Thirty years of density functional theory in computational chemistry: an overview and extensive assessment of 200 density functionals

Figures & data

Figure 1. Perdew's metaphorical Jacob's Ladder, composed of five rungs corresponding to increasingly sophisticated models for the unknown exchange-correlation functional of DFT. Since each rung contains new physical content that is missing in lower rungs, improved accuracy should be attainable at each higher level. This is illustrated by reporting the average ranking of the best-performing functional from each rung out of the total of 200 functionals assessed later on in the review (which covers functionals occupying Rungs 1–4).

Table 1. Mean signed errors (MSE) and mean absolute deviations (MAD) in kcal/mol for Hartree–Fock (HF) and nine density functionals. Statistics for three distinct interaction types are provided: 124 single-reference atomisation energies (AE), 12 dispersion-bound (DB) alkane dimers, and 38 hydrogen-bonded (HB) water clusters (dimers through decamers).

Data-set	HF	SPW92	PBE	TPSS	PBE0	TPSSh	PBE-D3(BJ)	TPSS-D3(BJ)	PBE0-D3(BJ)	TPSSh-D3(BJ)
AE MSE	112.79	−58.11	−12.17	−2.72	1.12	1.46	−13.13	−3.91	0.31	0.39
DB MSE	6.01	−1.10	3.36	4.67	3.37	4.55	−0.11	0.23	−0.02	0.18
HB MSE	18.29	−30.74	−1.26	3.21	0.20	3.52	−5.86	−2.44	−3.86	−1.77
AE MAD	112.79	58.11	13.61	4.40	3.22	4.37	14.32	5.15	3.37	4.40
DB MAD	6.01	1.11	3.36	4.67	3.37	4.55	0.11	0.30	0.10	0.29
HB MAD	18.29	30.74	1.30	3.21	0.67	3.52	5.86	2.45	3.86	1.80

Figure 2. Inhomogeneity correction factors (ICF) for a variety of GGA exchange functionals as a function of . The subfigure on the left contains exchange functionals with ICFs that are PBE-like, while the subfigure on the right contains exchange functionals with ICFs that differ substantially from the PBE form. The local density approximation is equivalent to a horizontal line at g_x = 1.

Figure 3. Inhomogeneity correction factors (ICF) for a variety of meta-GGA exchange functionals as a function of . The subfigure on the left plots the α = 0 limit (covalent bonds), while the subfigure on the right plots the α = ∞ limit (weak bonds). The local density approximation is equivalent to a horizontal line at g_x = 1.

Figure 4. Exact exchange attenuation plots for various long-range corrected (ωB97X-D, CAM-B3LYP, LC-ωPBE, and M11), screened-exchange (HSE-HJS), and global hybrid (B3LYP and M06-2X) density functionals. As a point of reference, Hartree–Fock is equivalent to a horizontal line at y = 1.

Figure 5. Stacked root-mean-square deviations (RMSD) in kcal/mol for a set of six local/hybrid functional pairs (e.g. BLYP/B3LYP). NCD contains 91 non-covalent interactions that are sensitive to electron delocalisation, ID contains 155 isomerisation energies that are sensitive to electron delocalisation, while BH contains 206 barrier heights. Section 4.1 provides a more complete description of NCD, ID, and BH. These 452 data points are representative of cases where the addition of a fraction of exact exchange should improve the performance of a local functional. For example, the performance of PBE is improved by about a factor of 2 upon the introduction of 25% exact exchange (PBE0).

Figure 6. Log– log plot of the helium dimer potential energy curve with Hartree–Fock (HF) and revPBE, as well as their dispersion-corrected counterparts. The post-equilibrium portion of the curve (5.1 Å and larger) is displayed in order to demonstrate that the long range behaviour of the dispersion-corrected functionals is approximately R⁻⁶.

Figure 7. Stacked root-mean-square deviations (RMSD) in kcal/mol for four density functionals and their D2 and D3(BJ) dispersion-corrected counterparts. The four parent functionals are BLYP (local GGA), B3LYP (hybrid GGA), TPSS (local meta-GGA), and TPSSh (hybrid meta-GGA). NCED contains 1744 non-covalent dimer interactions and IE contains 755 isomerisation energies. Section 4.1 provides a more complete description of NCED and IE. These 2499 data points are representative of cases where dispersion corrections should be very effective. Indeed, the performance of B3LYP is greatly improved by the addition of the older D2 dispersion tail, and even further refined when D2 is replaced by D3(BJ).

Figure 8. RMSD in kcal/mol for the 12 local and global hybrid Minnesota density functionals. SR contains 712 single-reference bonded interactions (124 atomisation energies, 505 heavy atom transfer reaction energies, and 83 bond dissociation energies), while MR contains 234 multi-reference bonded interactions (16 atomisation energies, 202 heavy atom transfer reaction energies, and 16 bond dissociation energies). The percentage of global exact exchange is displayed under the name of each functional. These 946 data points demonstrate the tradeoff between exact exchange and performance for single- and multi-reference interactions. For example, M06, with 27% exact exchange, performs equally well for SR and MR, while M06-2X, with 54% exact exchange, performs nearly 30% better than M06 for SR, but more than 30% worse than M06 for MR.

Figure 9. A graphical representation of elements that can be combined to define most existing density functionals. The circle contains the names of the 200 density functionals that are considered in this review, with the 20 featured functionals appearing in larger letters.

Figure 10. Ingredients that are conventionally used in the functional forms of semi-empirical density functionals. The primary GGA variable is the density gradient, ∇ρ, while the primary meta-GGA variable is the kinetic energy density, τ.

Figure 11. (Colour online) A visual depiction of parameter counting in semi-empirical density functionals. The middle (red) bars indicate the number of fitted parameters, while the right (blue) bars count the total number of borrowed or preset parameters. Furthermore, the left (black) bars that begin at zero and proceed left represent the number of constraints which subtract from the total number of fitted parameters. The values off to the right indicate the total number of fitted parameters (middle, red) minus the number of constraints (left, black), thus excluding borrowed or preset parameters. For example, ωB97X has 17 fitted parameters, 3 borrowed or preset parameters, and utilises 3 constraints, for a total of 14 parameters.

Figure 12. (Colour online) A visualisation of an (eighth-order meta-GGA) x (fourth-order GGA) functional space, which can be constructed by the two-dimensional power series in Equation (13(13) ) with N′= 2N= 8. The three separate blocks (from left to right) contain exchange, same-spin correlation, and opposite-spin correlation coefficients, respectively. Different truncations of the parameter space are shown, with the red coefficients representing up to quadratic fits in all three components, the highlighted coefficients representing the GGA-only subspace, and the blue boxes representing third order in exchange and sixth/third order in the meta-GGA/GGA dimension in both correlation components.

Table 2. Summary of the 84 data-sets that comprise MGCDB84. The datatypes are explained in Section 4.1. The fifth column contains the root-mean-squares of the data-set reaction energies. PEC stands for potential energy curve, SR stands for single-reference, MR stands for multi-reference, Bz stands for benzene, Me stands for methane, and Py stands for pyridine.

Name	Datatype	#	Description	ΔE (kcal/mol)	Ref.
A24	NCED	24	Binding energies of small non-covalent complexes	2.65	154
DS14	NCED	14	Binding energies of complexes containing divalent sulfur	3.70	155
HB15	NCED	15	Binding energies of hydrogen-bonded dimers featuring ionic groups common in biomolecules	19.91	156
HSG	NCED	21	Binding energies of small ligands interacting with protein receptors	6.63	157,158
NBC10	NCED	184	PECs for BzBz (5), BzMe (1), MeMe (1), BzH2S (1), and PyPy (2)	1.91	158–161
S22	NCED	22	Binding energies of hydrogen-bonded and dispersion-bound non-covalent complexes	9.65	141,158
X40	NCED	31	Binding energies of non-covalent interactions involving halogenated molecules	5.26	162
A21x12	NCED	252	PECs for the 21 equilibrium complexes from A24	1.43	163
BzDC215	NCED	215	PECs for benzene interacting with two rare-gas atoms and eight first- and second-row hydrides	1.81	164
HW30	NCED	30	Binding energies of hydrocarbon-water dimers	2.34	165
NC15	NCED	15	Binding energies of very small non-covalent complexes	0.95	166
S66	NCED	66	Binding energies of non-covalent interactions found in organic molecules and biomolecules	6.88	142,167
S66x8	NCED	528	PECs for the 66 complexes from S66	5.57	142
3B-69-DIM	NCED	207	Binding energies of all relevant pairs of monomers from 3B-69-TRIM	5.87	168
AlkBind12	NCED	12	Binding energies of saturated and unsaturated hydrocarbon dimers	3.14	47
CO2Nitrogen16	NCED	16	Binding energies of CO2 to molecular models of pyridinic N-doped graphene	3.84	169
HB49	NCED	49	Binding energies of small- and medium-sized hydrogen-bonded systems	14.12	170–172
Ionic43	NCED	43	Binding energies of anion-neutral, cation-neutral, and anion-cation dimers	69.94	173
H2O6Bind8	NCEC	8	Binding energies of isomers of (H2O)6	46.96	143,174
HW6Cl	NCEC	6	Binding energies of Cl^−(H2O)n (n = 1 − 6)	57.71	143,174
HW6F	NCEC	6	Binding energies of F^−(H2O)n (n = 1 − 6)	81.42	143,174
FmH2O10	NCEC	10	Binding energies of isomers of F^−(H2O)10	168.50	143,174
Shields38	NCEC	38	Binding energies of (H2O)n (n = 2 − 10)	51.54	48
SW49Bind345	NCEC	31	Binding energies of isomers of SO4^2 −(H2O)n (n = 3 − 5)	28.83	175
SW49Bind6	NCEC	18	Binding energies of isomers of SO4^2 −(H2O)6	62.11	175
WATER27	NCEC	23	Binding energies of neutral and charged water clusters	67.48	68,176
3B-69-TRIM	NCEC	69	Binding energies of trimers, with three different orientations of 23 distinct molecular crystals	14.36	168
CE20	NCEC	20	Binding energies of water, ammonia, and hydrogen fluoride clusters	30.21	177,178
H2O20Bind10	NCEC	10	Binding energies of isomers of (H2O)20 (low-energy structures)	198.16	143
H2O20Bind4	NCEC	4	Binding energies of isomers of (H2O)20 (dod, fc, fs, and es)	206.12	68,176,179,180
TA13	NCD	13	Binding energies of dimers involving radicals	22.00	144
XB18	NCD	8	Binding energies of small halogen-bonded dimers	5.23	181
Bauza30	NCD	30	Binding energies of halogen-, chalcogen-, and pnicogen-bonded dimers	23.65	182,183
CT20	NCD	20	Binding energies of charge-transfer complexes	1.07	184
XB51	NCD	20	Binding energies of large halogen-bonded dimers	6.06	181
AlkIsomer11	IE	11	Isomerisation energies of n = 4 − 8 alkanes	1.81	185
Butanediol65	IE	65	Isomerisation energies of butane-1,4-diol	2.89	186
ACONF	IE	15	Isomerisation energies of alkane conformers	2.23	68,187
CYCONF	IE	11	Isomerisation energies of cysteine conformers	2.00	68,188
Pentane14	IE	14	Isomerisation energies of stationary points on the n-pentane torsional surface	6.53	189
SW49Rel345	IE	31	Isomerisation energies of SO4^2 −(H2O)n (n = 3 − 5)	1.47	175
SW49Rel6	IE	18	Isomerisation energies of SO4^2 −(H2O)6	1.22	175
H2O16Rel5	IE	5	Isomerisation energies of (H2O)16 (boat and fused cube structures)	0.40	146
H2O20Rel10	IE	10	Isomerisation energies of (H2O)20 (low-energy structures)	2.62	143
H2O20Rel4	IE	4	Isomerisation energies of (H2O)20 (dod, fc, fs, and es)	5.68	68,176,179,180
Melatonin52	IE	52	Isomerisation energies of melatonin	5.54	190
YMPJ519	IE	519	Isomerisation energies of the proteinogenic amino acids	8.33	145
EIE22	ID	22	Isomerisation energies of enecarbonyls	4.97	191
Styrene45	ID	45	Isomerisation energies of C8H8	68.69	147
DIE60	ID	60	Isomerisation energies of reactions involving double-bond migration in conjugated dienes	5.06	192
ISOMERIZATION20	ID	20	Isomerisation energies	44.05	46
C20C24	ID	8	Isomerisation energies of the ground state structures of C20 and C24	36.12	193
AlkAtom19	TCE	19	n = 1 − 8 alkane atomisation energies	1829.31	185
BDE99nonMR	TCE	83	Bond dissociation energies (SR)	114.98	46
G21EA	TCE	25	Adiabatic electron affinities of atoms and small molecules	40.86	68,137
G21IP	TCE	36	Adiabatic ionisation potentials of atoms and small molecules	265.35	68,137
TAE140nonMR	TCE	124	Total atomisation energies (SR)	381.05	46

Table 2. Continued

Name	Datatype	#	Description	ΔE (kcal/mol)	Ref.
AlkIsod14	TCE	14	n = 3 − 8 alkane isodesmic reaction energies	10.35	185
BH76RC	TCE	30	Reaction energies from HTBH38 and NHTBH38	30.44	68,150,151
EA13	TCE	13	Adiabatic electron affinities	42.51	194
HAT707nonMR	TCE	505	Heavy-atom transfer energies (SR)	74.79	46
IP13	TCE	13	Adiabatic ionisation potentials	256.24	194
NBPRC	TCE	12	Reactions involving NH3/BH3 and PH3/BH3	30.52	68,195,196
SN13	TCE	13	Nucleophilic substitution energies	25.67	46
BSR36	TCE	36	Hydrocarbon bond separation reaction energies	20.06	148,196
HNBrBDE18	TCE	18	Homolytic N–Br bond dissociation energies	56.95	197
WCPT6	TCE	6	Tautomerisation energies for water-catalysed proton-transfer reactions	7.53	198
BDE99MR	TCD	16	Bond dissociation energies (MR)	54.51	46]
HAT707MR	TCD	202	Heavy-atom transfer energies (MR)	83.41	46
TAE140MR	TCD	16	Total atomisation energies (MR)	147.20	46
PlatonicHD6	TCD	6	Homodesmotic reactions involving platonic hydrocarbon cages, CnHn (n = 4, 6, 8, 10, 12, 20)	136.71	149
PlatonicID6	TCD	6	Isodesmic reactions involving platonic hydrocarbon cages, CnHn (n = 4, 6, 8, 10, 12, 20)	96.19	149
PlatonicIG6	TCD	6	Isogyric reactions involving platonic hydrocarbon cages, CnHn (n = 4, 6, 8, 10, 12, 20)	356.33	149
PlatonicTAE6	TCD	6	Total atomisation energies of platonic hydrocarbon cages, CnHn (n = 4, 6, 8, 10, 12, 20)	2539.27	149
BHPERI26	BH	26	Barrier heights of pericyclic reactions	23.15	68,199
CRBH20	BH	20	Barrier heights for cycloreversion of heterocyclic rings	46.40	200
DBH24	BH	24	Diverse barrier heights	28.34	201,202
CR20	BH	20	Cycloreversion reaction energies	22.31	203
HTBH38	BH	38	Hydrogen transfer barrier heights	16.05	151
NHTBH38	BH	38	Non-hydrogen transfer barrier heights	33.48	150
PX13	BH	13	Barrier heights for proton exchange in water, ammonia, and hydrogen fluoride clusters	28.83	177,178
WCPT27	BH	27	Barrier heights of water-catalysed proton-transfer reactions	38.73	198
AE18	–	18	Absolute atomic energies of hydrogen through argon	148,739.00	204
RG10	–	569	PECs for the 10 rare-gas dimers involving helium through krypton	1.21	152

Figure 13. Categorised list of the 200 benchmarked density functionals. The 20 highlighted functionals are featured in the paper and details regarding these functionals can be found in Table 3. Underlined functionals are range-separated hybrids. Italicised functionals are excluded from the box-and-whiskers plots in Section 5. The first level of separation pertains to hybridisation, the second level pertains to the density functional ingredients, and the third level pertains to the presence of dispersion corrections.

Table 3. Details for the 20 density functionals featured in the review, taken from the full list of 200 shown in Figure 13. The second column lists the number of parameters, counted according to the strategy in Figure 11. The third column indicates the percentage of exact exchange: a single, non-zero value indicates that the functional is a global hybrid (GH), while a range (e.g. 15–100) indicates that the functional is a range-separated hybrid (RSH), with the first value being the percentage of short-range exact exchange, and the second value being the percentage of long-range exact exchange (the value in parentheses is ω). The fourth column indicates the type of hybridisation. The fifth column indicates the ingredients (ρ, ∇ρ, and τ) contained in the functional. The sixth column indicates the type of dispersion correction, with D3(BJ) referring to Grimme's D3 method using the Becke– Johnson damping function, VV10 and rVV10 referring to the Vydrov and Van Voorhis nonlocal correlation functional and its revised version by Sabatini and co-workers, respectively, and CHG referring to the Grimme's D2 method using the Chai–Head–Gordon damping function.

Functional	#	cx · 100 (ω)	Hyb.	Ing.	Disp.	Year	Ref.
SPW92	0	0	None	LSDA	None	1992	23
PBE	0	0	None	GGA	None	1996	7
TPSS	0	0	None	meta-GGA	None	2003	8
B3LYP	3	20	GH	GGA	None	1993	9,29,35
PBE-D3(BJ)	3	0	None	GGA	D3(BJ)	2011	7,113
revPBE-D3(BJ)	4	0	None	GGA	D3(BJ)	2011	31,113
BLYP-D3(BJ)	4	0	None	GGA	D3(BJ)	2011	29,35,113
B97-D3(BJ)	12	0	None	GGA	D3(BJ)	2011	111,113
TPSS-D3(BJ)	3	0	None	meta-GGA	D3(BJ)	2011	8,113
SCAN-D3(BJ)	2	0	None	meta-GGA	D3(BJ)	2016	59,205
M06-L	34	0	None	meta-GGA	None	2006	62
B97M-rV	12	0	None	meta-GGA	rVV10	2017	124
PBE0-D3(BJ)	4	25	GH	GGA	D3(BJ)	2011	79,113
B3LYP-D3(BJ)	6	20	GH	GGA	D3(BJ)	2011	9,29,35,113
ωB97X-D	15	22.2–100 (0.2)	RSH	GGA	CHG	2008	119
ωB97X-V	10	16.7–100 (0.3)	RSH	GGA	VV10	2014	18
TPSSh-D3(BJ)	4	10	GH	meta-GGA	D3(BJ)	2011	84,113
M06-2X	29	54	GH	meta-GGA	None	2006	91
MN15	59	44	GH	meta-NGA	None	2016	94
ωB97M-V	12	15–100 (0.3)	RSH	meta-GGA	VV10	2016	20

Figure 14. Box-and-whisker plots for eight classes of density functionals, covering the three datatypes (NCED, NCEC, and IE) that are most sensitive to the inclusion of dispersion corrections. Each plot is generated from functional RMSDs (in kcal/mol) for the specified datatype. The eight plots together contain data for 188 of the 200 functionals benchmarked in this review (the italicised functionals in Figure 13 are excluded), with each plot containing (from left to right) 18, 39, 16, 19, 19, 33, 19, and 25 functional RMSDs, respectively.

Figure 15. Box-and-whisker plots for eight classes of density functionals, covering the three datatypes (NCD, ID, and BH) that are most sensitive to the inclusion of exact exchange. Each plot is generated from functional RMSDs (in kcal/mol) for the specified datatype. The eight plots together contain data for 188 of the 200 functionals benchmarked in this review (the italicised functionals in Figure 13 are excluded), with each plot containing (from left to right) 18, 39, 16, 19, 19, 33, 19, and 25 functional RMSDs, respectively.

Figure 16. Box-and-whisker plots for eight classes of density functionals, covering the two datatypes (TCE and TCD) that are associated with bonded interactions. Each plot is generated from functional RMSDs (in kcal/mol) for the specified datatype. The eight plots together contain data for 188 of the 200 functionals benchmarked in this review (the italicised functionals in Figure 13 are excluded), with each plot containing (from left to right) 18, 39, 16, 19, 19, 33, 19, and 25 functional RMSDs, respectively.

Figure 17. RMSDs in kcal/mol for the eight datatypes (as well as 81 equilibrium bond lengths (EBL in Ångström) and 81 equilibrium binding energies (EBE in kcal/mol)) for the 20 density functionals featured in this review, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The eight datatypes contain 1744, 243, 91, 755, 155, 947, 258, and 206 data points, respectively. NCED stands for non-covalent dimers (easy), NCEC stands for non-covalent clusters (easy), NCD stands for non-covalent dimers (difficult), IE stands for isomerisation energies (easy), ID stands for isomerisation energies (difficult), TCE stands for thermochemistry (easy), TCD stands for thermochemistry (difficult), and BH stands for barrier heights. ‘Difficult’ interactions involve either self-interaction error or strong correlation, while ‘easy’ interactions are not significantly characterised by either.

Figure 18. RMSDs in kcal/mol for the 18 data-sets that belong to the NCED datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 1744 interactions that are categorised as NCED are conventional, closed-shell, non-covalent dimers.

Figure 19. RMSDs in kcal/mol for the 12 data-sets that belong to the NCEC datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 243 interactions that are categorised as NCEC are conventional, closed-shell, non-covalent clusters.

Figure 20. RMSDs in kcal/mol for the five data-sets that belong to the NCD datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 91 interactions that are categorised as NCD are either open-shell non-covalent dimers or non-covalent dimers characterised by significant electron delocalisation.

Figure 21. RMSDs for equilibrium bond lengths (EBL) in Ångström and equilibrium binding energies (EBE) in kcal/mol for four data-sets that contain potential energy curves. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional.

Figure 22. RMSDs in kcal/mol for the 12 data-sets that belong to the IE datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 755 interactions that are categorised as IE are conventional, closed-shell, isomerisation energies.

Figure 23. RMSDs in kcal/mol for the five data-sets that belong to the ID datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 155 interactions that are categorised as ID are isomerisation energies characterised by significant electron delocalisation.

Figure 24. RMSDs in kcal/mol for the 15 data-sets that belong to the TCE datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 947 interactions that are categorised as TCE are conventional, single-reference bonded interactions.

Figure 25. RMSDs in kcal/mol for the seven data-sets that belong to the TCD datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 258 interactions that are categorised as TCD are bonded interactions that are either very sensitive to self-interaction error or significantly multi-reference in nature.

Figure 26. RMSDs in kcal/mol for the eight data-sets that belong to the BH datatype. Results for the 20 density functionals featured in this review are presented, along with the minimum RMSD from across all 200 benchmarked functionals, and the name of the corresponding functional. The 206 interactions that are categorised as BH are barrier heights.

Figure 27. RMSDs in kcal/mol for PBE-D3(BJ), B3LYP-D3(BJ), and ωB97M-V for the eight datatypes (as well as 81 equilibrium bond lengths (EBL) in Ångström and 81 equilibrium binding energies (EBE) in kcal/mol) computed with four different basis sets: 6-31G*, def2-SVPD, def2-TZVPPD, and def2-QZVPPD. Results for the ‘low-cost’ PBEh-3c method by Grimme and co-workers are given in the last row, in order to demonstrate that if modern functionals are to be used with small basis sets like 6-31G*, it is better to use a method developed for use with a small basis set.

Figure 28. Mean absolute percent integration grid errors for 20 density functionals, relative to the 15 binding energies from the NC15 data-set computed with a (500,974) grid. A regularisation value of 0.01 kcal/mol is included in the denominator in order to cap excessively large percentage errors for very weakly bound systems. A value of 1% or smaller is deemed acceptable for these interactions.

Figure 29. Root-mean-square integration grid errors in kcal/mol for 20 density functionals, relative to the 19 atomisation energies from the AlkAtom19 data-set computed with a (500,974) grid. A value of 0.05 kcal/mol or smaller is deemed acceptable for these interactions.

Figure 30. Benzene–neon dimer potential energy curves (PEC) computed with 20 density functionals with a (99,590) integration grid. The top subfigure shows LSDA, local GGA, and hybrid GGA functionals, and a majority of these functionals are converged with respect to the employed grid. The middle subfigure shows local meta-GGA functionals, while the bottom subfigure shows hybrid meta-GGA functionals. It is evident that the PECs of many of the meta-GGAs are sensitive to the integration grid.

Figure 31. Visual representations of the three case studies considered in Section 7. The top panel corresponds to the Grubbs case study, the middle panel corresponds to the PDB12 case study, and the bottom panel corresponds to the Water2510 case study.

Figure 32. Relative energies in kcal/mol for the Grubbs case study for 13 density functionals. Mean signed errors (MSE) and root-mean-square deviations (RMSD) are shown at the bottom. The top panel of Figure 31 contains more information regarding these interactions.

Figure 33. Binding energies in kcal/mol for the PDB12 case study for 13 density functionals. Mean signed errors (MSE) and root-mean-square deviations (RMSD) in kcal/mol are shown at the bottom. The middle panel of Figure 31 contains more information regarding these interactions.

Figure 34. Mean signed errors (MSE) and root-mean-square deviations (RMSD) in kcal/mol for the Water2510 case study for 13 density functionals. The statistics are further separated into unbound configurations (E ≥ 0) and bound configurations (E < 0). The bottom panel of Figure 31 contains more information regarding these interactions.

Figure 35. Sorted NCED RMSDs in kcal/mol for all 200 benchmarked density functionals. NCED contains 1744 non-covalent dimer interactions.

Figure 36. Sorted NCEC RMSDs in kcal/mol for all 200 benchmarked density functionals. NCEC contains 243 non-covalent cluster interactions.

Figure 37. Sorted NCD RMSDs in kcal/mol for all 200 benchmarked density functionals. NCD contains 91 difficult non-covalent dimer interactions.

Figure 38. Sorted IE RMSDs in kcal/mol for all 200 benchmarked density functionals. IE contains 755 isomerisation energies.

Figure 39. Sorted ID RMSDs in kcal/mol for all 200 benchmarked density functionals. ID contains 155 difficult isomerisation energies.

Figure 40. Sorted TCE RMSDs in kcal/mol for all 200 benchmarked density functionals. TCE contains 947 thermochemistry data points.

Figure 41. Sorted TCD RMSDs in kcal/mol for all 200 benchmarked density functionals. TCD contains 258 difficult thermochemistry data points.

Figure 42. Sorted BH RMSDs in kcal/mol for all 200 benchmarked density functionals. BH contains 206 barrier heights.

Figure 43. Average rankings for the best of the 200 functionals studied in this review. The functionals are ranked for each of the 82 data-sets that correspond to the eight datatypes, the rankings are averaged by datatype, and the functionals are partitioned by type (local/hybrid and GGA/mGGA). The three functionals with the best (smallest) rank populate the first eight rows of this table, and the last ‘Overall’ row is an average over the eight average datatype ranks. Functionals that differ only in their dispersion component are only listed once by selecting the best performer.

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