The Nonlocal Kernel in van der Waals Density Functionals as an

Subscriber access provided by Kaohsiung Medical University

Quantum Electronic Structure

The non-local kernel in van-der-Waals density functionals as an additive correction — an extensive analysis with special emphasis on the B97M-V and #B97M-V approaches Asim Najibi, and Lars Goerigk J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00842 • Publication Date (Web): 09 Oct 2018 Downloaded from http://pubs.acs.org on October 10, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Journal of Chemical Theory and Computation

The non-local kernel in van-der-Waals density functionals as an additive correction — an extensive analysis with special emphasis on the B97M-V and ωB97M-V approaches Asim Najibi and Lars Goerigk∗ School of Chemistry, The University of Melbourne, Parkville, Victoria 3010, Australia E-mail: [email protected]

Abstract The development of van-der-Waals density functional approximations (vdW-DFAs) has gained considerable interest over the past decade. While in a strictest sense, energy calculations with vdW-DFAs should be carried out fully self-consistently, we demonstrate conclusively for a total of 11 methods that such a strategy only increases the computational time effort without having any significant effect on energetic properties, electron densities or orbital-energy differences. The strategy to apply a non-local vdW-DFA kernel as an additive correction to a fully converged conventional DFA result is therefore justified and more efficient. As part of our study, we utilize the extensive GMTKN55 database for general main-group thermochemistry, kinetics and noncovalent interactions [Phys. Chem. Chem. Phys. 2017, 19, 32184], which allows us to analyze the very promising B97M-V [J. Chem. Phys. 2015, 142, 074111] and ωB97M-V [J. Chem. Phys. 2016, 144, 214110] DFAs. We also present new DFT-D3(BJ) based counterparts of these two methods and of ωB97X-V [J. Chem. Theory Comput 2013,

1

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

9, 263], which are faster variants with similar accuracy. Our study concludes with updated recommendations for the general method user, based on our current overview of 325 dispersion-corrected and -uncorrected DFA variants analyzed for GMTKN55. vdW-DFAs are the best representatives of the three highest rungs of Jacob’s Ladder, namely B97M-V, ωB97M-V, and DSD-PBEP86-NL.

1

Introduction

Many conventional computational-chemistry models based on Kohn-Sham Density Functional Theory (KS-DFT) 1 are unable to completely incorporate non-local electron-correlation (NLC) effects, such as London-dispersion interactions, on their own. 2–6 This shortcoming of conventional density functional approximations (DFAs) has sparked tremendous interest in developing schemes to rectify this issue. One popular strategy to fix the dispersion problem of conventional DFT has been the development of additive corrections, which use a molecular geometry as an input and sometimes also its electron density. 7–20 One of the most popular schemes representing this strategy is the “DFT-D3” correction, developed by Grimme and co-workers, 9,10 where the majority of the missing dispersion energy is accounted for by summing up the dispersion contributions of each atom pair, while three-body interactions can be additionally requested. In DFT-D3, not only the molecular geometry is used as an input, but the chemical environment of each individual atom modifies its C6 dispersion coefficient; see Ref. 9 for details. While this introduces some kind of system-dependence, DFT-D3 is still an addition to the molecular electronic energy, and as such it can only influence energetic properties or properties derived thereof, such as energy gradients or Hessians. A direct influence on molecular-orbital (MO) energies and, thus, electron densities is not achieved. In practice, however, the computational efficiency of DFT-D3 and its flexibility, which makes it applicable to almost all DFAs (see Refs. 21–24 for notable exceptions), made it a highly successful scheme for applications. In fact, some of the currently most accurate DFAs for thermochemistry, kinetics, noncovalent interactions, and geometries make use of this correc2


Page 2 of 45

Page 3 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


tion. 10,24–32 From a purist’s point of view, it is desirable to fix the dispersion problem at its root, and indeed another popular strategy has emerged, which adds NLC energy effects as a function of the electron density to an already-existing exchange-correlation DFA; this is known as vander-Waals density functional theory (vdW-DFT). The first example of this idea — the vdWDF functional — was proposed in 2004, 33,34 and its successor vdW-DF2 35 in 2010. In 2009, Vydrov and Van Voorhis developed the VV09 method, 36 which was further modified in 2010 to become the VV10 functional. 37 A revised version, called rVV10, was published by Sabatini et al. in 2013, 38 which turned out to be more efficient than VV10 in a plane-wave framework, but was later shown to provide little improvements for noncovalent interactions, kinetics and thermochemistry calculations in atomic-orbital (AO) based molecular calculations. 39 Most recent works in the area of vdW-DFT have focussed on approaches utilizing the VV10 idea, which is why the present study will do the same. The initial VV10 functional is based on the semi-local, generalized-gradient approximation (GGA) rPW86PBE, 40,41 which was enhanced with an NLC kernel, henceforth called “VV10 kernel”. Hujo and Grimme generalized the use of the VV10 kernel for other semi-local DFT methods, including hybrid ones, coining the approach as “DFT-NL”, where the suffix “NL” stands for “non-local”. 42,43 The probably most successful implementation of the VV10 kernel consists of a series of combinatorially optimized vdW-DFAs developed by Mardirossian and Head-Gordon, namely the B97M-V 44 meta-GGA, the ωB97X-V 45 and ωB97M-V 46 range-separated hybrids, and the ωB97M(2) 47 range-separated double hybrid. The combinatorially optimized B97-based vdW-DFAs have shown very promising potential in a recent comprehensive thermochemistry benchmark study that comprised 200 dispersioncorrected and uncorrected versions of 91 unique exchange-correlation functionals. 25 Given their success, both the developer and user communities have become increasingly interested in these approaches and it is therefore warranted to have a closer look at methods based on non-local vdW-DFT kernels, which is what we aim to achieve with the present work. The

3



VV10 NLC energy contribution — and in principle that of any other vdW-DFA — can be included in a fully self-consistent-field (SCF) procedure (“full-SCF”), which would be in the spirit of the KS-DFT idea, or after the SCF procedure of the semi-local exchange-correlation component (“post-SCF”), which would make it an additive correction that does not influence MO energies or electron densities. There have been a handful of studies that applied vdWDFAs in their post-SCF form for thermochemistry and geometries, without any reported shortcomings of this strategy, see e.g. Refs. 24,42,43,48–50. In fact, the quantum-chemistry packages ORCA 51 and PSI4 52,53 offer the user to choose between both strategies for any DFA. However, based on their papers, the full-SCF variant is presumably in the spirit of vdWDFT developers, as it promises a “true” solution to the dispersion problem of conventional DFT. 25,33,34,36–38,44–47,54 In fact, the QCHEM electronic structure program, 55 which was used for the development of the “B97(M)-V type” methods by Mardirossian and Head-Gordon, only offers the VV10 kernel in its full-SCF version. Similarly, the ERKALE program 56 offers only the full-SCF version for a limited number of DFAs, including the ones mentioned in the previous sentence. While the post-SCF version of some DFT-NL approaches has been used in applications, only one study in 2012 noticed for ionic-liquid model systems that the full- and post-SCF variants of revPBE38 21 -NL had negligible differences for noncovalent interaction energies. 48 To the best of our knowledge there have not been any studies directly comparing the full-SCF and post-SCF additions of the NLC energy term in different vdW-DFAs for a broader range of energetic and other orbital- or density-based properties. However, given the increased interest in this area and given the differences on how the VV10 kernel may be implemented in various electronic-structure codes, we aim to carry out the first detailed comparison between both the full and post-SCF implementations to guide method developers and users in the future. In this work, we thoroughly compare the accuracy of the full-SCF and post-SCF addition of the VV10 kernel for 11 vdW-DFAs across different rungs of Jacob’s Ladder. 57 As the basis for our analysis we use the recently established extensive GMTKN55 database of general main-

4


Page 4 of 45

Page 5 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


group thermochemistry, kinetics and noncovalent interactions developed by Goerigk et al. 24 To date, two articles that constitute some of the largest DFT benchmark studies conducted have provided insights into the robustness and applicability of 313 dispersion-corrected and -uncorrected approaches, with detailed analyses of 105 dispersion-corrected methods, incl. five VV10-corrected approaches in their post-SCF variant. To base our present work on GMTKN55 will not only allow us to gain reliable data to answer our main question, but it has the additional benefit to use the data published for GMTKN55 as a benchmark for additional methods that we assess on GMTKN55 for the first time, in particular the very promising B97M-V and ωB97M-V approaches. Whilst many studies using VV10 or similar kernels are based on energetic properties, we will go beyond those and briefly assess the influence of the NLC kernel on electron densities and MOs, and thus indirectly on molecular properties derived from them. Finally, we present DFT-D3 based modifications of Mardirossian and Head-Gordon’s vdW-DFAs as useful and faster alternatives.

2 2.1

Computational and technical details The GMTKN55 database

The GMTKN55 database is one of the largest quantum-chemical benchmark databases to date. Its 55 benchmark sets can be divided into five categories — basic properties and reaction energies of small systems (18 sets), reaction energies of large systems and isomerization energies (9 sets), reaction barrier heights (7 sets), intermolecular (12 sets) and intramolecular noncovalent interactions (9 sets). The entire database contains 2462 single point calculations, which give rise to a total of 1505 relative energies. Compared to its predecessor GMTKN30 58 and other databases, it relies on reference values of significantly higher quality and contains a wider range of different properties and larger chemical systems (up to C60 ). The benchmark sets in the GMTKN55 database are shown in Table 1. For the present work, we adopt the geometries and zero-point-vibrational-energy exclusive, non-relativistic 5



Page 6 of 45

and (mostly) all-electron based reference values as presented in Ref. 24.

2.2

Brief theoretical background

In this work, we augment various exchange-correlation DFAs with the “VV10” NLC energy term: 37 DF T −N L V V 10 EXC = EX + EC + EC,N L

(1)

where we adopted Grimme’s “DFT-NL” notation 42,43 with (in atomic units)

V V 10 EC,N L

Z =

"

1 drρ (r) 32

3 b2

34

1 + 2

#

Z

0

0

0

dr ρ (r ) Φ (r, r )

(2)

where ρ is the density at each position ri , and Φ is the NLC kernel describing the correlation between two points in space (see Ref. 37 for a detailed discussion on the kernel). b is an adjustable parameter which controls the short-range damping of the R−6 asymptote. The kernel also contains the parameter b, as well as another adjustable parameter C, which can give accurate asymptotic van der Waals C6 coefficients. Both parameters are adjustable to make the VV10 kernel compatible with any DFA, however, with some Minnesota-type DFAs being exceptions. 23 If all three components in Eq. 1 are used during the SCF procedure, we obtain the “fullSCF” variant, which is in the spirit of vdW-DFT developers. If the SCF calculation is only V V 10 carried out for the first two components in Eq. 1, the EC,N L term becomes a mere additive

correction, which is evaluated with the MOs and electron density from the previous SCF step, but which essentially has no potential influence on them. This is what we call herein the “post-SCF” variant.

2.3

Technical details

All calculations were performed on ORCA 51,115 in its versions 3.0.0, 3.0.3, 4.0.0, and 4.0.1. A local version was used for B97M-V, ωB97X-V, and ωB97M-V, as well as for their mod6


Page 7 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Description of the benchmark sets in the GMTKN55 database set description basic properties and reaction energies for small systems W4-11 59 total atomization energies G21EA 60,61 adiabatic electron affinities G21IP 60,61 adiabatic ionization potentials DIPCS10 24 double-ionization potentials of closed-shell systems PA26 24,62,63 adiabatic proton affinities (incl. of amino acids) SIE4x4 24 self-interaction error related problems ALKBDE10 64 dissociation energies in group-1 and -2 diatomics YBDE18 24,65 bond-dissociation energies in ylides AL2X6 24,61,66 dimerization energies of AlX3 compounds HEAVYSB11 24 dissociation energies in heavy-element compounds NBPRC 24,58,61,67 oligomerizations and H2 fragmentations of NH3 /BH3 systems; H2 activation reactions with PH3 /BH3 systems ALK8 24 dissociation and other reactions of alkaline compounds RC21 24 fragmentations and rearrangements in radical cations G2RC 24,61,68 reaction energies of selected G2/97 systems BH76RC 24,61 reaction energies of the BH76 set FH51 69,70 reaction energies in various (in-)organic systems TAUT15 24 relative energies in tautomers DC13 24,61,71–81 13 difficult cases for DFT methods reaction energies for large systems and isomerization reactions MB16-43 24 decomposition energies of artificial molecules DARC 24,61,66 reaction energies of Diels-Alder reactions RSE43 24,82 radical-stabilization energies BSR36 24 bond-separation reactions of saturated hydrocarbons CDIE20 24,83,84 double-bond isomerization energies in cyclic systems ISO34 24 isomerization energies of small and medium-sized organic molecules ISOL24 24,85 isomerization energies of large organic molecules C60ISO 86 relative energies between C60 isomers PArel 24 relative energies in protonated isomers reaction barrier heights BH76 24,61,87,88 barrier heights of hydrogen transfer, heavy atom transfer, nucleophilic substitution, unimolecular and association reactions BHPERI 61,89–92 barrier heights of pericyclic reactions BHDIV10 24 diverse reaction barrier heights INV24 93 inversion/racemization barrier heights BHROT27 24 barrier heights for rotation around single bonds PX13 24,94 proton-exchange barriers in H2 O, NH3 , and HF clusters WCPT18 24,95 proton-transfer barriers in uncatalyzed and water-catalyzed reactions intermolecular noncovalent interactions RG18 24 interaction energies in rare-gas complexes ADIM6 9,24 interaction energies of n-alkane dimers S22 96,97 binding energies of noncovalently bound dimers S66 98 binding energies of noncovalently bound dimers HEAVY28 9,24 noncovalent interaction energies between heavy element hydrides WATER27 99,100 binding energies in (H2 O)n , H+ (H2 O)n and OH− (H2 O)n CARBHB12 24 hydrogen-bonded complexes between carbene analogues and H2 O, NH3 , or HCl PNICO23 24,101 interaction energies in pnicogen-containing dimers HAL59 24,102,103 binding energies in halogenated dimers (incl. halogen bonds) AHB21 104 interaction energies in anion-neutral dimers CHB6 104 interaction energies in cation-neutral dimers IL16 104 interaction energies in anion-cation dimers

b

#a

| ∆E |

140 (152) 25 (50) 36 (71) 10(20) 26 (52) 16 (23) 10 (20) 18 (29) 6 (11) 11 (22) 12 (21)

306.91 33.62 257.61 654.26 189.05 33.72 100.69 49.28 35.88 58.02 27.71

8 (17) 21 (41) 25 (47) 30 51 (87) 15 (25) 13 (30)

62.60 35.70 51.26 21.39 31.01 3.05 54.98

43 (58) 14 (22) 43 (88) 36 (38) 20 (36) 34 (63) 24 (48) 9 (10) 20 (31)

414.73 32.47 7.60 16.20 4.06 14.57 21.92 98.25 4.63

76 (86)

18.61

26 10 24 27 13 18

20.87 45.33 31.85 6.27 33.36 34.99

(61) (20) (48) (40) (29) (28)

18 (25) 6 (12) 22 (57) 66 (198) 28 (38) 27 (30) 12 (36) 23 (69) 59 (105) 21 (63) 6 (18) 16 (48)

0.58 3.36 7.30 5.47 1.24 81.14 6.04 4.27 4.59 22.49 26.79 109.04

intramolecular noncovalent interactions IDISP 24,58,61,85,105,106 intramolecular dispersion interactions 6 (13) 14.22 ICONF 24 relative energies in conformers of inorganic systems 17 (27) 3.27 ACONF 107 relative energies of alkane conformers 15 (18) 1.83 AMINO20x4 24,108 relative energies in amino acid conformers 80 (100) 2.44 PCONF21 24,109,110 relative energies in tri- and tetrapeptide conformers 18 (21) 1.62 MCONF 24,111 relative energies in melatonin conformers 51 (52) 4.97 SCONF 24,61,112 relative energies of sugar conformers 17 (19) 4.60 UPU23 113 relative energies between RNA-backbone conformers 23(24) 5.72 BUT14DIOL 24,114 relative energies in butane-1,4-diol conformers 64 (65) 2.80 a number of relative energies (and single-point calculations in parentheses) for each benchmark set, except for BH76RC, for which the single-point calculations carried out for the BH76 benchmark set are sufficient. b averaged absolute relative energy (kcal/mol) for each benchmark set, according to reference values. Information on the reference values can be found in the original GMTKN55 study. 24

7



ified, herein introduced DFT-D3 corrected versions B97M-D3(BJ), ωB97X-D3(BJ), and ωB97M-D3(BJ). These DFAs will be included in the next release of ORCA and become, thus, freely available in both their full- and post-SCF implementations to academics. Unless otherwise stated, the def2-QZVP 116 AO quadruple-ζ basis set was used for all calculations to make our results directly comparable with those previously published for GMTKN55. Dunning’s diffuse functions (taken from the aug-cc-pVQZ 117 basis set) were added to def2QZVP for various benchmark sets: diffuse s and p functions were applied to all non-hydrogen atoms and diffuse s functions to hydrogen for the G21EA, AHB21 and IL16 benchmark sets, while diffuse s and p functions were applied to oxygen for the WATER27 benchmark set. We adopt the notation (aug’)-def2-QZVP from previous GMTKN studies for this modified basis set. 21,24,58,61 The def2-ECP 116 effective-core potentials (ECPs) were added to Bi, I, Pb, Rn, Sb, Te and Xe. A multi-grid approach was chosen for the numerical integration over the exchange-correlation potential, namely ORCA’s “GRID3” for the SCF followed by a single step with the larger “GRID4”. ORCA’s default grid “VDWGRID2” was chosen for the evaluation of the VV10 kernel. The SCF convergence criterion was set to 10−7 Eh . The resolution-of-the-identity approximation (RI-J) was applied to the evaluation of all Coulomb integrals and of the second-order perturbative-correlation part of double-hybrid DFAs, 118,119 while the chain-of-spheres 118 approximation (COSX) was applied to all exchange integrals, with the exception of range-separated hybrids. Appropriate auxiliary basis sets were applied in all cases. 118,119 A handful of single-point calculations did not reach convergence with ORCA’s various convergence strategies. Similar convergence problems were found with the QCHEM software. The convergence problems are discussed in detail in Section SI.11 in the Supporting Information (SI). In this work, we utilize a number of functionals from different rungs of Jacob’s Ladder which were chosen for their popularity or because they were shown to be robust and accurate for their respective DFA class: 1. GGA: revPBE 120 and VV10 37,40,41 (note that the VV10 functional uses rPW86 ex8


Page 8 of 45

Page 9 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


change and PBE correlation with the VV10 non-local correlation kernel). 2. meta-GGA: B97M-V 44 3. Hybrid: B3LYP, 121,122 PW6B95, 123 ωB97X-V, 45 and ωB97M-V 46 (where ω represents range-separation of the exchange energy and potential). 4. Double-hybrid: PWPB95, 58 DSD-BLYP, 124 and DSD-PBEP86. 125 Thus, the range of functionals included covers the four highest rungs of Jacob’s Ladder, and additionally it includes the vdW-DFAs of Mardirossian and Head-Gordon. The values for the parameters b and C in Eq. 2 were taken from Ref. 42 for revPBE and B3LYP, from Ref. 126 for PW6B95, and from Ref. 127 for PWPB95, DSD-PBEP86 and DSD-BLYP. In this work, the results of revPBE-D3(BJ), the VV10 functional (in the post-SCF version), rPW86PBE-D3(BJ), B3LYP-D3(BJ), B3LYP-NL (in the post-SCF version), PW6B95D3(BJ), PWPB95-D3(BJ), DSD-PBEP86-D3(BJ), and DSD-BLYP-D3(BJ) are taken from the original GMTKN55 study. 24 Note that the suffix “D3(BJ)” stands for Grimme’s DFT-D3 correction with Becke-Johnson 13,128,129 damping. 9,10 Additionally, we also use the results of the recommended meta-GGA functional SCAN-D3(BJ) 130,131 and the recommended hybrids ωB97X-D3 132 and M052X-D3(0) 21,133 [both based on DFT-D3 with “zero damping” 9 (DFTD3(0)] from the original GMTKN55 study to compare with the methods used in this work. The remainder of results are presented in this work. In order to assess the robustness of a DFA over the GMTKN55 database and its categories, the WTMAD-1 and WTMAD-2 (weighted total mean absolute deviation) schemes presented in the original GMTKN55 study were used. 24 These schemes are shown in detail in the SI and in Refs. 24 and 27. Both schemes give an indication of the overall performance of a method over several benchmark sets by weighting the individual mean absolute deviations (MADs) of each benchmark set, depending on the average reference absolute energy of the benchmark set, and in the case of WTMAD-2, also on the number of relative energies in the benchmark set. Note that eleven additional schemes based on MADs or root-mean-square 9



deviations were assessed in the original GMTKN55 study, all giving the same trends. 24 While this manuscript presents WTMAD-2 values, additional WTMAD-1 values are shown in the SI; both schemes allow drawing the same conclusions. The two WTMAD schemes allow a consistent ranking of all assessed methods, which is why we adopted them for the present study. In fact, we will use the previously published WTMAD values as threshold values that have to be surpassed in order to identify if B97M-V and ωB97M-V are indeed more robust than other approaches belonging to the same class of DFAs. For a discussion on the pros and cons of MAD-based analyses of thermochemical data, see e. g. Ref. 134.

3 3.1

Results and discussion DFT-D3(BJ) versus post-SCF non-local correlation

To date, GMTKN55 results have mostly been analyzed in detail for DFT-D3 corrected approaches, usually with the DFT-D3(BJ) variant, while only a handful of approaches have been assessed with an NLC correction based on the VV10 kernel: 24,27 VV10, B3LYP, ωB97XV, PBE-QIDH, 135 and SOS1-PBE-QIDH. 136 In all cases the NLC correction has only been used in a post-SCF fashion. Herein, we will analyze additional methods and discuss their performance with respect to DFT-D3(BJ). We include a number of functionals from different rungs of Jacob’s Ladder. The resulting WTMAD-2 values are shown in Table 2. We see that across all methods and all categories of the GMTKN55 database, there is no general trend as to which of the DFT-D3(BJ) and NLC corrections perform better than the other. However, there are many cases where the post-SCF vdW-DFT counterpart significantly outperforms the DFT-D3(BJ) counterpart and vice versa; in the following, we only discuss some selected examples. For the revPBE method, the ratio of the MAD of revPBE-NL (post-SCF NL) to the MAD of revPBE-D3(BJ) for CHB6 (interaction energies in cation-neutral dimers) is only 0.43, and that for WATER27 (interaction energies of water clusters) is 0.49 — see Table S15. However, this does not imply that the NLC correction 10


Page 10 of 45

Page 11 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: WTMAD-2 values (kcal/mol) over all subsets of the GMTKN55 database and over the entire GMTKN55 database for various functionals augmented with the DFT-D3(BJ) or post-SCF NLC corrections. rPW86PBEa

revPBE

basic properties and reaction energies for small systems reaction energies for large systems and isomerization energies barrier heights intermolecular noncovalent interactions intramolecular noncovalent interactions all noncovalent interactions all GMTKN55 a

B3LYP

PW6B95

PWPB95

DSD-BLYP

DSD-PBEP86

D3(BJ)

NL

D3(BJ)

NL

D3(BJ)

NL

D3(BJ)

NL

D3(BJ)

NL

D3(BJ)

NL

D3(BJ)

NL

5.54

5..93

6.71

6.33

4.36

4.27

3.28

3.32

2.23

2.33

1.88

2.06

1.69

1.73

10.50

9.79

14.95

12.76

10.28

8.58

9.06

8.39

5.41

4.97

4.32

4.81

3.91

4.30

15.79

17.77

15.17

16.43

9.04

9.92

6.70

6.57

3.39

3.68

3.04

2.86

3.52

3.25

6.19

6.31

8.33

9.71

5.56

5.86

4.22

5.46

4.02

4.62

3.92

3.52

4.25

2.94

7.99

7.65

8.21

9.33

5.68

6.21

6.69

6.29

5.98

5.84

3.15

2.83

3.46

3.06

7.07

6.96

8.27

9.52

5.62

6.03

5.43

5.87

4.98

5.22

3.55

3.18

3.86

3.00

8.27

8.49

9.75

9.93

6.42

6.39

5.50

5.57

3.98

4.07

3.08

3.05

3.14

2.84

rPW86PBE augmented with the VV10 non-local correlation kernel is the VV10 functional

is always better for these benchmark sets, for instance, the ratio is 1.80 for B3LYP and 2.20 for ωB97X-V for WATER27 (Tables S18 and S20). Similarly, for PWPB95, the ratio is only 0.32 for the AL2X6 (dimerization energies of AlX3 compounds), but 1.50 for the sugar conformers in SCONF (Table S22). Lastly, for DSD-PBEP86, the ratio is 0.50 for HEAVY28 (noncovalent interaction energies between heavy element hydrides) — see Table S24. On the other hand, there are also other examples with very large ratios, meaning that DFT-D3(BJ) is better than the NLC correction — it is 2.31 for revPBE for DIPCS10 (double ionization potentials for closed-shell systems) and 2.18 for B3LYP for AHB21 (interaction energies in anion-neutral dimers). While there is no clear trend regarding which of the two corrections is better, one significant finding is that for the intra- and intermolecular noncovalent interaction categories, for which the NL approach performs better with the two DSD-type double-hybrids, DSD-BLYP and DSD-PBEP86, for noncovalent interactions. Here, the NLC correction performs better by at least 0.32 kcal/mol (DSD-BLYP for intramolecular noncovalent interactions) and by as much as 1.31 kcal/mol (DSD-PBEP86 for intermolecular interactions) according to WTMAD-2 values (Table 2). Based on the two WTMAD schemes, DSD-BLYP-D3(BJ) and DSD-PBEP86-D3(BJ) were 11



identified as the two best out of 217 dispersion-corrected and -uncorrected DFAs in the original GMTKN55 study 24 and as the second and third best out of 313 DFT variants in a recent update on GMTKN55. 27 Here, we can report that they can be even more accurate for noncovalent interactions upon switching from the DFT-D3(BJ) to the NLC correction. The same trends can be observed with WTMAD-1 values (see Table S1), and additionally, we present ratios of MADs of the post-SCF DFT-NL to the DFT-D3(BJ) approaches in Section SI.4. An overall comparison with other published data and a re-evaluation of the GMTKN55-based DFA ranking is carried out later in Section 3.5.

3.2

post-SCF v full-SCF non-local correlation

Next, we compare the application of the VV10 NLC contribution in a full-SCF and post-SCF fashion for a number of DFAs from different rungs of Jacob’s Ladder. We exclude double hybrids, as they do not normally rely on full-SCF treatments — the exception would be orbital-optimized double hybrids, 137 which we did not consider for efficiency and accuracy reasons, as discussed in Ref. 26. The WTMAD-2 values for GMTKN55 and its categories are shown in Table 3. We find that the accuracy of the post-SCF approach basically matches that of the full-SCF approach, with the largest positive difference in WTMAD-2 values [WTMAD2(post-SCF approach) − WTMAD-2(full-SCF approach)] of 0.12 kcal/mol (PW6B95 for intramolecular noncovalent interactions). The same trends can be observed with WTMAD1 values (see Table S1). Additionally, we present ratios of MADs of the post-SCF to the full-SCF approaches in Section SI.5 in the SI. With the exception of very few cases, the ratio of the MAD of the post-SCF method to that of the full-SCF method is always approximately equal to unity. The few exceptions include: 1. the revPBE method for the ADIM6 benchmark set (interaction energies of n-alkane dimers) with a ratio of 0.56 and for UPU23 (relative energies between RNA-backbone conformers) with a ratio of 0.89, with the post-SCF version being better by 0.11 and 12


Page 12 of 45

Page 13 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3: WTMAD-2 values (kcal/mol) over all subsets of the GMTKN55 database and over the entire GMTKN55 database for various functionals augmented with the VV10 non-local correlation (NLC) energy in a full-SCF or post-SCF fashion. revPBE

basic properties and reaction energies for small systems reaction energies for large systems and isomerization energies barrier heights intermolecular noncovalent interactions intramolecular noncovalent interactions all noncovalent interactions all GMTKN55

VV10

B97M-V

B3LYP

PW6B95

ωB97X-V

ωB97M-V

full

post

full

post

full

post

full

post

full

post

full

post

full

post

5.94

5.93

6.35

6.33

3.68

3.68

4.26

4.27

3.32

3.32

3.35

3.34

2.74

2.73

9.79

9.79

12.75

12.76

9.30

9.31

8.54

8.58

8.39

8.39

6.66

6.68

4.77

4.79

17.74

17.77

16.36

16.43

7.52

7.53

9.93

9.92

6.57

6.57

4.21

4.21

3.40

3.40

6.47

6.31

9.80

9.71

3.56

3.56

5.80

5.86

5.46

5.46

3.03

3.03

2.86

2.90

7.92

7.65

9.35

9.33

5.76

5.74

6.32

6.21

6.17

6.29

3.66

3.63

4.57

4.53

7.18

6.96

9.58

9.52

4.64

4.63

6.05

6.03

5.81

5.87

3.34

3.32

3.70

3.70

8.57

8.49

9.97

9.93

5.46

5.46

6.39

6.39

5.54

5.57

3.99

3.98

3.53

3.53

0.05 kcal/mol, respectively. 2. the B97M-V method for ADIM6 with a ratio of 0.87, with the post-SCF version being better by 0.03 kcal/mol. 3. the B3LYP method for ACONF (relative energies of alkane conformers) with a ratio of 0.86 (both versions are within the 0.1 kcal/mol accuracy window of the reference method) and for ADIM6 with a ratio of 1.42, with the post-SCF version being worse by 0.05 kcal/mol. 4. the ωB97X-V method for ADIM6 with a ratio of 0.84, with the post-SCF version being better by 0.03 kcal/mol. In summary, we can reassure users, who may have applied the NLC correction in the postSCF version in the past, 24,42,43,48–50 that they can continue to do so in the future. Our findings are also of interest to method developers, showing that the fact that the NLC correction is used fully self-consistently is no essential requirement for a vdW-DFA’s good performance for energetic properties. In fact, favoring the post-SCF over the full-SCF approach is beneficial from an application point of view, as the first requires less computational effort than the latter. To demonstrate this, we calculated the single-point energy of the C60 buckyball taken from the C60ISO set on a single CPU of an Apple MacPro workstation (3.0GHz 8-Core 13



Page 14 of 45

Intel Xeon E5 with 25MB L3 cache and 64GB of 1866MHz DDR3 ECC memory) with the full-SCF and post-SCF variants of the VV10 GGA approach and the def2-QZVP basis set. It turns out that the post-SCF variant was 1.8 times faster than the full-SCF one, showing the benefits of using the VV10 (or similar) kernels merely as an additive correction.

3.3

Non-local correlation correction and molecular properties

Having established that vdW-DFAs do not benefit from a full-SCF treatment of energetic properties, the question remains if the VV10 kernel has a direct influence on the MOs and hence the electron density and potentially other molecular properties. As a first example of a density-based analysis we analyze “NCI plots” generated with the “NCIPLOT” program 138,139 for the water, methane and parallel-displaced benzene dimers, with geometries taken from the S22 benchmark set. 96 NCI plots are a very useful way of qualitatively analyzing the nature of noncovalent interaction as shown in Figure 1 for the three dimers. For readers unfamiliar with this idea, it suffices to know that the plots in Figure 1 show how the reduced density gradient

cS ∇ρ/ρ4/3 =

1 1

2(3π 2 ) 3

∇ρ/ρ4/3

(3)

varies with respect to the product of the density ρ and the sign of the value λ2 , which is one of three eigenvalues of the electron-density Hessian, which are connected with the density Laplacian ∇2 ρ as:

∇2 ρ = λ1 + λ2 + λ3

(4)

A strongly negative sign of λ2 indicates a mostly electrostatic (hydrogen-bonded) interaction, a slightly negative sign a dispersion interaction, and a positive sign no bonded interactions (see Ref. 138 for details). The peaks in all plots in Figure 1 demonstrate the type of interaction that can be expected in the various dimers. In the context of our study, it is 14


Page 15 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1: Variation of the reduced density gradient cS ∇ρ/ρ4/3 with respect to the product of the density ρ and the sign of the value λ2 (see Eqs. 3 and 4) for three different noncovalently bound dimers: a) water dimer, b) methane dimer and c) benzene dimer. For each system, the blue plot (first row) is the result using revPBE and the red plot (second row) is the result using revPBE-NL (full-SCF NLC). The def2-TZVP 116 triple-ζ AO basis set was used, which is sufficient for the sake of comparison between the two plots.

relevant to compare the plots for a dispersion-uncorrected DFA (in our case revPBE) with that of a fully self-consistently obtained results of a vdW-DFA (in our case revPBE-NL). We observe that there is no qualitative change in the plots with the VV10 kernel, indicating that the full-SCF addition of the NLC kernel does not change electron-density distributions in molecules. Little or no changes in an electron-density based scheme, may indicate only marginal influence on the MOs and their energies. We investigate this by having a look at the gaps between the highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (LUMO) of each of the three small dimers used for the analysis in Figure 1 as well as for a large merocyanine dimer shown in Figure 2. It contains 84 non-hydrogen atoms and 66 hydrogen atoms and the geometry is taken from computational work that investigated its exciton-coupled electronic circular-dichroism spectrum. 140 Exciton coupling is an important process for technological applications involving electron-transfer processes, for which 15



Figure 2: A merocyanine dimer dye of 150 atoms (84 non-hydrogen atoms), used for analysis in this study. Grey — C, white — H, blue — N and red — O.

computational studies, often based on linear-response time-dependent DFT (TD-DFT), are carried out. As TD-DFT also depends on MO energies and given the noncovalently bound nature of the system, we included it herein as an insightful example. Table 4 shows HOMO-LUMO gaps for all four dimers for revPBE and revPBE-NL in its full-SCF variant. There are only very small differences between both methods with the largest difference being for the water dimer, namely 0.14 eV, a value comparable to the expected error of the currently most accurate TD-DFT methods. 141,142 For the larger and technologically more relevant merocyanine dimer there is in fact no significant difference to report and any TD-DFT treatment would be completely unaffected by NLC correction to obtain ground-state MOs. Our results presented in this section clearly indicate that when a full-SCF NLC energy term is added to a DFT-based method, it gives the same qualitative MO energy gap as the uncorrected method. There is also no change in electron densities. Consequently, we expect the

16


Page 16 of 45

Page 17 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4: HOMO-LUMO gaps (eV) for various noncovalently bound dimers given by the revPBE and revPBE-NL (full-SCF NLC) methods.a

revPBE-NLb

revPBE water dimer methane dimer benzene dimer a merocyanine (a dimer) a The

5.63 9.25 4.78 1.63

5.77 9.17 4.75 1.63

def2-TZVP AO basis set was used, which is sufficient for the sake

of comparison between the HOMO-LUMO gaps.

b

Full-SCF version.

full-SCF addition of the NLC term having very little effect on impact on molecular properties derived from MO energies or electron densities. This signifies that vdW-DFT, exemplified herein by the VV10 kernel, is ultimately only an energy correction and not a methodology that has to be applied in the strictest KS-DFT sense. Moreover, our findings have practical implications and they may eventually be important for the usage and implementation of the very recently proposed ωB97M(2) 47 double-hybrid DFA. According to Ref. 47, it relies on fully converged ωB97M-V orbitals that are then used for a post-SCF step involving the actual double-hybrid energy expression. As orbital energies do not seem to be affected by the VV10 kernel, we suggest this functional to be used with mere ωB97M orbitals, an idea that we are currently assessing for a related study. 143

3.4

DFT-D3 variants for B97M-V, ωB97X-V and ωB97M-V

Having established that it is sufficient to add the NLC term in vdW-DFAs in a post-SCF fashion, we now turn to the question of the performance of B97M-V, ωB97X-V, and ωB97M-V augmented with the DFT-D3(BJ) correction instead of the NLC term, which would be useful, faster variants of those three DFAs and particularly handy for larger systems. Additive dispersion corrections can be parametrized for a given DFA in one of two ways: either reparametrizing the entire DFA and the dispersion correction simultaneously (examples include B97-D, 8 ωB97X-D, 144 ωB97X-D3 132 and PWPB95-D3 58 ), or only reparametrizing the dispersion correction without changing the underlying DFA [examples include PBE-D3, 145 17



B3LYP-D3, 121,122 and PWPB95-D3(BJ) 21 ]. Another example is the DSD-BLYP 124 method that was initially developed using the first strategy together with the DFT-D2 correction, 8 but then later the DFT-D3 correction was parametrized for it by Goerigk and Grimme using the latter strategy, without any apparent negative impact on its performance; in fact, results improved due to the application of the more accurate DFT-D3. 21 The success of the latter strategy is based on the hypothesis that the DFT-D3 damping parameters adjust themselves to the underlying exchange-correlation DFA. It has the advantage to keep one functional name, requires only changing the suffix indicating the type of dispersion correction, and ensures better consistency. Herein we follow the second strategy, using a least-squares fit (details are shown in Section SI.4) to optimize DFT-D3 damping parameters for B97M-V, ωB97X-V and ωB97M-V without their NLC terms. The training set used comprises intermolecular interaction energies of dimers in their equilibrium and non-equilibrium geometries. The optimized DFT-D3(BJ) damping parameters for the three functionals are shown in Table S25. DFT-D3(0) zerodamping parameters, i.e. the DFT-D3 variant with zero damping, were also optimized for B97M-V and ωB97M-V (see the Table S26). We call the resulting approaches B97M-D3(BJ), ωB97X-D3(BJ), and ωB97M-D3(BJ). Note that ωB97X-D3(BJ) should not be confused with ωB97X-D3. The latter was parametrized together with the underlying exchange-correlation DFA for usage with the zero-damping version of DFT-D3, 132 while the first is based on the parametrization of the exchange-correlation component of ωB97X-V. It is worth noting that both the post-SCF and full-SCF variants of B97M-V and ωB97M-V have very small numerical quadrature-grid dependencies for the evaluation of their exchangecorrelation potentials; note that by default all DFT-D3 variants must have the same dependencies as the post-SCF versions (see Tables S39-S44). We see that the full-SCF variants of these functionals have a small dependence at shorter intermolecular distances, but none for longer ones. Small, but for practical purposes negligible grid-based differences in interaction energies can be observed for the post-SCF and DFT-D3 variants. Both the post-SCF and

18


Page 18 of 45

Page 19 of 45

Table 5: WTMAD-2 values (kcal/mol) over all subsets of the GMTKN55 database and over the entire GMTKN55 database for the leading meta-GGA/NGA and hybrid functionals.

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


basic properties and reaction energies for small systems reaction energies for large systems and isomerization energies barrier heights intermolecular noncovalent interactions intramolecular noncovalent interactions all noncovalent interactions all GMTKN55 a

B97M-Va

B97M-D3(BJ)

SCAN-D3(BJ)b

ωB97X-Va,b

ωB97X-D3(BJ)

M052X-D3(0)b

ωB97M-Va

ωB97M-D3(BJ)

ωB97X-D3b,c

3.68

4.54

5.31

3.34

3.47

3.13

2.73

2.79

3.32

9.31

9.08

7.86

6.68

5.64

5.20

4.79

5.87

7.85

7.53

7.95

14.96

4.21

3.68

4.78

3.40

3.15

4.67

3.56

6.83

8.50

3.03

4.96

5.46

2.90

4.30

4.54

5.74

6.15

6.61

3.62

4.53

5.52

4.53

4.72

4.86

4.63

6.49

7.58

3.32

4.75

5.49

3.70

4.50

4.70

5.46

6.49

7.86

3.98

4.35

4.61

3.53

4.01

4.77

post-SCF version. b taken from Ref. 24. c The exchange-correlation functional and DFT-D3 correction of ωB97X-D3 were parametrized together with zero damping. 132



full-SCF variants of B97M-V and ωB97M-V are influenced to a similar extent by the chosen grid to evaluate the NLC, but again those depend on the underlying kernel and not on the way the entire functional is treated (post- or full-SCF). Table 5 shows the WTMAD-2 values of the post-SCF NLC versions of B97M-V, ωB97X-V and ωB97M-V methods and their DFT-D3(BJ) corrected counterparts, as well as for ωB97XD3 (values taken from Ref. 24). We see that the DFT-D3(BJ) counterparts of B97M-V can exceed the accuracy of the original vdW-DFT method in some cases. For example, for reaction energies of large systems and isomerization energies, we report WTMAD-2(B97MV) = 9.31 kcal/mol and WTMAD-2[B97M-D3(BJ)] = 9.08 kcal/mol. Similarly, we observe WTMAD-2(ωB97X-V) = 6.68 kcal/mol and WTMAD-2[ωB97X-D3(BJ)] = 5.64 kcal/mol for the isomerization energies and reaction energies of large systems, and WTMAD-2(ωB97X-V) = 4.21 kcal/mol and WTMAD-2[ωB97X-D3(BJ)] = 3.68 kcal/mol for barrier heights. The DFT-D3 counterpart of ωB97M-V performs only slightly better than the original method for barrier heights: WTMAD-2(ωB97M-V) = 3.40 kcal/mol and WTMAD-2[ωB97M-D3(BJ)] = 3.15 kcal/mol. While there are the aforementioned cases for which the DFT-D3(BJ) counterparts of B97M-V, ωB97X-V and ωB97M-V perform better than their original VV10-based formulations, it is important to note that the originally proposed methods outperform their DFT-D3(BJ) counterparts in both noncovalent interaction categories, for instance for all noncovalent interactions we report WTMAD-2(ωB97M-V) = 3.70 kcal/mol and WTMAD-2[ωB97M-D3(BJ)] = 4.50 kcal/mol. When all 55 benchmark sets are combined we also obtain better WTMADs for the original methods: WTMAD-2(B97M-V) = 5.46 kcal/mol and WTMAD-2[B97M-D3(BJ)] = 6.49 kcal/mol, WTMAD-2(ωB97X-V) = 3.98 kcal/mol and WTMAD-2[ωB97X-D3(BJ)] = 4.35 kcal/mol, WTMAD-2(ωB97M-V) = 3.53 kcal/mol and WTMAD-2[ωB97M-D3(BJ)] = 4.01 kcal/mol (see Table 5). However, while this may overall look like a negative outcome, one also has to put these result into the broader context. In the original GMTKN55 study, M052X-D3(0) was identified as the second-best hybrid DFA among 48 assessed dispersion-

20


Page 20 of 45

Page 21 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


corrected hybrids [WTMAD-2(M052X-D3(0)] = 4.61 kcal/mol]. It is worth noting that both the ωB97X-D3(BJ) and ωB97M-D3(BJ) methods perform better than M052X-D3(0) for GMTKN55, making them competitive methods and good alternatives to the original VV10based approaches. Additionally, ωB97X-D3(BJ) provides a viable alternative to ωB97X-D3 [WTMAD-2(ωB97X-D3) = 4.77 kcal/mol], overall outperforming it. We further put the results discussed so far into the broader context of GMTKN55 in the following section.

3.5

Update on previous GMTKN55 studies

With the addition of our study to previous GMTKN55 studies, 24,27 we increase the total number of dispersion-corrected and -uncorrected DFAs assessed on GMTKN55 with a quadruple-ζ AO basis set from 313 to 325, with 115 of them being thoroughly assessed dispersion-corrected methods, 13 of which use the VV10 NLC correction. We thoroughly analyzed the dispersion-corrected DFAs according to class of Jacob’s Ladder. The updated number of dispersion-corrected DFAs analyzed is now 20 for GGAs/NGAs (GGAs or nonseparable 146 gradient approximations), 11 for meta-GGAs/NGAs, 52 for hybrids, and 32 for double hybrids. We conclude our study, by giving an update on the numerical results of previous GMTKN55 benchmark studies. Firstly, we compare the performance of B97M-V and B97M-D3(BJ) to SCAN-D3(BJ), 130,131 which was shown to have the smallest WTMADs of all the meta-GGA/NGAs. 24 Similarly, we compare the performance of the ωB97M-V, ωB97M-D3(BJ) and ωB97X-D3(BJ) functionals to the ωB97X-V and M052X-D3(0) functionals, which were shown to have the smallest WTMADs of all the hybrid functionals over GMTKN55. 24 Table 5 shows the WTMAD-2 values for these functionals. The values for the SCAN-D3(BJ) and M052X-D3(0) are taken from the original GMTKN55 study. We find that B97M-V and B97M-D3(BJ) have a lower WTMAD-2 than SCAN-D3(BJ) for most subsets of GMTKN55, with an overall WTMAD2 value of 5.46 kcal/mol and 6.49 kcal/mol, respectively, compared to 7.86 kcal/mol for SCAN-D3(BJ), which is a considerable reduction. Similarly, ωB97M-V performs better 21



than ωB97X-V for GMTKN55 with a WTMAD-2 value of 3.53 kcal/mol compared to 3.98 kcal/mol for ωB97X-V; ωB97M-D3(BJ) is comparable with the latter with WTMAD-2 = 4.01 kcal/mol. It is also worth noting that the performance of ωB97X-D3(BJ) is better than that of the M052X-D3(0) with WTMAD-2 values over GMTKN55 of 4.35 kcal/mol and 4.61 kcal/mol, respectively. Table S28 is an updated list of the best DFAs for each individual benchmark set. Additionally, Table S29 shows the metaGGA/NGA and hybrid functionals that are the the best functionals (based on MADs) the greatest number of times, in each subcategory of GMTKN55 and the entire GMTKN55 database. This also shows that both B97M-V and ωB97M-V outperform other functionals for numerous cases across all categories of GMTKN55. Overall, B97M-V is the best meta-GGA/NGA functional for 20 benchmark sets (the most of any meta-GGA/NGA) and ωB97M-V is the best hybrid functional for 9 sets (the most of any hybrid functional). Note that both these functionals are never the worst for any test set in their respective rungs of Jacob’s Ladder. In fact, ωB97M-V is now the best performing functional across all rungs of Jacob’s Ladder for the DARC (reaction energies of Diels-Alder reactions) benchmark set, with an MAD of 0.64 kcal/mol compared to the previous 24,27 best of 0.87 kcal/mol for the double-hybrid functional B2NC-PLYP-D3(BJ). 27,147 Additionally, for the S66 benchmark set (interaction energies of noncovalently bound dimers), ωB97M-V matches the previous best of all rungs of Jacob’s Ladder, ωB97X-V, with an MAD of 0.12 kcal/mol. Overall, the average WTMAD-2 values for each rung on Jacob’s Ladder for the entire 115 dispersion-corrected DFAs assessed on GMTKN55 preserve the Jacob’s Ladder hierarchy. The updated average WTMAD-2 value is 10.37 kcal/mol for GGAs, 9.24 kcal/mol for metaGGAs, 6.63 kcal/mol for hybrids and 4.32 kcal/mol for double hybrids. The same trends can be observed with both WTMAD-1 and WTMAD-2 values for the GMTKN55 database and its subsets (see Tables S37 and S38). Note that B97M-V and B97M-D3(BJ) outperform many hybrid functionals, as their WTMAD-2 values for GMTKN55 are 5.46 kcal/mol

22


Page 22 of 45

Page 23 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and 6.49 kcal/mol, respectively, whereas the new average WTMAD-2 value for hybrids is 6.63 kcal/mol. Similarly, ωB97M-V and ωB97M-D3(BJ) outperform many double-hybrid functionals, as their WTMAD-2 values for GMTKN55 are 3.53 kcal/mol and 4.01 kcal/mol, respectively, compared to the average WTMAD-2 value for double-hybrids of 4.32 kcal/mol. In fact, ωB97M-V can outperform the best non-empirical double-hybrid, SOS0-PBE0-2D3(BJ), 27,148 which has a WTMAD-2 value of 3.86 kcal/mol. 27 Thus, we can recommend B97M-V and B97M-D3(BJ) as leading meta-GGA/NGA functionals, and ωB97M-V and ωB97M-D3(BJ) as leading hybrid functionals for general main group thermochemistry, kinetics and noncovalent interactions. The B97M-V and ωB97M-V functionals were also recommended in their respective rungs for the MGCDB84 database for noncovalent interactions, isomerization energies, thermochemistry, and barrier heights developed by Head-Gordon and co-workers, 25 which shows nicely how both large databases can be used for cross validation. Updated lists of the best three DFAs according to rungs and category of GMTKN55 can be found in the SI. Herein, we present such a ranking only for the entire database in Table 6. The data presented therein constitute our new recommendations for method users and should be used as a benchmark for the assessment of new developments. Continuously updated DFA rankings will be provided on our website. 149

4

Overall discussion and conclusion

We used the extensive GMTKN55 database for general main-group thermochemistry, kinetics, and noncovalent interactions for various questions related to the non-local correlation (NLC) energy term in van-der-Waals density functionals. We compared the VV10 NLC correction in a post self-consistent-field (post-SCF) version with the reliable DFT-D3(BJ) dispersion correction and found that the NLC correction is more beneficial than DFT-D3(BJ) for the two recommended double hybrids DSD-PBEP86 and DSD-BLYP, for the energetics of intra- and intermolecular noncovalent interactions. We compared the full-SCF and

23



Page 24 of 45

Table 6: Updated list of best three density functional approximations for each of the four highest rungs of Jacob’s Ladder for GMTKN55 according to WTMAD-1 and WTMAD-2 values (kcal/mol).a WTMAD-1

WTMAD-2

GGA/NGA

revPBE-D3(BJ) OLYP-D3(BJ) 21,150–152 revPBE-NL meta-GGA/NGA B97M-V B97M-D3(BJ) SCAN-D3(BJ) hybrid ωB97M-V ωB97M-D3(BJ) ωB97X-V double-hybrid DSD-PBEP86-NL ωB97X-2-D3(BJ) B2NC-PLYP-D3(BJ) a compare with Refs. 24 and 27. b TQZ version

(4.66) revPBE-D3(BJ) (4.75) revPBE-NL (4.80) B97-D3(BJ) 8,10 (3.11) B97M-V (3.78) B97M-D3(BJ) (4.67) SCAN-D3(BJ) (2.00) ωB97M-V (2.23) ωB97X-V (2.32) ωB97M-D3(BJ) (1.63) DSD-PBEP86-NL (1.66) ωB97X-2-D3(BJ) 27,153,b (1.71) DSD-BLYP-NL (see Ref. 153).

(8.27) (8.49) (8.55) (5.46) (6.49) (7.86) (3.53) (3.98) (4.01) (2.84) (2.97) (3.05)

post-SCF addition of the NLC energy term and found that the post-SCF addition provides basically the same results as the full-SCF addition for the entire GMTKN55 database, using a number of functional approximations from different rungs of Jacob’s Ladder. This is beneficial, as the full-SCF addition requires extra computational effort. We found that the full-SCF NLC term does not have any observable effect on the electron-density based NCI plots or HOMO-LUMO gaps of noncovalently bound dimers, so that we can indeed use the post-SCF NLC term to calculate energies, without affecting related molecular properties. We parametrized the DFT-D3(BJ) correction for the B97M-V, ωB97X-V and ωB97M-V functionals (without their NLC terms), and coined these new functional variants as B97MD3(BJ), ωB97X-D3(BJ) and ωB97M-D3(BJ). We assessed the performance of these new functionals with respect to their original vdW-DFT counterparts over the GMTKN55 database and found that they can be a viable and computationally less expensive alternative for some cases of general main group thermochemistry, kinetics and noncovalent interactions. In particular, the B97M-D3(BJ) and ωB97X-D3(BJ) outperform B97M-V and ωB97X-V, respectively, for the reaction energies of large systems

24


Page 25 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and isomerization energies subset of GMTKN55. Additionally, ωB97X-D3(BJ) and ωB97MD3(BJ) outperform ωB97X-V and ωB97M-V, respectively, for the barrier heights subset. Finally, we re-evaluated which functionals are the best in the meta-GGA and hybrid classes relative to the original GMTKN55 study, since we assessed the B97M-V and ωB97M-V functionals in this study. We found that B97M-V and ωB97M-V are indeed leading functionals in their respective classes, with WTMAD-2 values over GMTKN55 of 5.46 kcal/mol and 3.53 kcal/mol respectively, compared to 7.86 kcal/mol for the previous best meta-GGA SCAND3(BJ) and 3.98 kcal/mol for the previous best hybrid ωB97X-V. Additionally, ωB97M-V can compete with some double-hybrid functionals and it is the ninth best DFA in the overall comparison over GMTKN55 [with ωB97M-D3(BJ) being the 14th ] according to WTMAD-2 values. In fact, ωB97M-V gives the lowest WTMAD-2 value for the intermolecular noncovalent interactions subset out of all assessed methods in this study (WTMAD-2 = 2.90 kcal/mol, followed by 2.94 kcal/mol for DSD-PBEP86-NL). Overall, our study gives a comprehensive analysis of the use of van-der-Waals density functionals in the post-SCF manner with respect to the full-SCF manner, and the DFT-D3 dispersion correction. Our main conclusion is that van-der-Waals density functionals can be used with the post-SCF non-local correlation for energetic properties for main-group chemistry. Additionally, we recommend use of B97M-V, ωB97M-V, and their herein introduced DFT-D3(BJ) variants for main group thermochemistry, kinetics and noncovalent interactions.

Supporting Information Available The WTMAD schemes; WTMAD-1 values of all assessed methods; WTMAD-2 values of the DFT-D3 methods optimized in this study; statistical results for all methods; ratios of mean absolute deviations; DFT-D3 parametrizations; comparison of B97M-V and ωB97M-V to the best meta-GGA and hybrid functionals; updated recommendations for GMTKN55 and its categories; average WTMAD values for the four highest rungs of Jacob’s ladder;

25



grid-dependence study; convergence problems.

5

Author information

Corresponding author E-mail: [email protected] ORCID Asim Najibi: 00000001-9199-7560 Lars Goerigk: 0000-0003-3155-675X Notes The authors declare no competing financial interest.

Acknowledgement The authors would like to thank Dr Narbe Mardirossian for helpful discussions regarding the implementation of the B97M-V type of functionals, and Mr Marcos Casanova-Páez for technical assistance with maintaining our local version of ORCA. We are grateful for computing time provided by Melbourne Bioinformatics (project ID: RA0005) and by the National Computational Infrastructure (NCI) National Facility within the National Computational Merit Allocation Scheme (project ID: fk5). AN acknowledges a Melbourne Research Scholarship and an Australian Research Training Program Scholarship.

References (1) Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140, A1133. (2) Kristyán, S.; Pulay, P. Can (semi)local density functional theory account for the London dispersion forces? Chem. Phys. Lett. 1994, 229, 175 – 180. (3) Pérez-Jordá, J.; Becke, A. D. A density-functional study of van der Waals forces: rare gas diatomics. Chem. Phys. Lett. 1995, 233, 134 – 137.

26


Page 26 of 45

Page 27 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


˘ (4) Hobza, P.; Sponer, J.; Reschel, T. Density functional theory and molecular clusters. J. Comput. Chem. 1995, 16, 1315–1325. ˘ (5) Sponer, J.; Leszczynski, J.; Hobza, P. Base stacking in cytosine dimer. A comparison of correlated ab initio calculations with three empirical potential models and density functional theory calculations. J. Comput. Chem. 1996, 17, 841–850. (6) Grimme, S.; Hansen, A.; Brandenburg, J. G.; Bannwarth, C. Dispersion-corrected mean-field electronic structure methods. Chem. Rev. 2016, 116, 5105–5154. (7) Grimme, S. Accurate description of van der Waals complexes by density functional theory including empirical corrections. J. Comput. Chem. 2004, 25, 1463–1473. (8) Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem. 2006, 27, 1787–1799. (9) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (10) Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the damping function in dispersion corrected density functional theory. J. Comput. Chem. 2011, 32, 1456–1465. (11) Caldeweyher, E.; Bannwarth, C.; Grimme, S. Extension of the D3 dispersion coefficient model. J. Chem. Phys. 2017, 147, 034112. (12) Becke, A. D.; Johnson, E. R. A density functional model of dispersion interaction. J. Chem. Phys. 2005, 123, 154101. (13) Johnson, E. R.; Becke, A. D. A post-Hartree-Fock model of intermolecular interactions: Inclusion of higher-order corrections. J. Chem. Phys. 2006, 124, 174104. (14) Becke, A. D.; Johnson, E. R. Exchange-hole dipole moment and the dispersion interaction revisited. J. Chem. Phys. 2007, 127, 154108. 27



(15) Tkatchenko, A.; Scheffler, M. Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data. Phys. Rev. Lett. 2009, 102, 073005. (16) Tkatchenko, A.; DiStasio, R. A.; Car, R.; Scheffler, M. Accurate and efficient method for many-body van der Waals interactions. Phys. Rev. Lett. 2012, 108, 236402. (17) Ambrosetti, A.; Reilly, A. M.; DiStasio, R. A.; Tkatchenko, A. Long-range correlation energy calculated from coupled atomic response functions. J. Chem. Phys. 2014, 140, 18A508. (18) Steinmann, S. N.; Corminboeuf, C. A system-dependent density-based dispersion correction. J. Chem. Theory Comput. 2010, 6, 1990–2001. (19) Steinmann, S. N.; Corminboeuf, C. A generalized-gradient approximation exchange hole model for dispersion coefficients. J. Chem. Phys. 2011, 134, 044117. (20) Steinmann, S. N.; Corminboeuf, C. Comprehensive benchmarking of a densitydependent dispersion correction. J. Chem. Theory Comput. 2011, 7, 3567–3577. (21) Goerigk, L.; Grimme, S. A thorough benchmark of density functional methods for general main group thermochemistry, kinetics, and noncovalent interactions. Phys. Chem. Chem. Phys. 2011, 13, 6670–6688. (22) Goerigk, L.; Kruse, H.; Grimme, S. Benchmarking density functional methods against the S66 and S66x8 datasets for non-covalent interactions. Chem. Phys. Chem. 2011, 12, 3421–3433. (23) Goerigk, L. Treating London-dispersion effects with the latest Minnesota density functionals: problems and possible solutions. J. Phys. Chem. Lett. 2015, 6, 3891–3896. (24) Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main 28


Page 28 of 45

Page 29 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


group thermochemistry, kinetics and noncovalent interactions. Phys. Chem. Chem. Phys. 2017, 19, 32184–32215. (25) Mardirossian, N.; Head-Gordon, M. Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals. Mol. Phys. 2017, 115, 2315–2372. (26) Najibi, A.; Goerigk, L. A comprehensive assessment of the effectiveness of orbital optimization in double-hybrid density functionals in the treatment of thermochemistry, kinetics, and noncovalent interactions. J. Phys. Chem. A 2018, 122, 5610–5624. (27) Mehta, N.; Casanova-Páez, M.; Goerigk, L. Semi-empirical or non-empirical doublehybrid density functionals: which are more robust? Phys. Chem. Chem. Phys. 2018, 20, 23175–23194. (28) Grimme, S.; Steinmetz, M. Effects of London dispersion correction in density functional theory on the structures of organic molecules in the gas phase. Phys. Chem. Chem. Phys. 2013, 15, 16031–16042. (29) Goerigk, L.; Reimers, J. R. Efficient methods for the quantum chemical treatment of protein structures: the effects of London-dispersion and basis-set incompleteness on peptide and water-cluster geometries. J. Chem. Theory Comput. 2013, 9, 3240–3251. (30) Goerigk, L.; Collyer, C. A.; Reimers, J. R. Recommending Hartree-Fock theory with London-dispersion and basis-set-superposition corrections for the optimization or quantum refinement of protein structures. J. Phys. Chem. B 2014, 118, 14612– 14626. (31) Grimme, S.; Brandenburg, J. G.; Bannwarth, C.; Hansen, A. Consistent structures and interactions by density functional theory with small atomic orbital basis sets. J. Chem. Phys. 2015, 143, 054107.

29



(32) Brandenburg, J. G.; Bannwarth, C.; Hansen, A.; Grimme, S. B97-3c: A revised lowcost variant of the B97-D density functional method. J. Chem. Phys. 2018, 148, 064104. (33) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. van der Waals density functional for general geometries. Phys. Rev. Lett. 2004, 92, 246401. (34) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. Erratum: van der Waals density functional for general geometries. Phys. Rev. Lett. 2005, 95, 109902. ´ D.; Kong, L.; Lundqvist, B. I.; Langreth, D. C. Higher-accuracy (35) Lee, K.; Murray, E. van der Waals density functional. Phys. Rev. B 2010, 82, 081101. (36) Vydrov, O. A.; Van Voorhis, T. Nonlocal van der Waals density functional made simple. Phys. Rev. Lett. 2009, 103, 063004. (37) Vydrov, O. A.; Van Voorhis, T. Nonlocal van der Waals density functional: the simpler the better. J. Chem. Phys. 2010, 133, 244103. (38) Sabatini, R.; Gorni, T.; de Gironcoli, S. Density Functional Made Simple and Efficient. Phys. Rev. B 2013, 87, 041108. (39) Mardirossian, N.; Ruiz Pestana, L.; Womack, J. C.; Skylaris, C.-K.; Head-Gordon, T.; Head-Gordon, M. Use of the rVV10 Nonlocal Correlation Functional in the B97M-V Density Functional: Defining B97M-rV and Related Functionals. J. Phys. Chem. Lett. 2017, 8, 35–40. (40) Murray, E. D.; Lee, K.; Langreth, D. C. Investigation of exchange energy density functional accuracy for interacting molecules. J. Chem. Theory Comput. 2009, 5, 2754–2762. (41) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865–3868. 30


Page 30 of 45

Page 31 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(42) Hujo, W.; Grimme, S. Performance of the van der Waals Density Functional VV10 and (hybrid)GGA Variants for Thermochemistry and Noncovalent Interactions. J. Chem. Theory Comput. 2011, 7, 3866–3871. (43) Hujo, W.; Grimme, S. Comparison of the performance of dispersion-corrected density functional theory for weak hydrogen bonds. Phys. Chem. Chem. Phys. 2011, 13, 13942–13950. (44) Mardirossian, N.; Head-Gordon, M. Mapping the genome of meta-generalized gradient approximation density functionals: The search for B97M-V. J. Chem. Phys. 2015, 142, 074111. (45) Mardirossian, N.; Head-Gordon, M. ωB97X-V: a 10-parameter, range-separated hybrid, generalized gradient approximation density functional with nonlocal correlation, designed by a survival-of-the-fittest strategy. Phys. Chem. Chem. Phys. 2014, 16, 9904–9924. (46) Mardirossian, N.; Head-Gordon, M. ωB97M-V: A combinatorially optimized, rangeseparated hybrid, meta-GGA density functional with VV10 nonlocal correlation. J. Chem. Phys. 2016, 144, 214110. (47) Mardirossian, N.; Head-Gordon, M. Survival of the most transferable at the top of Jacob’s ladder: defining and testing the ωB97M(2) double hybrid density functional. J. Chem. Phys. 2018, 148, 241736. (48) Grimme, S.; Hujo, W.; Kirchner, B. Performance of dispersion-corrected density functional theory for the interactions in ionic liquids. Phys. Chem. Chem. Phys. 2012, 14, 4875–4883. (49) Goerigk, L. How do DFT-DCP, DFT-NL, and DFT-D3 compare for the description of London-dispersion effects in conformers and general thermochemistry? J. Chem. Theory Comput 2014, 10, 968–980. 31



(50) Hujo, W.; Grimme, S. Performance of non-local and atom-pairwise dispersion corrections to DFT for structural parameters of molecules with noncovalent interactions. J. Chem. Theory Comput. 2013, 9, 308–315. (51) Neese, F. The ORCA program system. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012, 2, 73–78. (52) Parrish, R. M.; Burns, L. A.; Smith, D. G. A.; Simmonett, A. C.; DePrince, A. E.; Hohenstein, E. G.; Bozkaya, U.; Sokolov, A. Y.; Di Remigio, R.; Richard, R. M. et al. PSI4 1.1: an open-source electronic structure program emphasizing automation, advanced libraries, and interoperability. J. Chem. Theory Comput. 2017, 13, 3185– 3197. (53) Turney, J. M.; Simmonett, A. C.; Parrish, R. M.; Hohenstein, E. G.; Evangelista, F. A.; Fermann, J. T.; Mintz, B. J.; Burns, L. A.; Wilke, J. J.; Abrams, M. L. et al. PSI4: an open-source ab initio electronic structure program. WIREs Comput. Mol. Sci. 2, 556–565. (54) Vydrov, O. A.; Van Voorhis, T. Improving the accuracy of the nonlocal van der Waals density functional with minimal empiricism. J. Chem. Phys. 2009, 130, 104105. (55) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X. et al. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys. 2015, 113, 184– 215. (56) Lehtola, J.; Hakala, M.; Sakko, A.; Hämäläinen, K. ERKALE-a flexible program package for X-ray properties of atoms and molecules. J. Comput. Chem. 2012, 33, 1572– 1585. (57) Perdew, J. P.; Schmidt, K. Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conference Proceedings 2001, 577, 1–20. 32


Page 32 of 45

Page 33 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(58) Goerigk, L.; Grimme, S. Efficient and accurate double-hybrid-meta-GGA density functionals — Evaluation with the extended GMTKN30 database for general main group thermochemistry, kinetics and noncovalent interactions. J. Chem. Theory Comput. 2011, 7, 291–309. (59) Karton, A.; Daon, S.; Martin, J. M. L.; Ruscic, B. W4-11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data. Chem. Phys. Lett. 2011, 510, 165–178. (60) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 theory for molecular energies of first- and second-row compounds. J. Chem. Phys. 1991, 94, 7221–7230. (61) Goerigk, L.; Grimme, S. A general database for main group thermochemistry, kinetics, and noncovalent interactions — Assessment of common and reparameterized (meta)GGA density functionals. J. Chem. Theory Comput. 2010, 6, 107–126. (62) Parthiban, S.; Martin, J. M. L. Assessment of W1 and W2 theories for the computation of electron affinities, ionization potentials, heats of formation, and proton affinities. J. Chem. Phys. 2001, 114, 6014–6029. (63) Zhao, Y.; Truhlar, D. G. Assessment of density functionals for π systems: energy differences between cumulenes and poly-ynes; proton affinities, bond length alternation, and torsional potentials of conjugated polyenes; and proton affinities of conjugated Schiff bases. J. Phys. Chem. A 2006, 110, 10478–10486. (64) Yu, H.; Truhlar, D. G. Components of the bond energy in polar diatomic molecules, radicals, and ions formed by group-1 and group-2 metal atoms. J. Chem. Theory Comput. 2015, 11, 2968–2983. (65) Zhao, Y.; Ng, H. T.; Peverati, R.; Truhlar, D. G. Benchmark database for ylidic bond

33



dissociation energies and its use for assessments of electronic structure methods. J. Chem. Theory. Comput. 2012, 8, 2824–2834. (66) Johnson, E. R.; Mori-Sánchez, P.; Cohen, A. J.; Yang, W. Delocalization errors in density functionals and implications for main-group thermochemistry. J. Chem. Phys. 2008, 129, 204112. (67) Grimme, S.; Kruse, H.; Goerigk, L.; Erker, G. The mechanism of dihydrogen activation by frustrated Lewis pairs revisited. Angew. Chem. Int. Ed. 2010, 49, 1402–1405. (68) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Pople, J. A. Assessment of Gaussian2 and density functional theories for the computation of enthalpies of formation. J. Chem. Phys. 1997, 106, 1063–1079. (69) Friedrich, J.; Hänchen, J. Incremental CCSD(T)(F12*)|MP2: A black box method to obtain highly accurate reaction energies. J. Chem. Theory Comput. 2013, 9, 5381– 5349. (70) Friedrich, J. Efficient calculation of accurate reaction energies — Assessment of different models in electronic structure theory. J. Chem. Theory Comput. 2015, 11, 3596–3609. (71) Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation. J. Chem. Phys. 2006, 124, 034108. (72) Grimme, S.; M¨ uck-Lichtenfeld, C.; W¨ urthwein, E.; Ehlers, A.; Goumans, K., T P M and Lammertsma Consistent theoretical description of 1,3-dipolar cycloaddition reactions. J. Phys. Chem. A 2006, 110, 2583–2586. (73) Piacenza, M.; Grimme, S. Systematic quantum chemical study of DNA-base tautomers. J. Comput. Chem. 2004, 25, 83–99.

34


Page 34 of 45

Page 35 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(74) Woodcock, H. L.; Schaefer III, H. F.; Schreiner, P. R. Problematic energy differences between cumulenes and poly-ynes: Does this point to a systematic improvement of density functional theory? J. Phys. Chem. A 2002, 106, 11923–11931. (75) Schreiner, P. R.; Fokin, A. A.; Pascal, R. A.; de Meijere, A. Many density functional theory approaches fail to give reliable large hydrocarbon isomer energy differences. Org. Lett. 2006, 8, 3635–3638. (76) Lepetit, C.; Chermette, H.; Gicquel, M.; Heully, J. L.; Chauvin, R. Description of carbo-oxocarbons and assessment of exchange-correlation functionals for the DFT description of carbo-mers. J. Phys. Chem. A 2007, 111, 136–149. (77) Lee, J. S. Accurate ab initio binding energies of alkaline earth metal clusters. J. Phys. Chem. A 2005, 109, 11927–11932. (78) Karton, A.; Martin, J. M. L. Explicitly correlated benchmark calculations on C8H8 isomer energy separations: How accurate are DFT, double-hybrid, and composite ab initio procedures? Mol. Phys. 2012, 110, 2477–2491. (79) Zhao, Y.; Tishchenko, O.; Gour, J. R.; Li, W.; Lutz, J. J.; Piecuch, P.; Truhlar, D. G. Thermochemical kinetics for multireference systems: Addition reactions of ozone. J. Phys. Chem. A 2009, 113, 5786–5799. (80) Zhao, Y.; Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: Two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor. Chem. Acc. 2008, 120, 215–241. (81) Manna, D.; Martin, J. M. L. What are the ground state structures of C20 and C24? An explicitly correlated ab initio approach. J. Phys. Chem. A 2016, 120, 153–160.

35



(82) Neese, F.; Schwabe, T.; Kossmann, S.; Schirmer, B.; Grimme, S. Assessment of orbitaloptimized, spin-component scaled second-order many-body perturbation theory for thermochemistry and kinetics. J. Chem. Theory Comput. 2009, 5, 3060–3073. (83) Steinmann, G.; Csonka, G.; Carminboeuf, C. Unified inter- and intramolecular dispersion correction formula for generalized gradient approximation density functional theory. J. Chem. Theory Comput. 2009, 5, 2950–2958. (84) Krieg, H.; Grimme, S. Thermochemical benchmarking of hydrocarbon bond separation reaction energies: Jacob’s ladder is not reversed! Mol. Phys. 2010, 108, 2655–2666. (85) Grimme, S.; Steinmetz, M.; Korth, M. How to compute isomerization energies of organic molecules with quantum chemical methods. J. Org. Chem. 2007, 72, 2118– 2126. (86) Sure, R.; Hansen, A.; Schwerdtfeger, P.; Grimme, S. Comprehensive theoretical study of all 1812 C60 isomers. Phys. Chem. Chem. Phys. 2017, 19, 14296–14305. (87) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Multi-coefficient extrapolated density functional theory for thermochemistry and thermochemical kinetics. Phys. Chem. Chem. Phys. 2005, 7, 43–52. (88) Zhao, Y.; González-Garcia, N.; Truhlar, D. G. Benchmark database of barrier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions and its use to test theoretical methods. J. Phys. Chem. A 2005, 109, 2012–2018. (89) Guner, V.; Khuong, K. S.; Leach, A. G.; Lee, P. S.; Bartberger, M. D.; Houk, K. N. A standard set of pericyclic reactions of hydrocarbons for the benchmarking of computational methods: The performance of ab initio, density functional, CASSCF, CASPT2, and CBS-QB3 methods for the prediction of activation barriers, reaction energetics, and transition state geometries. J. Phys. Chem. A 2003, 107, 11445–11459.

36


Page 36 of 45

Page 37 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(90) Ess, D. H.; Houk, K. N. Activation energies of pericyclic reactions: performance of DFT, MP2, and CBS-QB3 methods for the prediction of activation barriers and reaction energetics of 1,3-dipolar cycloadditions, and revised activation enthalpies for a standard set of hydrocarbon pericyclic reactions. J. Phys. Chem. A 2005, 109, 9542–9553. (91) Dinadayalane, T. C.; Vijaya, R.; Smitha, A.; Sastry, G. N. Diels-Alder reactivity of butadiene and cyclic five-membered dienes ((CH)4X, X = CH2, SiH2, O, NH, PH, and S) with ethylene: A benchmark study. J. Phys. Chem. A 2002, 106, 1627–1633. (92) Karton, A.; Goerigk, L. Accurate reaction barrier heights of pericyclic reactions: Surprisingly large deviations for the CBSQB3 composite method and their consequences in DFT benchmark studies. J. Comput. Chem. 2015, 36, 622–632. (93) Goerigk, L.; Sharma, R. The INV24 test set: how well do quantum-chemical methods describe inversion and racemization barriers? Can. J. Chem. 2016, 94, 1133–1143. (94) Karton, A.; O’Reilly, R. J.; Chan, B.; Radom, L. Determination of barrier heights for proton exchange in small water, ammonia, and hydrogen fluoride clusters with G4(MP2)-Type, MPn, and SCS-MPn procedures — A caveat. J. Chem. Theory Comput. 2012, 8, 3128–3136. (95) Karton, A.; O’Reilly, R. J.; Radom, L. Assessment of theoretical procedures for calculating barrier heights for a diverse set of water-catalyzed proton-transfer reactions. J. Phys. Chem. A 2012, 116, 4211–4221. ˇ (96) Jureˇcka, P.; Sponer, J.; Cerny, J.; Hobza, P. Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs. Phys. Chem. Chem. Phys. 2006, 8, 1985–1993. (97) Marshall, M. S.; Burns, L. A.; Sherrill, C. D. Benchmark database of barrier heights for

37



heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions and its use to test theoretical methods. J. Chem. Phys. 2011, 135, 194102. ˇ aˇc, J.; Riley, K. E.; Hobza, P. S66: A Well-balanced Database of Benchmark (98) Rez´ Interaction Energies Relevant to Biomolecular Structures. J. Chem. Theory Comput 2011, 7, 2427–2438. (99) Bryantsev, V. S.; Diallo, M. S.; van Duin, A. C. T.; Goddard, W. A. Evaluation of B3LYP, X3LYP, and M06-Class density functionals for predicting the binding energies of neutral, protonated, and deprotonated water clusters. J. Chem. Theory Comput 2009, 5, 1016–1026. (100) Manna, D.; Kesharwani, M. K.; Sylvetsky, N.; Martin, J. M. L. Conventional and explicitly correlated ab initio benchmark study on water clusters: revision of the BEGDB and WATER27 data sets. J. Chem. Theory Comput. 2017, 13, 3136–3152. (101) Setiawan, D.; Kraka, E.; Cremer, D. Strength of the pnicogen bond in complexes involving group Va elements N, P, and As. J. Phys. Chem. A 2015, 119, 1642–1656. (102) Kozuch, S.; Martin, J. M. L. Halogen bonds: Benchmark and theoretical analysis. J. Chem. Theory Comput. 2013, 9, 1918–1931. ˇ aˇc, J.; Riley, K. E.; Hobza, P. Benchmark calculations of noncovalent interactions (103) Rez´ of halogenated molecules. J. Chem. Theory Comput. 2012, 8, 4285–4292. (104) Lao, K. U.; Schäffer, R.; Jansen, G.; Herbert, J. M. Accurate description of intermolecular interactions involving ions using symmetry-adapted perturbation theory. J. Chem. Theory Comput 2015, 11, 2473–2486. (105) Schwabe, T.; Grimme, S. Double-hybrid density functionals with long-range dispersion corrections: Higher accuracy and extended applicability. Phys. Chem. Chem. Phys. 2007, 9, 3397–3406. 38


Page 38 of 45

Page 39 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(106) Grimme, S. Seemingly simple stereoelectronic effects in alkane isomers and the implications for Kohn-Sham density functional theory. Angew. Chem. Int. Ed. 2006, 45, 4460–4464. (107) Gruzman, D.; Karton, A.; Martin, J. M. L. Performance of ab initio and density functional methods for conformational equilibria of CnH2n+2 alkane isomers (n = 4-8). J. Phys. Chem. A 2009, 113, 11974–11983. (108) Kesharwani, M. K.; Karton, A.; Martin, J. M. L. Benchmark ab initio conformational energies for the proteinogenic amino acids through explicitly correlated methods. Assessment of density functional methods. J. Chem. Theory Comput 2016, 12, 444–454. ˇ (109) Reha, D.; Valdés, H.; Vondráˇsek, J.; Hobza, P.; Abu-Riziq, A.; Crews, B.; de Vries, M. S. Structure and IR spectrum of phenylalanyl-glycyl-glycine tripetide in the gas-phase: IR/UV experiments, ab initio quantum chemical calculations, and molecular dynamic simulations. Chem. Eur. J. 2005, 11, 6803–6817. (110) Goerigk, L.; Karton, A.; Martin, J. M. L.; Radom, L. Accurate quantum chemical energies for tetrapeptide conformations: why MP2 data with an insufficient basis set should be handled with caution. Phys. Chem. Chem. Phys. 2013, 15, 7028–7031. (111) Fogueri, U. R.; Kozuch, S.; Karton, A.; Martin, J. M. The melatonin conformer space: Benchmark and assessment of wave function and DFT methods for a paradigmatic biological and pharmacological molecule. J. Phys. Chem. A 2013, 117, 2269–2277. (112) Csonka, G. I.; French, A. D.; Johnson, G. P.; Stortz, C. A. Evaluation of density functionals and basis sets for carbohydrates. J. Chem. Theory Comput. 2009, 5, 679– 692. (113) Kruse, H.; Mladek, A.; Gkionis, K.; Hansen, A.; Grimme, S.; Sponer, J. Quantum chemical benchmark study on 46 RNA backbone families using a dinucleotide unit. J. Chem. Theory Comput. 2015, 11, 4972–4991. 39



(114) Kozuch, S.; Bachrach, S. M.; Martin, J. M. Conformational equilibria in butane-1,4diol: A benchmark of a prototypical system with strong intramolecular H-bonds. J. Phys. Chem. A 2014, 118, 293–303. (115) ORCA — an ab initio, density functional and semiempirical program package, V. 4.0.1, F. Neese, MPI f¨ ur Chemische Energiekonversion, M¨ ulheim a. d. Ruhr (Germany), 2017. (116) Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297–3305. (117) Woon, D. E.; Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358– 1371. (118) Izsak, R.; Neese, F. An overlap fitted chain of spheres exchange method. J. Chem. Phys. 2011, 135, 144105. (119) Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys. 2002, 116, 3175–3183. (120) Zhang, Y.; Yang, W. Comment on “Generalized gradient approximation made simple”. Phys. Rev. Lett. 1998, 80, 890. (121) Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648–5652. (122) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields. J. Phys. Chem. 1994, 98, 11623–11627.

40


Page 40 of 45

Page 41 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(123) Zhao, Y.; Truhlar, D. G. Design of density functionals that are broadly accurate for thermochemistry, thermochemical kinetics, and nonbonded interactions. J. Phys. Chem. A 2005, 109, 5656–5667. (124) Kozuch, S.; Gruzman, D.; Martin, J. M. L. DSD-BLYP: A general purpose double hybrid density functional including spin component scaling and dispersion correction. J. Phys. Chem. C 2010, 114, 20801–20808. (125) Kozuch, S.; Martin, J. M. L. DSD-PBEP86: in search of the best double-hybrid DFT with spin-component scaled MP2 and dispersion corrections. Phys. Chem. Chem. Phys. 2011, 13, 20104–20107. (126) Aragó, J.; Ort´ı, E.; Sancho-Garc´ıa, J. C. Nonlocal van der Waals approach merged with double-hybrid density functionals: Toward the accurate treatment of noncovalent interactions. J. Chem. Theory Comput. 2013, 9, 3437–3443. (127) Yu, F. Spin-component-scaled double-hybrid density functionals with nonlocal van der Waals correlations for noncovalent interactions. J. Chem. Theory Comput. 2014, 10, 4400–4407. (128) Becke, A. D.; Johnson, E. R. Exchange-hole dipole moment and the dispersion interaction. J. Chem. Phys. 2005, 122, 154104. (129) Johnson, E. R.; Becke, A. D. A post-Hartree-Fock model of intermolecular interactions. J. Chem. Phys. 2005, 123, 024101. (130) Sun, J.; Ruzsinszky, A.; Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 2015, 115, 036402. (131) Brandenburg, J. G.; Bates, J. E.; Sun, J.; Perdew, J. P. Benchmark tests of a strongly constrained semilocal functional with a long-range dispersion correction. Phys. Rev. B 2016, 94, 115144. 41



(132) Lin, Y.-S.; Li, G.-D.; Mao, S.-P.; Chai, J.-D. Long-range corrected hybrid density functionals with improved dispersion corrections. J. Chem. Theory Comput. 2013, 9, 263–272. (133) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Design of density functionals by combining the method of constraint satisfaction with parametrization for thermochemistry, thermochemical kinetics, and noncovalent interactions. J. Chem. Theory Comput. 2006, 2, 364–382. (134) Branko, R. Uncertainty quantification in thermochemistry, benchmarking electronic structure computations, and Active Thermochemical Tables. Int. J. Quantum Chem. 114, 1097–1101. ´ Sancho-Garc´ıa, J. C.; Pérez-Jiménez, A. ´ J.; Adamo, C. Communication: (135) Brémond, E.; Double-hybrid functionals from adiabatic-connection: The QIDH model. J. Chem. Phys. 2014, 141, 031101. ´ Savarese, M.; Sancho-Garc´ıa, J. C.; Pérez-Jiménez, A. ´ J.; Adamo, C. (136) Brémond, E.; Quadratic integrand double-hybrid made spin-component-scaled. J. Chem. Phys. 2016, 144, 124104. (137) Peverati, R.; Head-Gordon, M. Orbital optimized double-hybrid density functionals. J. Chem. Phys. 2013, 139, 024110. (138) Johnson, E. R.; Keinan, S.; Mori-Sánchez, P.; Contreras-Garc´ıa, J.; Cohen, A. J.; Yang, W. Revealing noncovalent interactions. J. Am. Chem. Soc. 2010, 132, 6498– 6506. (139) Contreras-Garcıa, J.; Johnson, E. R.; Keinan, S.; Chaudret, R.; Piquemal, J.-P.; Beratan, D. N.; Yang, W. NCIPLOT: A program for plotting noncovalent interaction regions. J. Chem. Theory Comput. 2011, 7, 625–632.

42


Page 42 of 45

Page 43 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(140) Goerigk, L.; Grimme, S. Quantum chemical investigation of exciton coupling: supermolecular calculations of a merocyanine dimer aggregate. ChemPhysChem. 2008, 9, 2467–2470. (141) Goerigk, L.; Grimme, S. Assessment of TD-DFT methods and of various spin scaled CIS(D) and CC2 versions for the treatment of low-lying valence excitations of large organic dyes. J. Chem. Phys. 2010, 132, 184103. (142) Schwabe, T.; Goerigk, L. Time-dependent double-hybrid density functionals with spincomponent and spin-opposite scaling. J. Chem. Theory Comput. 2017, 13, 4307–4323. (143) As a part of that study, we will also present an analysis of this DFA on GMTKN55. (144) Chai, J.-D.; Head-Gordon, M. Long-range corrected hybrid density functionals with damped atom-atom dispersion corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615– 6620. (145) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (146) Peverati, R.; Truhlar, D. G. Exchange-correlation functional with good accuracy for both structural and energetic properties while depending only on the density and its gradient. J. Chem. Theory Comput. 2012, 8, 2310–2319. (147) Yu, F. Double-hybrid density functionals free of dispersion and counterpoise corrections for non-covalent interactions. J. Phys. Chem. A 2014, 118, 3175–3182. (148) Alipour, M. On the opposite-spin to same-spin ratio of absolute and interaction MP2 correlation energy in parameter-free spin-opposite-scaled double hybrids. Chemical Physics Letters 2017, 684, 423 – 426. (149) http://goerigk.chemistry.unimelb.edu.au/research/the-gmtkn55-database/.

43



(150) Handy, N. C.; Cohen, A. J. Left-right correlation energy. Mol. Phys. 2001, 99, 403– 412. (151) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785–789. (152) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Results obtained with the correlation energy density functionals of Becke and Lee, Yang and Parr. Chem. Phys. Lett. 1989, 157, 200 – 206. (153) Chai, J.-D.; Head-Gordon, M. Long-range corrected double-hybrid density functionals. J. Chem. Phys. 2009, 131, 174105.

44


Page 44 of 45

Page 45 of 45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Graphical TOC Entry

45


The Nonlocal Kernel in van der Waals Density Functionals as an

Recommend Documents