other methods. Nevertheless, we will show for the larger systems HF is similarly accurate for the quantities of interest here. The compounds of Scheme 1 provide computationally tractable models but cannot provide experimental data on chair versus boat activation energies. While one can imagine thermal activation inducing reactions through both pathways, the products are identical in either case and relative transitionstate information is lost as a result. The compounds in Scheme 3 provide different stereochemical products for the two transition-state pathways despite originating from thermally equilibrated conformers of a given reactant. Thus, these compounds provide an independent method for calibrating the methods used here. Tables 5 and 6 provide data for transition-state energy differences for the compounds of Scheme 3. On the basis of the

Table 4. Relative Energies of the B3LYP/cc-pVDZ Transition States Using Various Methods in Extended Basis Setsa method

basis

ΔETS parent


ΔΔE

errorb

−3.61 −3.57 −3.55

−4.09 −3.98 −3.93

−0.48 −0.41 −0.38

1.37/1.45 1.41/1.56 1.43/1.61

M06-2X

cc-pVTZ cc-PVQZ aug-ccPVQZ cc-pVTZ

−5.64

−6.45

−0.81

M06-2X

cc-PVQZ

−5.58

−6.32

−0.74

B3LYPGD3BJ B3LYPGD3BJ HF

cc-pVTZ

−3.76

−4.23

−0.47

−0.66/0.91 −0.60/0.78 1.22/1.31

cc-PVQZ

−3.72

−4.12

−0.40

1.26/1.42

cc-pVTZ

−4.31

−4.65

−0.34

MP2

cc-pVTZ

−6.39

−7.25

−0.86

MP2

Extrap(d/t)

−6.38

−7.15

−0.77

CCSD(T) CCSD(T) CCSD(T) est. RPA/BLYP RPA/BLYP RPA/BLYP RPA/BLYP RPA/PBE RPA/PBE RPA/PBE RPA/PBE

cc-pVTZ Extrap(d/t) cc-pVQZ

−4.93 −4.98 −4.77

−5.57 −5.54 −

−0.64 −0.56 −

−0.67/0.31 −1.41/1.71 −1.40/1.61 0.05/-0.03 − 0.21

cc-pVTZ cc-PVQZ Extrap(d/t) Extrap(t/q) cc-pVTZ cc-PVQZ Extrap(d/t) Extrap(t/q)

−4.44 −4.21 −4.48 −4.16 −4.51 −4.32 −4.55 −4.28

−4.96 −4.68 −4.90 −4.60 −5.01 −4.74 −4.95 −4.65

−0.52 −0.47 −0.42 −0.44 −0.50 −0.42 −0.40 −0.37

0.54/0.58 0.77/0.86 0.50/0.64 0.82/0.94 0.47/0.53 0.67/0.80 0.43/0.59 0.70/0.89

B3LYP B3LYP B3LYP

Table 5. Relative Transition-State Energies for Chair and Boat Transition States of the 5-Ring, 5-Ring-Methyl, 6-Ring, and 6-Ring-Methyl Compounds Optimized and Evaluated using B3LYP and the cc-pVDZ Basisa

a

Results are in kilocalories per mole based on the chair and boat B3LYP/cc-pVDZ optimized geometries. Energy differences are for the given method and basis set. CCSD(T) est. results are based on ccpVQZ CCSD results corrected by the cc-pVTZ (T) energy increment. Energy differences are as defined in Table 1, and the extrapolation method is defined in the text. bError is defined relative to the extrapolated CCSD(T) results, and the pair of entries correspond to the parent and 1,4 substituted differences in ΔETS relative to the extrapolated CCSD(T) results, respectively.

a

species

ΔETS

5-ring 5-ring-methyl 6-ring 6-ring-methyl

−0.96 −1.61 −1.30 −2.18

Quantitites defined in Table 1.

Table 6. Bond Distances (Angstroms) between Terminal Atoms in the 5-Ring, 5-Ring-Methyl, 6-Ring, 6-Ring-Methyl Claisen Rearrangement for Optimized Transition Statesa system 5-ring 5-ring-methyl 6-ring 6-ring-methyl

geometry

O−C

C−C

chair boat chair boat chair boat chair boat

1.95 1.98 1.96 2.01 1.97 2.00 1.97 2.02

2.53 2.66 2.54 2.68 2.53 2.64 2.54 2.66

a

Results based on optimizations for each transition state with the B3LYP functional and the cc-pVDZ basis. Each stationary point was confirmed by vibrational analysis and was shown to possess a single imaginary frequency corresponding to the 3,3 sigmatropic shift characteristic of Claisen rearrangements. Figure 1. Error in ΔΔE in kilocalories per mole at the B3LYP/ccpVDZ transition states using various methods relative to the CCSD(T) (d/t) extrapolated results (see Table 4). The DFT and HF results are in the cc-pVTZ basis, and the correlated results are from the (d/t) extrapolated values for the given method.

results presented above, we have focused on B3LYP/cc-pVDZ optimized geometries. For all four compounds, the O−C bond distances are comparable to those of compounds 1 and 2, but the C−C bonds are approximately 0.2 Å longer than those observed in Table 2. Perhaps more surprisingly, despite the greater bulk of ligands in compounds 3−6, the ΔETS between chair and boat transition states for a given compound are F


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insensitive to orbital choice, and we felt the comparison of B3LYP and RPA/BLYP was fairer (the same nonhybrid portion of the functionals) because we were not making an exhaustive survey of methods. Once again it is seen that the B3LYP and M06-2X results bracket the RPA results, with GD3BJ providing ΔΔE results similar to those of RPA. However, the GD3BJ ΔETS results tend to be nearly as far below the RPA results as the M06-2X results are above them. Finally, the HF results are in fairly good agreement with the RPA results, though somewhat below. Sadly, compounds 3−6 are too large to treat using CCSD(T), and even if they were accessible in small basis sets, the results of Table 4 indicate the important differential effects on ΔΔE of basis set extension. However, experimental results are available for these compounds (ΔGTS and ΔΔG) that shed some light on the relative accuracy of the methods. We used unscaled frequencies from the B3LYP/cc-pVDZ transition-state optimizations to correct our ΔETS values for thermal effects (yielding estimated ΔGTS values), and these results are presented in Table 9 (see also Figure 2). Experimental values from one of our laboratories83 are presented for comparison.

smaller than those for either compound 1a or 1b. On the other hand, addition of a single methyl group has a larger impact on ΔΔE for compounds 3−6 than addition of two methyl groups did for compounds 1a or 1b. Extended basis set results are presented in Tables 7 and 8, where we treat basis sets only up to cc-pVTZ. On the basis of Table 7. Relative Energies of the Chair and Boat B3LYP/ccpVDZ 5-Ring and 5-Ring-Methyl Transition States Using Various Methods in Extended Basis Setsa method

basis

ΔETS 5ring

ΔETS 5-ringmethyl

ΔΔE

B3LYP M06-2X B3LYP-GD3BJ HF MP2 RPA/BLYP RPA/BLYP

cc-pVTZ cc-pVTZ cc-pVTZ cc-pVTZ cc-pVTZ cc-pVTZ Extrap(d/t)

−0.72 −1.65 −0.29 −1.43 −1.82 −1.04 −0.88

−1.67 −3.50 −1.48 −2.43 −4.10 −2.35 −2.22

−0.95 −1.85 −1.19 −1.00 −2.28 −1.31 −1.34

a

Results are in kilocalories per mole based on the chair and boat B3LYP/cc-pVDZ optimized geometries. Energy differences are for the given method and basis set. Energy differences are as defined in Table 1, and the extrapolation method is defined in the text.

Table 8. Relative Energies of the Chair and Boat B3LYP/ccpVDZ 6-Ring and 6-Ring-Methyl Transition States Using Various Methods in Extended Basis Setsa method

basis

ΔETS 6ring

ΔETS 6-ringmethyl

ΔΔE

B3LYP M06-2X B3LYP-GD3BJ HF MP2 RPA/BLYP RPA/BLYP

cc-pVTZ cc-pVTZ cc-pVTZ cc-pVTZ cc-pVTZ cc-pVTZ Extrap(d/t)

−1.14 −2.74 −0.97 −2.15 −2.98 −1.90 −1.76

−2.28 −4.68 −2.20 −3.34 −5.35 −3.39 −3.23

−1.14 −1.94 −1.23 −1.19 −2.37 −1.49 −1.47

Figure 2. Mean unsigned error (MUE) in kilocalories per mole of various methods relative to the experimental results for ΔGTS at 80 °C (see Table 9).

a

Results are in kilocalories per mole based on the chair and boat B3LYP/cc-pVDZ optimized geometries. Energy differences are for the given method and basis set. Energy differences are as defined in Table 1, and the extrapolation method is defined in the text.

No method yields exact agreement with experiment, but the following observations characterize the data: (1) M06-2X results for ΔETS and ΔΔE are consistently larger than the experimental results. (2) MP2 results are qualitatively similar to the M06-2X results but are even farther away from the experimental results. Interestingly, the HF values are relatively close to the RPA results.

the comparable extrapolated results for compounds 1a and 1b in Table 4, we felt this was an adequate level of treatment for the present systems. In addition, we present results based on only RPA/BLYP. While RPA/PBE is more common in the literature, our results (Table 4) showed the RPA was relatively

Table 9. Estimated Free Energy Differences for Chair and Boat Transition States for the 5-Ring, 5-Ring-Methyl, 6-Ring, and 6Ring-Methyl Compounds at the B3LYP/cc-pVDZ Stationary Pointsa method

ΔGTS 5-ring

ΔGTS 5-ring-methyl

ΔGTS 6-ring

ΔGTS 6-ring-methyl

MSE/MUEb

B3LYP cc-pVTZ M06-2X cc-pVTZ B3LYP-GD3BJ cc-pVTZ HF cc-pVTZ MP2 cc-pVTZ RPA/BLYP cc-pVTZ RPA/BLYP Extrap(d/t) exptl

−0.27/-0.22 −1.20/−1.15 0.17/0.22 −0.98/−0.93 −1.37/−1.32 −0.59/−0.54 −0.43/−0.38 −0.80/−0.63

−1.88/−1.91 −3.70/−3.73 −1.70/−1.72 −2.62/−2.67 −4.28/−4.31 −2.56/−2.59 −2.43/−2.46 −1.79/−1.65

−0.80/−0.76 −2.43/−2.39 −0.63/−0.59 −1.80/−1.77 −2.64/−2.62 −1.56/−1.52 −1.42/−1.38 −1.31/−1.44

−1.91/−1.86 −4.36/−4.30 −1.82/−1.78 −2.97/−2.92 −4.98/−4.93 −3.02/−2.97 −2.86/−2.81 −2.40/−2.51

0.36/0.41 −1.35/1.35 −0.58/0.58 −0.49/0.49 −1.74/1.74 −0.36/0.46 −0.21/0.40

a

Results are in kilocalories per mole and use vibrational frequencies from the B3LYP/cc-pVDZ calculations to estimate thermal and entropic corrections. The two values for each entry are based on temperatures of 80° and 125 °C. bMean signed error (MSE) and mean unsigned error (MUE) relative to experimental results at 80 °C. G


Article

The Journal of Physical Chemistry B (3) B3LYP and GD3BJ results for ΔETS and ΔΔE are consistently lower than the experimental results, and the GD3BJ results are somewhat more erratic than B3LYP results in comparison with experiment. (4) The RPA results are somewhat closer to experiment than other methods. (5) All methods yield similar trends for relative reactivity and substitution effects on chair versus boat energetics. While it is unlikely to be the dominant source of the theory− experiment discrepancy here, it is worth noting that the B3LYP/cc-pVDZ geometries and frequencies capture the temperature dependence of only the unsubstituted 5membered ring compound. In all other cases the theoretical results predict temperature dependences in the opposite direction from the experimental results, though the effect is modest in all cases.

quantitative but not qualitative. The effect is even smaller when one considers ΔΔE values where B3LYP and RPA results are quite similar. Comparing B3LYP and GD3BJ results, it appears that inclusion of the GD3BJ correction marginally decreases the agreement with experiment. (2) Are there trends among the results of these methods for these reactions? A few appear, though further generalization should await new results. We found that inclusion of correlation with extended basis sets tended to reduce the difference between chair and boat transition-state energies. In addition, all methods did a fair to good job of predicting the impact of substitution on relative transition-state energies. (3) Do activation energies tell the same story as activation free energies for these reactions? Yes, for gas-phase species. One expects a similar story for solution-phase reactants. (4) How do the results of the various methods compare with experimental results for these systems? All methods gave good qualitative predictions of the transition-state energy variations with substitution and ring structure. RPA and B3LYP were somewhat more accurate when compared with experiment, but similar conclusions would have been drawn from M06-2X and GD3BJ for the systems examined here. More work is required to probe the limits and strengths of the RPA, but on the basis of the present study we feel this work is merited. It is a fast, correlated approach with strengths that are needed for the treatment of transition states of differentially substituted reactions. It appears to be at least as accurate for the reactions considered here as any of the most common (and believed accurate) available DFT methods. In coming work we will examine a considerably wider range of substituted pericyclic systems and compare further with experiment. We will also examine activation energies (in addition to activation energy differences as was done here) to probe for the presence of error cancellation as a source of the RPA’s relatively good accuracy.

4. DISCUSSION Perhaps the most important take-home message from the present results is that all of the methods we considered provide useful semiquantitative estimates of ΔETS and ΔΔE when compared with CCSD(T) or experimental data. Organic chemists have regularly used B3LYP and M06-2X in modest basis sets62,64−66 with good success to estimate activation energies. Our second-order comparison (comparing dif ferences of activation energies for similar reactions) tends to minimize methodological differences. The focus in this study then shifts to nonbonded interactions and their impacts in different transition-state structures. Given the demonstrable success of conventional DFT for organic reaction chemistry, these effects are not expected to be large. However, it is clear they are not negligible. If we take as standards the extrapolated CCSD(T) results for compounds 1a and 1b and the experimental results for compounds 3−6, then M06-2X appears to overestimate the relative steric interactions. This is a separate question from how well it treats, for example, protobranching of carbon chains or Diels−Alder activation energies.84−86 Its errors are not catastrophic but do disagree with the “standard” ΔΔE values for these compounds to a greater degree than the other methods examined. The addition of BSSE- and dispersion-corrections via the GD3BJ correction yields modest changes relative to B3LYP results in a given basis. In addition, the changes are not uniformly for the better or worse, just modestly different. Both B3LYP and GD3BJ yield trends in agreement with experiment and higher-level methods. The RPA results are by no means perfect but are relatively systematic. Basis set extension tends to lessen both ΔETS and ΔΔE, and the agreement with higher-level calculations or experiment is generally at least as good as any of the other methods. The observed decrease in ΔETS and ΔΔE for these reactions are inconsistent with a dominant role for BSSE in the energy changes but suggests that differential correlation effects are somewhat more important here. It is fair to say, however, the effect is modest (on the order of 0.3−0.7 kcal/mol for ΔETS). Interestingly, the ΔΔE values (changes in relative transition-state energies upon substitution) are less basis-setdependent. We now return to the questions posed in the Introduction: (1) How important is it to include dispersion treatment for the relative activation energies of this series of reactions? Judging by the comparison of B3LYP and RPA ΔETS values, the effect is

5. CONCLUSIONS We examined several DFT-based methods for calculating relative activation energies for a series of pericyclic reactions. We found that all of the methods yield qualitative agreement with CCSD(T) or experimental results for these systems, with somewhat better performance for relative energy differences by B3LYP and RPA. Oddly enough, one could interpret this as suggesting dispersion is unimportant (B3LYP success) or it is quantitatively important (RPA success). The success of B3LYP may be due to error cancellation of BSSE and lack of dispersion as is suggested by the trend in relative activation energies in the RPA results (decreasing activation energy difference). At the very least our results for differential activation energies suggest that the RPA is competitive with conventional, commonly used methods for predicting reaction thermochemistry and merits further study.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: 909-607-2629. *E-mail: [email protected]. Phone: 909-607-3505. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.K.H. gratefully acknowledges support from the Harvey Mudd College Stauffer Summer Research Endowment. G.W.D. H


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and R.J.C. acknowledge generous institutional support from Harvey Mudd College in the pursuit of this work, and R.J.C. acknowledges a grant of computer time from XSEDE.

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Comparison of Relative Activation Energies Obtained by Density

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