Benchmarking Density Functional Theory Methods for Metalloenzyme Reactions: The Introduction of the MME55 Set

Abstract

We present a new benchmark set of metalloenzyme model reaction energies and barrier heights that we call MME55. The set contains 10 different enzymes, representing eight transition metals, both open and closed shell systems, and system sizes of up to 116 atoms. We use four DLPNO–CCSD(T)-based approaches to calculate reference values against which we then benchmark the performance of a range of density functional approximations with and without dispersion corrections. Dispersion corrections improve the results across the board, and triple-ζ basis sets provide the best balance of efficiency and accuracy. Jacob’s ladder is reproduced for the whole set based on averaged mean absolute (percent) deviations, with the double hybrids SOS0-PBE0-2-D3(BJ) and revDOD-PBEP86-D4 standing out as the most accurate methods for the MME55 set. The range-separated hybrids ωB97M-V and ωB97X-V also perform well here and can be recommended as a reliable compromise between accuracy and efficiency; they have already been shown to be robust across many other types of chemical problems, as well. Despite the popularity of B3LYP in computational enzymology, it is not a strong performer on our benchmark set, and we discourage its use for enzyme energetics.

1. Introduction

Across most areas of chemistry, Kohn–Sham Density Functional Theory^1,2 (KS-DFT, often shortened to just DFT) has become a common part of understanding reactions and structures, either alongside experiments or on its own. One particular application is the study of enzymes, where computational results can be used to understand the mechanisms of action and gain structural insights that can then be applied to enzyme design or general catalysis applications. Generally, these studies are done by treating a curated model of the active site with DFT and calculating the indirect effects of the remaining enzyme either as an electrostatic field, as in cluster model studies, or with force fields, as part of a hybrid quantum mechanics/molecular mechanics (QM/MM) scheme; for reviews of QM/MM approaches for enzyme modeling, see refs (3) and (4).

While DFT is often a good choice for computational studies due to its accessibility and cost-to-accuracy ratio, there are an overwhelming number of density functional approximations (DFAs) available to users, each having been determined from different sets of data or first-principles boundary conditions and therefore having different strengths. The choice of DFA can thus alter the conclusions drawn about possible mechanisms or structural features; therefore, it is important to consider the choice carefully. This process is assisted by benchmarking studies that test the performance of density functionals either on comprehensive databases of chemical reactions in order to find generally robust functionals or test sets that are more specific to the chemical problem at hand. Some examples of broad benchmark sets that represent general main group chemistry are GMTKN55⁵ and MGCDB84,⁶ across which hundreds of DFAs have been tested—more than 350 by our group alone.^5,7–11 There are also smaller test sets that focus on organic biochemical reactions,^12–17 including work by us,¹⁸ as well as sets that represent functional groups and reaction types often found in, but not exclusive to, enzyme chemistry.^19–23

However, biochemistry is not limited to main group elements. It is estimated that around one-third of the human proteome requires metals to function,²⁴ and metalloenzymes are crucial in many other living species too. Unfortunately, transition metals are harder to treat accurately with computational methods, as their electronic structure is more complex, so recommendations of DFAs from tests on organic enzymes are not necessarily applicable to metalloenzymes. Due to difficulties in getting reliable reference values, there are significantly fewer benchmark studies that represent transitional metal chemistry. Most existing test sets feature inorganic dimers or small organometallic complexes,^25–39 or only include a small number of systems,^40–47 and these are all typically closed shell species. Various groups are working on filling the gaps of larger complexes and open-shell species, particularly with the WCCR10,^48,49 MOR41,⁵⁰ MOBH35⁵¹ and ROST61⁵² sets, and recent studies assessing coupled cluster approaches for calculating reference values for these types of systems.^53–55 Chan et al. have also compiled the TMC151 database⁵⁶ from sets of organometallic reaction energies (REs), organometallic barrier heights (BHs), and inorganic dimer bond energies, to be used as a (smaller) counterpart to the existing broad main group databases and find functionals that are robust across multiple types of transition-metal chemistry.

Within computational bioinorganic chemistry, DFT has been tested for a range of properties, including spin state splitting,^57–59 metal–ligand binding and interaction energies,^60,61 excited states,^62,63 relative energies of Cu₂O₂ isomers,^42,64,65 redox potentials,⁶⁶ and bond dissociation energies.⁶⁷ Many studies of the mechanisms or properties of a specific enzyme/family of enzymes also validate the chosen DFT methodology with a comparison to ab initio methods^68–71 or other density functionals.^{58,70,72–74} Most of these data, however, are small scale, in both the range of systems included and the methods tested, and were not tested against reliable references. We also note that some studies use small simplified models,^60,66 an approach which does not reliably represent the range of interactions found in an enzyme. In our recent guide to benchmarking enzymatically catalyzed reactions,⁷⁵ we have shown that oversimplified models can have very different benchmarking outcomes compared to larger models for organic enzymes, as well as how the quality of the references will change the perceived performance of the tested functionals—a fact that has also been shown before for pericyclic and inorganic reactions.^5,76

We have, thus, sought to create a benchmark set for metalloenzyme reactions to sit between our previous set of mostly organic enzyme models¹⁸ and the MOBH35, MOR41, and ROST61 sets for transition metals, to explore whether the recommendations from these previous studies hold for bioinorganic systems. We additionally hope that by adding more data in the overlap of biochemical and organometallic benchmarking, this set can assist bioinorganic chemists in choosing reliable functionals for their calculations or method developers in creating new functionals for accurate transition metal calculations.

In the following sections, we present a description of the set, which we call MME55—mechanisms of metalloenzymes, containing 55 data points including both BHs and REs. This is followed by an analysis of the multireference character of the included models and details on how the reference values were calculated. Finally, we test a range of density functionals as well as the still popular second-order Møller–Plesset perturbation theory⁷⁷ (MP2), analyze their performance, and compare the results to other related benchmarking studies. We also reiterate the importance of London dispersion corrections for DFT methods, which has been stated in many studies including for bioorganic^18,78,79 and bioinorganic^80–83 systems, although the recommendation is still not always followed.

2. Computational Details

All calculations in this work were done using the ORCA^84–86 quantum chemistry package (versions 4.21, 5.0.1, 5.0.2, and 5.0.3). The Ahlrichs-type def2-nZVPP family of basis sets⁸⁷ was used, with the default def2-ECP^87,88 effective core potentials for the systems containing molybdenum and tungsten. The resolution of the identity approximation for Coulomb integrals (RI-J)⁸⁹ was used for generalized gradient approximation (GGA), meta-GGA and meta-NGA (nonseparable gradient approximation) functionals, while RI-J with the chain of spheres approximation for exchange integrals (RIJCOSX)⁹⁰ was used for the hybrid and double-hybrid functionals, as well as the Hartree–Fock steps of MP2 and coupled cluster calculations. MP2 steps—including in double hybrids—and coupled cluster calculations were sped up with the RI-C approximation⁹¹ and used ORCA’s default frozen core settings. These approximations were used with the def2/J⁹² and def2-nZVPP/C⁹³ auxiliary basis sets and the “GridXS2” setting for RIJCOSX. Most calculations were done with the default grids and self-consistent-field (SCF) convergence thresholds.

Geometries of all structures were optimized at the PBEh-3c⁹⁴ level of theory in ORCA v4.2.1 with tight SCF and default geometry convergence settings, and the “grid3 finalgrid5” grids. PBEh-3c is a PBE⁹⁵-based hybrid functional, combined with a special double-ζ basis set and corrections for London dispersion [DFT-D3(BJ)^96,97] and basis set superposition error (gCP⁹⁸). For the multireference diagnostics, a test of the weighted fractional occupational electron density (FOD)⁹⁹ was done using the default settings, which are TPSS¹⁰⁰/def-TZVP with TightSCF convergence and an occupation number smearing temperature of 5000 K. To calculate the A_λ diagnostic, PBE and PBE0^101,102 calculations were done with the def2-TZVPP basis set.

DLPNO–CCSD (domain-based local pair natural orbital coupled cluster with singles and doubles) and DLPNO–CCSD(T) (with additional perturbative triples)^103,104 calculations were done in ORCA versions 5.0.1, 5.0.2, and 5.0.3, using either the quasiperturbative (T₀)¹⁰³ or the iterative (T₁)^105,106 approach for the triples correction. For the initial multireference screening, def2-SVP was used with the NormalPNO thresholds.¹⁰⁷ For the benchmark energies, the def2-TZVPP and def2-QZVPP basis sets were used with TightPNO thresholds, and these results were extrapolated to the complete basis set (CBS) limit using the standard two-point extrapolation schemes with individual extrapolations of the SCF¹⁰⁸ and correlation energies¹⁰⁹

1

and

2

where X and Y are the cardinal numbers of the basis sets, and α and β are optimized constants specific to the basis sets used. For the def2-TZVPP/def2-QZVPP CBS(3,4) extrapolation used here, α = 7.88 and β = 2.97.¹¹⁰

Upon a reviewer’s request, we briefly assessed the impact of spin–orbit coupling, but any such effects were canceled for the REs and BHs in our tests, and as such, it was not necessary to include them in our reference value calculations.

All methods listed in Table 1 were applied in ORCA v5.0.2 with the def2-QZVPP basis set, except for the composite DFT (−3c) methods, which used their own specific basis sets. Additional calculations with B3LYP, M06, PWPB95, and ωB2PLYP were done with the def2-SVP and def2-TZVPP basis sets. The ORCA implementation of the LibXC density functional library¹⁵⁵ was used to run MN15-L, MN15, and MPW1B95 calculations. DOD-SCAN, revDSD-PBEP86, and revDOD-PBEP86 were run in both the DFT-D3(BJ) and DFT-D4^115,156 parametrized forms. For B97M-V, ωB97M-V, and ωB97X-V, the nonlocal VV10 kernel was used in its post-SCF implementation, as this strategy does not impact the results but can lead to a considerable speedup in calculations.⁸ The DFT-D3(BJ) dispersion correction with Becke–Johnson damping was used preferentially to the older, zero-damping DFT-D3(0)⁹⁶ variant for all functionals except M06L, MN15-L, M06, and M062X—the Minnesota functionals are parametrized to have better descriptions of midrange interactions, but this leads to some double-counting effects when the D3(BJ) correction is applied.⁵ Specific dispersion correction damping parameters for B3LYP* have not been determined yet, so we follow the common approach of applying the standard B3LYP damping parameters.^157–161

Table 1. Methods Tested on the MME55 Set^a.

		references
type	name	method	D3	D4	% HF	% MP2
GGA	B97-3c	(111)	(111)
GGA	BLYP	(112–114)	(97)	(115)
GGA	BP86	(112,116,117)	(97)	(115)
GGA	OLYP	(113,114,118)	(119)	(115)
GGA	OPBE	(95 and 118)	(119)	(115)
GGA	PBE	(95)	(97)	(115)
GGA	PW91	(120)	(121)	(115)
GGA	revPBE	(122)	(97)	(115)
meta-GGA	B97M-Vb	(123)	(8)	(10)
meta-GGA	M06L	(124)	(119)c	(115)
meta-NGA	MN15-L	(125)	(5)c
meta-GGA	r2SCAN	(126)	(127)	(127)
meta-GGA	r2SCAN-3c	(128)		(128)
meta-GGA	revTPSS	(129 and 130)	(5)	(115)
meta-GGA	TPSS	(100)	(97)	(115)
hybrid	B3LYP	(131 and 132)	(97)	(115)	20
hybrid	B3LYP*d	(133)	(97)	(115)	15
hybrid	BHLYP	(134)	(119)	(115)	50
hybrid	CAM-B3LYP	(135)	(119)	(115)	19–65e
hybrid	M06	(136)	(119)c	(115)	27
hybrid	M062X	(136)	(119)c	(115)	54
hybrid	MN15	(38)	(5)		44
hybrid	MPW1B95	(137)	(119)	(115)	31
hybrid	PBE0	(101 and 102)	(97)	(115)	25
hybrid	PBEh-3c	(94)	(94)		42
hybrid	PW6B95	(138)	(97)	(115)	28
hybrid	TPSS0	(139)	(97)	(115)	25
hybrid	TPSSh	(140)	(119)	(115)	10
hybrid	ωB97M-Vb	(141)	(8)	(10)	15–100e
hybrid	ωB97X-Vb	(142)	(8)	(10)	16.7–100e
double hybrid	B2PLYP	(143)	(119)	(115)	53	27
double hybrid	B2GP-PLYP	(144)	(119)	(115)	65	36
double hybrid	B2K-PLYP	(144 and 145)	(76)		72	42
double hybrid	DOD-SCAN-D3(BJ)f	(146)	(146)		66	0/62.83g
double hybrid	DOD-SCAN-D4f	(146)		(146)	66	0/63.44g
double hybrid	mPW2PLYP	(147)	(5)	(115)	55	25
double hybrid	PBE0-DH	(148)	(149)	(115)	50	12.5
double hybrid	PWPB95	(150)	(119)	(115)	50	0/26.9g
double hybrid	revDOD-PBEP86-D3(BJ)f	(146)	(146)		69	0/60.55g
double hybrid	revDOD-PBEP86-D4f	(146)		(146)	69	0/61.22g
double hybrid	revDSD-PBEP86-D3(BJ)f	(146)	(146)		69	7.9/57.85g
double hybrid	revDSD-PBEP86-D4f	(146)		(146)	69	6.36/59.22g
double hybrid	SOS0-PBE0-2	(151)	(7)		79.37	0/66g
double hybrid	ωB2PLYP	(152)	(11)	(11)	53–100e	27
double hybrid	ωB2GP-PLYP	(152)	(11)	(11)	65–100e	36
double hybrid	ωB88PP86	(153)			65–100e	42
double hybrid	ωPBEPP8	(153)			70–100e	48
ab initio	MP2	(77)	(7 and 154)		100	100

^aReferences for DFT-D3 and DFT-D4 are where the damping parameters for each functional were first presented, with D3 referring to the Becke–Johnson damping [D3(BJ)] variant unless otherwise stated.

^bThe DFT-D3(BJ)/4 dispersion correction replaces the VV10 kernel in this van der Waals functional, while its semilocal exchange–correlation component stays the same.

^cDFT-D3(0) correction.

^dDamping parameters for B3LYP* have yet to be defined, so we use the standard B3LYP damping parameters.

^eRange-separated functionals with variable amounts of Fock exchange.

^fThe underlying parameters for these functionals differ between their D3(BJ) and D4 versions.

^gScale factors for the same spin and opposite spin contributions to the MP2 correlation energy, respectively.

3. Development of the MME Set

3.1. Enzyme Models Included in the Set

Two main considerations were involved in the selection of the enzyme models. The first was that the set should represent a range of different transition metals, spin states, and reaction types; the second was that the systems should be of similar size to those used in typical enzyme modeling studies. The generation of reference data becomes more complicated as systems get larger or more metal ions are included; on the other hand, we have previously shown that using reduced models of enzyme active sites is inadequate for benchmarking,⁷⁵ and this is likely even more inappropriate in metalloenzymes where the coordination environments of the metal ions also need to be represented. We thus searched the literature for DFT and QM/MM studies of enzyme mechanisms, focusing on ones where the active site model or QM region contained up to 120 atoms and no more than two metal centers so that the entire model could be used without needing to be simplified. The selected enzyme models, which range in size from 51 to 116 atoms and cover eight different transition metals, are the following:

• (2R,3S)-Dimethylmalate lyase¹⁶² (DMML): A manganese-dependent enzyme that catalyzes the cleavage of dimethyl malate, forming propionate and pyruvate.

• Cysteine dioxygenase⁷² (CDO): An iron-dependent enzyme that catalyzes the metabolization of cysteine. The four-step process is modeled in the singlet, triplet, and quintet spin states.

• Nitrile hydratase¹⁶³ (Co-NHase): A cobalt-dependent enzyme that catalyzes the hydrolysis of organic nitriles into their amides.

• Superoxide dismutase¹⁶⁴ (NiSOD): A nickel-dependent enzyme that catalyzes the disproportionation of superoxide to molecular oxygen and hydrogen peroxide. One reaction step (4 → 4′) was modeled in two spin states (singlet and triplet), all others were modeled in one (see the Supporting Information).

• Hemocyanin⁷⁴ (Hc): The oxygen-binding process in the Cu₂O₂ core, modeled in both the singlet and triplet states.

• Aminopeptidase¹⁶⁵ (AAP): A zinc-dependent enzyme that catalyzes the cleavage of the N-terminal amino acid residues of polypeptides and proteins.

• Phosphotriesterase¹⁶⁶ (PTE): A zinc-dependent enzyme that catalyzes hydrolysis of organophosphate triesters. This model was also included in the mainly organic enzyme set that we have previously used to benchmark DFT.¹⁸

• Acetylene hydratase¹⁶⁷ (AH): a tungsten-dependent enzyme that catalyzes the nonredox hydration of acetylene to acetaldehyde.

• Formaldehyde ferredoxin oxidoreductase¹⁶⁸ (W-FOR): A tungsten-dependent enzyme that catalyzes the reduction of formaldehyde to formic acid.

• Carbon monoxide dehydrogenase¹⁶⁹ (Mo–Cu CODH): A binuclear molybdenum–copper enzyme that catalyzes the oxidation of CO to CO₂.

Reaction schemes for these 10 models are shown in Figure 1. An 11^th model (the molybdenum-dependent enzyme perchlorate reductase¹⁷⁰) was also tested, but had to be cut entirely due to its multireference character; details of this system are given in section S1 of the Supporting Information.

Figure 1.

Reaction schemes for the enzyme models, adapted from refs (72), (74), (162), and (169). For the sake of clarity, not all ligands for each metal center are shown. Steps that are included in the final MME55 set are labeled in red.

3.2. Geometry Optimizations

After taking the published structures for each enzyme included in the set, our first step was to reoptimize the geometries to ensure that a consistent level of theory was used. Any constraints that had been applied in the previous studies (see the Supporting Information for details) were maintained in our optimizations, and for some structures in the Hc model, ORCA’s fragment optimization feature of the ORCA was used to ensure that correct configurations were maintained. The original DMML structures were altered prior to optimization to replace the carbon atoms at the QM/MM crossover points with hydrogens so that they could be optimized as cluster models. For systems where multiple spin state surfaces were included, the structures were separately optimized in each spin state, and REs and BHs were only calculated between structures of the same multiplicity to avoid any spin-crossover effects. Not all transition states of each reaction were able to be successfully reoptimized, so the final set contains no BHs for DMML, Co-NHase, Mo–Cu CODH, CDO, and the singlet states of Hc and NiSOD.

The importance of dispersion corrections in geometry optimizations has been shown numerous times,^78,79 including for organometallic complexes.¹⁷¹ In our previous work on enzyme benchmarking,¹⁸ we showed some examples of how PBEh-3c structures improved the description of dispersion-supported interactions such as hydrogen bonding and aromatic π-stacking, including in the PTE model which we have also included here in the MME55 set. We see similar changes in some of the structures here, and we give two examples in Figure 2; see the figure caption for dispersion-uncorrected levels of theory of the original structures. For AAP Int1, the optimized structure has the substrate rotated so that the acetyl group can interact with one of the histidine residues. For Co-NHase ES, an off-center parallel π-stacking interaction between the benzonitrile substituent and tyrosine residue is seen only in the optimized structure, not the original structure.

Figure 2.

Comparisons between the original structures (red) and reoptimized structures (blue). For clarity, one histidine residue has been hidden in AAP Int1a, while for the original structure of Co-NHase ES, only the Co center, benzonitrile substituent, and tyrosine side chain are shown. Original AAP structure: B3LYP/6-31G(d,p)¹⁷² with Stuttgart–Dresden ECP⁸⁸ on Zn. Original Co-NHase structure: M06L/6-31G(d,p) with Stuttgart–Dresden ECP on Co. Reoptimized structures: PBEh-3c.

We note that in some cases, the optimizations have converged in arrangements that do not match the original crystal structures from which the models were developed and may be unfavorable for the overall mechanisms—for example, with amino acid side chains rotating away from where they will later react. While this is a problem when comparing to experimental data, here we use the same structures for both the theoretical reference values and the DFT tests, so there is less influence on the calculated deviations. The reoptimized systems should still represent the types and overall magnitude of noncovalent interactions that are seen in enzymes, even if they are configured differently, and therefore we believe that slight alterations in the structures are not harmful for the purpose of a benchmark study.

3.3. Testing Multireference Character

To ensure that reliable single reference benchmark energies could be calculated for the systems, the multireference character of each structure was checked with a range of diagnostics. First A_λ values¹⁷³ were obtained using the PBE and PBE0 density functionals. Then DLPNO–CCSD(T₀)/NormalPNO/def2-SVP calculations were done to test the % TAE(T),¹⁷⁴T₁ diagnostic,¹⁷⁵ and largest T₂ amplitude. The A_λ and % TAE(T) approaches to test the sensitivity of the total atomization energies to the level of theory (namely, Hartree–Fock and higher-order coupled cluster excitations, respectively). A_λ ≤ 0.1 and % TAE(T) ≤ 2% generally mean that a system is dominated by dynamic correlation, while values of A_λ ≈ 0.15 and % TAE(T) up to 5% indicate mild multireference character. For the T₁ diagnostic, typically a threshold of 0.02 is used, but Wilson and co-workers state that this can be increased up to 0.05 for systems containing transition metals.¹⁷⁶ While thresholds for the largest T₂ amplitude are not as clearly defined, generally values of 0.1–0.2 have been considered mild but not necessarily problematic when other diagnostics are low,^42,173 while some say up to 0.15 can be considered low.^52,176

The results of these diagnostic tests are summarized in Table 2. % TAE(T) values could not be calculated for Mo–Cu CODH due to convergence issues for the Mo atom calculation, but given that no systems have % TAE(T) ≥ 2%, even those which are indicated to have mild multireference character by the other diagnostics, it is likely that these values would also be low. Looking at the other diagnostics, we see that none of the T₁ values are greater than the revised 0.05 threshold for transition metals and even the largest A_λ values still indicate only mild multireference character. The trends across the diagnostics, however, are a more reliable indicator of problematic behavior than any individual one. We therefore use the more conservative T₁ = 0.02 and largest T₂ amplitude = 0.1 thresholds, along with A_λ = 0.1, and any structures that are above the thresholds on all three diagnostics are removed from the set.

Table 2. Summarized Results of Tests of the Multireference Character of Each System, with Values Indicating Mild or Moderate Multireference Character Shown in Bold and the Number of Structures above the Threshold for Each Diagnostic Given in Parentheses^a.

reaction	no. of structures	Aλ	% TAE(T)	T1 diagnostic	largest T2 amp	no. of structures cut	remaining no. of BHs	remaining no. of REs
DMML	6	0.10–0.10	1.18–1.20	0.012–0.013	0.054–0.056	0	0	6
CDO	15	0.11–0.14 (15)	1.51–1.76	0.013–0.039 (5)	0.053–0.229 (2)	2	0	8
Co-NHase	4	0.11–0.11 (4)	1.41–1.42	0.016–0.018	0.069–0.082	0	0	3
NiSOD	15	0.11–0.14 (15)	1.45–1.70b	0.014–0.048 (8)b	0.056–0.215 (8)b	10	2	2
Hc	8	0.13–0.14 (8)	1.66–1.97	0.013–0.018	0.045–0.196 (2)	0	2	4
AAP	10	0.09–0.10	1.23–1.25	0.013–0.013	0.061–0.063	0	4	5
PTE	5	0.12–0.12 (5)	1.53–1.54	0.013–0.013	0.055–0.057	0	2	2
AH	10	0.09–0.09	1.38–1.45	0.013–0.015	0.057–0.070	0	4	5
W-FOR	5	0.10–0.10	1.40–1.52	0.014–0.016	0.057–0.065	0	2	2
Mo–Cu CODH	6	0.13–0.14 (6)	b	0.016–0.016	0.064–0.084	0	0	5

^aOverall ranges for each reaction are given, while full results can be found in the Supporting Information. Any reaction steps involving structures that are flagged by three or more diagnostics are cut from the final set.

^bThe DLPNO–CCSD(T) calculation could not be converged for the Mo atom or NiSOD structures 3 and TS5, so some diagnostics could not be obtained. These two NiSOD structures were, thus, also cut from the set.

For DMML, AAP, AH, and W-FOR, all structures are shown to have a low multireference character on all diagnostics. All structures for Co-NHase, PTE, and Mo–Cu CODH, as well as the triplet state of Hc and quintet state of CDO, have A_λ values between 0.1 and 0.15 but are still low on the other three diagnostic tests. The A_λ values are similarly low-to-mild for the singlet state of Hc, while the largest T₂ amplitudes, ranging from 0.098 to 0.196, also indicate a mild multireference character. Of the 10 CDO structures across the singlet and triplet states, all have A_λ between 0.11 and 0.14, five have T₁ values between 0.02 and 0.05, and two have a largest T₂ amplitude greater than 0.1. NiSOD is the most problematic, with almost all structures being flagged as having mild-to-moderate multireference character by all three of the A_λ, T₁ diagnostic, and largest T₂ amplitudes—only the reaction step from 4 to 4’ (see scheme in Figure 1) in both its possible spin states can be considered to have low multireference character. Removal of the structures flagged on three diagnostics leads to four reaction steps being cut from CDO and 10 from NiSOD. The remaining numbers of data points in each reaction are listed in the final columns of Table 2, and the steps included in the final set are labeled in Figure 1. We note that the four steps shown for CDO (Figure 1) are modeled in three different spin states, so there were initially 12 REs; two steps were cut from the singlet state (1RE3 and 1RE4) and two from the triplet state (3RE2 and 3RE3).

The T₁ diagnostic and maximum T₂ amplitudes were also briefly checked for the coupled cluster approaches used to calculate the reference energies. The differences between the initial screening NormalPNO/def2-SVP and higher level TightPNO/def2-TZVPP and def2-QZVPP values were minimal in most cases, with only a few changes in which species were flagged as mild cases. Two of the surviving structures from NiSOD have T₁ values up to 0.022 at both of these higher levels of theory but still have reasonable maximum T₂ amplitudes. Three structures from Mo–Cu CODH are flagged on both these diagnostics with their TightPNO/def2-TZVPP values, but the TightPNO/def2-QZVPP results match with the initial screening where they were comfortably below the thresholds. Even in the TightPNO/def2-TZVPP results, the T₁ diagnostic values and largest T₂ amplitudes are still below the alternative thresholds of T₁ < 0.05 and max T₂ < 0.15, so we choose to leave the associated steps in the set. The diagnostics from all coupled cluster calculations are given in the Supporting Information.

We also calculated and visualized the weighted FOD. Overall the results are very similar to the other diagnostics. DMML, AAP, and PTE all have virtually no FOD, while for Co-NHase, AH, W-FOR, Mo–Cu CODH, and the five safe NiSOD structures, the FOD is small and mostly metal-centered. A few examples of some of the structures which were flagged as mildly multireference by at least one diagnostic are shown in Figure 3, while all others are provided in the Supporting Information (Figures S4–S13). All structures for CDO have similar levels of FOD, even between ones that are flagged by all the other diagnostics and ones that are low on both the T₁ and maximum T₂ amplitudes (like 3C and 5A, respectively). The same is seen for Hc, where the FOD is mild and spreads relatively evenly across the whole Cu₂O₂ core for all structures. For the sake of completeness, the Supporting Information also lists the N(FOD) values for all tested systems.

Figure 3.

FOD plots of the selected structures. Surfaces are plotted with σ = 0.005 e bohr^–3.

3.4. Calculation of Reference Values

We aimed to calculate the reference BHs and REs for this set using DLPNO–CCSD(T) with the TightPNO thresholds, which is regularly used in benchmarking as a cost-effective alternative to “gold standard” CCSD(T) results.^5,23,50,51 In our previous benchmark study on enzymatically catalyzed reactions,¹⁸ we did an analysis of various extrapolation and estimation schemes to determine a protocol for calculating reference values. Our first choice was a DLPNO–CCSD(T₀)/TightPNO/CBS(3,4) extrapolation with aug-cc-pVTZ/aug-cc-pVQZ,^177,178 followed by the equivalent extrapolation with the ma-def2-TZVPP/ma-def2-QZVPP¹⁷⁹ basis sets. Both of these levels of theory used the original semicanonical (T₀) procedure for triples correction. The third option was to use a composite scheme based on MP2, which is corrected with the difference in correlation energy between MP2 and DLPNO–CCSD(T₀) (CC) calculated with a small basis set

3

Due to the size of the systems in MME, we can already rule out the Dunning basis sets as being too expensive due to their higher number of primitive Gaussian-type orbitals. The def2 basis sets gave almost identical results to their minimally augmented equivalents in our previous tests of basis sets, while they were also shown to agree well with aug-cc-pVTZ/aug-cc-pVQZ results in the ROST61 benchmark study.⁵² Additionally, the improved iterative triples procedure for DLPNO–CCSD(T), typically referred to as (T₁), has been shown to perform better with respect to CCSD(T) for both open and closed shell species,^105,106 and therefore, we consider DLPNO–CCSD(T₁)/TightPNO/CBS(def2-TZVPP/def2-QZVPP) our first choice level of theory for calculation of the reference values. For systems where this is too expensive, we cannot use the MP2-based composite scheme again, as MP2 is known to be unreliable for transition metal chemistry,^42,51,52,150 so we have tested three additional alternative strategies here.

The first two approaches reduce cost in the triples correction of the quadruple-ζ calculation. This is typically the most costly part of the calculation, with the iterative (T₁) procedure taking significantly longer than (T₀). The first option is, thus, to correct DLPNO–CCSD(T₀)/CBS(3,4) energies with the difference between (T₁) and (T₀) calculated with a smaller basis set

4

where . This only requires the (T₁) iterative triples procedure to be used at the triple-ζ level, while the (T₀) correction is used at the quadruple-ζ level. While (T₀) triples are more efficient than (T₁), they are still a significantly lengthy part of the calculation, so the second approach involves eliminating them entirely from the quadruple-ζ calculation, and correcting DLPNO–CCSD/CBS(3,4) extrapolated results with the (T₁) triples energy calculated at the triple-ζ level

5

The third alternative level of theory, for systems where even DLPNO–CCSD/TightPNO/def2-QZVPP calculations were unfeasible, uses the PNO extrapolation scheme described by Altun et al.¹⁸⁰ This involves calculating results with increasing values of the threshold T_CutPNO, which defines the PNOs included in the final coupled cluster treatment while maintaining all other cutoffs at the same values. To approximate standard TightPNO results, where T_CutPNO = 10^–7, DLPNO–CCSD(T₁) calculations are done with the looser T_CutPNO values of 10^–5 and 10^–6, and the correlation energies are extrapolated for each basis set as such

6

where E_Corr^(X) is the correlation calculated with T_CutPNO = 10^–X. The PNO-extrapolated triple-ζ and quadruple-ζ energies are then extrapolated as usual to give CBS results. We refer to these three approaches as “estimated (T₁)” (eq 4), “estimated CBS(3,4)” (eq 5), and “estimated TightPNO” (eq 6), and we test them alongside DLPNO–CCSD(T)/TightPNO/CBS(3,4) with both (T₁) and (T₀) triples for some small systems in Table 3.

Table 3. Tests of Coupled Cluster Approaches for Calculating Benchmark Energies^a.

		DLPNO–CCSD(T)/CBS
system	step	with (T1)	with (T0)	est. (T1)b	est. CBS(3,4)c	est. TightPNOd
CDO	1RE1	–1.168	–1.145	–1.168	–1.179	–1.245
CDO	1RE2	–52.083	–52.601	–52.139	–52.193	–52.359
NiSOD	1RE4	8.043	8.220	8.038	8.298	8.829
Hc	1RE1	–26.506	–26.948	–27.003	–27.107	–27.188
Hc	1RE2	–9.933	–4.820	–10.159	–10.124	–8.011
mean absolute deviation			1.255	0.157	0.234	0.749

^aMean absolute deviations are calculated from the DLPNO–CCSD(T₁)/CBS values.

^bCalculated following eq 4.

^cCalculated following eq 5.

^dE_Corr for each basis set calculated following eq 6, then used for a standard CBS extrapolation.

We first look at the impact of the type of triples correction on the DLPNO–CCSD(T)/CBS(3,4) energies. The mean absolute deviation (MAD) of 1.255 kcal/mol for DLPNO–CCSD(T₀)/CBS(3,4) when compared to DLPNO–CCSD(T₁)/CBS(3,4) is heavily influenced by Hc 1RE2, for which there is a difference of over 5 kcal/mol. This is likely due to the mild multireference character of the structures involved in this step—Semidalas and Martin have noted a link between sensitivity to the triples correction and static correlation.⁵⁴ Across the other four reaction steps (CDO 1RE1 and 1RE2, NiSOD 1RE4, and Hc 1RE1), the MAD is only 0.290 kcal/mol. The majority of systems in the MME set, however, are large enough to necessitate an estimated level of theory for their reference values, and we do not wish to have the additional error of the (T₀) triples correction on top of the error associated with the estimation scheme. For example, the MAD of estimated CBS(3,4) using (T₁) triples is 0.234 kcal/mol as seen in Table 3, but when the (T₀) triples energy is used instead, the MAD becomes 1.650 kcal/mol. Even excluding Hc 1RE2 from this small test set again, the error of the estimated CBS(3,4) approach is still more than twice as high when (T₀) triples are used instead of (T₁) (0.548 kcal/mol vs 0.244 kcal/mol). We therefore use the (T₁) triples procedure to calculate all reference values for the set.

Across the singlet REs of CDO, NiSOD, and Hc, the estimated (T₁) approach is the best alternative to DLPNO–CCSD(T₁)/TightPNO/CBS(3,4), with deviations ranging from 0 (CDO 1RE1) to −0.497 kcal/mol (Hc 1RE1). Estimated CBS(3,4) also gives good results with a more significant cost reduction—none of the deviations are larger than 1 kcal/mol, the generally considered “chemical accuracy limit” for REs and BHs. For both of these approaches, we have also tested double-ζ and CBS(2,3) correction terms (Table S2 in the Supporting Information), but both perform worse than the triple-ζ-based corrections. This includes CBS(3,4) with CBS(2,3) triples, which is similar to a Weizmann-1 extrapolation¹⁸¹—it has a MAD of 0.744 kcal/mol, more than three times that of the triple-ζ-corrected version shown here.

The estimated TightPNO approach also cuts costs but comes with a greater reduction in accuracy—the MAD is still below 1 kcal/mol, but the error for Hc 1RE2 is almost twice that at 1.922 kcal/mol. However, this is likely to be an issue specific to the Hc enzyme. The DLPNO scheme involves treating only strongly interacting pairs at the coupled cluster level, while weakly interacting pairs are calculated with MP2. As Liakos and Neese have pointed out,⁴² the MP2 correlation energy of the Cu₂O₂ core is wildly inaccurate compared to CCSD(T) and LPNO–CCSD, particularly for the peroxo configuration which is formed in the second Hc RE. In the calculations with looser T_CutPNO values, more pairs are considered as weak pairs, and therefore MP2 has a stronger influence on the overall energy. This leads to larger-than-expected changes between the T_CutPNO = 10^–5 and 10^–6 results and the PNO extrapolation overshoots the standard TightPNO correlation energy. Across the other four REs, the results are much better (the MAD across these steps is only 0.455 kcal/mol).

Ideally, we would additionally test these estimated levels of theory against full DLPNO–CCSD(T₁)/TightPNO/CBS(3,4) for some open shell systems as well, but the open shell implementation of DLPNO–CCSD(T₁) is significantly more costly, so quadruple-ζ calculations with (T₁) triples were unfeasible. The models of Hc and NiSOD are small enough that the energies of the triplet models can be calculated at the estimated (T₁) level of theory, so we have used these for a brief assessment of estimated CBS(3,4) on open-shell systems (Table S3 of the Supporting Information). The MAD of estimated CBS(3,4) against estimated (T₁) across this small set is 0.336 kcal/mol, although we note that this is likely underestimated as the equivalent MAD_est.(T1) across the closed shell models tested here is 0.093 kcal/mol, significantly smaller than the MAD against DLPNO–CCSD(T₁)/CBS seen in Table 3 (0.234 kcal/mol).

For the final reference values for the following benchmark study, the DLPNO–CCSD(T₁)/TightPNO/CBS(3,4) values were used for the singlet REs of CDO, NiSOD, and Hc, while estimated (T₁) was used for the triplet states of NiSOD and Hc. Co-NHase, as the largest model in the set with a 3d metal center, required the estimated TightPNO approach. It is a closed-shell cobalt enzyme with no significant multireference character, therefore we believe that the error of this level of theory will not be unreasonable for this system. The estimated CBS(3,4) approach was used for all other systems. At this point, we have cut three additional steps from the set (one each from DMML, AAP, and PTE) that had reference REs of <0.5 kcal/mol, as these give unreliably high percent deviations for the tested DFAs. The reference values for the final 55 data points in the set are given in Table 4.

Table 4. Final Reference Values for the MME55 Set (kcal/mol).

reaction	stepg	ref. value	reaction	stepg	ref. value	reaction	stepg	ref. value
DMML	RE1	–0.847c	Hc	1RE1e	–26.506a	AH	RE1	–8.459c
DMML	RE3	65.056c	Hc	1RE2e	–9.933a	AH	RE2	9.954c
DMML	RE4	–9.166c	Hc	3RE1e	6.954b	AH	RE3	–22.538c
DMML	RE5	–31.746c	Hc	3RE2e	–6.035b	AH	RE4	–2.685c
DMML	RE6	24.144c	Hc	3BH1e	5.229b	AH	RE5	–3.418c
CDO	1RE1e	–1.168a	Hc	3BH2e	4.165b	AH	BH2	16.709c
CDO	1RE2e	–52.083a	AAP	RE2	4.139c	AH	BH3	14.660c
CDO	3RE1e	–2.643c	AAP	RE3	0.723c	AH	BH4	8.501c
CDO	3RE4e	–77.322c	AAP	RE4	0.739c	AH	BH5	17.105c
CDO	5RE1e	–43.307c	AAP	RE5	–3.412c	W-FOR	RE1	–3.210c
CDO	5RE2e	–67.762c	AAP	BH1	8.797c	W-FOR	RE2	–14.939c
CDO	5RE3e	–10.277c	AAP	BH3	2.339c	W-FOR	BH1	14.272c
CDO	5RE4e	–1.082c	AAP	BH4	1.706c	W-FOR	BH2	19.699c
Co-NHase	RE1	28.103d	AAP	BH5	1.483c	Mo–Cu CODH	RE1	10.721c
Co-NHase	RE2	–27.574d	PTE	RE1	2.492c	Mo–Cu CODH	RE2	–5.793c
Co-NHase	RE3	8.371d	PTE	BH1	6.652c	Mo–Cu CODH	RE3	–2.630c
NiSOD	1RE4e	8.043f	PTE	BH2	8.302c	Mo–Cu CODH	RE4	5.881c
NiSOD	3RE4e	25.440b				Mo–Cu CODH	RE5	1.800c
NiSOD	3BH4e	27.773b
NiSOD	3RBH4e	2.333b

^aDLPNO–CCSD(T₁)/CBS.

^bEstimated (T₁) (see eq 4).

^cEstimated CBS(3,4) (see eq 5).

^dEstimated TightPNO (uses E_Corr values calculated following eq 6 for the CBS extrapolation).

^eA number in front of the label indicates multiplicity when needed for clarity.

^fBarrier of the reverse reaction.

^gSpecific details of each step can be found in the Supporting Information.

4. Benchmark Study: Results and Discussion

Using the MME55 set, we tested the density functionals listed in Table 1 with and without dispersion corrections, resulting in a total of 117 unique combinations. Deviations were calculated as RE_method – RE_ref. or BH_method – BH_ref. for all data points in the set, and statistical analysis was done on these values. Additional analysis based on percent deviations can be found in the Supporting Information, Section 4.2. In Table 5 we present MADs for each functional, while violin plots of the deviations across each rung of Jacob’s Ladder¹⁸² are shown in Figure 4, along with the mean MAD (MMAD) of each category. In this section, we discuss the D3(BJ) and D3(0) results together generally under the label “D3,″ and we only name the specific variant for individual functionals.

Table 5. MADs (kcal/mol) for All Assessed Methods, Listed in Alphabetical Order within the Rungs of Jacob’s Ladder, as Well as Mean MADs for Each Rung of Jacob’s Ladder^a,b.

type	method	plain	D3	D4	V
GGA	B97-3c		6.5
GGA	BLYP	7.9	6.4	6.5
GGA	BP86	7.0	6.5	6.7
GGA	OLYP	8.3	5.6	6.1
GGA	OPBE	8.0	6.7	7.2
GGA	PBE	6.9	6.3	6.5
GGA	PW91	6.9	6.4	6.4
GGA	revPBE	7.6	5.9	6.3
meta-GGA	B97M		4.6	4.4	4.5
meta-GGA	M06L	4.9	4.9	4.8
meta-NGA	MN15-L	3.8	3.8
meta-GGA	r2SCAN	5.3	5.2	5.3
meta-GGA	r2SCAN-3c			6.0
meta-GGA	revTPSS	6.3	6.2	6.4
meta-GGA	TPSS	6.6	6.0	6.3
hybrid	B3LYP	5.3	4.0	4.0
hybrid	B3LYP*	5.7	4.4	4.5
hybrid	BHLYP	5.1	4.0	3.9
hybrid	CAM-B3LYP	4.0	3.3	3.1
hybrid	M06	3.2	3.2	3.2
hybrid	M062X	3.6	3.6	3.6
hybrid	MN15	3.3	3.3
hybrid	MPW1B95	3.1	2.7	2.9
hybrid	PBE0	3.9	3.3	3.5
hybrid	PBEh-3c		4.8
hybrid	PW6B95	3.4	2.9	3.0
hybrid	TPSS0	3.7	3.1	3.4
hybrid	TPSSh	5.3	4.7	5.0
hybrid	ωB97M		3.1	3.0	2.7
hybrid	ωB97X		3.4	3.1	2.8
double hybrid	B2PLYP	4.0	3.7	3.8
double hybrid	B2GP-PLYP	3.2	3.0	3.1
double hybrid	B2K-PLYP	2.9	2.8
double hybrid	DOD-SCANc		3.1	3.3
double hybrid	mPW2PLYP	3.5	3.2	3.2
double hybrid	PBE0-DH	2.8	2.6	2.6
double hybrid	PWPB95	2.7	2.5	2.6
double hybrid	revDOD-PBEP86c		2.4	2.4
double hybrid	revDSD-PBEP86c		2.6	2.6
double hybrid	SOS0-PBE0-2	2.2	1.9
double hybrid	ωB2PLYP	2.3	2.3	2.3
double hybrid	ωB2GP-PLYP	2.2	2.2	2.2
double hybrid	ωB88PP86	4.4
double hybrid	ωPBEPP86	5.5
ab initio	MP2	5.8	5.7
all GGAs		7.5	6.3	6.5
all meta-GGAs		5.4	5.1	5.5
all hybrids		4.1	3.6	3.5
all double hybrids		3.2	2.7	2.8

^aResults are for the def2-QZVPP basis set, except for the −3c methods, which use their own modified basis sets. The lowest MAD in each rung is in bold.

^bCalculated using the damping parameters for B3LYP.

^cSeparate functional parametrizations were used for the D3(BJ) and D4 versions of these functionals.

Figure 4.

Violin plots of deviations by rungs of Jacob’s Ladder based on all dispersion-corrected variants of each functional. The overall shape of each plot represents the density of the data across the range of deviation values, while the internal black boxes and white dots represent the interquartile range and median value, respectively. Mean MADs for each rung are given in kcal/mol.

The results in Table 5 and Figure 4 show two key trends—the general improvement in the performance of density functionals as one goes up the rungs of Jacob’s Ladder and the improvement of the results when dispersion corrections are applied—that match the findings of many other benchmark studies.^{5,18,21,23,49,50,52,183} The MMAD of each rung is lower than for the previous rung for both the plain and dispersion-corrected functionals, and the improvements from higher rungs are visually obvious in the changes in the shape seen in the violin plots. The density of the data narrows dramatically, and the interquartile ranges get smaller. The ranges of deviations (the length of each plot) also decrease, but there are still long tails in the plots for the hybrids and double hybrids, as some systems are particularly sensitive to the amount of Fock exchange or MP2 correlation in a functional. For all DFAs except revTPSS, the DFT-D3 and DFT-D4 dispersion corrections either lower or do not change the MAD, so we recommend their use as a default. For further analysis of the effect of different dispersion corrections, see the Supporting Information, Section 4.1.

Given that this benchmark set is inspired by our mostly organic enzyme benchmark set¹⁸ and the organometallic benchmark sets MOR41,⁵⁰ MOBH35,⁵¹ and ROST61,⁵² we briefly compare our top-performing functionals to those. Among the GGAs, OLYP-D3(BJ) has the lowest MAD, which was also the case for the organic enzymes. revPBE-D3(BJ) has the second lowest MAD, while its D4 variant was consistently among the top-performing GGAs for the three organometallic benchmark sets. MN15-L-D3(0) is the best third-rung (meta-GGA/NGA) functional based on MAD, as it was for MOBH35; it is followed by B97M-D4 and B97M-V. Those functionals and the related hybrids ωB97M-V and ωB97X-V, which are some of the best hybrids in this study, are consistently good performers in the previously mentioned studies as well as many other benchmark sets for main group^6,8 and transition-metal^43,56,184 chemistry. ωB97M-V shares first place among the hybrids with MPW1B95-D3(BJ), which was the second best hybrid on MOR41. Finally, the best double hybrid (and overall best DFA) is SOS0-PBE0-2-D3(BJ), which was also the best for the organic enzymes. ωB2GP-PLYP-D4 and ωB2PLYP-D4 are second and third, which is interesting as they were originally developed for excited-state time-dependent DFT, but ωB2PLYP has already been shown to perform well in predictions of UV–vis spectra for copper complexes.¹⁸⁵ None of these top three double hybrids were tested on the aforementioned organometallic benchmark sets, so we also mention the fourth and fifth best functionals, revDOD-PBEP86-D4 and PWPB95-D3(BJ); the former was recommended by the MOBH35 study, while the latter was the best DFA for both ROST61 and MOR41.

Taking a closer look at the hybrids and double hybrids, we present violin plots of the best variant of selected functionals in Figure 5, with each point colored by reaction; equivalent plots for the GGA and meta-GGA/NGA functionals are also given in Figure S15 in the Supporting Information. The majority of deviations lie between −10 and 5 kcal/mol for the hybrids, and −5 and 5 kcal/mol for the double hybrids. revDOD-PBEP86-D4 and revDSD-PBEP86-D4 have the tightest distributions, with interquartile ranges (range containing the middle 50% of data points) of only 1.5 and 1.6 kcal/mol, respectively. For the double hybrids, the trend in error ranges (distance between the maximum and minimum deviations, i.e., the height of each plot in Figure 5) roughly matches the functional ranking by MADs, with SOS0-PBE0-2-D3(BJ) and ωB2GP-PLYP-D4 being first and second again on this statistic. This is not always the case for the hybrid DFAs, as the error ranges are strongly influenced by the functional’s performance for CDO 3RE4 and 5RE1, which generally have large positive deviations that lie significantly higher than the others. While the absolute deviations for these steps are particularly large, neither stands out on an equivalent plot of percent deviations (PDs) (Figure S14 in the Supporting Information) as their reference values are some of the largest by magnitude in the set. CDO 3RE4 has a reference RE of −77.322 kcal/mol, so its average percent deviation (PD) across all hybrid functionals is only 29.5%; for 5RE1, these values are −43.307 kcal/mol and 37.8%, respectively.

Figure 5.

Violin plots of deviations of selected hybrid (top) and double hybrid (bottom) functionals. The functionals are presented in order of decreasing MAD, listed at the bottom in parentheses (values given in kcal/mol). The overall shape represents the density of the data along the range of deviation values, while the internal black boxes and white dots represent the interquartile range and median value, respectively. Individual points are colored by reaction, and the maxima and minima are labeled by the reaction step.

In general, it appears that higher amounts of Fock exchange lower the REs (and thus the deviations) for CDO. 3RE4’s deviation is smaller for B3LYP-D3(BJ) (20% Fock exchange) than for B3LYP*-D3(BJ) (15%), and it is no longer the maximum deviation for BHLYP-D4 (50%). For the Minnesota functionals, 3RE4 and 5RE1 also stand out for M06-D4 (27%), but not M062X-D4 (54%); the latter also has the largest negative deviation for 3RE1. The same is also seen for the double hybrids, with only three of the ten shown in Figure 5 not having CDO 3RE4 as their largest positive deviation: SOS0-PBE0-2-D3(BJ) (79.37%), B2K-PLYP-D3(BJ) (72%), and the range separated functional ωB2GP-PLYP (65–100%). Instead, these three have their largest deviations for REs from the copper-dependent enzyme Hc—as mentioned earlier, MP2 is particularly problematic for copper enzymes, and the PDs of MP2 itself for the two triplet Hc REs are 658.9 and 607.5%, respectively. 1RE2, which was particularly sensitive to MP2 in the reference value tests in the previous section, has an MP2 deviation of −26.6 kcal/mol (PD of −267.7%) and contributes the largest negative deviation to four of the double hybrids shown in Figure 5. Mo–Cu CODH RE1 is also regularly underestimated across all rungs.

Given that some systems seem more sensitive to the choice of functional than others, the DFAs have been additionally assessed on each reaction separately using mean absolute PDs (MAPDs) for a more fair comparison between them. The full analysis can be found in Section 4.3 in the Supporting Information, but here we briefly summarize the results of Table S6, which lists the top ten methods for each reaction. The best five DFAs on overall MAPDs are revDOD-PBEP86-D4 (30.6%), ωB97M-V (32.4%), SOS0-PBE0-2-D3(BJ) (32.7%), MPW1B95-D3(BJ) (33.1%), and ωB97X-V (33.1%). At least two of these (or their variants with other dispersion corrections) can be found in the top ten list for each enzyme, with the ωB97X functionals appearing most frequently in the top ten (8 out of 10 systems) and SOS0-PBE0-2[-D3(BJ)] appearing most frequently in the top three (5 systems, in the top ten for 7). This set of five DFAs thus provides a good starting point for finding a functional for a given metalloenzyme as at least one should give a reasonable result. While we have noted that MP2 causes issues for Hc, the double hybrid PBE0-DH is actually its best functional; PWPB95 is second best, suggesting that this system does benefit from the use of fifth-rung functionals as long as only a small amount of MP2 correlation is included. PWPB95’s good performance for transition metals and organometallic compounds has also been observed elsewhere.^50,52,150 PBE0-DH is also the third best for Mo–Cu CODH, so we also include this as a specific recommendation for copper-dependent enzymes.

Returning to the overall results presented here, we remind the reader that popularity is not necessarily a good indicator of robust performance. Ranking the DFAs, each in its best (dispersion corrected) form, B2PLYP-D3(BJ) is 15th out of 17 double hybrids, while the Minnesota functionals M06-D4 and M062X-D4 are seventh and 10th out of the 15 hybrids, respectively. M06 and M062X were developed simultaneously, but the fitting set for M06 contained an additional database of transition-metal bond energies¹³⁶ so it is generally better for organometallic systems. B3LYP—arguably the most popular hybrid DFA for molecular quantum chemistry—is regularly used for mechanistic studies of metalloenzymes,^{69,73,169,170,186–190} often uncorrected or with the older DFT-D2¹⁹¹ or -D3(0) dispersion corrections. B3LYP-D3(BJ), however, only comes in 12th place and is outperformed by the best functional in the rung below [MN15-L-D3(0)]. B3LYP* (B3LYP with 15% Fock exchange instead of the standard 20%) is sometimes recommended instead for metalloenzyme energetics,^159,192 but here it is slightly worse: B3LYP*-D3(BJ) comes in 13th place, with its MAD being 0.4 kcal/mol higher than B3LYP-D3(BJ). However, we do note that use of the older DFT-D3(0) correction does not make the results any worse; data for B3LYP-D3(0) and B3LYP*-D3(0) are additionally provided in the Supporting Information.

While changing the amount of global Fock exchange does not improve the results, introducing range separation does—CAM-B3LYP is noticeably better than B3LYP, and the same is seen for the double hybrids (ω)B2PLYP and (ω)B2GP-PLYP. Another feature that leads to better functional performance is spin-component-scaling or spin-opposite-scaling (SCS/SOS),^193,194 where different amounts of the same spin and opposite spin contributions to the MP2 energy are included. Many previous studies have shown that SCS- and SOS-MP2 are better than standard MP2 for transition metals,^{33,34,49–51,195} and here the SOS DFAs are particularly strong. SOS0-PBE0-2-D3(BJ) and PWPB95-D3(BJ) are first and fifth among the double hybrids, and revDOD-PBEP86-D4 (fourth best) performs slightly better than its SCS equivalent revDSD-PBEP86-D4.

Finally, we look at the choice of basis set, as large transition metal complexes and open-shell systems can be quite computationally demanding; therefore, faster methods that maintain accuracy are desirable for computational studies of metalloenzymes. We do not recommend cutting costs by using GGA or meta-GGA/NGA functionals, as the results in Figure 4 and Table 5 show that these are significantly less accurate. We specifically note that MN15-L, the best third-rung functional on MADs, was the slowest converging functional overall for most structures; therefore, it provided no cost benefit over a good hybrid or even double hybrid. However, this may be due to the LibXC¹⁵⁵ implementation evoked by ORCA, as MN15 and MPW1B95 were also slow. We also cannot recommend the −3c composite methods, as although they are fast, they perform poorly—PBEh-3c is the worst hybrid, while r²SCAN-3c and B97-3c are both the second worst in their respective rungs. We therefore test smaller basis sets for each of B3LYP-D3(BJ), M06-D4, PWPB95-D3(BJ), and ωB2PLYP-D4 (the best dispersion combination for each functional) in Figure 6. Between def2-SVP and def2-TZVPP, there is a significant reduction in the MAD, error range, and distribution of the data for all four DFAs, confirming that double-ζ basis sets are not adequate for calculating BHs and REs. Between def2-TZVPP and def2-QZVPP, however, the results are almost identical. All MADs are the same, and the error ranges only decrease slightly with the larger basis set (1.5 kcal/mol for ωB2PLYP-D4, and less than 1 kcal/mol for the other DFAs). PWPB95 in particular is known to have a weaker basis set dependence than other double hybrids, due to its relatively low fraction of SOS-MP2 correlation.¹⁵⁰ Thus, we can say that the DFT energies are adequately converged at the triple-ζ level, and it is not worth the extra cost to go up to a quadruple-ζ basis set for calculating REs and BHs in such systems.

Figure 6.

Violin plots of selected functionals with double-, triple- and quadruple-ζ basis sets. The overall shape represents the density of the data across the range of deviation values, while the internal black boxes and white dots represent the interquartile range and median value, respectively. Each plot is labeled with the MAD (kcal/mol) for that level of theory.

Based on these benchmarking results, our top recommendations for calculating metalloenzyme BHs and REs are SOS0-PBE0-2-D3(BJ)/def2-TZVPP and revDOD-PBEP86-D4/def2-TZVPP. SOS0-PBE0-2-D3(BJ) performs well across the range of metalloenzymes tested here, and we have previously shown its accuracy for organic enzymes as well.¹⁸ Double hybrid DFAs are more reliable than hybrids across the board, although the best hybrid functionals still perform well. With all DFAs, we strongly recommend the use of a dispersion correction. For copper-dependent enzymes, specifically, a double hybrid DFA with only a small amount of MP2 correlation is the most reliable option, so we also recommend PBE0-DH for these. Acknowledging that double hybrids can be prohibitively expensive, we also recommend the hybrid functionals ωB97M-V and ωB97X-V. MPW1B95-D3(BJ) also performs well, but the LibXC implementation makes it slower than other hybrid functionals directly implemented in ORCA. We do note, however, that functionals that are good for energetics are not always the best for geometries.¹⁶ A test of small organometallic complexes has shown that ωB97M-D3(BJ) and ωB97M-D4 perform similarly to B3LYP-D3(BJ) for geometry optimizations, while B97M-V is slightly better,¹⁰ but without further testing we can only definitively recommend these functionals for single-point energy calculations.

5. Summary and Conclusions

Here, we have presented and used our new set, MME55, for benchmarking metalloenzyme REs and BHs. Ten model reactions were taken from the literature, representing 8 different transition metals, and all geometries were reoptimized at the PBEh-3c level of theory. Some reaction steps described in the original studies were omitted from the set based on an analysis of the multireference character.

Five coupled cluster approaches were tested for the calculation of the reference values. We see a large enough difference between the (T₁) iterative triples procedure and the older (T₀) triples to select DLPNO–CCSD(T₁)/TightPNO/CBS(def2-TZVPP/def2-QZVPP) as our preferred level of theory. Due to the size of the models in the MME55 set, however, most systems required approximations to this level of theory—estimation of fully extrapolated DLPNO–CCSD(T₁) was done by either correcting DLPNO–CCSD(T₀)/TightPNO/CBS(3,4) with the difference between (T₁) and (T₀) for a smaller basis set or adding a lower-level (T₁) triples correction to DLPNO–CCSD/TightPNO/CBS results, while TightPNO thresholds for DLPNO–CCSD(T₁) were estimated by extrapolating results calculated with looser values of the T_CutPNO parameter following the scheme designed by Altun et al.

The final set contains 16 BHs and 39 REs for a total of 55 data points and was used to test a range of density functionals across the top four rungs of Jacob’s Ladder (GGAs up to double hybrids). SOS0-PBE0-2-D3(BJ) had the lowest MAD of all functionals, while ωB97M-V was the best hybrid functional; we also recommend revDOD-PBEP86-D4 and ωB97X-V based on an analysis of MAPDs across the whole set and then each enzyme individually. SOS0-PBE0-2 and revDOD-PBEP86-D4 are both spin-opposite-scaled double hybrids, and our results reaffirm previous studies that show that SCS-MP2 and SOS-MP2 are significantly better for transition metals than standard MP2. We also recommend PBE0-DH specifically for copper-dependent enzymes, which are more sensitive to the amount of MP2 correlation in double hybrid DFAs. B3LYP, despite its popularity in metalloenzyme modeling, is clearly outperformed by multiple other hybrids, even in its best dispersion-corrected form [B3LYP-D3(BJ)]. Tests of three different basis sets showed that def2-TZVPP gives almost identical results to def2-QZVPP, but double-ζ basis sets are insufficiently small.

While double hybrids generally give better results than hybrids, the additional computational cost can be prohibitive for large active site models/QM regions; our most cost-effective recommendations are therefore ωB97M-V/def2-TZVPP and ωB97X-V/def2-TZVPP, which give reliable and robust results across many different types of enzymes. We hope these recommendations help improve the results of computational mechanistic studies of enzymes and help the ongoing efforts to flesh out the field of DFT benchmarking for transition metal chemistry.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00558.

• Details of the cut model of perchlorate reductase, FOD plots for all structures, additional tests of estimated DLPNO–CCSD(T₁) approaches, and additional benchmarking analysis and discussion(PDF)

• System details, multireference diagnostics, statistics, deviations, and percent deviations of assessed methods(XLSX)

• Optimized structures(ZIP)

The authors declare no competing financial interest.