Canadian Journal of Chemistry. The INV24 Test Set: How Well do Quantum-Chemical Methods Describe Inversion and Racemization Barriers?

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1 The INV24 Test Set: How Well do Quantum-Chemical Methods Describe Inversion and Racemization Barriers? Journal: Manuscript ID cjc r1 Manuscript Type: Article Date Submitted by the Author: 15-Jul-2016 Complete List of Authors: Goerigk, Lars; The University of Melbourne, School of Chemistry Sharma, Rahul; The University of Melbourne, School of Chemistry; Indian Institute of Technology Roorkee, Department of Chemistry Keyword: barrier heights, chemical inversion, racemization barriers, benchmarking, density functional theory

2 Page 1 of 38 The INV24 Test Set: How Well do Quantum-Chemical Methods Describe Inversion and Racemization Barriers? Lars Goerigk, and Rahul Sharma, School of Chemistry, The University of Melbourne, Victoria 3010, Australia Present address: Department of Chemistry, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand , India lars.goerigk@unimelb.edu.au Abstract For years, there has been ongoing interest in experimentally and theoretically understanding inversion and racemization processes. However, to the best of our knowledge, there has been no computational study that systematically investigated how well low-level quantum-chemical methods predict inversion barriers. Herein, we provide an answer to this question and we present the INV24 benchmark set of 24 high-level, abinitio inversion barriers. INV24 covers inversion in triatomics, in pyramidal molecules, in one cyclic system, and in various helical and bowl-shaped compounds. Our results indicate that previously applied DFT approximations combined with small basis sets are not reliable enough and that at least a triple-ζ basis is needed for meaningful results. Moreover, we show that intramolecular London dispersion influences the barriers by 2 kcal/mol or more and that dispersion corrections should always be applied to DFT results. With our analysis of 34 DFT approximations we can reproduce the well-known 1

3 Page 2 of 38 Jacob s Ladder scheme with (meta-)generalized-gradient-approximation methods underestimating barriers and global-hybrid DFT functionals performing better. Rangeseparated hybrids or Minnesota-type hybrids are not particularly superior to more conventional methods, such as B3LYP-D3. The by far best results are achieved with dispersion-corrected double hybrids, which give results below the chemical-accuracy target of 1 kcal/mol. They also outperform wave-function second-order perturbation theory approaches and we recommend using them whenever possible. Given that our systematic study of INV24 is the first of its kind, our findings have the potential to change common practice in this field and they will guide future investigations of inversion processes. Keywords: barrier heights, chemical inversion, racemization barriers, benchmarking, density functional theory, wave function theory 1 Introduction Studying the energetic barriers of inversion processes has been of ongoing interest in the Quantum Chemistry community for decades. Even in recent times, where the general focus has shifted to the treatment of large systems, understanding fundamental inversion barriers in small molecules has been the target of various research groups. These studies were usually carried out with ab-initio methods, and they treated the barriers to linearity in triatomic molecules usually water 1 3 or dihydrogen sulfide 4 as well as inversion barriers in trigonal-pyramidal systems, such as ammonia, 3,5 9 trimethylamine 9,10 or general XH 3 compounds, with X standing for the group-15 elements from N to Bi. 11 In the context of this special issue dedicated to Professor Arvi Rauk, it is only appropriate to particularly highlight his contributions to this field. In 1970, his group analyzed the ammonia molecule and obtained a value for its inversion barrier that came very close to the Hartree-Fock (HF) complete-basis-set (CBS) limit. 5 In the same year, Rauk et al. published a seminal paper that unraveled the underlying physicochemical and theoretical concepts of 2

4 Page 3 of 38 pyramidal inversion; by combining these insights with experimental observations, they provided a unified description of the entire process. 12 Subsequent studies then changed the focus to other hydrides of first-, second-, 13 and third-row elements. 14 Given improved hardware capabilities, Rauk s research in the late 1980s then shifted to inversion barriers of larger molecules, such as azetidine 15 and tetrasilabicyclo[1.1.0]butane. 16 Most of Rauk s studies in this area were published in high-impact journals, which highlights the importance of this research and the general significance of understanding inversion processes. Also the racemization barriers of helical systems have been of interest to both experimentalists 17 and computational chemists alike, with one of the first computational studies on such systems being carried out with semi-empirical molecular-orbital (MO) theory already in The majority of papers on this topic have been published in the 1990s, however, also in very recent times joint experimental-computational studies have been conducted. 22,23 More recently, the experimental and computational interest in inversion barriers has movedtobowl-shapedmolecules,suchascorannulene, sumaneneanditsderivatives, and other bowl-shaped hydrocarbons, including some derived from fullerene fragments Also, bowl-shaped and warped ( pringle-shaped ) boron-nitride and aluminum-nitride analoguesofthosehydrocarbonshavebeenstudied. 37 Thesignificanceofunderstandinginversion in such systems has been demonstrated by Karton, who investigated the catalytic potential of graphene sheets on bowl-to-bowl inversion. 38 A closer look at the above mentioned studies on helices and bowls reveals that highlevel ab-initio studies have been rarely conducted. Instead, most computational analyses were limited to the semi-empirical MO, HF or density functional theory (DFT) levels. With rare exceptions, 38 the latter methodology has almost exclusively been employed with the B3LYP 39,40 hybrid density functional approximation and relatively small basis sets, even though there is ample evidence that this level of theory may be problematic for barriers of other reaction types Furthermore, the publication years of the cited studies demonstrate that understanding inversion processes is also nowadays of significance to the general 3

5 Page 4 of 38 community of chemists. Therefore, it is crucial to thoroughly investigate current quantumchemical methods in this context before they are applied further in related studies. This is particularly important as we are nowadays faced with a zoo of methods, which makes it hard for the mere method user to identify robust and accurate approaches that may have surpassed more popular, older strategies. In order to facilitate a better understanding of this method zoo, comprehensive benchmark databases have been developed that cover several chemical problems simultaneously, such as the Gn test sets, the GMTKN24 and GMTKN30 databases for general main group thermochemistry, kinetics and noncovalent interactions, 48,49 as well as a variety of other comprehensive sets, such as the ones reviewed in Ref. 50. In the context of reaction barrier heights, it is worthwhile to particularly highlight standalone test sets for hydrogen transfer, heavy-atom transfer, nucleophilic-substitution, unimolecular, and association reactions (the HTBH38, 51 NHTBH38, 52 and DBH24/ sets), for pericyclic reactions (BHPERI 48,56 and CRBH20 57 ), for water-catalyzed protontransfer reactions, 58 for proton exchange in water, ammonia, and hydrogen-fluoride clusters, 59 and rotational barriers in bifuranes and their analogues. 60 Except for a limited study of fullerene fragments in 1999, 61 no comprehensive benchmark study of inversion barriers has been published. Herein, we will close this gap and we present a benchmark set that comprises 24 high-level wave-function inversion barriers in triatomic, trigonal-pyramidal, cyclic, helical and bowl-shaped compounds, which we dub INV24 (see Fig. 1). The aim of this study is to gain insight into how modern methods compare to older, more popular strategies in the description of such barriers. In particular, we want to answer the question whether one has to go to demanding levels of theory, or whether error-cancellation effects would also favor computationally less demanding approaches. This question is particularly interesting for inversion barriers, as during this process no bonds are formally broken and as the number of electron pairs in the systems do not change. A second important topic will be the influence of London-dispersion forces on barriers. Given 4

6 Page 5 of 38 Figure 1: Lewis structures of the 24 molecules in the INV24 benchmark set. the ongoing interest in understanding inversion processes, our findings will ultimately be important and insightful for future experimental and computational studies conducted in this field. 2 The INV24 Test Set The Lewis structures of the 24 molecules in INV24 are shown in Fig. 1, while Tab. 1 lists their names, their calculated high-level barrier heights and the levels of theory at which those have been obtained. The reference values are zero-point-exclusive electronic energies. The barrier heights in INV24 range from 4.1 to 79.7 kcal/mol with an average value of 31.8 kcal/mol. Tab. 1 also shows how the 24 systems are divided into five categories covering 5

7 Page 6 of 38 Table 1: Reference values (kcal/mol) for the inversion barrier heights of the INV24 set. # system barrier level of theory triatomics 1 H 2 O 31.7 W2-F12 2 H 2 S 69.3 W2-F12 3 SO W2-F12 4 O(CH 3 ) W2-F12 5 S(CH 3 ) W2-F12 trigonal pyramids 6 N(CH 3 ) W2-F12 7 NCl W2-F12 8 P(CH 3 ) W2-F12 9 PCl W2-F12 10 PH 2 Ph 31.2 W2-F12 11 PPh DLPNO-CCSD(T)/CBS cyclic 12 dibenzocycloheptene 10.3 DLPNO-CCSD(T)/CBS helices 13 tetrahelicene 4.5 W1-F12 14 pentahelicene 24.7 DLPNO-CCSD(T)/CBS 15 hexahelicene 37.6 DLPNO-CCSD(T)/CBS 16 dibenzo[c,g]carbazole 4.1 DLPNO-CCSD(T)/CBS 17 monomethinecyanine 13.1 DLPNO-CCSD(T)/CBS bowls 18 corannulene 11.2 DLPNO-CCSD(T)/CBS 19 BN-analogue of corannulene 6.2 DLPNO-CCSD(T)/CBS 20 sumanene 21.3 DLPNO-CCSD(T)/CBS 21 triazasumanene 42.3 DLPNO-CCSD(T)/CBS 22 BN-analogue of sumanene 27.2 DLPNO-CCSD(T)/CBS 23 tetrabenzo[bc,ef,hi,kl]pyracylene 8.4 DLPNO-CCSD(T)/CBS 24 triindenotriphenylene 68.6 DLPNO-CCSD(T)/CBS triatomics, trigonal pyramids, one cyclic system, helices and bowls. In the remainder of this section, we briefly discuss these systems and comment on the calculation of their reference inversion barriers. Tabs. S1 and S2 in the Supporting Information (SI) provide details on all total and electron-correlation energies needed for the derivation of the reference values. 6

8 Page 7 of 38 INV24 includes five triatomic systems: water, dihydrogen sulfide, sulfur dioxide, dimethylether and dimethylthioether. Six systems in INV24 have a trigonal pyramidal shape; these are either ammonia or phosphane derivatives with methyl, chloro or phenyl substituents. With the exception of triphenylphosphane (molecule 11), all molecules have been small enough to obtain non-relativistic, all-electron reference values for their inversion barrier heights with the highly accurate W2-F12 62 level of theory. W2-F12 is a composite, thermochemical wavefunction protocol that recovers the Coupled Cluster Singles Doubles and Perturbative Triples [CCSD(T)] 63 CBS limit by a series of finite-basis-set calculations partially involving explicitly correlated 64 techniques; see the original reference for technical details. The accuracy of W2-F12 for heats of formation has been established as being smaller than 1 kcal/mol, 65 a value that is normally regarded as the chemical accuracy threshold. For relative energies, such as reaction energies or barrier heights, an even smaller error is expected. 62,65 Therefore this level of theory is ideal to obtain accurate benchmarks for the assessment of lower-level methods, as previously shown for barrier heights in different chemical processes. 56,57,59 In fact, it has been demonstrated that W2-F12 outperforms other popular composite wavefunction approaches for barrier heights of pericyclic reactions. 56 For the water molecule, we can directly compare the W2-F12 result with an estimate for the experimental barrier to linearity, which has been given as cm 1 (31.4 kcal/mol). 66 Our calculated W2-F12 value of 31.7 kcal/mol shows an error of only 0.3 kcal/mol (about 1%). Note that effects such as higher-order excitations in the coupled-cluster treatment, relativistic effects, or non-born-oppenheimer contributions have been discussed in detail elsewhere. 1 Given that herein only lower level methods will be assessed, our W2-F12 value is therefore an acceptable and very accurate reference. For triphenylphosphane, an elaborate treatment such as W2-F12 or the closely related W1-F12 62 protocol has not been feasible and a different strategy to obtain a CCSD(T)/CBS estimate for its inversion barrier had to be employed. Herein, we used Neese s Domain Based Local Pair Natural Orbital technique [DLPNO-CCSD(T)]. 67 The 7

9 Page 8 of 38 DLPNO approach depends on a set of three adjustable parameters and we followed recent suggestions by Neese, Martin and co-workers, and chose a setup dubbed TightPNO that provides an error of only 0.13 kcal/mol compared to conventional CCSD(T) calculations. 68 The value at the CBS limit was obtained from two-point extrapolations, where the basis-set convergence of the Hartree-Fock (HF) total energy is described as: 69 E HF (X) = E HF (CBS)+Aexp( α X) (1) and the convergence of the CCSD(T) electron-correlation energy as: 70 E corr (CBS) = Xβ E corr (X)+Y β E corr (Y) X β Y β (2) where X and Y are the cardinal numbers of the two involved basis sets, α and β basissets specific constants and A another constant, which is normally not determined but instead expressed through a second HF calculation with a basis set with cardinal number Y. Herein, we used the Ahlrichs-type quadruple-ζ (X=4) def2-qzvpp 71 and triple-ζ (Y=3) def2-tzvpp 71 atomic-orbital (AO) basis sets, with values of α=7.88 and β= For the smaller PH 2 Ph molecule, we have been able to compare the DLPNO-CCSD(T)/CBS result with the W2-F12 level. We obtained a barrier of 31.1 kcal/mol for DLPNO-CCSD(T)/CBS compared to 31.2 kcal/mol for W2-F12. Thus, DLPNO-CCSD(T)/CBS is an acceptable compromise for larger systems. Dibenzocycloheptene (molecule 12) is an example for the inversion between two different envelope conformations of a cyclic system. Our original intention was to include various cyclic systems in the benchmark set, however, most barrier heights turned out to be insignificantly small compared to the other systems and hence it seemed prudent to leave those out. The reference value for 12 is 10.2 kcal/mol and it has also been determined at the DLPNO- CCSD(T)/CBS level. The five helical systems in INV24 are the prototypical systems tetra-, penta-, and hexa- 8

10 Page 9 of 38 helicene, as well as a carbazole and a methinecyanine; the latter was inspired by a low-level computational study of similar systems. 21 It is worthwhile to note that the first and last two systems in this category have planar transition states, whereas penta- and hexahelicene have warped transition-state structures (see Fig. S1 in the SI for a depiction). 19,20 The geometries and W1-F12 reference value for tetrahelicene have been taken from Ref. 38; all other reference values have been determined for this work at the DLPNO-CCSD(T)/CBS level. The final seven systems in INV24 are bowl-shaped molecules. Inspired by Karton s study of bowl-to-bowl inversion catalyzed by graphene, 38 we made corannulene and sumanene part of the test set. Interestingly, substituting three CH moieties in sumanene by nitrogen atoms (triazasumane) doubles the DLPNO-CCSD(T)/CBS barrier (Tab 1). To determine whether low-level methods also capture this effect, we have decided to include it in INV24 (molecule 21). Recent works have also shown interest in inversion of bowl-shaped and warped boronnitride (BN) systems, which is why we also included BN analogues of corannulene and sumanene (molecules 19 and 22). Bowl-shaped hydrocarbons are of particular interest because they can be understood as fragments of fullerenes. In fact, molecules 23 and 24 are precursors in fullerene synthesis. 35,36 These fragments have also been studied with the B3LYP density functional approximation in an early computational study of bowl-to-bowl inversion. 61 In summary, the new INV24 set covers different inversion barriers in a wide variety of systems that represent those that both experimentalists and computational chemists have focussed on in the past and present. To the best of our knowledge, the reference values for the larger systems, i. e. beyond the triatomics and ammonia, are the most accurate published so far. They will serve as a reliable benchmark to assess more commonly applied low-level methods. 9

11 Page 10 of 38 3 Computational Details Non-relativistic,all-electronW2-F12 62 referencevalueswereobtainedwithmolpro ,74 ORCA was used for all geometry optimizations, Møller-Plesset type perturbation theory treatments and for the majority of density functional theory (DFT) calculations. An ORCA development version (the precursor of the upcoming ORCA4.0 release) was used for the DLPNO-CCSD(T) calculations. All geometries were obtained at the B3LYP 39,40 - D3(BJ) 76 /def2-tzvpp 71 level of theory. The structures for tetrahelicene were taken from Ref. 38, where they had been optimized at the same level of theory. The xyz coordinates of all structures are available for download (see SI). DLPNO-CCSD(T) 67 calculations can be adjusted by setting three separate truncation thresholds to select which electron-pair contributions should be calculated and which omitted, see Refs. 67 and 68 for details. Herein, we used the computationally most demanding TightPNO setup suggested by Neese, Martin and co-workers. 68 Rather than testing arbitrarily chosen methods, the DFT approximations tested in this study have either been shown to be generally reliable in previous studies on general thermochemistry and kinetics with the large GMTKN30 benchmark database, 42 they are extremely popular, or they are relatively new and, thus, of interest. Herein we assessed functionals that formally belong to rungs 2-5 on Perdew s Jacob s Ladder. 77 Rung-2 functionals follow the generalized gradient approximation (GGA) or Truhlar s nonseparable gradient approximation 78 (NGA) ideas and herein we tested the ten functionals B97-D3, 79 revpbe, 80 PBE, 81 PW91, 82 BP86, 83,84 BLYP, 83,85 mpwlyp, 85,86 OLYP, 85,87 rpw86pbe 81,88 the semi-local exchange-correlation component of the VV10 89 van-der-waals (vdw) density functional and the Minnesota functional N Rung-3 functionals (meta-ggas) contain higher-order derivatives of the electron density with respect to spatial coordinates. Herein we investigate the popular TPSS 90 functional, as well as the three highly parametrized Minnesota-type methods M06L, 91 M11L, 92 and MN12L

12 Page 11 of 38 In hybrid density functionals (rung 3), parts of the semi-local DFT exchange is replaced with nonlocal Fock-exchange (well known from HF theory). We have assessed the 13 functionals PW6B95, 94 B3LYP, B3PW91, 39 PBE0, 95,96 TPSSh, 97 BHLYP, 98 MPWB1K, 99 and theminnesotafunctionalsm052x, 100 M06, 101 M062X, 101 M11, 102 N12SX, 103 andmn12sx. 103 Note that most of these methods are global hybrid functionals and only the last two are relatively new range-separated methods. Other range-separated methods have been excluded from this study as previous works have shown that they are not necessarily better than their global counterparts for reaction energies or barrier heights. 42,56 Rung-5 functionals take into account information from unoccupied (virtual orbitals). Various formulations of this idea exist, with double-hybrid density functionals being the currently most practicable approach for general applications. 104 In double hybrids, parts of the semi-local DFT correlation are replaced by non-local second-order perturbation theory. Herein, welimitourassessmenttosixdoublehybridsthathaveshowntobeverypromisingin rigorous benchmark studies: B2PLYP, 105 B2GPPLYP, 54 DSD-BLYP, 106 DSD-PBEP86, 107 DSD-PBEB95, 108 and PWPB Additionally, we also include a B2PLYP variant that has been specifically parametrized for barrier heights and kinetics (B2KPLYP). 109 Most DFT calculations were carried out with ORCA3.0.2 with the self-consistent-field (SCF) convergence option TightSCF and a numerical integration grid grid5. All calculations also made use of the resolution of the identity (RI) approximation for the evaluation of Coulomb integrals, 110,111 and the chain-of-spheres approximation 112 for the evaluation of exchange integrals in hybrid and double-hybrid functionals. The second-order perturbation correction in double-hybrids was also assessed with the RI approximation. 110 All Minnesota functionals were assessed with Gaussian09 Rev. D with an SCF convergence criterion of 10 7 E h and the finegrid option. All density functionals have been dispersion-corrected withgrimme sdft-d3dispersioncorrection. 114 InmostcasesthelatestversionwithBecke- Johnson damping 115 has been applied [DFT-D3(BJ)]. 76 Note that it has been established thatminnesotafunctionalsalsobenefitfromdispersioncorrections, 116,117 however,depending 11

13 Page 12 of 38 on the functional, this can sometimes be the older DFT-D3 114 variant with zero-damping [DFT-D3(0)]. This is the case for the functionals N12, M06L, M11L, M052X, M06, and M062X. In the following discussion, the exact nature of the dispersion correction is not relevant for our main messages, which is why we will use the suffix -D3 in all cases. Both versions of the DFT-D3 correction have been implemented in ORCA and GAUSSIAN. An additional analysis is also carried out with the non-local vdw correlation kernel by Vydrov and van Voorhis. 89 This has originally been developed for the VV10 functional, 89 but it can also be combined with other density functionals. 118 In this case, this approach is sometimes highlighted with the suffix -NL for non-local. 118 The NL kernel was evaluated in ORCA in a post-scf fashion with the special numerical integration grid vdwgrid4. Most DFT calculations have been carried out with the def2-qzvp quadruple-ζ AO basis set. 71 However, we also assessed the basis-set dependence with the def2-tzvp triple-ζ and the def2-svp double-ζ basis sets; the latter was not used for double-hybrid density functionals as it has been established that results are not reliable with small basis sets. 42 While this study focuses mostly on DFT approximations, a comparison will also be made with HF, MP2, and its spin-component-scaled SCS-MP2 119,120 variant. In this paper, we will only discuss results at the CBS limit obtained from calculations with the def2-tzvpp and def2-qzvpp basis sets (see previous section for more details on the extrapolation to the CBS limit). 4 Results and Discussion 4.1 The Importance of London Dispersion The significance of taking into account London-dispersion forces in computational chemistry scenarios that go beyond the determination of mere noncovalent interaction energies has been pointed out numerous times; see Ref. 121 for a recent review. Due to the fact that conventional DFT approximations do not properly describe these forces, it has been 12

14 Page 13 of 38 Figure 2: Signed deviations (kcal/mol) for all 24 systems in the INV24 set for dispersioncorrected and -uncorrected versions of the BLYP (a), B3LYP (b), and PWPB95 (c) density functional approximations. Part d compares signed deviations for B3LYP corrected with either the DFT-D3 or VV10 (NL) approaches. A negative signed deviation indicates an underestimation of inversion barriers. demonstrated that applying a dispersion correction 121 is needed even for common DFT treatments, such as the optimization of molecular geometries, 76,79,125 and the calculation of reaction energies 42,126 or barrier heights. 42,56 It turns out that nearly all DFT calculations of inversion barrier heights with one noteworthyexception 38 haveexcludeddispersioncorrections. Thisisinlinewithageneral trend that can be observed in many computational (organic) chemistry applications, with the negative consequences of this trend being pointed out in Ref. 43. For the present study, it is therefore crucial to begin our discussion by investigating how intramolecular dispersion 13

15 Page 14 of 38 effects may influence inversion barriers. For that purpose, we restrict our analysis to the BLYP, B3LYP and PWPB95 functionals as representatives for rungs 2, 4, and 5 of Jacob s Ladder. Signed deviations from reference values for each of the 24 molecules for the dispersion-corrected and -uncorrected functionals are shown in parts a-c of Fig. 2; all results are based on calculations with the quadrupleζ def2-qzvp atomic-orbital (AO) basis set. While the actual magnitude of the DFT-D3 dispersioncorrectiondiffersforeachfunctional somethingthatiswell-known 76,79,114 one can nevertheless observe common trends for the barriers to linearity: dispersion corrections do not influence the barriers for the smaller molecules water, dihydrogen sulfide and sulfur dioxide (1-3), however, they already have a discernible effect on the ether and thioether (4 and 5). For instance, the absolute deviations for B3LYP decrease by 0.4 and 0.7 kcal/mol for these two molecules when a dispersion correction is applied (Fig. 2b and Tabs. S12 and S13 in the SI). Also for most of the trigonal-pyramidal systems, adding a dispersion correction leads to smaller absolute deviations. The importance of intramolecular dispersion becomes evident for larger substituents. Indeed, the barrier in triphenylphosphane (11) is affected the most: for BLYP the barrier increases by more than 2 kcal/mol from 22.3 to 24.6 kcal/mol when a dispersion correction is applied, while it increases from 24.9 kcal/mol for B3LYP to 26.8 kcal/mol for B3LYP-D3 (see Tabs. S3, S4, S12, and S13). Even in the cycloheptene system (12), dispersion affects the inversion barrier by about 1 kcal/mol, while dispersion effects can reach up to 2 kcal/mol for most barriers in helices and bowls. These contributions are well above the chemical-accuracy limit of 1 kcal/mol, and they should therefore not be neglected. Overall, parts a-c in Fig. 2 show that dispersion minimizes the absolute deviations in most of the 24 cases. This can also be seen from the mean deviations (MDs) and mean absolute deviations (MADs) for the three functionals. Adding a dispersion correction leads to a more robust treatment of inversion barriers with MDs being closer to zero: MD= 3.1 kcal/mol for BLYP vs. 1.6 kcal/mol for BLYP-D3, and MD= 1.7 kcal/mol for B3LYP vs

16 Page 15 of 38 Table 2: Mean deviations (MDs) and mean absolute deviations (MADs) in kcal/mol for DFT-D3- and DFT-NL-corrected density functionals averaged over INV24. Method MD MAD rpw66pbe-d VV revpbe-d revpbe-nl B3LYP-D B3LYP-NL B2PLYP-D B2PLYP-NL kcal/mol for B3LYP-D3 (Tab. 2). Our findings also reconfirm that dispersion corrections are needed for double-hybrid density functionals, 49,127 despite their nonlocal perturbative correlation portion (MD= 0.6 kcal/mol for PWPB95 vs. 0.3 kcal/mol for PWPB95-D3). The MADs improve from 3.4 to 2.3 kcal/mol for BLYP, from 1.9 kcal/mol to 1.0 kcal/mol for B3LYP, and from 1.0 kcal/mol to 0.8 kcal/mol for PWPB95 when the dispersion correction is applied. Even though the general trend shows the DFT-D3 correction improves the description of inversion barriers, outliers cannot always be ruled out and in some cases adding the correction may overshoot the barrier. This can for example be seen for the bowl-shaped fullerene fragment 24, whose barrier is underestimated by nearly 3 kcal/mol for B3LYP, but overestimated by a similar magnitude for B3LYP-D3 (Fig. 2b). Interestingly, applying the nonlocal vdw kernel originally developed for the VV10 functional to B3LYP (B3LYP- NL) fixes this problem somewhat and the deviation for molecule 24 is only 0.5 kcal/mol (Fig. 2d). Fig. 2d also provides a comparison between B3LYP-D3 and B3LYP-NL for the 23 other systems, but with the exception of some bowl-shaped systems the errors for B3LYP-D3 and B3LYP-NL are fairly similar. Averaged over the entire benchmark set, the MD for B3LYP-NL is 0.9 kcal/mol. In fact, the MAD for B3LYP-NL is the same as for B3LYP-D3 (1.0 kcal/mol); see also Tab. 2. Furthermore, Tab. S4 in the SI shows results for the vdw GGA functional VV10 and its DFT-D3-corrected semi-local exchange-correlation 15

17 Page 16 of 38 component ( rpw6p986-d3). Based on those results, one can conclude that the example given for B3LYP cannot be transferred directly to this GGA functional. For instance, VV10 underestimates the barrier for system 24 by about 0.5 kcal/mol, but also rpw6p86-d3 is close to the reference value with a deviation of 0.8 kcal/mol. Averaged over the entire set, the performance of rpw6p86-d3 is very similar to that of VV10 (Tab. 2). The same is also true when the VV10 kernel is combined with the revpbe GGA functional or the B2PLYP double-hybrid functional. Considering that the evaluation of the nonlocal kernel is computationally more demanding than estimating the DFT-D3 dispersion contribution, we recommend using the latter in routine treatments of large systems. The first main conclusion of this study is, therefore, that intramolecular London dispersion does indeed influence the height of inversion barriers and that dispersion-corrected DFT does provide statistically better results than the underlying density functionals alone. Even though there may be exceptions, one also observes in Fig. 2 that adding a dispersion correction usually does not make the results worse and that for smaller systems the influence of dispersion is nearly zero. Given that the DFT-D3 correction does not cost anything, we therefore recommend to always apply it to be on the safe side. For the sake of a compact discussion we will continue our analysis by applying the DFT-D3 correction to all tested density functionals. This also includes Minnesota functionals, as pointed out previously. 116,117 Dispersion-uncorrected results, however, are also listed in the SI. 4.2 Analysis of Separate Cases In the following, we will briefly analyze how dispersion-corrected DFT and some common wave-function methods behave for the separate categories of INV24 before we continue with an analysis of the entire set. The results discussed in the following subsections are based on calculations with the quadruple-ζ def2-qzvp AO basis set, which provides results close to the CBS limit for most DFT methods. Our analysis of dispersion-corrected DFT methods is supported by Fig. 3 and Tab. 3. The bars in Fig. 3 show the average MAD for each of the 16

18 Page 17 of 38 Figure 3: The bars represent mean absolute deviations(mads) in kcal/mol averaged for four different classes of density functionals: GGAs, meta-ggas, hybrids and double hybrids. The vertical lines depict the range of MADs for each functional class, with the two ends of each line showing the smallest and largest MAD within that class. Results are shown for four separate categories, as well as for the entire INV24 set. All MADs are based on def2-qzvp calculations, and all functionals have been dispersion corrected with the DFT-D3 approach. four assessed functional classes (or rungs) according to Jacob s Ladder, while the vertical lines indicate the energy range that the various MADs cover in each class. Tab. 3 lists the three best dispersion-corrected methods in each functional class based on their MADs. MADs are either calculated for the four separate main categories of INV24 (excluding the cycloheptene system) or they are based on all 24 systems in INV24. More detailed results, including MDs for each functional, are shown in the SI and all given numbers and examples can be verified therein in sections SI.4 and SI.5 (Tabs. S26-S51) Barriers to Linearity All ten GGA functionals underestimate the barriers to linearity in triatomic molecules. In fact, a look at Tabs. S27 and S28 reveals that the MADs are exactly the absolute values of their related MDs. MADs for GGAs range between 2.1 (OLYP-D3) and 4.4 kcal/mol (N12- D3) and the average MAD for second-rung functionals is 3.2 kcal/mol (Fig. 3). Meta-GGAs are on average more accurate. However, it has to be conceded that the average MAD of 2.2 kcal/mol is based on only four methods whose MADs differ by only 0.6 kcal/mol or less. Thatbeingsaid, themadforthebestmeta-gga,m11l-d3, isonly0.2kcal/mollowerthan 17

19 Page 18 of 38 Table 3: The top 3 functionals in each of the four functional classes for the various subcategories and the entire INV24 test set. The numbers in parentheses are the mean absolute deviations of the respective methods. a All functionals have been dispersion-corrected with the DFT-D3 approach, however, the suffix -D3 has been left out for better clarity. b GGA meta-gga hybrid double-hybrid triatomics 1. OLYP (2.1) M11L (1.9) B3LYP (1.5) DSD-PBEB95 (0.8) TPSSh (1.5) 2. B97-D3 (2.4) M06L (2.0) B3PW91 (1.6) PWPB95 (1.1) DSD-PBEP86 (1.1) 3. revpbe (2.9) TPSS (2.3) PBE0 (1.8) DSD-BLYP (1.2) pyramids 1. B97-D3 (2.8) M06L (2.7) B3LYP (1.3) B2PLYP (0.7) 2. OLYP (3.4) TPSS (3.0) TPSSh (1.5) DSD-PBEP86 (0.9) 3. BLYP (3.6) MN12L (4.2) PW6B95 (1.8) DSD-BLYP (1.0) M06 (1.8) B2GPPLYP (1.0) helices 1. OLYP (0.6) M11L (0.6) B3PW91 (0.1) B2PLYP (0.2) M06L (0.6) 2. B97-D3 (0.7) TPSS (1.3) PW6B95 (0.3) B2GPPLYP (0.3) MN12L (1.3) PBE0 (0.3) N12SX (0.3) 3. N12 (0.8) B3LYP (0.4) PWPB95 (0.4) DSD-PBEP86 (0.4) B2KPLYP (0.4) bowls 1. N12 (0.7) M06L (0.7) M06 (0.5) PWPB95 (0.4) 2. PBE (0.8) MN12L (0.9) PBE0 (0.6) B2PLYP (0.6) PW91 (0.8) N12SX (0.6) DSD-BLYP (0.6) MN12SX (0.6) 3. BP86 (1.0) TPSS (1.0) PW6B95 (0.7) DSD-PBEB95 (0.7) M11L (1.0) M062X (0.7) B2GPPLYP (0.7) M052X (0.7) B2KPLYP(0.7) INV24 1. revpbe (2.2) M06L (1.5) B3LYP (1.0) B2PLYP (0.7) 2. BLYP (2.3) TPSS (1.8) PW6B95 (1.1) PWPB95 (0.8) BP86 (2.3) B3PW91 (1.1) DSD-PBEP86 (0.8) TPSSh (1.1) B2GPPLYP (0.8) DSD-PBEB95 (0.8) DSD-BLYP (0.8) 3. PBE (2.4) M11L (2.0) PBE0 (1.2) B2KPLYP (0.9) rpw86pbe (2.4) a In some cases, various functionals have the same MADs. b All results are based on calculations with the def2-qzvp AO basis set. that for OLYP-D3. The MADs for the 13 assessed hybrid functionals show the largest spread with the best methods having an MAD of 1.5 kcal/mol (B3LYP-D3 and TPSSh-D3) and the worst 5.1 kcal/mol (BHLYP-D3). In fact, BHLYP-D3 is the worst of all tested functionals for barriers to linearity. The average MAD for hybrids is with 2.8 kcal/mol higher than 18

20 Page 19 of 38 that for meta-ggas. The two tested range-separated hybrids show mediocre performance with an MAD of 2.2 kcal/mol for MN12SX-D3 and an MAD of 3.7 kcal/mol for N12SX-D3. Double-hybrid density functionals show the best performance. Even the worst double hybrid has an MAD that is better than that of the best hybrids (1.4 kcal/mol for B2PLYP-D3). The average MAD for double hybrids is only 1.2 kcal/mol, and indeed some functionals are below or very close to the chemical-accuracy limit of 1 kcal/mol (DSD-PBEB85-D3, PWPB95-D3, DSD-PBEP86-D3). Low-level wave-function methods have also been traditionally used in the analysis of inversion barriers and, hence, they are also of interest for the present study. Results for HF, MP2 and its spin-component-scaled variant SCS-MP2 at the CBS limit are shown in Tab. 4. The results for both the MADs and MDs clearly show that electron correlation plays a decisive role in an accurate description of barriers to linearity in the five tested systems, with MP2 and SCS-MP2 having significantly lower MADs (MAD=2.8 and 2.0 kcal/mol, respectively) than HF (MAD=10.9). However, the results also indicate that the two tested perturbative methods provide results of only (meta-)gga DFT quality, and that hybrid and double-hybrid functionals are better suited for the description of these systems. Like conventional DFT approximations, HF is also incapable of describing London-dispersion effects. An interesting finding in previous works has been that HF corrected with DFT- D3(BJ) provides noncovalent interaction energies of dispersion-corrected GGA quality, 42 and that it also has the potential to sometimes provide structures that are of dispersioncorrected hybrid-dft quality or better. 76,125,128,129 Having demonstrated the importance for dispersion corrections at the DFT level for INV24, it is therefore logical to also assess dispersion-corrected HF. The findings in Tab. 4, however, indicate that in this instance, HF does not benefit from the correction and that results become worse. While results for HF-D3 are also shown for the other categories in Tab. 4, we will therefore not explicitly comment on this approach in the remainder of this paper. 19

21 Page 20 of 38 Table 4: Mean absolute deviations and mean deviations (values in parentheses) in kcal/mol for low-level wave-function methods at the complete-basis-set (CBS) limit for INV24 and four of its subcategories. triatomics pyramids helices bowls INV24 HF 10.9 (9.9) 9.9 (6.0) 1.7 (1.7) 1.4 ( 0.6) 5.5 (3.8) HF-D (10.9) 10.8 (7.6) 3.1 (3.1) 6.2 (6.2) 7.5 (6.7) MP2 2.8 ( 2.7) 0.6 ( 0.4) 0.9 (0.9) 1.7 (1.7) 1.5 (0.1) SCS-MP2 2.0 ( 1.1) 1.1 (1.1) 0.9 (0.9) 1.0 (1.1) 1.2 (0.6) Trigonal-Pyramidal Systems Contrary to the barriers to linearity in triatomic systems, the average MADs for trigonalpyramidal systems shown in Fig. 3 restore the Jacob s Ladder picture, with GGAs having on average the largest MAD (4.0 kcal/mol), closely followed by meta-ggas (3.7 kcal/mol). Hybrid functionals have on average an MAD of 2.4 kcal/mol, which is further halved for double hybrids (1.0 kcal/mol). Compared to the other three classes, the MADs for the various double hybrids lie closer to one another, with the best double hybrid having an MAD of 0.7 kcal/mol (B2PLYP-D3) and the worst an MAD of 1.3 kcal/mol (PWPB95-D3). Note that the MAD of PWPB95-D3 is the same as for B3LYP-D3, which is the best hybrid functional. All other MADs are usually higher, with the best meta-gga having a value of 2.7 kcal/mol (M06L-D3), which is very close to the best GGA functional (B97-D3); see Tab. 3. The worst functionals in the second, third, and fourth rungs of Jacob s Ladder are the Minnesota-type methods N12-D3 (MAD=5.3 kcal/mol), M11L-D3 (MAD=4.9 kcal/mol), and N12SX-D3 (MAD=4.3 kcal/mol). Contrary to the barriers to linearity, MP2-based methods yield MADs that are within the chemical-accuracy limit (Tab. 4). In fact, the value for MP2/CBS is slightly better than for the best double hybrid with an MAD of 0.6 kcal/mol. Again, HF theory is not suitable to describe these barriers with an unacceptably high MAD of about 10 kcal/mol. 20

22 Page 21 of Helical Systems Contrary to triatomic and trigonal-pyramidal systems, DFT approximations behave more homogeneously for racemization barriers in helical systems. While the Jacob s Ladder scheme is again reproduced, the average MADs are all close or within the chemical-accuracy limit. In fact, the three best second-rung functionals are competitive with higher-rung methods: OLYP-D3 has an MAD of only 0.6 kcal/mol, followed by B97-D3 with 0.7 kcal/mol, and N12-D3 with 0.8 kcal/mol. The best two meta-ggas are M11L-D3 and M06L-D3 with MADs of 0.6 kcal/mol. With the exception of MPWB1K-D3 (MAD=1.0 kcal/mol), M052X- D3 (MAD=1.1 kcal/mol) and M11-D3 (MAD=1.4 kcal/mol), all tested hybrid functionals have MADs below 1 kcal/mol, with B3PW91-D3 having nearly perfect mean and mean absolute deviations of 0.1 and 0.1 kcal/mol, respectively. Double hybrids combined have an average MAD of only 0.4 kcal/mol, and all their MADs are 0.7 kcal/mol or lower. This makes recommending particular double hybrids unnecessary, as they all lie close to the error expected for the high-level reference values. The results for wave-function methods also show that inversion barriers in helical systems are less sensitive to the underlying method, with the difference between HF and the MP2 methods being only 0.8 kcal/mol. Both MP2 variants have an MAD of 0.9 kcal/mol Bowl-Shaped Systems Also the average errors for inversion barriers are generally smaller than for the triatomic and pyramidal systems, however, the differences between the various methods are larger than for the helical systems. The average MAD for GGAs is 1.7 kcal/mol and only three methods have an MAD below 1 kcal/mol: N12-D3 (MAD=0.7 kcal/mol), PBE-D3 (MAD=0.8 kcal/mol), and PW91-D3 (MAD=0.8 kcal/mol). The worst second-rung functional in this case is interestingly B97-D3, which is one of the best performers in the other subcategories of INV24 (MAD= 4.7 kcal/mol). The average MADs for rungs 3 and 4 on Jacob s Ladder are both below 1.0 kcal/mol. With the exception of M11-D3, all tested hybrids have an MAD 21

23 Page 22 of 38 of 1 kcal/mol or lower. Again, all double hybrids are within the chemical-accuracy limit, with PWPB95-D3 having the lowest MAD (0.4 kcal/mol). This study therefore confirms Karton s assumption that double hybrids are the most robust density functionals to study bowl-to-bowl inversion. 38 Interestingly, MP2/CBS is outperformed by most DFT methods (MAD=1.7 kcal/mol). Moreover, HF/CBS has a smaller MAD of 1.4 kcal/mol. The MDs on the other hand differ, with HF/CBS having a negative MD of 0.6 kcal/mol, while MP2/CBS tends to overestimate the barrier heights (MD=1.7 kcal/mol). Mean and mean absolute deviations of SCS-MP2 are about 1 kcal/mol. 4.3 Overall Analysis We conclude the discussion of the def2-qzvp-based results with an analysis of the entire INV24 benchmark set, i.e. the previously discussed four categories are combined and the result for the inversion in the cycloheptene derivative (12) is included. Like for most of the previously discussed categories, we also see for the combined set that the Jacob s Ladder scheme is reproduced. The average error for rung-2 functionals is 2.4 kcal/mol, rung-3 functionals have a slightly reduced average MAD of 1.8 kcal/mol, while hybrid functionals have a value of 1.6 kcal/mol. The average error of double hybrids is half of that of hybrids with a value of 0.8 kcal/mol, indicating that overall, double hybrids are the most accurate functionals to describe inversion barrier heights. The differences between the MADs for the individual functionals are also the smallest for double hybrids. For instance, B2PLYP-D3 has the best MAD of the entire study (0.7 kcal/mol), while the MAD for the worst doublehybrids is only 0.2 kcal/mol higher(b2kplyp-d3). In fact, the MADs for all double hybrids are below the chemical-accuracy threshold of 1 kcal/mol and also their MDs (see Tab. S22) are close to the ideal value of zero, indicating that also for inversion barriers double hybrids show high robustness. Hybrid functionals show a starker contrast with individual MADs ranging from

24 Page 23 of 38 kcal/mol (B3LYP-D3) to 2.5 kcal/mol (M11-D3 and BHLYP-D3). Also, the MDs show a larger difference and they range from 1.5 kcal/mol (N12SX-D3) to 1.5 kcal/mol (BHLYP- D3); see Tabs. S13 and S14. A closer inspection of the top-3 density functionals for each separate category of INV24 and the discussion in the previous sections clearly draw a somewhat inconsistent picture of rung-2 to rung-4 functionals. As we have already seen, B97-D3 is in the top-3 for triatomic, pyramidal and helical molecules, but the worst GGA for bowlshaped systems. Another example is the range-separated N12SX-D3, which is one of the worst hybrids for triatomics and pyramidal systems, but one of the best for helices and bowls. Therefore, an analysis of the combined INV24 set also allows us to assess the robustness of a given method, which is why Tab. 4 also lists the best three DFT approximations for each assessed rung for the entire INV24 set. According to this assessment, revpbe-d3 is the best GGA functional for inversion barriers with an MAD of 2.2 kcal/mol. However, it is closely followed by BLYP-D3, BP86-D3, PBE-D3 and rpw86pbe-d3. Thus, with the exception of rpw86pbe-d3, the GGA functionals that perform best are relatively established methods. Note however, that the MDs for all of these methods are negative, indicating a general tendency to underestimate inversion barrier heights. The best meta-gga functional is M06L-D3 with an MAD of 1.5 kcal/mol, but it is closely followed by TPSS-D3 with an MAD of 1.8 kcal/mol. The two methods have an underbinding tendency with negative MDs of 0.6 and 1.5 kcal/mol, respectively. The best hybrid functional is B3LYP-D3 (MAD=1.0 kcal/mol) and it is surprisingly followed by TPSSh (MAD=1.1 kcal/mol), which in previous studies has been flagged as being relatively unreliable. 42,56 The hybrids PW6B95-D3 and B3PW91-D3 have the same MAD as TPSSh- D3, and also PBE0-D3 can be recommended with an MAD of 1.2 kcal/mol. However, none of those aforementioned methods can compete with double hybrid functionals. For the reader s convenience, Tab. 5 lists the MDs and MADs for all tested DFT-D3- corrected functionals. More detailed results are provided in the SI. 23

25 Page 24 of 38 Table 5: Mean deviations (MDs) and mean absolute deviations (MADs) in kcal/mol for the entire INV24 set for all 34 dispersion-corrected DFT methods. a MD MAD rung 2 revpbe-d BP86-D BLYP-D rpw86pbe-d PBE-D PW91-D mpwlyp-d OLYP-D N12-D B97-D rung 3 M06L-D TPSS-D M11L-D MN12L-D rung 4 B3LYP-D TPSSh-D PW6B95-D B3PW91-D PBE0-D MN12SX-D M06-D M062X-D M052X-D MPWB1K-D N12SX-D M11-D BHLYP-D rung 5 B2PLYP-D PWPB95-D DSD-PBEP86-D DSD-BLYP-D B2GPPLYP-D DSD-PBEB95D B2KPLYPD a All results are based on calculations with the def2-qzvp basis set. 24

26 Page 25 of 38 Finally, Tab. 4 allows us to assess HF/CBS, MP2/CBS and SCS-MP2/CBS for INV24. The MAD of 5.5 kcal/mol for HF is reduced to 1.5 kcal/mol by the MP2-correlation contribution. Allowing for a more balanced treatment of same- and opposite-spin electron pairs in SCS-MP2, brings the MAD further down to 1.2 kcal/mol. Thus, SCS-MP2 is competitive with dispersion-corrected hybrid functionals, but in general it cannot compete with doublehybrids. Given the same computational cost of MP2-type approaches and double hybrids, the latter are therefore recommended for a treatment of inversion barriers in future studies. 4.4 Dependence on Fock exchange and Basis-Set Effects We conclude our study by addressing whether there is any noteworthy dependence of the barrier on the amount of Fock exchange or nonlocal exchange mixed into a given density functional, and also by commenting on the general basis-set dependence of the DFT results. For usual reaction barrier heights that are characterized by elongated bonds, it is well known that the self-interaction error (SIE) leads to a general underestimation of barrier heights, while pure HF theory (100% of Fock exchange) usually overestimates barriers. 42 Table 6: Mean deviations (MDs) and mean absolute deviations (MADs) in kcal/mol for various DFT methods and HF compared to their portion of Fock-exchange. a % Fock exchange MD MAD BLYP-D TPSSh-D B3LYP-D B3PW91-D PBE0-D PW6B95-D BHLYP-D B2PLYP-D3 b HF/CBS a All results are based on calculations with the def2-qzvp basis set, except for HF, which is based on a CBS extrapolation. b Also contains 27% of nonlocal, perturbative correlation. 25

27 Page 26 of 38 It is worthwhile to investigate if similar trends can be observed for inversion barriers. Indeed, we have already seen above that GGAs and meta-ggas (0% of Fock exchange) generally have an underbinding tendency, indicated by negative MDs. As one representative forggas, themdandmadforblyp-d3areshownintab. 6andcomparedtotheB3LYP- D3 and BHLYP-D3 functionals, which contain the same semi-local DFT components, but have different nonlocal-exchange contributions (20 and 50%, respectively). The relatively small fraction of 20% nonlocal exchange in B3LYP-D3 improves both statistical values by more than 1 kcal/mol, with a negative MD of 0.5 kcal/mol. BHLYP-D3, on the other hand, clearly overshoots the barriers with an MD of 1.4 and a significantly larger MAD. HF theory complements the results accordingly and with 100% of Fock exchange it overestimates barriers with an MD of 3.8 kcal/mol. In this sense, general trends previously observed for barrier heights in different reactions can also be observed for inversion processes. One way of improving the result of a functional with high amounts of Fock exchange is to mix in nonlocal correlation, as done in the B2PLYP-D3 double hybrid. It has a nonlocalexchange contribution very similar to that of BHLYP-D3, however, it has a nearly perfect MD of 0.2 kcal/mol and the best MAD of the entire study. Tab. 6 also lists results for global hybrid functionals that vary in their Fock-exchange contributions between 10% and 28%, and in principle the statistical results are very close to those of B3LYP-D3. Hence, one can conclude that a global hybrid functional applied to inversion barriers should ideally have a nonlocal-exchange contribution in that percentage range. The previous analysis of DFT approximations has been carried out with the large quadrupleζ def2-qzvp basis set to ensure that any basis-set effects were reduced to a minimum. In practical applications, however, smaller basis sets have to be applied. In the case of inversion barriers, we see that the basic molecular composition is unchanged during the reaction and that we only have to deal with geometric differences between the minimum-energy and transition-state structures. Therefore, it is justified to ask if any possible basis-set effects 26