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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Assessment of Newest Meta-GGA Hybrids for Late Transition Metal Reactivity: Fractional Charge and Fractional Spin Perspective Marcin Modrzejewski,†,‡ Grzegorz Chalasinski,*,† and Malgorzata M. Szczesniak*,§ †

Faculty of Chemistry, University of Warsaw, Pasteura 1, Warsaw 02-093 Poland Department of Chemical Physics and Optics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, CZ-12116 Prague 2, Czech Republic § Department of Chemistry, Oakland University, Rochester, Michigan United States Downloaded via UNIV OF LOUISIANA AT LAFAYETTE on November 14, 2018 at 21:12:13 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: In recent years there has been a significant interest of DFT community in the development of new (hybrid) meta-generalized gradient approximations (metaGGAs), including range-separated hybrids. The new DFT methods, e.g., SCAN, SCAN0, ωB97M-V, and our own LCPBETPSS-D3 promise an improvement over well-established models, such as, PBE, PBE0, ωB97X-D, and the M06-type functionals. However, the tests published to date cover only main-group chemistry. This work fills this gap by examining reactivity of model systems, such as gold-ligand complexes, Pd and Ni insertion reactions into covalent bonds, and the pathway for olefin metathesis by a model Grubbs system, all of which include late transition metals. In the attempt to rationalize the performance of functionals, we study the fractional charge and fractional spin errors of the Au atom and the Au7 cluster. While we find the main qualitative issues of DFT are not yet solved, the introduction of meta-GGA ingredients yields a notable improvement makeing the new meta-GGAs the preferred choice for transition-metal chemistry.

I. INTRODUCTION Kohn−Sham density functional theory (DFT) accounts for the properties of a many-electron system via the Hamiltonian for independent particles in an effective one-electron potential which makes up for electron correlation. It is applicable to molecules, nanoparticles, solids, and interfaces due to favorable computational scaling.1 Depending on the type of the ingredients entering the exchange-correlation energy, a hierarchy of DFT approximations is established known as rungs on the Jacob’s ladder of approximations.2 The increasingly sophisticated ingredients allow for a more flexible form of a functional which obeys more physical constraints. The lowest rung is the local spin density approximation (LSDA) constructed to reproduce the energy of the uniform electron gas, an important limiting case for the solid state. The generalized gradient approximation (GGA) extends LSDA by incorporating dependence on the density gradient. GGAs improved dramatically some of the chemistry’s most important properties, such as atomization energies. Meta-GGAs occupy the third rung by incorporating the kinetic energy density and, optionally, the Laplacian. The functionals on this rung, owing to their more sophisticated form, acquire important new capabilities, e.g., of recognizing one-electron densities. LSDA, GGAs and meta-GGAs are semilocal functionals depending on the quantities computed at grid points. © XXXX American Chemical Society

The nonlocality is added on the next rung via the inclusion of the Hartree−Fock (HF) exchange to GGAs and metaGGAs. The hybrid functionals brought down dramatically the errors in atomization energies and improved the description of barrier heights. The HF exchange can be admixed either as a constant fraction of the exchange (global hybrids) or as a varying fraction of it dependent on the interelectron distance (range-separated hybrids). The latter kind proved successful in ameliorating many problems of pure semilocal functionals related to their handling of charge-transfer excitations, polarizabilities of long unsaturated chains, and the dissociation of symmetric radicals−all attributable to a large extent to the many-electron self-interaction error.3 A great deal of progress has also been achieved in recent years toward the understanding of how to incorporate the meta-GGA ingredients into more effective functionals.4,5 The first-principles strongly-constrained appropriately normed (SCAN)6 functional appears to be the crowning achievement of these efforts as it includes all known constraints applicable to the meta rung. Therefore, until some new constraints are Special Issue: Hans-Joachim Freund and Joachim Sauer Festschrift Received: July 31, 2018 Revised: November 2, 2018 Published: November 2, 2018 A

DOI: 10.1021/acs.jpcc.8b07394 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

performance on databases of reference data and statistical analysis of the errors. Such an approach calls for generating new and more sophisticated databases. Alternatively, one can focus on the errors themselves. Such perspective is offered by studying the behavior of density functional approximations (DFA) in the function of fractional charges and fractional spins.12 In the extension of DFT to fractional electrons Perdew et al.13 demonstrated that the exact energy of a system with noninteger charge, e.g. He0.5+, lies on the straight line segment between the adjacent integer electron values (1/2E(He) + 1 /2E(He+). The original statement was based on a canonical ensemble DFT. Later Yang et al. showed that it can be restated for pure states of molecules.14 Common functionals violate this condition by overstabilizing fractionally charged (FC) systems (convex behavior) whereas HF destabilizes them (concave behavior). In DFT this problem gives rise to the delocalization error which manifest itself in many problems (dissociation of symmetric cations, barrier heights, charge transfer complexes, CT excitations, band gaps, polarizabilities of long chains, and many others). A similar exact condition for DFT was subsequently derived for systems with fractional spins.10 That is, the energy is independent of spin polarization, e.g. E(1↑) = E(1↓) = E(0.5↑, 0.5↓) in Li. This condition is also violated by common functionals which destabilize fractional spin (FS) states (concave behavior). This problem manifests itself in overbinding in stretched chemical bonds where the static correlation becomes important. It also leads to a large class of problems experienced by DFT in the treatment of degenerate and quasi-degenerate systems, singlet−triplet splitting, the so-called strongly correlated solids, and transition metal complexes. The two errors are intertwined, as recently demonstrated by Janesko et al.15 in the simultaneous treatment of H2+ (FC error) and H2 (FS error) who showed that common functionals which minimize the former maximize the latter and vice versa. Both errors are especially pernicious in transition metals because they tend to form coordinate covalent bonds affected by FC errors and have high densities of states linked to the static correlation, or FS, errors. Our paradigm late transition metal system, the Au atom, is used here to analyze both errors for a representative group of functionals, with special emphasis on those newly designed from the meta-GGA rung, both pure and hybrid. To examine the size-dependence of these errors we choose Au7 cluster which retains the same D6h structure in its three charge states Au7+, Au70 and Au7−. The calculations with fractional charges and fractional spins were performed in NWChem16 using unrestricted KS scheme in the DFT module with a FON option and, for some functionals, with our in-house developed DFT program. Fractional charge/spin calculations are analogous to the traditional KS self-consistent field algorithm but with density matrices built using fractional occupation numbers of selected spin−orbitals. An extra-fine grid and tightened convergence criteria were employed. In the case of a non-spin-polarized system with 1/2 ↑ and 1/2 ↓ spins (in HOMO spin−orbitals), the restricted KS solution can be used. All calculations employed def2-TZVP basis set17 for Au which includes a 60 electron ECP. The departures from linearity expressed in eV have parabolic shapes in both cases. In the fractional charge case, the departure is, for the most part, convex whereas for the fractional spins it is always concave. The curvature coefficients (in eV) are thus used here to quantify static correlation and

discovered it will remain the best in its class of nonempirical, local meta-GGA functionals. On the empirical side, the new range-separated hybrid metaGGA of Mardirossian and Head-Gordon, ωB97M-V, also appears to be state-of-the-art as it was combinatorially optimized to a record number of data points - about five thousand.7 To date the tests of this functional have been confined to main-group chemistry for which it was trained. Finally, a new nonempirical range-separated hybrid metaGGA has recently been developed by our group. The LCPBETPSS exchange-correlation energy8 combines the shortrange PBE-based exchange, long-range HF exchange, and TPSS correlation: EXC(LC-PBETPSS) = EXSR (PBE) + EXLR(HF) + EC(TPSS). The short-range part of a long-range corrected (LC) hybrid exchange requires a model for the exchange hole. A unique feature of our model is that it is built on the Becke−Roussel exchange hole,9 which depends at each point of space not only on the GGA ingredients, the density ρ and its gradient, but also on the kinetic energy density τ and the Laplacian ∇2ρ. Consequently, in the LC-PBETPSS functional the PBE exchange is elevated to the long-range corrected hybrid meta-GGA rung. All details can be found in ref 8. The simple constructs employed in meta-GGAs and their hybrids invariably lead to errors in chemical applications. An important class of DFT limitations can be understood using the generalizations of DFT to fractional spins and fractional charges.10 This limitation was articulated by Ruzsinszky and Perdew11 in Twelve outstanding problems in ground state density f unctional theory as Problem 9: “Can we describe long-range charge transfer and static correlation both?” Incidentally, none of the 17 known exact constraints satisfied by SCAN mitigates fractional charge and fractional spin situations arising in strongly correlated systems and indeed SCAN by the authors own admission applies to cases where the exchange-correlation hole is sufficiently localized.6 In the present work, we examine the newly developed metaGGAs and their hybrids in application to transition metal chemistry. Transition metals pose much greater problems to DFT compared to the main group chemistry because they have more complex level structures related to much higher densities of states. For example, while carbon has only four electronically excited states within 7.65 eV of the ground state, Ni has an infinite number of them because its ionization energy is 7.64 eV. This fact, in combination with higher angular momentum, makes transition metals capable of forming a much wider variety of bonds from sextuple covalent to coordinate covalent to back-donation, etc. One of the objectives is to validate our new nonempirical functional in this challenging chemistry. In addition, we aim to understand why functionals fail if they do. To this end, we first discuss the problem of fractional charge and fractional spin errors in prototypical transition metal case. Next we examine the performance of these newly developed functionals as they apply to specific problems in transition metal chemistry: gold−ligand interactions, bond-activation by transition metals, and olefin metathesis. The performance of each tested method is rationalized, or attempted to be, as a compromise between the fractional-charge and fractional-spin errors.

II. RESULTS AND DISCUSSION II.1. Static Correlation and Delocalization Errors in Gold. We typically assess DFT methods by testing their B


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Figure 1 shows that both errors plotted vs one another correlate linearly. A similar dependence was demonstrated by

delocalization errors on the equal footing. For convenience, an absolute value of the FS curvature coefficient is used. Table 1 shows the FS error and FC errors for both ionization and attachment in Au along with ionization Table 1. Fractional Spin (FS) and Fractional Charge (FC) Errors in Au Atom in eV; I and A Denote Ionization Potential and Electron Affinity, Respectively Functional PBE RPBE BLYP BP86 M06-L TPSS MVS SCAN PBE0 B3LYP M06-2X TPSSh SCAN0 HSE06 LC-ωPBE (ω = 0.3) LC-ωPBE (ω = 0.4) CAM-B3LYP LC-PBETPSSa HF experimentb a

FS error

FC error attachment

FC error ionization

0.18 0.21 0.13 0.18 0.23 0.22 0.34 0.40 0.66 0.51 1.10 0.41 0.82 0.46 1.72 1.87 1.20 1.77 1.96

3.09 3.06 3.18 3.08 2.82 2.99 2.71 2.67 2.15 2.45 1.44 2.62 1.86 2.55 0.12 −0.17 1.12 0.07 −0.33

3.57 3.53 3.65 3.55 3.19 3.44 3.18 3.13 2.55 2.84 1.57 3.05 2.24 2.94 0.42 0.06 1.42 0.08 −0.25

I (eV)

A (eV)

9.53 8.74 9.50 9.70 8.89 9.35 8.87 9.67 9.21 9.41 8.95 9.24 9.37 9.21 9.10 8.90 9.23 8.98 7.68 9.22

2.14 1.27 2.13 2.32 1.94 2.01 1.61 2.24 1.87 2.08 1.56 1.92 1.98 1.87 1.69 1.52 1.62 1.70 0.43 2.31

ROKS was used for fractional charge calculations. webbook.nist.gov/.

b

Figure 1. Fractional charge vs fractional spin error in Au in Def2TZVP basis set calculations. The values of both errors are from Table 1 (see the text for definitions).

Janesko et al. for the dissociation of H2+ (FC-error) vs H2 (FSerror) with a variety of functionals. It is noteworthy that it is 100% HF exchange in the longrange that brings FS error to the Hartree−Fock territory. CAM-B3LYP which asymptotically includes 65% of the HF exchange has a smaller FS error. Thus the correct −1/r asymptotic behavior of the exchange-correlation potential which rectifies the effects of the delocalization error is also detrimental to the static correlation problem. Another possible scenario when FS error can be maximal and FC errors minimal involves short-range hybrids with 100% of the HF exchange in the short-range and semilocal DFT exchange in the long-range. Several such PBE-based screened hybrids with varying fraction of HF exchange are shown as a supplement to Table 1 (see Supporting Information). Interestingly, such short-range hybrids are very sensitive to the range separation parameter ω and the FS/FC error reversal can be seen when the HF exchange is switched off sufficiently early, e.g., at (ω = 0.4)−1 a0 interelectron distance. That even a small fraction of the HF exchange causes an increase in the static correlation error was explained by Janesko et al.15 as follows: A DFT exchange, e.g., GGA-exchange, accounts for the static or left−right correlation in addition to bona f ide exchange effect (see Handy and Cohen24); when some part of DFT exchange is replaced by the HF exchange the capability of DFT to account for the static correlation is gradually being lost. In Table 2 the FC and FS errors are shown for Au7. The aim here is to examine the scaling of these errors with the size of gold for selected functionals. It appears that the delocalization errors for both attachment and ionization have gone down by a ratio of ca. 1.8 for pure functionals and ca. 2 for global hybrids. The associated static correlation error also drops down by an even higher factor. Does it mean that the errors improve with the system size? As concerns the FC errors, the smaller curvature represents a smaller departure from linearity which may appear as a good thing. However, the slopes of the straight lines between the integer electrons, i.e., ionization potential and electron affinity are too low, as evidenced by systematically underestimated ionization potentials and electron affinities compared with

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potentials and electron affinities computed as energy differences between ions and the neutral. Pure functionals, PBE, BLYP, TPSS, M06-L, MVS,18 and SCAN, show small FS errors. The error variation with respect to the exchange functional, e.g. Becke88 vs PBE exchange, is quite small. These functionals also reproduce the ionization potential and electron affinity of Au reasonably well (except for RPBE, MVS). In contrast, their FC errors for both ionization and attachment are very large. Notably, newer meta-GGAs, MVS and SCAN, show a slight improvement in the delocalization error but at the expense of a slight increase in the static correlation error. Upon addition of a constant fraction of the HF exchange in hybrid functionals the FC errors drop roughly in proportion to the HF percent contribution, i.e. TPSSh19 error drops the least, M06-2X drops the most and PBE0 and SCAN020 by a similar ratio. The FS error simultaneously rises in the same proportion, i.e. the least for TPSSh and the most for M06-2X. The range-separated group includes a short-range (screened) hybrid, HSE06,21 and long-range hybrids, CAMB3LYP, LC-ωPBE and our own LC-PBETPSS. The shortrange hybrid HSE06 shows reduced FS error compared to PBE0 but the FC errors rise slightly. The long-range hybrids reduce the delocalization error practically to zero, but the static correlation error balloons to the Hartree−Fock size (1.96 eV). To complement these findings, we might add that previous studies have shown22 that mid-range hybrids (e.g., HISS23) have similar appealing characteristics as the short-range hybrid HSE06. C


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thus showing too much of a HF behavior. The static correlation error is some of the largest of the studied group. Concluding this section, one may attempt to offer an exchange-correlation hole perspective on the properties of these functionals. Long-range corrected hybrids by design have exceptionally small FC errors and particularly large FS errors (Table 1). This must be so as long as the range-separated exchange is applied without nonlocal correlation. For weakly interacting subsystems (e.g., NaCl) with fluctuating number of electrons, the HF exchange correctly keeps the number of electrons integer as the subsystems dissociate.29 However, in systems like stretched H2, the HF exchange hole becomes too diffuse. In the case of the exact wave function, the nonlocal correlation hole cancels the Fermi hole so that the sum of the two, the exchange-correlation hole, is localized around the reference electron (see Figure 1 in ref 30). This static correlation effect is simulated by the localized exchangecorrelation holes of pure functionals, but not by the holes of long-range corrected hybrids, where the semilocal correlation cannot systematically cancel the long-range exact exchange. As expected, long-range corrected functionals, including the new meta-GGAs, have the largest fractional spin errors. By this reasoning, the combination of fully semilocal long-range exchange and hybrid short- and middle-range parts in HSE makes the characteristics of this functional superior to traditional global hybrids.31 II.2. Gold−Ligand Test Cases. This section focuses on comparing the performance of our LC-PBETPSS functional with other functionals in a class of gold-ligand complexes. Gold forms strong donor−acceptor interactions with soft-base ligands. These interactions are challenging for DFT for two reasons: (1) susceptibility to a ground-state charge transfer (2) importance of dispersion interactions.32 The complexes considered here are formed between Au4 and the following ligands: SCN−, pyridine, benzothiol (HSC6H5), trimethylphosphine (P(CH3)3). In addition, a (HAuPH3)2 dimer with a prototypical aurophilic bond33 is included in the test set. The structure of (HAuPH3)2 is that of ref.;34 the remaining structures are optimized using ωB97X-D with def2-TZVPP17 basis set keeping the Au4 unit rigid. The coordinates can be found in the Supporting Information.

Table 2. Fractional Spin (FS) and Fractional Charge (FC) Errors in Au7 in eV, Where Iv and Av Denote Vertical Ionization Potential and Electron Affinity, Respectively Functional PBE M06-L TPSS MVS SCAN PBE0 B3LYP M06-2X TPSSh SCAN0 HSE06 LC-ωPBE(0.4) CAM-B3LYP LC-PBETPSS experimenta

FS error

FC error attachment

FC error ionization

Iv (eV)

Av (eV)

0.06 0.03 0.08 0.12 0.08 0.28 0.27 0.55 0.13 0.42 0.11 1.06 0.64 1.11

1.81 1.73 1.78 1.69 1.66 1.26 1.40 0.71 1.56 1.13 1.58 −0.31 0.53 −0.11

1.87 1.80 1.84 1.76 1.73 1.32 1.46 0.71 1.63 1.20 1.65 −0.29 0.57 −0.15

6.55 6.10 6.42 6.00 6.73 6.27 6.40 5.97 6.32 6.44 6.27 6.01 6.24 5.95 7.8

2.60 2.27 2.47 2.08 2.70 2.31 2.46 1.96 2.39 2.41 2.33 1.96 2.27 2.15 3.415

a

https://webbook.nist.gov/.

experimental values. Such improper scaling of the delocalization error was recently demonstrated by Yang’s group.25 Concerning the static correlation error, the decrease of the FS error with the system size was previously observed by Haunschild et al.22 in alkali metals: from Li to Cs the FS error decreased dramatically. Moving on to an even larger system, Au7, let us examine the HOMO spin orbitals of Au7 in which fractional spins can reside. HOMOα and HOMOβ are displayed below in Figure 2 HOMOα delocalizes on the entire cluster and shows nodal properties of a d-type orbital of a superatomic structure of Au7, whereas HOMOβ resembles a p-type orbital of the same superatom. The energetic effect thus corresponds to the fractional spin being distributed among all seven Au atoms. Pure functionals in Table 2 show noticeably small values of the FS error regardless of rung. This explains their (relatively) good performance in solids.26−28 Among the hybrids only TPSSh and HSE06 have relatively small FS errors. LCPBETPSS and LC-ωPBE reverse the curvature of the FC error,

Figure 2. Shapes of spin−orbitals (a) HOMOα and (b) HOMOβ in Au7 cluster. These orbitals have nodal structures of d-type and p-type orbitals, respectively, encompassing the whole cluster. D


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The Journal of Physical Chemistry C Table 3. Interaction Energies and Mean Absolute Errors (MAE) of Gold−Ligand Complexes in kcal/mola method\system

(AuH−PH3)2

Au4−SCN−

Au4−pyridine

Au4−benzothiol

Au4−P(CH3)3

MAE

reference LC-PBETPSS-D3/QZVP LC-PBETPSS-D3/AVTZ LC-wPBE-D3/AVTZ B3LYP-D3/AVTZ CAM-B3LYP-D3/QZVP M06-L/AVTZ M06-2X-D3/AVTZ ωB97M-V/QZVP SCAN/AVTZ SCAN0/QZVP DLPNO−CCSD(T)/AVTZ

−6.1 −5.6 −5.9 −6.9 −5.8 −4.9 −4.4 −3.4 −5.0 −7.1 −5.2

−54.6c −51.4 −51.8 −51.8e −50.7 −51.3 −48.6 −43.8 −49.3 −59.5 −55.1 −51.7

−33.2c −31.9 −32.5 −33.1 −30.1 −31.6 −26.2 −24.5 −29.3 −36.3 −32.2 −31.3

−32.7c −32.8 −33.5 −34.7 −30.5 −30.8 −26.0 −23.1 −29.4 −36.0 −33.0 −29.6

−58.0c −60.8 −61.7 −59.9f −54.3 −55.4 −48.6 −45.5 −52.9 −63.8g −60.1 −53.9

1.60 1.65 1.52 2.66 2.16 6.18 8.89 3.79 3.58 0.95 3.68

b

a

AVTZ denotes aug-cc-pVTZ basis set (aug-cc-pVTZ-PP in the case of Au) which in second row atoms includes an extra d function. QZVP stands for def2-QZVP basis sets.17 Both basis sets yield very similar results (see LC-PBETPSS-D3 entries). bFrom ref.32 two-point CBS extrapolation using canonical CCSD(T). cTwo-point CBS extrapolation using DLPNO−CCSD(T). dError of DLPNO−CCSD(T) vs canonical CCSD(T) in AVTZ basis set is 0.9 kcal/mol. eValue without D3 correction is −48.8 kcal/mol. fValue without D3 correction is −53.9 kcal/mol. gSCAN energies evaluated on the Hartree−Fock densities lead to −62.88 kcal/mol.

The reference values for interaction energies are based on the complete-basis-set (CBS)- extrapolated DPLNO−CCSD(T), except for the case of the aurophilic system for which the reference value was previously provided32 from a basis set extrapolated canonical CCSD(T). DFT computations involve NWChem 6.8, Q-Chem 5.0,35 and our own code, depending on the availability of functionals of interest. We employ two basis sets, aug-cc-pVTZ (aug-cc-pVTZ-PP for Au) and def2QZVP, with their appropriate pseudopotential variants in Au.17 We were compelled to use two basis sets because of convergence problems in Q-Chem in the former. As Table 3 demonstrates, in the first two entries, the results in LCPBETPSS are qualitatively similar. Grimme’s D3 dispersion terms36 accompany most functionals whereas ωB97M-V is equipped with the nonlocal dispersion.37 No dispersion corrections are employed for M06-L, SCAN and SCAN0. The results of Table 3 show that the range-separated functionals give reasonable predictions of interaction energies (MAE in the range of 1.5 to 2.12 kcal/mol) with the exception of ωB97M-V (MAE 3.79). Our range-separated meta-GGA functional LC-PBETPSS performs very well but not better than the range-separated GGA, LC-ωPBE-D3. A disappointing performance of ωB97M-V is surprising given the sophistication of this functional, i.e. meta rung, involvement of range separation and nonlocal dispersion. The M06-L functional, a reputed strong performer for transition metal chemistry,38 systematically underbinds and is clearly not suited for this class of complexes. Its exchangeenhanced variant M06-2X-D3 is even worse. Another surprise is an excellent overall performance of the hybrid SCAN0, despite its parent’s, SCAN’s, systematic overbinding. Does any of this performance correlate with FS and FC errors discussed earlier? One can expect that in the present complexes the fractional charge errors for electron addition to gold should matter. Indeed, in Figure 3 we see the subset of functionals where MAE correlates well with FC error for electron addition to Au7. The subset includes good performers LC-PBETPSS-D3 and LC-ωPBE-D3, an intermediate B3LYPD3 one, as well as a poor performer, SCAN. Therefore, the fractional charge error in gold may serve as a rough predictor of the performance of these functionals in gold-ligand interactions. The outliers are SCAN0−an excellent performer,

Figure 3. Fractional charge errors for electron addition to Au7 against MAE for Au4-ligand complexes. A subset of functionals shows a good correlation.

and ωB97M-V − a poor performer. In the case of SCAN0, one should recall that this functional is used without a dispersion correction (D3 has not been optimized yet for this functional) so its performance may be somewhat fortuitous. As far as ωB97M-V is concerned, we note that it systematically underbinds compared to the reference values. Furthermore, MAE is dominated by the two complexes: Au4−P(CH3)3 and Au4−SCN−. To establish whether the reason for this underbinding is related to dispersion, one may try to compute the interaction energies with and without dispersion. To do that we chose a good performer from Table 3, LC-ωPBE-D3, in which the dispersion term is not hard-wired to the rest of the parametrization, as it is in ωB97M-V, and performed calculations without the dispersion correction. The values obtained without dispersion (see footnotes to Table 3) are close to the ωB97M-V results. From this we hypothesize that the reason for underperformance of ωB97M-V has to do with an underestimation of dispersion in this class of complexes. It should be reiterated that this functional was not optimized for compounds of metals and heavy elements. It is also important to ascertain whether the overbinding by SCAN is an energy-expression error or a density-related error.39 For Au4−P(CH3)3, where the error is especially severe, we computed SCAN energies on the HF densities. The rationale is that if the delocalization error of SCAN leads to excessive charge transfer from P(CH3)3 to Au4, this excess will E


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The Journal of Physical Chemistry C be mitigated at the Hartree−Fock level. The SCAN interaction energy evaluated on the HF density is smaller in magnitude by only 1.1 kcal/mol. From this we conclude that the SCAN error is more likely energy-expression-related rather than densityrelated. II.3. Transition Metal Catalysis Test Cases. II.3.A. Bond Insertion Reactions Case. In this section we focus on a more difficult problem for DFT: Selective bond activation of common covalent bonds by transition metals which is at the core of homogeneous catalysis mediated by transition metal complexes.40 This is the area where computational insights could be transformative to real-world applications if the theory is found to be sufficiently predictive.41 To assess the average performance of new meta-GGAs, we employ the wellestablished database of Steinmetz and Grimme.42 The benchmark set is derived from the earlier works of Quintal et al.43 and de Jong et al.44 and contains reaction profiles where a transition-metal catalyst, Pd, PdCl−, PdCl2, or Ni, is inserted into covalent bonds: C−H (aliphatic and aromatic), O−H, N− H, and B−H in simple molecules containing these bonds. Overall, 164 data points of this database include prereactive complex-formation energies, forward and backward barriers, and reaction energies defined as a difference between the product and prereactive complex energies. The energy difference between the transition state and the prereactive complex is the forward barrier energy; the energy difference between the transition state and the product is the backward barrier. Prior studies on the performance of DFT for transition metal-catalyzed reactions have been rather inconclusive regarding the advantage of using (hybrid) meta-GGAs over simpler (hybrid) GGAs.42,44,45 The most clear conclusion that has been formulated to date is that the percentage of exact exchange is the crucial factor affecting the reaction profile.42,46 Of the four computed quantities characterizing a reaction, the complex dissociation energy is especially challenging for DFT and sensitive to the amount of HF exchange in global hybrids.42 Seth et al. have found that range-separated hybrids yield an improvement over pure functionals but fail to surpass the traditional global hybrids developed in the 1990s, B3LYP and PBE0, which happen to contain just the right amount of exact exchange.46 The meta-GGAs tested thus far in the literature are nonempirical TPSS, global hybrid TPSSh, and a variety of Minnesota empirical meta-GGAs, e.g., M06-2X, M06, and M06L, which are extensively parametrized for different contents of HF exchange: 54%, 27%, and 0%, respectively. The latter two are recommended by Peverati and Truhlar for transition-metal chemistry.27 Table 4 shows mean-absolute errors (MAE) for the dissociation energies, forward and backward reaction barriers and reaction energies for Pd- and Ni-catalyzed bond-activation reactions. Our results are intended to update the data of Steinmetz and Grimme by including range-separated hybrid GGAs and meta-GGAs and newly developed meta-GGAs hence the basis set employed here is the same, def2-QZVPP,17 as in their work. Minnesota functionals and B3LYP-D3 are included for comparison. Our first observation confirms the above-mentioned recommendation of Peverati and Truhlar that the low percentage of HF exchange correlates with better performance of empirical functional for Pd- and Ni-catalyzed reactions. Of the two tested Minnesota functionals, M06-L and M06-2X, only the former approaches the level of accuracy of

Table 4. Mean−Absolute Errors for the Dissociation Energies (De), Forward Reaction Barriers (Eforw), Backward Reaction Barriers, and Reaction Energies for Pd- and NiCatalyzed Bond-Activation Reactions (kcal/mol)a method

full set

Eforw (full set)

De (Ni complexes)

LC-PBETPSS-D3 LC-ωPBE-D3 ωB97M-V ωB97X-D B3LYP-D3 M06-2X-D3 M06-L SCAN SCAN0 LC-PBETPSS-D3/RHF LC-PBETPSS-D3/SCAN0

2.3 3.1 2.6 2.6 2.7 6.3 2.2 4.8 2.0 6.3 2.5

1.8 2.6 1.9 2.3 1.5 4.4 2.2 4.1 1.1 4.3 2.0

1.9 2.2 5.2 2.8 10.5 16.7 5.4 2.2 6.1 22.1 2.1

The errors averaged over all systems are denoted “full set.” The LCPBETPSS-D3 energies using orbitals converged with method X are denoted LC-PBETPSS-D3/X.

a

simple GGA hybrids, e.g., B3LYP-D3. The performance of the M06-type functionals reveals one challenging aspect of developing new meta-GGAs by empirical fitting. Adjusting the parameters requires many accurate data points, but sufficiently representative training sets do not yet exist. This obstacle has been recognized by several authors as a challenge for future DFT development.7,27 The case in point is ωB97MV designed to achieve the limits of accuracy for main-group chemistry, but whose training set lacks transition metals entirely.7 As seen in Table 4 the performance of ωB97M-V is decent but no better than nonempirical meta-GGAs. Furthermore, for the entire set of bond activation reactions, ωB97M-V performs on a par with its older range-separated GGA counterpart, ωB97X-D. MAE over the entire set for both functionals are the same 2.6 kcal/mol (Table 4). By contrast, the nonempirical range-separated meta-GGA, LC-PBETPSSD3, performs much better (MAE = 2.3 kcal/mol). Our results suggest that there is currently little to no advantage of using highly empirical DFT for the considered type of systems. A pure semilocal SCAN underestimates forward barrier heights and overestimates backward barrier heights. Therefore, in this respect, SCAN is qualitatively no different from other pure DFT functionals considered by Steinmetz and Grimme42 but has a disadvantage of being used without dispersion. Adding a portion of HF exchange to SCAN improves this model dramatically, so that it becomes statistically the best method in our test set. A sizable chunk of this improvement originates from a better description of barrier heights (see Supporting Information). Transition state may be viewed, according to Zhang and Yang,47 as a system composed of noninteger electron subsystems. This is where approximate functionals suffer from self-interaction errors. Therefore, one might be tempted to follow a heuristic recommendation offered for chargetransfer complexes of using 40−60% of HF-exchange in global hybrids or full long-range HF in range-separated hybrids.48,49 For transition metals this strategy would be completely wrong based on our results. The low HF exchange global hybrids, SCAN0 and B3LYP-D3, yield barrier heights about as accurate as the best range-separated hybrids, LC-PBETPSS-D3, ωB97M-V, and ωB97X-D, whereas M06-2X-D3, which includes 54% of exact exchange, is the worst performer. F


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The Journal of Physical Chemistry C Indeed, the case of M06-2X-D3 nicely illustrates why a large portion of HF exchange is incompatible with transition-metal compounds. The most difficult cases for M06-2X-D3 involve the nickel subset with the most pronounced multireference character (as demonstrated by high T1 diagnostics in ref 42). The error in the Ni subset has a substantial density component which can be traced back to the density error due to the Hartree−Fock part of the hybrid exchange energy. To demonstrate this point, we computed LC-PBETPSS dissociation energies for Ni systems on the HF density. As seen in Table 4, the results are 1 order of magnitude less accurate than in the self-consistent DFT variant. Conversely, substituting the SCAN0 density in LC-PBETPSS no-SCF computation leads back to as accurate values of De as in the self-consistent case. A further point illustrating the relationship between the static correlation error and MAEs for the Ni subset is the increase of the error from SCAN to SCAN0, from 2.2 to 6.1 kcal/mol, which makes perfect sense from the perspective of FS errors. Less sense makes the excellent performance for this subset of LC-PBETPSS-D3. Our results indicate that SCAN0 even without a dispersion correction appears to be a robust method for transition metal insertion reactions as well as for gold ligand interactions studied in the previous section. However, one should keep in mind that the 25% HF exchange contribution represents a compromise between the delocalization and the static correlation errors that appears to be right only for transition metals (and perhaps only for the late transition metals). In a paradigm case of main-group charge-transfer complex, NH3− ClF, SCAN0 (without dispersion) yields the binding curve too deep by 35% (see Supporting Information). The substitution of the HF density reduces the error 2-fold, but the systematic overbinding still remains. It appears, therefore, that it is currently impossible to construct a functional in a globalhybrid or range-separated form that is truly robust against the major types of errors in both main-group and organometallic chemistry. II.3.B. Model Grubbs Catalyst Test Case. Another important homogeneous catalysis reaction is olefin metathesis using Grubbs catalytic process. A model and benchmarks for this reaction were initially developed by Zhao and Truhlar.50 The reaction has five steps and branches out to cis and trans pathways (see Figure S1 in Supporting Information). These benchmarks were recently revised using state-of-the-art CCSD(T)-F12 explicitly correlated wave function methodology by Kesharwani and Martin.51 They also tested a number of DFT methods but the only range-separated functional in their work was ωB97X-D. The steps of this reaction cover a broad range of different bonding situations involving Ru metal: donor−acceptor, Ru−P, RuC, Ru−C, Ru···π, and stretched bonds in transition states; altogether, they cover the range of bonding to a metal from covalent to noncovalent that should present a demanding test for DFT methods. Table 5 shows the mean absolute errors averaged over the reactions steps for a number of DFT methods with respect to benchmarks of Kesharwani and Martin.51 The new meta-GGAs are clear leaders with our LCPBETPSS leading the group with ωB97M-V following as close second. SCAN0 is also a very good performer. To gain insights into the performance of DFT for various steps of this reaction, Table 6 shows the relative energies for each step relative to step 2 (see Figure S1). The results demonstrate that the excellent performance of LC-PBETPSS is not due to a

Table 5. Mean Absolute Errors (MAE) (in kcal/mol) Averaged over Steps of the Grubbs Model System with Respect to Benchmarks Denoted Ru(4s,4p) from Kesharwani and Martin51 method

MAE

LC-PBETPSS-D3 ωB97M-V SCAN0 wB97X-D LC-wPBE-D3 M06-L M06-2X-D3 B3LYP-D3

1.56 1.63 2.12 2.61 3.15 3.27 4.10 6.43

fortuitous error cancellation but because it is among the top three performers for a largest number of reaction steps. SCAN0 distinguishes itself by outperforming for the two transition states and for the prereactive complex. Overall, in this reaction the meta-GGAs show clear advantage over GGAs both in the global hybrids category, SCAN0 vs B3LYP, and in the long-range hybrid category, LCPBETPSS and ωB97M-V vs LC-ωPBE. M06-L is less erratic than M06-2X but both are relatively poor performers. Finally, in the present test, we did not include semilocal functionals, except for M06-L. For those, the recommendation of Kesharwani and Martin51 still holds: the semilocal GGAs, PBE, BP86, (but not BLYP) and meta-GGA TPSS, give the best error statistics, provided they are matched with a dispersion term. These are indeed the functionals with some of the smallest static correlation errors.

III. CONCLUSIONS A challenge for DFT in transition metal reactivity is to perform well for a wide variety of bonding situations, much wider than in main-group chemistry. This work has examined many types of them in model systems such as gold-ligand complexes, model metal insertion reactions into covalent bonds, and the pathway for olefin metathesis by a model Grubbs system, all of which include late transition metals. Our focus has been on the performance of newly developed hybrid meta-GGA functionals including range-separated hybrids. In the attempt to rationalize the performance of these methods we have studied the fractional charge and fractional spin errors of Au and Au7 for a variety of functionals. Fractional charge and fractional spin errors for a late transition metal atom Au are negatively correlated, confirming previously observed trend for H2+/H2 systems exhibiting these errors.15 No major dependence on the functional rung or inclusion of meta ingredients was found. The only significant factor was the percentage of HF exchange in a functional. For the long-range corrected functionals with 100% of HF exchange in the asymptotic region the fractional charge error is minimized but the fractional spin error is maximized. The large fractional spin errors here result from the fact that the long-range correction causes the exchange hole to be delocalized and uncompensated by the correlation hole. In gold-ligand complexes where the interactions are essentially donor−acceptor, thus presumed to be sensitive to the fractional charge errors, the range separated functionals perform very well. The best performance, however, was that of SCAN020 which includes 25% of HF exchange. Its parent SCAN showed strong systematic overbinding. However, when G


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Table 6. Energetics of Steps of Grubbs Model System (See Figure S1 in Supporting Information) in kcal/mol with Respect to Step 2a

a

Reference values are benchmarks denoted Ru(4s,4p) of Kesharwani and Martin.51

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the SCAN energy was evaluated on the HF density without self-consistency, the overbinding was mitigated. This strategy proved successful for a main-group charge-transfer complex as well. In the prototypical donor−acceptor complex NH3−ClF, the strong overbinding by SCAN was reduced 2-fold when its energy was evaluated on the HF density. In bond insertion reactions by Pd, PdCl−, PdCl2, and Ni, a low percentage of HF exchange leads to best results as evidenced by SCAN0 being the best performer over the entire set. The 20−25% of HF exchange contributions appears to be the right compromise between the fractional charge and fractional spin errors. In the Ni subset, the case of presumptive static correlation importance, a correlation, albeit a tenuous one, with the fractional spin error was observed. That is, adding the HF exchange to SCAN is detrimental to performance. Notably, when static correlation is important, performing a DFT calculations on the HF density leads to catastrophic results, as shown in the LC-PBETPSS computation on the HF density. In the model Grubbs-catalyst reaction the primary focus was on the range separated functionals and of these the newly developed LC-PBETPSS and ωB97M-V are clear leaders. SCAN0 is also a good performer particularly for two of its transition states and for the prereactive complex. Overall, our new functional LC-PBETPSS8 is the best or the second best over the whole range of systems examined in this work. Notably, SCAN0 shows to be a very robust method in all the cases studied. Furthermore, there is a distinct advantage of using nonempirical functionals over the empirical ones. None of the functionals studied in this work are simultaneously free from the static correlation and delocalization errors, i.e., can respond to the challenge posed by Ruzsinszky and Perdew11 (see Introduction). Rather, a good performance results from the compromise between these two errors. This is not to say that these errors cannot be mitigated. For example, in instances of where delocalization error might be important, performing DFT calculations on the HF density results in a much improved energy value. We should mention at this point that the development of functionals which are free from these errors is currently a hotly pursued goal (see Kong and Proynov52 and references therein). There already exist functionals which do just that but they are either too difficult to apply (B13)53 or too new to be implemented in popular packages.52 The approximate hyper-GGAs of the rung 3.5 proposed by Janesko are also promising in this regard.54 Yet another approach based on fractionally occupied localized orbitals has just been proposed by Yang’s group.55

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b07394.

■

Supplementary data for Table 1, coordinates of goldligand complexes, spreadsheets with data for bond activation energies and NH3-ClF potential energy curves, and Grubbs catalyst all energy data, and Figure S1 (ZIP) Table of contents of the zip file (PDF)

AUTHOR INFORMATION

Corresponding Authors

*(G.C.) E-mail: [email protected]. *(M.M.S.) E-mail: [email protected]. ORCID

Marcin Modrzejewski: 0000-0001-9979-8355 Grzegorz Chalasinski: 0000-0002-3535-5034 Notes

The authors declare no competing financial interest.

■

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