Self-consistent (SC) and range-separated (RS) functionals

The generalized Kohn-Sham potential ($GKS$) is given by the sum of the Hartree ($H$), exchange ($x$), correlation ($c$) and external ($ext$) potentials:

$\begin{equation} v_{GKS}(r,r') = v_{H}(r)+v_{x}(r,r')+v_{c}(r)+v_{ext}(r) \end{equation}$

where

$\begin{equation} v_{x}(r,r') = \alpha v_{x}^{lr-ex}(r,r';\mu)+\beta v_{x}^{sr-ex}(r,r';\mu)+(1-\alpha)v_{x}^{lr}(r;\mu)+(1-\beta)v_{x}^{sr}(r;\mu) \end{equation}$

and

$\begin{equation} v_{x}^{lr-ex}(r,r';\mu) = -\rho(r,r')\frac{erf(\mu |r-r'|)}{|r-r'|} \end{equation}$

and

$\begin{equation} v_{x}^{sr-ex}(r,r';\mu) = -\rho(r,r')\frac{erf(\mu |r-r'|)}{|r-r'|} \end{equation}$

$\rho$ is the density matrix and $\mu$ is chosen as:

$\begin{equation} \mu_{WS} = \frac{1}{r_s} = (\frac{4\pi n_v}{3})^\frac{1}{3} \end{equation}$

$\begin{equation} \mu_{TF} = \frac{1}{2}k_{TF} = (\frac{3 n_v}{\pi})^\frac{1}{6} \end{equation}$

Here $n_v$ is the number of valence electrons; we also considered $\mu$ = $\mu_{PDEP}$ = long range decay of the diagonal elements of the dielectric matrix.

Note that:

Functional	α	β	μ (bohr^-1)
PBE	0	0	0
EXXc	1	0	∞
PBE0	0.25	0	∞
SC Hybrid	1/ϵ_∞	0	∞
HSE06	0	0.25	0.11
sX-LDA	0	1	Thomas-Fermi
CAM-B3LYP	0.46	0.19	0.33
LC-μPBE	1	0	0.4

TABLE I. The dielectric constants (ϵ_∞) determined self-consistently as described in Skone et al. 2014 [Ref.] for the set of semiconductors and insulators listed in the first column is given in column 2. The screening parameters (μ) used in the RSH functional form proposed in Skone et al. 2016 [Ref.] are listed in units of bohr^-1 in columns 3-6.

				μ_PDEP
	ϵ_∞	μ_WS	μ_TF	μ_erfc-fit	μ_TF-fit
Si	11.76	0.50	0.55	0.64	0.64
AlP	7.23	0.50	0.55	0.65	0.64
SiC	6.50	0.62	0.62	0.77	0.77
TiO₂	6.56	0.68	0.65
NiO	5.49	0.82	0.71
C	5.61	0.76	0.68	0.93	0.97
CoO	4.92	0.78	0.69
GaN₂	5.14	0.60	0.61
ZnS	4.95	0.65	0.63
MnO	4.45	0.72	0.66
WO₃	4.72	0.66	0.63
BN	4.40	0.75	0.68	0.91	0.95
HfO₂	3.97	0.66	0.63
AlN	4.16	0.49	0.55
ZnO	3.46	0.78	0.69
Al₂O₃	3.01	0.71	0.66
MgO	2.81	0.64	0.63
LiCl	2.77	0.53	0.57
NaCl	2.29	0.49	0.54	0.63	0.64
LiF	1.86	0.68	0.64	0.80	0.83
H₂O₂	1.68	0.55	0.58	0.52	0.53
Ar	1.66	0.52	0.56	0.72	0.73
Ne	1.21	0.61	0.61	0.83	0.89

TABLE II. The Kohn-Sham (KS) energy gaps (eV) evaluated with hybrid functionals are compared with the experimental (Exp.) electronic gaps for a wide range of semiconductors ansd insulators. The experimental values correspond to either photoemission measurements or to optical measurements where the excitonic contributions were removed, with alumina being the only exception. The KS gaps were computed as the single-particule energy difference of the conduction band minimum and the valence band maximum. The sc-hybrid heading refers to hybrid calculations where the fraction of exact exchange is the self-consistent ϵ_∞ (see Skone et al. 2014 [Ref.]). The RSH columns correspond to the electronic gap evaluated with the range separation scheme described in Skone et al. 2016 [Ref.] from Shishkin et al. [Ref.]. ME, MAE, MRE, and MARE are the mean, mean absolute, mean relative, and mean absolute relative error, respectively. The experimental geometry was used in all calculations. Note that CoO, NiO, and MnO are magnetic with AFM-II magnetic ordering. The structure/polytype used for each system is the same as Table I of Skone et al. 2014 [Ref.].

	PBE0	sc-hybrid	RSH μ_WS	RSH μ_TF	RSH μ_erfc-fit	scGW [62]	Exp.
Si	1.75	0.99	1.03	1.02	1.01	1.24	1.17 [see C. Kittel, Introduction to Solid State Physics (Wiley, NewYork, 2005)]
AlP	2.98	2.37	2.43	2.42	2.40	2.57	2.51 [Ref.]
SiC	2.91	2.29	2.32	3.32	2.31	2.53	2.39 [Ref.]
TiO₂	3.05	3.16	0.65	3.17			3.3 [Ref.]
NiO	5.28	4.11	4.45				4.3 [Ref.]
C	5.95	5.42	5.44	5.45	5.43	5.79	5.48 [Ref.]^a
CoO	4.53	3.62	3.92	3.98			2.5 [Ref.]
GaN₂	3.26	3.30	0.61	3.30		3.27	3.29 [Ref.]
ZnS	4.18	3.82	3.85	3.86		3.60	3.91 [see C. Kittel, Introduction to Solid State Physics (Wiley, NewYork, 2005)]
MnO	3.87	3.60	3.65	3.49			3.9 [Ref.]
WO₃	3.76	3.47	3.49	3.49			3.38 [Ref.]
BN	6.51	6.33	6.33	6.33	6.33	6.59	6.4 [see M. Levinshtein et al., Properties of Advanced Semiconductor Materials: GaN, AlN, InN, BN, SiC, and SiGe (Wiley, New York, 2001)]
HfO₂	6.65	6.68	6.67	6.67			5.84 [Ref.]
AlN	6.31	6.23	6.22	6.23			6.28 [Ref.]
ZnO	3.41	3.78	3.75			3.2	3.44 [Ref.]
Al₂O₃	8.84	9.71	9.63	9.61			8.8 [Ref.]
MgO	7.25	8.33	8.23	8.22	8.27	8.12	7.83 [Ref.]
LiCl	8.66	9.62	9.52	9.54			9.4 [Ref.]
NaCl	7.26	8.84	8.60	8.66	8.71		8.6 [Ref.]
LiF	12.18	15.69	15.24	15.18	15.42	14.5	14.2 [Ref.]
H₂O₂	7.92	11.49	10.89	10.94	10.84		10.9 [Ref.]
Ar	11.20	14.67	14.12	14.20	14.41	13.9	14.2 [Ref.]
Ne	15.20	23.67	21.44	21.44	22.28	21.4	21.7 [Ref.]
ME (eV)	-0.40	0.32	0.18	0.19
MAE (eV)	1.08	0.42	0.29	0.30
MRE (%)	6.2	3.7	3.4	3.5
MARE (%)	17.1	7.5	6.4	6.5

^aThe exp. QP gap reported here does not account for the zero-point vibrational gap renormalization, which has been shown to be non-negligible for diamond (see Cardona [Ref.], Giustino et al. [Ref.], and van Elp et al. [Ref.].

TABLE III. The dielectric contant (ϵ_∞) determined self-consistently as described in Skone et al. 2014 [Ref.] for the set of molecular crystals listed in the first column is given in column 2. The screening parameters (μ) usd in the RSH functional form (see Skone et al. 2016 [Ref.]) of Eq. (17) are listed in units of bohr^-1 in columns 3 and 4. All μ have units of bohr^-1.

		Crystalline
	ϵ_∞	μ_TF	μ_erfc-fit
C₁₄H₈S₄-C₁₂H₄N₄ (DBTTF-TCNQ)	11.07	0.58
C₆₀ (buckminsterfullerene)	4.29	0.61	0.57
C₃₂H₁₈N₈ (phtalocyanine)	3.97	0.60
C₂₄H₈O₆ (α-PTCDA)	3.45	0.61
C₂₂H₁₄ (pentacene)	3.36	0.59
C₂₀H₁₂O₂ (β-quinacridone)	3.15	0.60
C₁₈H₁₂ (tetracene)	3.15	0.59
C₁₄H₁₀ (anthtracene)	3.02	0.58
C₄₂H₂₈ (rubrene)	2.88	0.58	0.50
C₁₀H₈ (naphtalene)	2.70	0.58
C₆H₆ (benzene)	2.40	0.57	0.54
NH₃ (ammonia)	2.00	0.57	0.53
C₂H₄O₂ (acetic acid)	1.88	0.60
H₂O (ice)	1.68	0.58	0.52

TABLE IV. The xx, yy, zz components of the dielectric tensor (ϵ_∞) of the systems listed on the first row, computed at the PBE, PBE0, and sc-hybrid levels of theory, and the corresponding experimental results.

	Anthracene			PTCDA			DBTTF-TCNQ
	ϵ_xx	ϵ_yy	ϵ_zz	ϵ_xx	ϵ_yy	ϵ_zz	ϵ_xx	ϵ_yy	ϵ_zz
PBE	2.28	2.93	4.30	2.25	4.40	4.50	42.38	3.67	2.90
PBE0	2.22	2.83	4.10	2.12	4.10	4.12	11.08	5.82	2.72
sc-hybrid	2.21	2.80	4.02	2.11	4.06	4.08	26.28	3.54	2.81
Exp. [Ref., Ref.]	2.42 ± 0.05	2.90 ± 0.05	4.07 ± 0.05	2.40	5.29	5.02
Exp. [Ref., Ref.]	2.62 ± 0.03	2.94 ± 0.03	4.08 ± 0.03	1.90		4.09
Exp. [Ref., Ref.]	2.51	2.99	4.11	2.28		3.73

TABLE V. The Kohn-Sham (KS) energy gaps (eV) evaluated with the dielectric-dependent hybrid functionals PBE and PBE0 are compared with the experimental electronic gaps for several molecular crystals. The experimental values are from photoemission measurements.

	PBE α=0	PBE0 α=0.25	Hybrid α=1/ϵ^PBE_∞	Hybrid α=1/ϵ^PBE0_∞	Hybrid α=1/ϵ^sc_∞	RSH μ_TF	Exp.
DBTTF-TCNQ	0.16	0.74	0.26	0.45	0.31	0.33
C60	1.27	2.34	2.11	2.27	2.26	2.26	2.3 ± 0.1 [Ref.]
phtalocyanine (H₂Pc)	1.22	1.85	1.84	1.85	1.85	1.85	2.2 ± 0.2 [Ref.]
PTCDA	1.41	2.53	2.62	2.72	2.73	2.73	2.74 ± 0.2 [Ref.]
pentacene	0.76	1.83	1.95	2.04	2.05	2.05	2.1 [Ref., Ref.]
quinacridone	1.43	2.76	3.02	3.13	3.13	3.11
tetracene	1.26	2.46	2.72	2.78	2.80	2.79	3.3 [Ref., Ref.]
anthracene	2.05	3.45	3.82	3.89	3.91	3.89	3.72 [Ref.]
rubrene	1.15	2.32	2.70	2.77	2.79	2.77	2.67 [Ref.]
naphtalene	3.05	4.64	5.32	5.39	5.41	5.37	5.29 [Ref., Ref.]
benzene	4.57	6.37	7.48	7.56	7.58	7.47	7.58 [Ref., Ref.]
ammonia	4.52	6.82	8.76	9.00	9.17	8.92
acetic acid	5.18	7.96	10.76	10.95	11.12	10.62
H₂O (ice^a)	5.42	7.92	10.96	11.23	11.49	10.94	10.9 [Ref.]
ME (eV)	-2.05	-0.69	-0.12	-0.02	0.02	-0.06
MAE (eV)	2.05	0.70	0.19	0.18	0.21	0.16
MRE (%)	-49.6	-13.5	-4.9	-2.3	-1.7	-2.6
MARE (%)	49.6	13.8	6.1	5.1	5.4	4.9

^aSee the Supplemental Material of Skone et al. 2016 [Ref.] for details on the cell of ice used. The experimental photoemission gap shown is for proton-disordered ice Ih at 80 K.

TABLE VI. The vertical ionization potential (vIP_s), in units of eV, of several solid molecular crystals evaluated with PBE, PBE0, sc-hybrid ans RSH functionals. The experimental values are listed for comparison. Note that for rubrene the surface listed corresponds to the orthorhombic cell. The screening parameters μ_TF and μ_αM are defined in (see Skone et al. 2016 [Ref.]) Eq. (9) and Eq. (13), respectively.

	surface	PBE	PBE0	sc-hybrid	RSH μ_TF	Exp.
rubrene	(100)	3.85	4.45	4.69	4.68	4.85 [Ref.]
benzene	(001)	6.08	7.02	7.63	7.61	7.58 [Ref.]
ice^a	(1010)	7.2	8.7	11.2	10.7	11.8 [Ref., and D. Nordlund, H. Ogasawara, and A. Nilsson, Maxlab Annual Report, p. 236, 2001]

^aThe prism surface of ice is used. See Pan et al. [Ref.] for further details on the common surfaces of ice.

TABLE VII. Screening parameters for isolated molecules. The second and third columns list the screening paramaters obtained from the fit of the RPA dielectric function of the isolated molecules and obtained from the molecular polarizability radius [see Eq. (13) of Skone et al. 2016 [Ref.]]. We also give in column 4 the screening parameters obtained from the OT-RSH functional defined in Refaely-Abramson et al. [Ref.].

	μ_erfc-fit	μ_αM	μ_OT-RSH
C₆₀ (buckminsterfullerene)	0.64	0.12	0.14
C₂₄H₈O₆ (α-PTCDA)	0.62	0.14	0.14
C₂₂H₁₄ (pentacene)	0.56	0.14	0.15
C₂₀H₁₂O₂ (β-quinacridone)	0.59	0.15	0.15
C₁₈H₁₂ (tetracene)	0.58	0.16	0.16
C₁₄H₁₀ (anthtracene)	0.59	0.18	0.18
C₄₂H₂₈ (rubrene)	0.61	0.12	0.11
C₁₀H₈ (naphtalene)	0.61	0.21	0.21
C₆H₆ (benzene)	0.63	0.24	0.21
NH₃ (ammonia)	0.63	0.40	0.33
C₂H₄O₂ (acetic acid)	0.69	0.31	0.27
H₂O (water)	0.67	0.46	0.38

TABLE VIII. The gas phase vertical ionization potential (vIP_s), in units of eV, of several molecules that compose the molecular crystals evaluated using the RSH-DDH with μ = μ_erfc-fit and μ = μ_αM, are shown in column 2 and column 3, respectively. Also shown are values determined by using the OT-RSH functional defined in Refaely-Abramson et al. [Ref.], column 4. The experimental values (Exp.), taken from the NIST WebBook [see Linstrom et al., NIST Chemistry WebBook, NIST Standard Reference Database No. 69 (National Institute of Standards and Technology), Gaithersburg, MD, 2001], are listed in column 5.

	vIP_g (eV)
	RSH μ_erfc-fit	RSH μ_αM	RSH μ_OT-RSH	Exp.
C60	8.74	7.40	7.76	7.60
PTCDA	9.23	8.20	8.20	8.20
pentacene	7.08	6.29	6.29	6.61
quinacridone	8.44	7.35	7.35	7.23
tetracene	7.51	6.72	6.72	6.97
anthracene	8.14	7.30	7.30	7.44
rubrene	7.28	6.28	6.16	6.52
naphtalene	8.87	8.07	8.07	8.14
benzene	10.15	9.37	9.24	9.25
ammonia	11.85	11.07	10.7	10.8
acetic acid	12.61	11.08	10.78	10.9
H₂O	13.78	13.01	12.55	12.62
ME (eV)	0.95	-0.01	-0.10
MAE (eV)	0.95	0.19	0.14
MRE (%)	11.1	-0.6	-1.3
MARE (%)	11.1	2.3	1.9

Method	MAE [eV]
G₀W₀@PBE0^‡	0.19
SX	0.28
G₀W₀@SX	0.38
G₀W₀@PBE^‡	0.44
EXXc^‡	1.50
B3LYP^‡	2.78
PBE₀^‡	2.87
HSE06^‡	3.27
PBE^‡ (ΔSCF^†)	4.29 (0.24)

"Dielectric-dependent hybrid functionals for heterogeneous materials", H. Zheng, M. Govoni, and G. Galli, Phys. Rev. Mat. 3, 073803 (2019)
"Performance and self-consistency of the generalized dielectric dependent hybrid functional", N. P. Brawand, M. Govoni, Márton Vörös and G. Galli, J. Chem. Theory. Comp. 13, 3318 (2017)
"Nonempirical range-separated hybrid functionals for solids and molecules", J. Skone, M. Govoni, and G. Galli, Phys. Rev. B 93, 235106 (2016)
"Photoelectron spectra of aqueous solutions from first principles", A. P. Gaiduk, M. Govoni, R. Seidel, J. Skone, B. Winter, and G. Galli, J. Am. Chem Soc. Comm. 138, 6912 (2016)
"Self-consistent hybrid functional for condensed systems", J. H. Skone, M. Govoni, and G. Galli, Phys. Rev. B 89, 195112 (2014)