r²SCANX-D4

Functional Implementation

r²SCANX-D4 functionals can be build in Orca and Turbomole via the LibXC
r²SCAN can be used with libxc version 5.1.0 as MGGA_X_R2SCAN (id=497), MGGA_C_R2SCAN (id=498)
XCFun 2.1.0 also implements support for the r²SCAN functional
routines for r²SCAN functional implementation can be found at https://gitlab.com/dhamil/r2scan-subroutines (Fortran), routines for Vasp are included
Libraries implementing DFT-D4 can be found at https://github.com/dftd4/dftd4 (Fortran) and https://github.com/dftd4/cpp-d4 (C++)

In the following * indicates that D4 Parameters from https://doi.org/10.3390/molecules27010141 were employed.

Functional parameters

method	s6	s8	s9	a1	a2/Bohr
r²SCANh-D4(EEQ)-ATM/def2-QZVP	1	0.8324	1	0.4944	5.9019
r²SCANh-D3(BJ)-ATM/def2-QZVP	1	1.1236	1	0.4709	5.9157
r²SCAN0-D4(EEQ)-ATM/def2-QZVP	1	0.8992	1	0.4778	5.8779
r²SCAN0-D3(BJ)-ATM/def2-QZVP	1	1.1846	1	0.4534	5.8972
r²SCAN50-D4(EEQ)-ATM/def2-QZVP	1	1.0471	1	0.4574	5.8969
r²SCAN50-D3(BJ)-ATM/def2-QZVP	1	1.3294	1	0.4311	5.9240

method	s6	s8	s9	a1	a2/Bohr
r²SCANh-D4(EEQ)-ATM/def2-TZVPP	1	0.9119	1	0.4832	6.2073
r²SCANh-D3(BJ)-ATM/def2-TZVPP	1	1.1493	1	0.4761	6.0947
r²SCAN0-D4(EEQ)-ATM/def2-TZVPP	1	0.9397	1	0.4578	6.1864
r²SCAN0-D3(BJ)-ATM/def2-TZVPP	1	1.1859	1	0.4567	6.0583
r²SCAN50-D4(EEQ)-ATM/def2-TZVPP	1	1.0576	1	0.4232	6.2378
r²SCAN50-D3(BJ)-ATM/def2-TZVPP	1	1.2980	1	0.4314	6.0662

method	b
r²SCANh-VV10	11.9
r²SCAN0-VV10	11.4
r²SCAN50-VV10	10.8

Benchmark Results

NCIBLIND10

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	–0.05	0.17	0.29	0.29
r²SCANh-D4-ATM/def2-QZVP	-0.06	0.18	0.32	0.31
r²SCAN0-D4-ATM/def2-QZVP	-0.07	0.20	0.36	0.35
r²SCAN50-D4-ATM/def2-QZVP	-0.10	0.26	0.46	0.45

S22x5

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	–0.12	0.28	0.51	0.50
r²SCANh-D4-ATM/def2-QZVP	-0.16	0.30	0.57	0.55
r²SCAN0-D4-ATM/def2-QZVP	-0.20	0.35	0.66	0.63
r²SCAN50-D4-ATM/def2-QZVP	-0.28	0.47	0.86	0.82

S66x8

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	–0.09	0.23	0.38	0.36
r²SCANh-D4-ATM/def2-QZVP	-0.11	0.24	0.41	0.39
r²SCAN0-D4-ATM/def2-QZVP	-0.13	0.27	0.46	0.44
r²SCAN50-D4-ATM/def2-QZVP	-0.18	0.35	0.60	0.57

GMTKN55 / def2-QZVP

Statistics in kcal/mol.

method	WTMAD-2	basic	react.	barriers	inter. NCI	intra. NCI	NCI
r²SCAN-D4/def2-QZVP	7.54	5.55	8.26	14.29	7.46	5.74	6.62
r²SCANh-D4/def2-QZVP	6.22	4.42	7.22	11.10	6.34	4.97	5.67
r²SCAN0-D4/def2-QZVP	5.63	4.93	6.76	7.60	5.63	4.53	5.09
r²SCAN50-D4/def2-QZVP	6.27	6.60	8.43	5.97	5.32	5.14	5.23

method	WTMAD-1	basic	react.	barriers	inter. NCI	intra. NCI	NCI
r²SCAN-D4/def2-QZVP	4.43	4.46	4.85	6.15	3.87	3.33	3.64
r²SCANh-D4/def2-QZVP	3.80	3.89	4.21	4.95	3.41	2.82	3.15
r²SCAN0-D4/def2-QZVP	3.47	3.81	3.94	3.73	3.19	2.48	2.89
r²SCAN50-D4/def2-QZVP	3.81	4.40	4.86	3.10	3.22	2.91	3.09

*With D4 Parameters from https://doi.org/10.3390/molecules27010141

method	WTMAD-2	basic	react.	barriers	inter. NCI	intra. NCI	NCI
r²SCANh-D4/def2-QZVP*	6.04	4.4	7.40	10.99	5.46	4.86	5.17
r²SCAN0-D4/def2-QZVP*	5.44	4.91	6.82	7.53	4.75	4.51	4.63
r²SCAN50-D4/def2-QZVP*	6.16	6.6	8.36	5.97	4.76	5.22	4.99

method	WTMAD-1	basic	react.	barriers	inter. NCI	intra. NCI	NCI
r²SCANh-D4/def2-QZVP*	3.77	3.88	4.27	4.91	3.32	2.78	3.09
r²SCAN0-D4/def2-QZVP*	3.43	3.8	3.94	3.7	3.05	2.48	2.8
r²SCAN50-D4/def2-QZVP*	3.79	4.41	4.86	3.1	3.1	2.94	3.03

The MADs of all GMTKN55 subsets in kcal/mol are given here. Values for TPSS-D4, PBE-D4 and PBE0-D4/def2-QZVPP are taken from Caldeweyher et al. J. Chem Phys, 2019, 150, 154122. DOI: 10.1063/1.5090222. Values for r²SCAN-D4 are taken from Ehlert et al. J. Chem. Phys. 2021, 154, 061101. DOI: 10.1063/5.0041008

subset	r²SCAN-D4	r²SCANh-D4	r²SCAN0-D4	r²SCAN50-D4	TPSS-D4	PBE-D4	PBE0-D4
W4-11	3.86	4.58	11.35	22.62	5.89	15.62	3.67
G21EA	5.64	3.93	4.61	6.00	2.12	3.59	2.57
G21IP	4.77	4.57	4.76	5.32	3.87	3.80	3.64
DIPCS10	5.05	4.64	4.55	5.71	3.28	4.22	2.84
PA26	2.51	2.46	2.49	2.73	4.08	1.85	2.42
SIE4x4	18.13	15.19	10.93	4.56	21.81	23.62	14.31
ALKBDE10	5.93	5.42	7.17	14.35	4.22	6.36	5.69
YBDE18	3.34	3.11	3.41	5.99	4.49	4.87	0.99
AL2X6	1.59	1.64	1.79	2.05	2.28	1.59	1.33
HEAVYSB11	3.24	3.82	4.93	6.60	2.59	3.53	1.4
NBPRC	1.55	1.80	2.34	3.38	1.37	2.41	3.08
ALK8	2.84	2.79	2.70	2.90	0.80	2.21	2.36
RC21	4.85	4.65	4.51	4.83	4.41	6.75	5.37
G2RC	5.65	5.37	5.93	8.13	7.23	6.94	6.71
BH76RC	3.17	2.83	2.94	4.19	3.59	4.17	2.45
FH51	4.58	2.12	2.67	4.11	6.34	5.07	4.68
TAUT15	1.57	1.34	1.09	1.07	1.60	1.80	1.13
DC13	7.85	7.18	7.91	10.69	8.11	8.93	8.43
MB16-43	14.62	13.56	14.77	22.17	25.84	25.10	15.99
DARC	2.72	2.06	2.90	7.40	4.64	3.13	3.94
RSE43	1.6	1.29	0.94	0.80	2.08	2.93	1.57
BSR36	0.44	0.96	1.65	2.76	3.09	2.30	2.73
CDIE20	1.62	1.44	1.20	0.93	1.72	1.70	1.29
ISO34	1.3	1.19	1.31	1.83	2.31	1.49	1.43
ISOL24	4.04	3.16	2.46	3.46	5.30	4.19	2.08
C60ISO	5.56	3.14	4.16	12.84	9.74	11.80	2.11
PArel	1.54	1.32	1.23	1.47	1.53	1.77	1.2
BH76	7.34	2.83	2.94	4.19	9.25	9.61	4.99
BHPERI	4.67	3.44	2.01	1.86	5.89	6.82	3.3
BHDIV10	6.12	4.89	3.17	1.71	7.00	8.89	4.81
INV24	2.33	1.03	1.52	2.53	1.89	2.06	1.14
BHROT27	0.76	0.77	0.80	0.87	0.53	0.47	0.58
PX13	8.93	7.28	4.97	1.94	8.91	11.95	6.49
WCPT18	6.11	4.66	2.87	2.58	6.36	9.29	4.29
RG18	0.17	0.15	0.11	0.09	0.13	0.26	0.09
ADIM6	0.36	0.35	0.36	0.32	0.38	0.09	0.19
S22	0.29	0.31	0.38	0.57	0.33	0.42	0.41
S66	0.3	0.32	0.35	0.44	0.33	0.38	0.36
HEAVY28	0.29	0.27	0.26	0.22	0.38	0.47	0.28
WATER27	6.72	6.27	5.57	5.10	4.16	8.28	5.35
CARBHB12	1.13	1.07	0.99	0.96	1.36	1.84	1.36
PNICO23	0.82	0.73	0.62	0.55	1.15	1.36	0.89
HAL59	0.81	0.65	0.47	0.33	1.07	1.29	0.62
AHB21	3.5	1.37	1.51	1.83	0.73	1.24	1.31
CHB6	0.46	0.49	0.83	1.44	0.71	0.67	1.13
IL16	0.76	0.43	0.32	0.42	0.35	0.96	0.5
IDISP	2.56	2.42	2.16	2.71	2.77	2.55	1.47
ICONF	0.3	0.30	0.34	0.46	0.16	0.31	0.27
ACONF	0.18	0.20	0.21	0.22	0.12	0.08	0.06
AMINO20x4	0.19	0.20	0.23	0.31	0.36	0.33	0.25
PCONF21	0.42	0.30	0.28	0.37	0.83	0.99	0.74
MCONF	0.45	0.38	0.29	0.21	0.44	0.47	0.27
SCONF	0.54	0.36	0.14	0.18	1.22	0.88	0.32
UPU23	0.39	0.34	0.39	0.52	0.45	0.51	0.54
BUT14DIOL	0.27	0.21	0.13	0.08	0.47	0.59	0.31

GMTKN55 / def2-TZVPP

Statistics in kcal/mol.

method	WTMAD-2	basic	react.	barriers	inter. NCI	intra. NCI	NCI
r²SCANh-D4/def2-TZVPP	6.80	4.54	7.40	11.72	7.64	5.81	6.75
r²SCAN0-D4/def2-TZVPP	6.12	5.04	6.90	8.17	6.76	5.20	6.00
r²SCAN50-D4/def2-TZVPP	6.51	6.56	8.59	5.98	6.02	5.58	5.81

method	WTMAD-1	basic	react.	barriers	inter. NCI	intra. NCI	NCI
r²SCANh-D4/def2-TZVPP	4.07	3.97	4.28	5.18	4.06	3.24	3.71
r²SCAN0-D4/def2-TZVPP	3.72	3.88	3.98	3.93	3.8	2.83	3.38
r²SCAN50-D4/def2-TZVPP	3.92	4.36	4.86	3.1	3.56	3.22	3.41

L7 / def2-QZVPP

Statistics in kcal/mol.

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	0.36	1.59	1.70	1.66
r²SCANh-D4/def2-QZVPP	0.34	1.54	1.69	1.66
r²SCAN0-D4/def2-QZVPP	0.45	1.60	1.82	1.76
r²SCAN50-D4/def2-QZVPP	0.73	1.86	2.20	2.08
ωB97X-V/def2-QZVPP	-0.86	0.86	1.31	0.99
PBE0-D4/def2-QZVPP	–0.27	0.56	0.63	0.61
PW6B95-D4/def2-QZVPP	0.95	1.11	1.52	1.28
r²SCANh-D4/def2-QZVPP*	-0.33	1.05	1.46	1.42
r²SCAN0-D4/def2-QZVPP*	-0.65	1.10	1.59	1.45
r²SCAN50-D4/def2-QZVPP*	-0.35	1.15	1.75	1.72

L7 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	0.62	1.90	2.07	1.97
r²SCAN0-D4/def2-TZVPP	0.71	1.93	2.17	2.05
r²SCAN50-D4/def2-TZVPP	0.99	2.17	2.54	2.35

S30L / def2-QZVPP

ωB97X-D3 taken from https://doi.org/10.1021/acs.jctc.5b00296

Statistics in kcal/mol.

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-1.83	1.92	2.60	1.84
r²SCANh-D4/def2-QZVPP	-1.93	2.01	2.61	1.76
r²SCAN0-D4/def2-QZVPP	-1.78	2.01	2.50	1.75
r²SCAN50-D4/def2-QZVPP	-1.58	2.04	2.46	1.88
wB97X-D3/def2-QZVPP		2.6
wB97X-V/def2-QZVPP	-3.83	3.83	4.68	2.70
PBE0-D4/def2-QZVPP	-1.81	2.59	3.34	2.81
PW6B95-D4/def2-QZVPP	1.53	2.45	2.87	2.47
r²SCANh-D4/def2-QZVPP*	-3.86	3.86	4.78	2.82
r²SCAN0-D4/def2-QZVPP*	-4.36	4.36	5.29	2.99
r²SCAN50-D4/def2-QZVPP*	-3.62	3.74	4.33	2.38

S30L / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-1.67	1.87	2.45	1.79
r²SCAN0-D4/def2-TZVPP	-1.57	1.85	2.35	1.76
r²SCAN50-D4/def2-TZVPP	-1.35	1.89	2.41	2.00

X40x10(2018) / def2-QZVPP

Statistics in kcal/mol.

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-0.09	0.30	0.57	0.56
r²SCANh-D4/def2-QZVPP	-0.07	0.30	0.57	0.56
r²SCAN0-D4/def2-QZVPP	-0.06	0.31	0.58	0.57
r²SCAN50-D4/def2-QZVPP	-0.07	0.34	0.62	0.62
ωB97X-V/def2-QZVPP	0.17	0.26	0.47	0.44
PBE0-D4/def2-QZVPP	-0.16	0.32	0.55	0.52
PW6B95-D4/def2-QZVPP	0.07	0.21	0.31	0.30
r²SCANh-D4/def2-QZVPP*	-0.13	0.30	0.56	0.54
r²SCAN0-D4/def2-QZVPP*	-0.13	0.30	0.56	0.54
r²SCAN50-D4/def2-QZVPP*	-0.15	0.32	0.60	0.59

X40x10(2018) / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-0.17	0.35	0.66	0.63
r²SCAN0-D4/def2-TZVPP	-0.15	0.35	0.66	0.64
r²SCAN50-D4/def2-TZVPP	-0.16	0.37	0.70	0.68

CHAL336 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-1.67	1.90	2.81	2.25
r²SCANh-D4/def2-QZVPP	-1.54	1.75	2.62	2.12
r²SCAN0-D4/def2-QZVPP	-1.48	1.67	2.54	2.06
r²SCAN50-D4/def2-QZVPP	-1.50	1.69	2.70	2.24
ωB97X-V/def2-QZVPP	-0.11	0.89	1.31	1.31
PBE0-D4/def2-QZVPP	-2.01	2.05	2.80	1.95
PW6B95-D4/def2-QZVPP	-0.93	1.28	1.90	1.65
r²SCANh-D4/def2-QZVPP*	-1.77	1.85	2.71	2.06
r²SCAN0-D4/def2-QZVPP*	-1.72	1.78	2.64	2.00
r²SCAN50-D4/def2-QZVPP*	-1.73	1.80	2.82	2.22

CHAL336 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-2.23	2.42	3.82	3.09
r²SCAN0-D4/def2-TZVPP	-2.09	2.26	3.51	2.82
r²SCAN50-D4/def2-TZVPP	-2.06	2.20	3.36	2.66

HB300SPX / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-0.58	0.62	1.03	0.85
r²SCANh-D4/def2-QZVPP	-0.54	0.57	0.95	0.78
r²SCAN0-D4/def2-QZVPP	-0.50	0.53	0.87	0.72
r²SCAN50-D4/def2-QZVPP	-0.49	0.52	0.85	0.70
ωB97X-V/def2-QZVPP	0.01	0.23	0.34	0.34
PBE0-D4/def2-QZVPP	-0.69	0.70	0.99	0.72
PW6B95-D4/def2-QZVPP	-0.08	0.24	0.33	0.32
r²SCANh-D4/def2-QZVPP*	-0.60	0.61	0.98	0.78
r²SCAN0-D4/def2-QZVPP*	-0.57	0.58	0.91	0.71
r²SCAN50-D4/def2-QZVPP*	-0.57	0.58	0.90	0.70

HB300SPX / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-0.67	0.69	1.07	0.83
r²SCAN0-D4/def2-TZVPP	-0.63	0.65	0.99	0.76
r²SCAN50-D4/def2-TZVPP	-0.61	0.64	0.96	0.74

R160x6 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-0.02	0.24	0.35	0.35
r²SCANh-D4/def2-QZVPP	-0.00	0.22	0.31	0.31
r²SCAN0-D4/def2-QZVPP	0.02	0.20	0.28	0.28
r²SCAN50-D4/def2-QZVPP	0.04	0.22	0.32	0.32
ωB97X-V/def2-QZVPP	0.11	0.18	0.26	0.24
PBE0-D4/def2-QZVPP	-0.19	0.27	0.39	0.34
PW6B95-D4/def2-QZVPP	0.02	0.18	0.25	0.25
r²SCANh-D4/def2-QZVPP*	0.03	0.22	0.30	0.30
r²SCAN0-D4/def2-QZVPP*	0.04	0.20	0.28	0.27
r²SCAN50-D4/def2-QZVPP*	0.02	0.20	0.31	0.30

R160x6 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-0.04	0.25	0.35	0.35
r²SCAN0-D4/def2-TZVPP	-0.01	0.23	0.32	0.32
r²SCAN50-D4/def2-TZVPP	0.01	0.24	0.35	0.35

MOR41 / def2-QZVPP

Statistics in kcal/mol

ωB97X-V data taken from https://doi.org/10.1021/acs.jctc.7b01183

PBE0-D4 data taken from https://doi.org/10.1063/1.5090222

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-0.13	3.32	4.29	4.29
r²SCANh-D4/def2-QZVPP	-0.52	2.54	3.47	3.47
r²SCAN0-D4/def2-QZVPP	0.09	2.31	2.91	2.90
r²SCAN50-D4/def2-QZVPP	0.19	3.19	4.20	4.20
ωB97X-V/def2-QZVPP		2.21
PBE0-D4/def2-QZVPP	–0.25	2.28	3.08	3.11
PW6B95-D4/def2-QZVPP	2.68	3.20	3.99	2.98
r²SCANh-D4/def2-QZVPP*	-0.74	2.26	3.26	3.18
r²SCAN0-D4/def2-QZVPP*	-0.64	2.10	2.81	2.74
r²SCAN50-D4/def2-QZVPP*	-0.42	3.22	4.23	4.21

MOR41 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-0.04	2.59	3.55	3.55
r²SCAN0-D4/def2-TZVPP	0.08	2.26	2.90	2.90
r²SCAN50-D4/def2-TZVPP	0.12	3.10	4.11	4.11

ROST61 / def2-QZVPP

Statistics in kcal/mol

ωB97X-V, PW6B95-D4, and PBE0-D4 data taken from https://doi.org/10.1021/acs.jctc.1c00659

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	0.89	3.33	4.48	4.38
r²SCANh-D4/def2-QZVPP	0.31	2.64	3.61	3.59
r²SCAN0-D4/def2-QZVPP	-0.53	2.96	3.98	3.95
r²SCAN50-D4/def2-QZVPP	-1.94	4.84	7.12	6.87
ωB97X-V/def2-QZVPP		2.8
PBE0-D4/def2-QZVPP		2.6
PW6B95-D4/def2-QZVPP		2.5
r²SCANh-D4/def2-QZVPP*	0.05	2.57	3.49	3.49
r²SCAN0-D4/def2-QZVPP*	-0.84	2.94	3.93	3.84
r²SCAN50-D4/def2-QZVPP*	-2.21	4.88	7.11	6.76

ROST61 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	0.23	3.17	4.29	4.28
r²SCAN0-D4/def2-TZVPP	-0.62	3.42	4.72	4.68
r²SCAN50-D4/def2-TZVPP	-2.05	5.16	7.63	7.35

WCCR10 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	1.22	2.74	3.78	3.57
r²SCANh-D4/def2-QZVPP	0.91	1.96	2.69	2.53
r²SCAN0-D4/def2-QZVPP	0.44	0.88	1.22	1.14
r²SCAN50-D4/def2-QZVPP	-0.21	1.30	1.58	1.57
ωB97X-V/def2-QZVPP	0.49	1.18	1.49	1.41
PBE0-D4/def2-QZVPP	-0.11	0.83	0.96	0.95
PW6B95-D4/def2-QZVPP	-1.82	1.82	2.13	1.11
r²SCANh-D4/def2-QZVPP*	1.77	2.24	2.85	2.23
r²SCAN0-D4/def2-QZVPP*	1.50	1.53	1.82	1.04
r²SCAN50-D4/def2-QZVPP*	0.66	1.53	2.11	2.00

WCCR10 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	1.05	1.20	2.82	2.61
r²SCAN0-D4/def2-TZVPP	0.60	1.05	1.36	1.22
r²SCAN50-D4/def2-TZVPP	-0.04	1.26	1.57	1.57

MOBH35 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-2.97	3.71	4.86	3.85
r²SCANh-D4/def2-QZVPP	-1.97	2.79	3.76	3.21
r²SCAN0-D4/def2-QZVPP	-0.56	2.07	2.81	2.75
r²SCAN50-D4/def2-QZVPP	1.57	2.86	4.03	3.71
ωB97X-V/def2-QZVPP	1.09	2.00	2.60	2.36
PBE0-D4/def2-QZVPP	-1.24	2.38	3.20	2.95
PW6B95-D4/def2-QZVPP	-1.06	2.07	2.84	2.63
r²SCANh-D4/def2-QZVPP*	-2.06	2.89	3.98	3.41
r²SCAN0-D4/def2-QZVPP*	-0.67	2.18	3.01	2.93
r²SCAN50-D4/def2-QZVPP*	1.46	2.78	4.00	3.72

MOBH35 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-1.97	2.78	3.76	3.21
r²SCAN0-D4/def2-TZVPP	-0.56	2.06	2.80	2.74
r²SCAN50-D4/def2-TZVPP	1.57	2.87	4.01	3.69

TMBH / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-2.59	3.24	4.04	3.10
r²SCANh-D4/def2-QZVPP	-1.60	2.33	2.92	2.46
r²SCAN0-D4/def2-QZVPP	-0.10	1.88	2.43	2.43
r²SCAN50-D4/def2-QZVPP	2.13	2.89	4.64	4.12
ωB97X-V/def2-QZVPP	1.85	2.11	3.40	2.85
PBE0-D4/def2-QZVPP	-0.87	2.32	2.87	2.74
PW6B95-D4/def2-QZVPP	0.10	1.26	1.66	1.65
r²SCANh-D4/def2-QZVPP*	-1.56	2.33	2.90	2.45
r²SCAN0-D4/def2-QZVPP*	-0.10	1.89	2.43	2.43
r²SCAN50-D4/def2-QZVPP*	2.08	2.89	4.63	4.14

TMBH / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-1.63	2.39	3.00	2.52
r²SCAN0-D4/def2-TZVPP	-0.16	1.92	2.49	2.48
r²SCAN50-D4/def2-TZVPP	2.07	2.93	4.65	4.16

MLA24 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	1.37	4.81	5.42	5.24
r²SCANh-D4/def2-QZVPP	1.95	4.48	5.19	4.81
r²SCAN0-D4/def2-QZVPP	2.73	4.09	5.08	4.28
r²SCAN50-D4/def2-QZVPP	3.88	4.21	5.32	3.64
ωB97X-V/def2-QZVPP	-1.49	1.69	2.23	1.65
PBE0-D4/def2-QZVPP	0.92	3.24	3.69	3.57
PW6B95-D4/def2-QZVPP	0.08	2.62	2.87	2.87
r²SCANh-D4/def2-QZVPP*	1.77	4.50	5.15	4.84
r²SCAN0-D4/def2-QZVPP*	2.46	4.12	4.96	4.31
r²SCAN50-D4/def2-QZVPP*	3.61	4.03	5.16	3.69

MLA24 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	2.15	4.82	5.27	4.81
r²SCAN0-D4/def2-TZVPP	2.91	4.09	5.17	4.27
r²SCAN50-D4/def2-TZVPP	4.03	4.29	5.42	3.62

IONPI19 / def2-QZVPP

Statistics in kcal/mol

ωB97X-V, PW6B95-D4, and PBE0-D4 data taken from https://pubs.rsc.org/en/content/articlelanding/2021/CP/D1CP01333E

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-0.06	0.72	0.85	0.85
r²SCANh-D4/def2-QZVPP	-0.07	0.67	0.80	0.80
r²SCAN0-D4/def2-QZVPP	-0.13	0.71	0.82	0.81
r²SCAN50-D4/def2-QZVPP	-0.27	0.84	1.12	1.09
ωB97X-V/def2-QZVPP		0.73
PBE0-D4/def2-QZVPP		0.88
PW6B95-D4/def2-QZVPP		0.83
r²SCANh-D4/def2-QZVPP*	-0.18	0.77	0.93	0.91
r²SCAN0-D4/def2-QZVPP*	-0.33	0.85	1.04	0.99
r²SCAN50-D4/def2-QZVPP*	-0.54	0.95	1.26	1.14

IONPI19 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-0.92	1.14	1.38	1.03
r²SCAN0-D4/def2-TZVPP	-0.95	1.04	1.28	0.86
r²SCAN50-D4/def2-TZVPP	-1.02	1.04	1.35	0.88

ACONF12 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	0.19	0.19	0.20	0.07
r²SCANh-D4/def2-QZVPP	0.18	0.18	0.20	0.07
r²SCAN0-D4/def2-QZVPP	0.16	0.16	0.17	0.07
r²SCAN50-D4/def2-QZVPP	0.11	0.12	0.14	0.08
ωB97X-V/def2-QZVPP	-0.26	0.26	0.28	0.12
PBE0-D4/def2-QZVPP	-0.19	0.19	0.19	0.05
PW6B95-D4/def2-QZVPP	-0.67	0.67	0.70	0.17
r²SCANh-D4/def2-QZVPP*	0.25	0.25	0.27	0.09
r²SCAN0-D4/def2-QZVPP*	0.19	0.19	0.20	0.09
r²SCAN50-D4/def2-QZVPP*	0.05	0.09	0.10	0.09

ACONF12 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	0.24	0.24	0.25	0.07
r²SCAN0-D4/def2-TZVPP	0.20	0.20	0.21	0.07
r²SCAN50-D4/def2-TZVPP	0.13	0.15	0.16	0.09

MPCONF196 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	0.40	0.75	1.06	0.99
r²SCANh-D4/def2-QZVPP	0.49	0.77	1.09	0.97
r²SCAN0-D4/def2-QZVPP	0.61	0.83	1.19	1.02
r²SCAN50-D4/def2-QZVPP	0.59	1.34	2.40	2.33
ωB97X-V/def2-QZVPP	0.43	0.62	1.00	0.91
PBE0-D4/def2-QZVPP	0.53	0.85	1.16	1.03
PW6B95-D4/def2-QZVPP	-0.01	0.90	1.31	1.31
r²SCANh-D4/def2-QZVPP*	0.52	0.80	1.12	0.99
r²SCAN0-D4/def2-QZVPP*	0.68	0.89	1.26	1.06
r²SCAN50-D4/def2-QZVPP*	0.67	1.37	2.44	2.35

MPCONF196 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	0.53	0.81	1.15	1.02
r²SCAN0-D4/def2-TZVPP	0.66	0.88	1.25	1.07
r²SCAN50-D4/def2-TZVPP	0.87	1.06	1.52	1.25

37CONF8 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	0.09	0.50	0.69	0.69
r²SCANh-D4/def2-QZVPP	0.11	0.46	0.64	0.63
r²SCAN0-D4/def2-QZVPP	0.13	0.46	0.62	0.61
r²SCAN50-D4/def2-QZVPP	0.18	0.54	0.72	0.70
ωB97X-V/def2-QZVPP	-0.13	0.31	0.41	0.39
PBE0-D4/def2-QZVPP	0.03	0.52	0.67	0.67
PW6B95-D4/def2-QZVPP	-0.28	0.64	0.82	0.77
r²SCANh-D4/def2-QZVPP*	0.12	0.46	0.64	0.63
r²SCAN0-D4/def2-QZVPP*	0.14	0.46	0.61	0.59
r²SCAN50-D4/def2-QZVPP*	0.18	0.53	0.70	0.68

37CONF8 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	0.16	0.47	0.64	0.62
r²SCAN0-D4/def2-TZVPP	0.18	0.46	0.61	0.58
r²SCAN50-D4/def2-TZVPP	0.22	0.54	0.71	0.67

TMCONF16 / def2-QZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	-0.10	0.22	0.30	0.28
r²SCANh-D4/def2-QZVPP	-0.13	0.22	0.29	0.26
r²SCAN0-D4/def2-QZVPP	-0.17	0.23	0.31	0.26
r²SCAN50-D4/def2-QZVPP	-0.25	0.28	0.39	0.31
ωB97X-V/def2-QZVPP	-0.47	0.50	0.68	0.48
PBE0-D4/def2-QZVPP	-0.39	0.39	0.49	0.29
PW6B95-D4/def2-QZVPP	-0.40	0.43	0.54	0.36
r²SCANh-D4/def2-QZVPP*	-0.15	0.22	0.30	0.26
r²SCAN0-D4/def2-QZVPP*	-0.20	0.25	0.33	0.27
r²SCAN50-D4/def2-QZVPP*	-0.27	0.31	0.42	0.32

TMCONF16 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	-0.09	0.19	0.27	0.25
r²SCAN0-D4/def2-TZVPP	-0.14	0.21	0.28	0.24
r²SCAN50-D4/def2-TZVPP	-0.22	0.26	0.37	0.30

SIE8 / def2-QZVPP (remaining systems of old GMTKN SIE11 subset)

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCAN-D4/def2-QZVPP	9.24	9.79	13.44	9.75
r²SCANh-D4/def2-QZVPP	7.76	8.26	11.23	8.15
r²SCAN0-D4/def2-QZVPP	7.08	7.48	10.14	7.23
r²SCAN50-D4/def2-QZVPP	5.15	5.58	8.73	7.06
ωB97X-V/def2-QZVPP	1.70	6.20	8.64	8.48
PBE0-D4/def2-QZVPP	6.70	7.75	9.79	9.79
PW6B95-D4/def2-QZVPP	6.13	7.26	9.91	7.79
r²SCANh-D4/def2-QZVPP*	7.76	8.24	11.26	8.15
r²SCAN0-D4/def2-QZVPP*	7.08	7.47	10.15	7.28
r²SCAN50-D4/def2-QZVPP*	5.17	5.60	8.77	7.08

SIE8 / def2-TZVPP

Statistics in kcal/mol

method	MD	MAD	RMSD	SD
r²SCANh-D4/def2-TZVPP	7.54	8.24	10.93	7.91
r²SCAN0-D4/def2-TZVPP	6.86	7.46	9.87	7.10
r²SCAN50-D4/def2-TZVPP	4.96	5.68	8.66	7.11

Geometry Benchmark Results

LMGB35

Statistics in pm.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	–0.12	0.68	1.01	1.02
r²SCANh-D4-ATM/def2-QZVP	–0.07	1.29	2.27	2.30
r²SCANh-D4-ATM/def2-QZVP*	–0.07	1.29	2.27	2.30
r²SCANh-D4-ATM/def2-TZVPP	0.01	1.25	2.27	2.30
r²SCAN0-D4-ATM/def2-QZVP	–0.59	1.59	2.33	2.28
r²SCAN0-D4-ATM/def2-QZVP*	–0.59	1.59	2.33	2.28
r²SCAN0-D4-ATM/def2-TZVPP	–0.59	1.59	2.33	2.28
r²SCAN50-D4-ATM/def2-QZVP	–1.36	2.09	2.78	2.45
r²SCAN50-D4-ATM/def2-QZVP*	–1.36	2.09	2.78	2.45
r²SCAN50-D4-ATM/def2-TZVPP	–1.36	2.09	2.78	2.45
PBE0-D4/def2-QZVP	–0.55	0.92	1.41	1.31
PW6B95-D4/def2-QZVP	–0.34	1.41	2.24	2.25
ωB97X-V/def2-QZVP	0.01	1.33	2.27	2.30

HMGB11

Statistics in pm.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	0.51	1.17	1.34	1.29
r²SCANh-D4-ATM/def2-QZVP	–0.11	1.01	1.16	1.21
r²SCANh-D4-ATM/def2-QZVP*	–0.11	1.01	1.16	1.21
r²SCANh-D4-ATM/def2-TZVPP	0.11	0.97	1.14	1.19
r²SCAN0-D4-ATM/def2-QZVP	–1.16	1.16	1.53	1.06
r²SCAN0-D4-ATM/def2-QZVP*	–1.15	1.15	1.53	1.06
r²SCAN0-D4-ATM/def2-TZVPP	–1.14	1.15	1.53	1.07
r²SCAN50-D4-ATM/def2-QZVP	–2.71	2.71	2.92	1.15
r²SCAN50-D4-ATM/def2-QZVP*	–2.73	2.73	2.94	1.15
r²SCAN50-D4-ATM/def2-TZVPP	–2.69	2.69	2.91	1.15
PBE0-D4/def2-QZVP	–0.49	1.07	1.21	1.16
PW6B95-D4/def2-QZVP	–0.09	1.09	1.22	1.27
ωB97X-V/def2-QZVP	–1.44	1.54	1.77	1.09

TMC32

Statistics in pm.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	–1.57	1.89	2.35	1.77
r²SCANh-D4-ATM/def2-QZVP	–1.90	2.18	2.61	1.80
r²SCANh-D4-ATM/def2-QZVP*	–1.89	2.17	2.60	1.80
r²SCANh-D4-ATM/def2-TZVPP	–1.94	2.22	2.65	1.82
r²SCAN0-D4-ATM/def2-QZVP	–2.43	2.71	3.11	1.96
r²SCAN0-D4-ATM/def2-QZVP*	–2.42	2.70	3.10	1.96
r²SCAN0-D4-ATM/def2-TZVPP	–2.42	2.70	3.10	1.96
r²SCAN50-D4-ATM/def2-QZVP	–2.91	3.37	3.97	2.73
r²SCAN50-D4-ATM/def2-QZVP*	–2.92	3.37	3.97	2.73
r²SCAN50-D4-ATM/def2-TZVPP	–2.90	3.36	3.97	2.73
PBE0-D4/def2-QZVP	–2.04	2.40	2.78	1.90
PW6B95-D4/def2-QZVP	–1.11	2.60	4.53	4.44
ωB97X-V/def2-QZVP	–1.56	2.09	2.56	2.05

ROT34

Statistics in MHz.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	–3.22	4.64	5.88	4.99
r²SCANh-D4-ATM/def2-QZVP	1.92	3.00	4.33	3.94
r²SCANh-D4-ATM/def2-QZVP*	1.39	2.81	3.98	3.78
r²SCANh-D4-ATM/def2-TZVPP	1.10	2.53	3.58	3.46
r²SCAN0-D4-ATM/def2-QZVP	10.46	10.46	12.76	7.40
r²SCAN0-D4-ATM/def2-QZVP*	10.10	10.10	12.45	7.39
r²SCAN0-D4-ATM/def2-TZVPP	10.28	10.28	12.63	7.46
r²SCAN50-D4-ATM/def2-QZVP	24.15	24.15	28.99	16.28
r²SCAN50-D4-ATM/def2-QZVP*	24.13	24.13	28.94	16.21
r²SCAN50-D4-ATM/def2-TZVPP	23.96	23.96	28.87	16.35
PBE0-D4/def2-QZVP	2.66	3.86	5.79	5.22
PW6B95-D4/def2-QZVP	11.71	11.71	14.92	9.38
ωB97X-V/def2-QZVP	–7.71	7.71	10.40	7.09

LB12

Statistics in pm.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	1.81	3.57	5.67	5.61
r²SCANh-D4-ATM/def2-QZVP	2.33	3.88	6.00	5.77
r²SCANh-D4-ATM/def2-QZVP*	2.68	4.11	6.19	5.82
r²SCANh-D4-ATM/def2-TZVPP	1.92	4.29	6.07	6.01
r²SCAN0-D4-ATM/def2-QZVP	0.67	3.16	4.89	5.05
r²SCAN0-D4-ATM/def2-QZVP*	0.75	3.22	5.01	5.15
r²SCAN0-D4-ATM/def2-TZVPP	0.90	3.22	5.02	5.15
r²SCAN50-D4-ATM/def2-QZVP	–2.24	5.48	9.03	9.14
r²SCAN50-D4-ATM/def2-QZVP*	–2.48	5.45	9.00	9.03
r²SCAN50-D4-ATM/def2-TZVPP	–2.05	5.98	9.08	9.24
PBE0-D4/def2-QZVP	–0.17	3.16	5.09	5.31
PW6B95-D4/def2-QZVP	–3.05	5.96	8.25	8.00
ωB97X-V/def2-QZVP	–0.67	6.33	10.56	11.01

CCse21

Bond lengths. Statistics in pm.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	0.03	0.38	0.56	0.57
r²SCANh-D4-ATM/def2-QZVP	–0.26	0.43	0.62	0.57
r²SCANh-D4-ATM/def2-QZVP*	–0.26	0.43	0.62	0.57
r²SCANh-D4-ATM/def2-TZVPP	–0.18	0.41	0.58	0.56
r²SCAN0-D4-ATM/def2-QZVP	–0.69	0.70	0.99	0.71
r²SCAN0-D4-ATM/def2-QZVP*	–0.69	0.70	0.98	0.71
r²SCAN0-D4-ATM/def2-TZVPP	–0.69	0.70	0.99	0.71
r²SCAN50-D4-ATM/def2-QZVP	–1.34	1.34	1.68	1.02
r²SCAN50-D4-ATM/def2-QZVP*	–1.34	1.34	1.68	1.02
r²SCAN50-D4-ATM/def2-TZVPP	–1.34	1.34	1.68	1.02
PBE0-D4/def2-QZVP	–0.23	0.53	0.75	0.72
PW6B95-D4/def2-QZVP	–0.60	0.60	0.81	0.56
ωB97X-V/def2-QZVP	0.01	0.51	0.73	0.74

Bond angles. Statistics in deg.

method	MD	MAD	RMSD	SD
r²SCAN-D4-ATM/def2-QZVP	0.01	0.29	0.37	0.37
r²SCANh-D4-ATM/def2-QZVP	0.00	0.24	0.28	0.29
r²SCANh-D4-ATM/def2-QZVP	0.00	0.24	0.28	0.29
r²SCANh-D4-ATM/def2-TZVPP	–0.01	0.24	0.29	0.30
r²SCAN0-D4-ATM/def2-QZVP	0.02	0.22	0.28	0.28
r²SCAN0-D4-ATM/def2-QZVP*	0.02	0.22	0.28	0.29
r²SCAN0-D4-ATM/def2-TZVPP	0.02	0.22	0.28	0.29
r²SCAN50-D4-ATM/def2-QZVP	0.07	0.28	0.38	0.38
r²SCAN50-D4-ATM/def2-QZVP*	0.07	0.28	0.38	0.38
r²SCAN50-D4-ATM/def2-TZVPP	0.07	0.28	0.38	0.38
PBE0-D4/def2-QZVP	0.03	0.31	0.38	0.38
PW6B95-D4/def2-QZVP	0.03	0.24	0.29	0.29
ωB97X-V/def2-QZVP	0.06	0.18	0.23	0.22

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haneug/r2scanx-D4

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r²SCANX-D4

Functional Implementation

Functional parameters

Benchmark Results

NCIBLIND10

S22x5

S66x8

GMTKN55 / def2-QZVP

GMTKN55 / def2-TZVPP

L7 / def2-QZVPP

L7 / def2-TZVPP

S30L / def2-QZVPP

S30L / def2-TZVPP

X40x10(2018) / def2-QZVPP

X40x10(2018) / def2-TZVPP

CHAL336 / def2-QZVPP

CHAL336 / def2-TZVPP

HB300SPX / def2-QZVPP

HB300SPX / def2-TZVPP

R160x6 / def2-QZVPP

R160x6 / def2-TZVPP

MOR41 / def2-QZVPP

MOR41 / def2-TZVPP

ROST61 / def2-QZVPP

ROST61 / def2-TZVPP

WCCR10 / def2-QZVPP

WCCR10 / def2-TZVPP

MOBH35 / def2-QZVPP

MOBH35 / def2-TZVPP

TMBH / def2-QZVPP

TMBH / def2-TZVPP

MLA24 / def2-QZVPP

MLA24 / def2-TZVPP

IONPI19 / def2-QZVPP

IONPI19 / def2-TZVPP

ACONF12 / def2-QZVPP

ACONF12 / def2-TZVPP

MPCONF196 / def2-QZVPP

MPCONF196 / def2-TZVPP

37CONF8 / def2-QZVPP

37CONF8 / def2-TZVPP

TMCONF16 / def2-QZVPP

TMCONF16 / def2-TZVPP

SIE8 / def2-QZVPP (remaining systems of old GMTKN SIE11 subset)

SIE8 / def2-TZVPP

Geometry Benchmark Results

LMGB35

HMGB11

TMC32

ROT34

LB12

CCse21

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