# Dielectric dependent hybrid functionals

Method developments include deriving and benchmarking hybrid density functionals to compute ground and excited state electronic properties of both condensed and finite systems.

We derived new classes of hybrid functionals with nonempirical parameters that depend on the dielectric screening of the system. These hybrid functionals are referred to as dielectric-dependent hybrid (DDH) functionals.

### Screened exchange (SXC) hybrid functional

Can a unique dielectric constant be defined for molecules or nanocrystals? Dielectric dependent global hybrid functionals (DDH) have remained limited to bulk systems due to this unsolved conceptual problem. We generalized DDH functionals to finite systems by defining an orbital dependent screened exchange (SXC) parameter as a mixing fraction of exact and local exchange. The proposed SXC functional yields accurate fundamental and optical gaps of many, diverse molecular systems in excellent agreement with experiment, including organic and inorganic molecules and semiconducting nanocrystals.

• "Generalization of dielectric dependent hybrid functionals to finite system", N. Brawand, M. Vörös, M. Govoni, and Giulia Galli, Phys. Rev. X 6, 041002 (2016)

### Self-consistent global and range-separated hybrid functionals

We proposed a self-consistent (SC) global hybrid functional and a range separated (RS) one. In the definition of the former, applicable to solids, the dielectric constant of the material (ϵ), computed self-consistently from first principles, is used to determine the global fraction of exact exchange. In the RS version of DDH functionals , applicable to both molecules and solids, the short- and long-range components are matched using system dependent, nonempirical parameters. Both functionals yield accurate electronic properties of inorganic and organic solids, including energy gaps and absolute ionization potentials. The evaluation of SC and RS functionals is computationally less expensive to than that of GW self-energies and they can be efficiently utilized to study trends in various properties of solids and molecules; in addition they provide an excellent starting point for G0W0 calculations of specific systems, for example in the case of aqueous solutions.

#### Definition of SC and 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:

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

where

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

and

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

and

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

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

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

or

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

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)
PBE000
EXXc10
PBE00.250
SC Hybrid1/ϵ0
HSE0600.250.11
sX-LDA01Thomas-Fermi
CAM-B3LYP0.460.190.33
LC-μPBE100.4

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.

μPDEP
ϵ μWS μTF μerfc-fit μTF-fit
Si11.760.500.550.640.64
AlP7.230.500.550.650.64
SiC6.500.620.620.770.77
TiO26.560.680.65
NiO5.490.820.71
C5.610.760.680.930.97
CoO4.920.780.69
GaN25.140.600.61
ZnS4.950.650.63
MnO4.450.720.66
WO34.720.660.63
BN4.400.750.680.910.95
HfO23.970.660.63
AlN4.160.490.55
ZnO3.460.780.69
Al2O33.010.710.66
MgO2.810.640.63
LiCl2.770.530.57
NaCl2.290.490.540.630.64
LiF1.860.680.640.800.83
H2O21.680.550.580.520.53
Ar1.660.520.560.720.73
Ne1.210.610.610.830.89

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
μWS
RSH
μTF
RSH
μerfc-fit
scGW [62] Exp.
Si1.750.991.031.021.011.241.17 [see C. Kittel, Introduction to Solid State Physics (Wiley, NewYork, 2005)]
AlP2.982.372.432.422.402.572.51 [Ref.]
SiC2.912.292.323.322.312.532.39 [Ref.]
TiO23.053.160.653.173.3 [Ref.]
NiO5.284.114.454.3 [Ref.]
C5.955.425.445.455.435.795.48 [Ref.]a
CoO4.533.623.923.982.5 [Ref.]
GaN23.263.300.613.303.273.29 [Ref.]
ZnS4.183.823.853.863.603.91 [see C. Kittel, Introduction to Solid State Physics (Wiley, NewYork, 2005)]
MnO3.873.603.653.493.9 [Ref.]
WO33.763.473.493.493.38 [Ref.]
BN6.516.336.336.336.336.596.4 [see M. Levinshtein et al., Properties of Advanced Semiconductor Materials: GaN, AlN, InN, BN, SiC, and SiGe (Wiley, New York, 2001)]
HfO26.656.686.676.675.84 [Ref.]
AlN6.316.236.226.236.28 [Ref.]
ZnO3.413.783.753.23.44 [Ref.]
Al2O38.849.719.639.618.8 [Ref.]
MgO7.258.338.238.228.278.127.83 [Ref.]
LiCl8.669.629.529.549.4 [Ref.]
NaCl7.268.848.608.668.718.6 [Ref.]
LiF12.1815.6915.2415.1815.4214.514.2 [Ref.]
H2O27.9211.4910.8910.9410.8410.9 [Ref.]
Ar11.2014.6714.1214.2014.4113.914.2 [Ref.]
Ne15.2023.6721.4421.4422.2821.421.7 [Ref.]
ME (eV)-0.400.320.180.19
MAE (eV)1.080.420.290.30
MRE (%)6.23.73.43.5
MARE (%)17.17.56.46.5

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 μTF μerfc-fit
C14H8S4-C12H4N4 (DBTTF-TCNQ)11.070.58
C60 (buckminsterfullerene)4.290.610.57
C32H18N8 (phtalocyanine)3.970.60
C24H8O6 (α-PTCDA)3.450.61
C22H14 (pentacene)3.360.59
C20H12O2 (β-quinacridone)3.150.60
C18H12 (tetracene)3.150.59
C14H10 (anthtracene)3.020.58
C42H28 (rubrene)2.880.580.50
C10H8 (naphtalene)2.700.58
C6H6 (benzene)2.400.570.54
NH3 (ammonia)2.000.570.53
C2H4O2 (acetic acid)1.880.60
H2O (ice)1.680.580.52

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
ϵxx ϵyy ϵzz ϵxx ϵyy ϵzz ϵxx ϵyy ϵzz
PBE2.282.934.302.254.404.5042.383.672.90
PBE02.222.834.102.124.104.1211.085.822.72
sc-hybrid2.212.804.022.114.064.0826.283.542.81
Exp. [Ref., Ref.]2.42 ± 0.052.90 ± 0.054.07 ± 0.052.405.295.02
Exp. [Ref., Ref.]2.62 ± 0.032.94 ± 0.034.08 ± 0.031.904.09
Exp. [Ref., Ref.]2.512.994.112.283.73

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/ϵPBE
Hybrid
α=1/ϵPBE0
Hybrid
α=1/ϵsc
RSH
μTF
Exp.
DBTTF-TCNQ0.160.740.260.450.310.33
C601.272.342.112.272.262.262.3 ± 0.1 [Ref.]
phtalocyanine (H2Pc)1.221.851.841.851.851.852.2 ± 0.2 [Ref.]
PTCDA1.412.532.622.722.732.732.74 ± 0.2 [Ref.]
pentacene0.761.831.952.042.052.052.1 [Ref., Ref.]
quinacridone1.432.763.023.133.133.11
tetracene1.262.462.722.782.802.793.3 [Ref., Ref.]
anthracene2.053.453.823.893.913.893.72 [Ref.]
rubrene1.152.322.702.772.792.772.67 [Ref.]
naphtalene3.054.645.325.395.415.375.29 [Ref., Ref.]
benzene4.576.377.487.567.587.477.58 [Ref., Ref.]
ammonia4.526.828.769.009.178.92
acetic acid5.187.9610.7610.9511.1210.62
H2O (icea)5.427.9210.9611.2311.4910.9410.9 [Ref.]
ME (eV)-2.05-0.69-0.12-0.020.02-0.06
MAE (eV)2.050.700.190.180.210.16
MRE (%)-49.6-13.5-4.9-2.3-1.7-2.6
MARE (%)49.613.86.15.15.44.9

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 (vIPs), 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 μTF and μαM are defined in (see Skone et al. 2016 [Ref.]) Eq. (9) and Eq. (13), respectively.

surface PBE PBE0 sc-hybrid RSH
μTF
Exp.
rubrene(100)3.854.454.694.684.85 [Ref.]
benzene(001)6.087.027.637.617.58 [Ref.]
icea(1010)7.28.711.210.711.8 [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.].

μerfc-fit μαM μOT-RSH
C60 (buckminsterfullerene)0.640.120.14
C24H8O6 (α-PTCDA)0.620.140.14
C22H14 (pentacene)0.560.140.15
C20H12O2 (β-quinacridone)0.590.150.15
C18H12 (tetracene)0.580.160.16
C14H10 (anthtracene)0.590.180.18
C42H28 (rubrene)0.610.120.11
C10H8 (naphtalene)0.610.210.21
C6H6 (benzene)0.630.240.21
NH3 (ammonia)0.630.400.33
C2H4O2 (acetic acid)0.690.310.27
H2O (water)0.670.460.38

TABLE VIII. The gas phase vertical ionization potential (vIPs), in units of eV, of several molecules that compose the molecular crystals evaluated using the RSH-DDH with μ = μerfc-fit and μ = μαM, 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.

vIPg (eV)
RSH
μerfc-fit
RSH
μαM
RSH
μOT-RSH
Exp.
C608.747.407.767.60
PTCDA9.238.208.208.20
pentacene7.086.296.296.61
quinacridone8.447.357.357.23
tetracene7.516.726.726.97
anthracene8.147.307.307.44
rubrene7.286.286.166.52
naphtalene8.878.078.078.14
benzene10.159.379.249.25
ammonia11.8511.0710.710.8
acetic acid12.6111.0810.7810.9
H2O13.7813.0112.5512.62
ME (eV)0.95-0.01-0.10
MAE (eV)0.950.190.14
MRE (%)11.1-0.6-1.3
MARE (%)11.12.31.9
Method MAE [eV]
G0W0@PBE00.19
SX0.28
G0W0@SX0.38
G0W0@PBE0.44
EXXc1.50
B3LYP2.78
PBE02.87
HSE063.27
PBE (ΔSCF)4.29 (0.24)