r²SCAN-Based Double-Hybrids

Functional Implementation

• Pr²SCAN 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

• $\kappa$ Pr²SCAN functionals employ the regularized MP2 implemented in a development version of ORCA 5.0

• $\omega$ Pr²SCAN functionals utilize the error function splitting of the two-electron operator implemented in a development version of ORCA 5.0

• Libraries implementing DFT-D4 can be found at https://github.com/dftd4/dftd4 (Fortran) and https://github.com/dftd4/cpp-d4 (C++)

• NL dispersion correction DFT-NL (VV10, by Vydrov and Van Voorhis) can be used in Orca and Turbomole. For program-suits that do not support the usage of a scaling factor of the NL energy, one can scale the obtained NL energy by $NLScal$ manually.

ORCA Input Files

Double-hybrids:

• r²SCAN0-DH-D4

• r²SCAN-CIDH-D4

• r²SCAN-QIDH-D4

• r²SCAN0-2-D4

• Pr²SCAN50-D4 / NL

• Pr²SCAN69-D4 / NL

• kPr²SCAN50-D4 / NL

• wPr²SCAN50-D4 / NL

Range-separated hybrid:

• wr²SCAN-D4 / NL

Functional Parameters

Functional

All functionals employ the following fundamental relation.

$E_{XC}^{DH} = (1-a_{X})E_{X}^{DFT} + a_{X}E_{X}^{HF} + (1-a_{C})E_{C}^{DFT} + a_{C}(a_{SS}E_{C}^{SS-MP2}+a_{OS}E_{C}^{OS-MP2})$

Functional	$a_X$	$a_C$	$a_{OS}$	$a_{SS}$	$\kappa$	$\omega$
r²SCAN0-DH	$1/2$ (0.50)	$1/8$ (0.13)	$4/3$	0
r²SCAN-CIDH	$6^{-1/3}$ (0.55)	$1/6$ (0.17)	$4/3$	0
r²SCAN-QIDH	$3^{-1/3}$ (0.69)	$1/3$ (0.33)	$4/3$	0
r²SCAN0-2	$2^{-1/3}$ (0.79)	$1/2$ (0.50)	$4/3$	0
Pr²SCAN50	$1/2$ (0.50)	$1/4$ (0.25)	$4/3$	0
Pr²SCAN69	$3^{-1/3}$ (0.69)	$4/9$ (0.44)	$4/3$	0
$\kappa$ Pr²SCAN50	$1/2$ (0.50)	$3/10$ (0.30)	$4/3$	0	2.7500
$\omega$ Pr²SCAN50	$1/2$ (0.50)	$7/20$ (0.35)	$4/3$	0		0.2140

Dispersion Correction

Dispersion correction parameters were fitted against the S22x5, NCIBLIND10, and S66x8 (2022 revision by Martin et al.) benchmark sets for non-covalent interaction energies by least-squares Levenberg-Marquardt minimization. All parameters were fitted close to the complete basis set limit using the large def2-QZVPP quadruple-zeta basis set.

D4

Functional	$s_6$	$s_8$	$s_9$	$a_1$	$a_2$
r²SCAN0-DH	0.9424	0.3856	1.0000	0.4271	5.8565
r²SCAN-CIDH	0.8666	0.5336	1.0000	0.4171	5.9125
r²SCAN-QIDH	0.7867	0.2955	1.0000	0.4001	5.8300
r²SCAN0-2	0.7386	0.0000	1.0000	0.4030	5.5142
Pr²SCAN50	0.7964	0.3421	1.0000	0.4663	5.7916
Pr²SCAN69	0.7167	0.0000	1.0000	0.4644	5.2563
$\kappa$ Pr²SCAN50	0.8402	0.1212	1.0000	0.4382	5.8232
$\omega$ Pr²SCAN50	0.8143	0.3842	1.0000	0.4135	5.8773

This Markdown table includes the provided data with appropriate formatting for GitHub. Adjustments

NL

Non-local dispersion correction makes use of a scaling factor $a_{NL}$, that is given by $a_{NL}=1-a_C$.

Method	$a_{NL}$	$NL_{b}$
Pr²SCAN50	0.7500	10.9207
Pr²SCAN69	0.5556	9.0691
$\kappa$ Pr²SCAN50	0.7000	10.6723
$\omega$ Pr²SCAN50	0.6500	9.4149