HLE17: An Improved Local ExchangeâCorrelation Functional for

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HLE17: An Improved Local Exchange–Correlation Functional for Computing Semiconductor Band Gaps and Molecular Excitation Energies Pragya Verma, and Donald G. Truhlar J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b01066 • Publication Date (Web): 09 Mar 2017 Downloaded from http://pubs.acs.org on March 17, 2017

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The Journal of Physical Chemistry

revised for JPC C, 3/9/2017

HLE17: An Improved Local Exchange–Correlation Functional for Computing Semiconductor Band Gaps and Molecular Excitation Energies Pragya Verma* and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0431

ABSTRACT: The local approximations to exchange–correlation functionals that are widely used in Kohn-Sham density functional theory usually underestimate band gaps and molecular excitation energies, and therefore, it becomes necessary to use more expensive hybrid functionals or more empirical DFT+U functionals for accurate predictions and modeling of these properties. This work presents a meta-generalized gradient approximation (meta-GGA) called High Local Exchange 2017 (HLE17), and illustrates how it can be useful for obtaining accurate semiconductor band gaps and molecular excitation energies. Unlike the conventional way of using the DFT+U method, one does not need to determine new parameters for every property or system studied. The HLE17 functional builds upon our earlier work (HLE16) where we had shown that by increasing the coefficient of local exchange and simultaneously decreasing the coefficient of local correlation with a GGA, the two properties could be significantly improved without significantly degrading the ground-state molecular energetic properties. However, for almost every database tested in this work, HLE17 shows improvement over HLE16, and the improvement is particularly notable for solid-state lattice constants. In addition, this provides a strategy for calculating properties that are otherwise difficult to calculate by a local functional. Keywords: band gap, density functional theory, excitation energy, Hubbard U correction, orbital energy.

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1. Introduction It has long been accepted by practitioners of Kohn-Sham density functional theory (KS-DFT) that the accurate determination of band gaps and molecular excitation energies requires one to include nonlocal elements such as those brought in by the nonlocal self-energy of GW theory or by nonlocal exchange in hybrid density functionals.1,2,3,4,5,6,7,8,9,10,11,12,13,14 Such treatments can considerably raise the cost for extended systems. Alternatively one can introduce empirical Coulomb and exchange integrals for selected subshells, as in the DFT+U method,15,16,17 but in the conventional way of applying this method, a parameter U (or two parameters U and J) need to be redetermined for every property and every system studied. These approaches are widely used, despite the higher cost and complexity of nonlocal methods and despite the need for reparametrization of DFT+U for each new application, because conventional local functionals often have failings that make them useless for specific applications, e.g., they may predict semiconductors to be metals, predict large-gap semiconductors to be small-gap ones, or have very large errors in predictions of molecular excitation spectra. In a recent letter18 we presented a counterexample to the accepted thinking by showing how to calibrate a local density functional (in a system-independent, property-independent way) to obtain much more accurate values for band gaps and molecular excitation energies, competitive with those obtained via hybrid functionals. A key element in that study is that we showed, by examining its performance for diverse ground-state databases, that the resulting exchangecorrelation functional (called HLE16 to emphasize that it has high local exchange) was not unphysical since for several of these databases (main-group atomization energies, transition metal bond energies, diverse barrier heights, and solid-state cohesive energies) it gave more accurate results than one or more of the popular local functionals, BLYP,19,20 PBE,21 and PBEsol.22 That work built on a generalized gradient approximation (GGA) as a starting point; here we show that we can obtain similarly improved band gaps and molecular excitation energies by building on a meta-GGA functional. In particular, we start with the TPSS functional23 because it is relatively simple and is widely available in many DFT packages; thus the resulting recalibrated functional (HLE17) is very portable among computer programs, and in many cases it will allow calculations of realistic band gaps and molecular excitation energies with much greater affordability than was possible with previous functionals. A key result to be shown is that HLE17 shows improvement over HLE16 for almost every database examined.

2. Databases The databases used for validating the performance of the new functional are shown in Table 1;24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41 these databases represent a diverse set of data, and include


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ground-state and excited-state properties as well as molecular and solid-state properties. Table 1 also lists the references for the experimental (or reference) values of each database used in this work. For databases TMOBG4, AE6, TMABE10, DBH24, DBH18, PA3, EA13, IP21, NCCE31, EE23, and EEA11, single-point calculations were performed, and for the rest of the databases geometries were consistently optimized with the functional being tested. References for the geometries that were used for single-point calculations are in Table S1 of the Supporting Information (SI). Table 1. The Databases Used in This Work. Databases Description Reference(s) Solid-state properties SBG31 Semiconductor Band Gaps 24, 25 SSCE8 Solid-State Cohesive Energies 26, 27 TMOBG4 Transition-Metal Oxide Band Gaps 28 MGLC4 Main Group Lattice Constants 26, 27 ILC5 Ionic Lattice Constants 26, 27 TMLC4 Transition Metal Lattice Constants 26, 27 SLC34 Semiconductor Lattice Constants 24, 25 Molecular properties AE6 Atomization Energies (of main-group molecules) 29 TMABE10 Transition-Metal Average Bond Energies 30, 31 DBH24 Diverse Barrier Heights 32 a Diverse Barrier Heights of Neutrals 28 DBH18 PA3 Proton Affinities 33 EA13 Electron Affinities 31 IP21 Ionization Potentials 42 NCCE31 Noncovalent Complexation Energies 34, 35 EE23 Excitation Energies (of molecules) 36, 37 EEA11 Excitation Energies of Atoms 39, 41 DM20 Dipole Moments 40 MGHBL9 Main-Group Hydrogenic Bond Lengths 27 MGNHBL11 Main-Group Non-Hydrogenic Bond Lengths 27, 38 DGH4 Bond lengths for diatomic molecules (geometries) with 31 one or more heavy atoms a DBH24 has 24 barrier heights, 18 of which involve only neutral species, and 6 of which involve ions; DBH18 is the subset of 18 barrier heights for the neutral reactions.

3. Computational Details All databases were calculated using Gaussian 0943 suite of quantum mechanical programs or a locally modified version44 of Gaussian 09, and some were also calculated using VASP.5.3.5.45,46 Both Gaussian and VASP were used for performing both molecular and solid-state calculations.



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All VASP calculations used periodic boundary conditions; the Gaussian 09 calculations used Gaussian basis sets for molecules and periodic boundary condition methods47 for calculations on solids. As usual, band gaps for solids were computed as the difference between the lowest unoccupied crystal orbital and the highest occupied crystal orbital,24,48 and excitation energies of molecules were computed by linear-response time-dependent DFT49 (LR-TDDFT) starting from closed-shell ground state. For molecular triplet states, we also calculated the excitation energy by the ΔSCF method,50,51 in which one performs separate SCF calculations on the singlet and the triplet. Gaussian 09 calculations. Because this work aims at improving a local density functional for properties such as band gaps and excitation energies, we chose for comparison with the new density functional some of the most popular local functionals and one of the most widely used hybrid functionals for solid-state properties. Altogether nine density functionals were tested using Gaussian 09; they are shown with references in Table 2. The basis sets used with each database are given in the SI. The UltraFine grid, which has 99 radial shells and 590 angular points per shell, was used for both molecular and solid-state calculations. The “stable = opt” keyword was used for open-shell atoms or molecules in order to reoptimize the Slater determinant to a stable solution if it was found unstable. Scalar relativistic effects were not included in our calculations. In order to understand some of the trends in some of the results for molecules, we computed B1 diagnostics52 of multireference character. A “multireference” molecule is an intrinsically multiconfigurational one (sometimes called strongly correlated). A B1 diagnostic value below 10 kcal/mol indicates that a molecule or bond is classified as a single-reference molecule or bond, and a value above 10 kcal/mol usually indicates that the molecule or bond is multireference.52 VASP calculations. The databases that do not involve charged species (PA3, EA13, and IP21), TDDFT calculations (EE23 and EEA11), or dipole moments (DM20) were calculated using PBE+U in VASP. In addition to these databases being left out for PBE+U, six reactions of DBH24 database were also left out as they involved charged species, resulting in the DBH18 database. The +U correction was applied to valence subshells of atoms following the Dudarev et al. approach,16 where only the difference U – J matters (specified with LDAUTYPE = 2), and where U and J are on-site Coulomb and exchange energies, respectively. The difference U – J will henceforth be denoted by U. Our earlier work28 involved some of the databases tested here (AE6, TMABE10, DBH18, a subset of SBG31 (SBG14), and TMOBG4), where we had tried to find an optimum value of U, and the results showed that the optimum value could vary significantly with


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the database. However, in this work, instead of optimizing our own value of U for each database, we will obtain a single value of U from the literature. To see how well DFT+U can do with a single U, we looked at ten highly cited articles in the literature53,54,55,56,57,58,59,60,61,62 that use PBE+U and averaged the U value they used or recommended. This yields a value around 4 eV. (Our own previous work28 also suggested that a value of 4 eV was often (but certainly not always) close to optimum.) Therefore we will test PBE+U with a U value of 4 eV. The reason for using a single value of U is to provide a comparison to HLE17, which uses the same parameters for all calculations. Following the language sometimes used in the literature, the +U correction will be called the Hubbard correction.15,63,64,65 For cases where there is a transition metal, we applied the Hubbard correction only to the transition metal. For solids without a transition metal, the Hubbard correction was applied only to the more electropositive element (or to the only element if it is unary). For molecules that do not have a transition metal, the Hubbard correction was applied only to the electronegative element (in particular, N, O, F, S, Cl, and Br). We never apply the Hubbard correction to a C atom or an H atom or inert gases. Although it is possible that in some cases the Hubbard correction would work better when applied to all the elements in a molecule or solid, these choices on selective application of the correction are based on tests done with GAM+U (and PBE+U in a few cases) made in our earlier work.28 For running calculations the same settings were not applied to all the databases. The projector-augmented-wave (PAW) potentials,66,67 energy cut-offs, k-points, SCF energy and force convergence criteria that were used are provided in the SI (Tables S2–S4). For most of the cases we used the PAW potentials recommended on the VASP website, and in some cases (mainly for molecules) we used a harder PAW potential.



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Table 2. The Exchange–Correlation Functionals Tested in This Work. Type GGA GGA GGA GGA NGA GGA GGA meta-GGA meta-GGA hybrid-GGA

Method PBE PBE+U BLYP PBEsol GAM HCTH/407 HLE16 TPSS HLE17 HSE06

Reference(s) 21 15,16,68 19,20 22 31 69 18 23 this work 70, 71

4. The New Functional After preliminary investigation of the effect of various calibrations, we decided to accept the same calibration factors for HLE17 as we used for HLE16. Therefore, HLE17 is obtained from TPSS by multiplying the exchange functional by 5/4 and the correlation functional by 1/2. To run the HLE17 functional, the keywords in Gaussian 09 are TPSSTPSS/def2TZVP IOp(3/76=1250000000) IOp(3/77=1000010000) IOp(3/78=0500005000)

The functional can also be added in a straightforward way to other programs.

5. Results and Discussion Table 3 presents the calculated band gaps for the SBG31 database consisting of band gaps of 31 semiconductors. In our earlier work,18,25 we tested several local functionals and a hybrid functional (HSE06) on the same database and Table 3 presents results for additional functionals that are meta-GGAs, in particular, TPSS and HLE17, and for the PBE+U (U = 4.0 eV) method. Table 4 compares the mean signed errors (MSEs) and the mean unsigned errors (MUEs) for the results in the previous paper and the present results. The local functionals have a tendency to underestimate band gaps,72,73,74 and we saw in ref. 18 and ref. 25 that the functionals that depend only on the density and the gradient of the density (except HLE16) do underestimate band gaps compared to the experimental values (as shown by the mean signed errors being negative). Tables 3 and 4 show that with the inclusion of kinetic energy density in TPSS, band gaps are still underestimated. However, with HLE17, which is obtained by modifying TPSS, and which is very similar to HLE16 in the way it has been modified, one obtains an MUE of 0.32 eV on this database. This is comparable with the performance of HLE16 (MUE = 0.30 eV) and the computationally more expensive hybrid GGA, HSE06 (MUE = 0.31 eV), both of which were described in ref. 18. Although HLE17 does not


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have an overall advantage over HLE16 for SBG31, we show next in Tables 5 and 6 that HLE17 is better for the prediction of lattice constants than HLE16, and we shall show in later tables that it is also better for other properties such as atomization energies of main group and transitionmetal containing molecules, barrier heights, and solid-state cohesive energies. For comparison to HLE17, Tables 3 and 4 also shows band gaps calculated by PBE+U, which is widely used in solid-state chemistry and materials research. It can be seen that PBE+U with a universal value of 4 eV for U (as specified in Section 3) does not give a good prediction of band gaps (the MUE is greater than 1 eV). Of course, it might be possible to obtain good band gaps by adjusting U for each solid, but that is not the goal of the present work. When compared with the PBE results of ref. 18, which were computed using Gaussian 09, PBE+U (U = 4.0 eV) shows no improvements (if we instead reported the PBE results obtained using VASP, our conclusions would not change).



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Table 3. Semiconductor Band Gaps for Database SBG31 (eV). Hubbard TPSS HLE17 Ref. PBE+Ub Typec correction ona C p of C 3.55 4.19 5.30 5.48 I Si p of Si 0.27 0.74 1.41 1.17 I Ge p of Ge 0.06 0.01 0.55 0.74 I SiC p of Si 1.45 1.35 2.16 2.42 I BP p of B 1.03 1.41 2.06 2.40 I BAs p of B 0.98 1.24 1.82 1.46 I AlP p of Al 1.62 1.83 2.66 2.51 I AlAs p of Al 1.41 1.64 2.38 2.23 I AlSb p of Al 0.88 1.49 2.04 1.68 I GaN p of Ga 0.88 1.74 3.19 3.50 D β-GaN p of Ga 0.72 1.56 2.95 3.30 D GaP p of Ga 0.46 1.90 2.61 2.35 I GaAs p of Ga 0.50 0.52 1.27 1.52 D GaSb p of Ga 0.11 0.08 0.51 0.73 D InN p of In 0.04 0.00 0.58 0.69 D InP p of In 0.65 0.91 1.50 1.42 D InAs p of In 0.35 0.00 0.22 0.41 D InSb p of In 0.15 0.00 0.00 0.23 D ZnO d of Zn 1.29 0.94 2.61 3.40 D ZnS d of Zn 2.34 2.40 3.54 3.66 D ZnSe d of Zn 1.39 1.48 2.41 2.70 D ZnTe d of Zn 1.26 1.45 2.06 2.38 D CdS d of Cd 1.27 1.34 2.27 2.55 D CdSe d of Cd 0.68 0.73 1.49 1.90 D CdTe d of Cd 0.74 0.88 1.38 1.92 D MgS s of Mg 3.48 3.67 5.12 5.40 D MgSe s of Mg 2.23 2.01 2.75 2.47 I MgTe s of Mg 2.47 2.96 3.83 3.60 I BaS s of Ba 2.17 2.55 3.32 3.88 I BaSe s of Ba 1.96 2.19 2.81 3.58 I BaTe s of Ba 1.61 1.73 2.37 3.08 I a A Hubbard correction of 4.0 eV is applied to the valence subshell of only the first element in case of binary compounds and to the element in case of unary compounds. Semiconductor

b

The PBE+U (U = 4.0 eV) values were computed using VASP and the rest of the table is calculated using Gaussian 09. c

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D = direct band gap, I = indirect band gap.


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Table 4. Mean Errors for Semiconductor Band Gaps of Database SBG31 (eV). b

MSE MUEc MSEb MUEc

PBE –1.11 1.11 HCTH/407 –0.89 0.89

PBE+Ua –1.19 1.19 HLE16 0.04 0.30

BLYP –1.15 1.15 TPSS –0.96 0.96

PBEsol –1.13 1.13 HLE17 –0.18 0.32

GAM –0.99 0.99 HSE06 –0.24 0.31

a

The PBE+U (U = 4.0 eV) values were computed using VASP and the rest of the table is calculated using Gaussian 09. b MSE = mean signed error, cMUE = mean unsigned error

The performance of the new functional, HLE17, on semiconductor lattice constants (SLC34 database) is shown in Tables 5 and 6. The SLC34 database has the same semiconductors as the SBG31 database, but for three of the solids (GaN, InN, and ZnO), there are two independent lattice parameters, and this raises the data count to 34. The PBEsol functional, which was specifically parametrized to predict accurate solid-state lattice constants, was the best performing functional (MUE = 0.031 Å; see Table 6) in ref. 18, although it does not do well for band gaps. The hybrid GGA, HSE06, was the second best performing functional (MUE = 0.052 Å) for this database. In comparison with the functionals tested in ref. 18 as well as Table 5, the HLE17 functional is the third best performing functional with an MUE of 0.077 Å, and it gives a significant improvement over the recently developed HLE16, which gave an MUE of 0.157 Å (see Table 6). While most of the functionals in ref. 18 and Table 5 overestimate lattice constants, HLE16 and HLE17 tend to underestimate lattice constants.



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Table 5. Semiconductor Lattice Constants for Database SLC34 (Å). Semiconductor C Si Ge SiC BP BAs AlP AlAs AlSb GaN

Hubbard correction ona p of C p of Si p of Ge p of Si p of B p of B p of Al p of Al p of Al p of Ga

PBE+Ub

TPSS

HLE17

Ref.

3.646 3.579 3.424 3.555 5.633 5.465 5.269 5.422 6.079 5.743 5.563 5.644 4.434 4.394 4.206 4.348 4.644 4.566 4.382 4.527 4.917 4.820 4.636 4.764 5.564 5.497 5.313 5.450 5.795 5.712 5.531 5.649 6.302 6.193 6.023 6.126 3.342 3.221 3.103 3.179 5.415 5.246 5.035 5.160 β-GaN p of Ga 4.716 4.552 4.379 4.523 GaP p of Ga 5.729 5.522 5.357 5.441 GaAs p of Ga 5.977 5.744 5.578 5.641 GaSb p of Ga 6.465 6.182 6.035 6.086 InN p of In 3.750 3.584 3.473 3.527 6.011 5.776 5.577 5.679 InP p of In 6.253 5.961 5.813 5.858 InAs p of In 6.482 6.171 6.026 6.048 InSb p of In 6.956 6.585 6.461 6.473 ZnO d of Zn 3.243 3.278 3.194 3.223 5.227 5.240 5.070 5.194 ZnS d of Zn 5.415 5.464 5.366 5.399 ZnSe d of Zn 5.713 5.737 5.632 5.658 ZnTe d of Zn 6.169 6.175 6.089 6.079 CdS d of Cd 5.923 5.943 5.841 5.808 CdSe d of Cd 6.195 6.195 6.094 6.042 CdTe d of Cd 6.622 6.610 6.529 6.470 MgS s of Mg 5.712 5.717 5.535 5.612 MgSe s of Mg 5.517 5.522 5.336 5.375 MgTe s of Mg 6.525 6.503 6.341 6.410 BaS s of Ba 6.436 6.434 6.307 6.364 BaSe s of Ba 6.666 6.662 6.536 6.570 BaTe s of Ba 7.069 7.022 6.935 6.982 a A +U correction of 4.0 eV is applied to the valence subshell of only the first element in the case of binary compounds and to the only element in the case of unary compounds. b

The PBE+U (U = 4.0 eV) values were computed using VASP and the rest of the table is calculated using Gaussian 09.


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Table 6. Mean Errors for Semiconductor Lattice Constants of Database SLC34 (Å).

a

PBE

PBE+Ua

BLYP

PBEsol

GAM

MSE MUE

0.095 0.095 HCTH/407

0.184 0.184 HLE16

0.178 0.178 TPSS

0.025 0.031 HLE17

0.157 0.158 HSE06

MSE MUE

0.155 0.155

–0.157 0.157

0.080 0.080

–0.068 0.077

0.052 0.052

The PBE+U (U = 4.0 eV) values were computed using VASP and the rest of the table is calculated using Gaussian 09.

In our earlier work18 we showed that HLE16, which showed good performance for band gaps, also performs reasonably well for valence and Rydberg excitation energies of molecules (EE23 database). Table 7 shows the performance of HLE17 on the EE23 database in which we look at vertical excitation energies. The EE23 database consists of (1) 18 valence excitations of 14 organic molecules – for four of these molecules (benzene, naphthalene, furan, and hexatriene) excitations for the triplet state are also considered; (2) two Rydberg excitations of water molecule for singlet and triplet states; and (3) three charge transfer (CT) excitations – two intramolecular (para-nitroaniline (PNA) and dimethylaminobenzonitrile (DMABN)) and one intermolecular (benzene–tetracyanoethylene complex (B-TCNE)). Two approaches have been used here for computing the excitation energies. In the first approach, LR-TDDFT is used to calculate excitation energies of all the states. In the second approach, LR-TDDFT is used to calculate excitation energies of singlet states and the ΔSCF method is used for the excitation energies of triplet state. Table 7 shows that for both valence and Rydberg excitations, HLE17 is not as good as HLE16 if we consider the first approach, although for Rydberg excitations it does better than all the other local functionals. The main contributions to the error in HLE17 are from triplet naphthalene, furan, and hexatriene, for which we see significantly lower valence excitations compared to the rest of the functionals. This can be attributed to these systems being highly multireference, which possibly leads to breakdown of TDDFT, and has been investigated by other coworkers.75 Therefore for all the five triplet cases in Table 7, we test the ΔSCF method. The ΔSCF method gives improved results with MUEs for the valence and Rydberg excitations now being 0.32 and 0.35 eV, respectively, as opposed to 0.47 and 0.37 eV, respectively, when LR-TDDFT is done for all the states. The ΔSCF method also improves the prediction of valence and Rydberg excitations by some of the other functionals, for example, HSE06. The performance of HLE17 for CT excitations is not improved over other local functionals.



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In Table 7, for most of the molecules, the excitation energies increase as one goes from HCTH/407 to HLE16 or as one goes from TPSS to HLE17. Similarly we saw above that the band gaps increase for most of the semiconductors for HLE17 and also for HLE16 in ref. 18. To understand this situation we looked at the HOMO and LUMO energies of the molecules in the EE23 database with TPSS and HLE17; the results of this examination are given in Table 8. Both the HOMO and the LUMO are more strongly bound with HLE17 than with TPSS, but the key point for the calculation of excitation energies is that the LUMO–HOMO difference tends to be greater for HLE17 for most of the cases. We also compared LUMO–HOMO differences to LRTDDFT results; the “diff.” columns in Table 8 are obtained by taking an unsigned difference between the LUMO–HOMO and the LR-TDDFT columns. Upon averaging the “diff.” columns we see that TPSS gives a small value compared to HLE17 indicating that the LUMO–HOMO gaps in TPSS are closer to the LR-TDDFT results than with HLE17. We conclude that the HLE strategy does not succeed mainly by improving the orbital energies. The last two rows of Table 8 show the mean errors one would get if one computed excitation energies simply from the HOMO–LUMO gaps; we see that the linear response treatment gives more accurate results than the orbital energy gaps, as expected. Altogether, Tables 7 and 8 show that in improving the semiconductor band gaps, we have also improved the molecular excitation energies, although the improvement is smaller than for the band gaps.


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Table 7. Excitation Energies (eV) for EE23 Using LR-TDDFT for All the States and the ΔSCF Method Only for the Triplet State. Molecule acetaldehyde acetone formaldehyde pyrazine pyridazine pyridine pyrimidine s-tetrazine ethylene butadiene benzene ” ” naphthalene ” ” furan ” ” hexatriene ” ” water ” ” pNA DMABN B-TCNE

Transition 1 A″ n → π* 1 A2 n → π* 1 A2 n → π* 1 B3u n → π* 1 B1 n → π* 1 B1 n → π* 1 B1 n → π* 1 B3u n → π* 1 B1u π → π* 1 Bu π → π* 1 B2u, π → π* 3 B1u π → π* ΔSCF 1 B3u π → π* 3 B2u π → π* ΔSCF 1 B2 π → π* 3 B2 π → π* ΔSCF 1 Bu π → π* 3 Bu π → π* ΔSCF Singlet, 2px → 3s Triplet, 2px → 3s ΔSCF Intramolecular CT,1A1, π → π* Intramolecular CT,1A1, π → π* Intermolecular CT,1A, π → π* MSE valence (LR)b MUE valence (LR) MSE valence (LR/Δ)c MUE valence (LR/Δ) MSE Rydberg (LR) MUE Rydberg (LR) MSE Rydberg (LR/Δ) MUE Rydberg (LR/Δ) MSE charge transfer (LR)

PBE 4.10 4.20 3.77 3.52 3.11 4.32 3.75 1.84 7.35 5.41 5.14 3.91 4.33 4.02 2.79 2.96 5.88 3.90 4.08 4.42 2.27 2.47 6.36 6.01 7.00 3.55 4.36 1.35 –0.33 0.36 –0.28 0.33 –1.01 1.01 –0.52 0.52 –1.07

BLYP 4.12 4.21 3.80 3.57 3.15 4.35 3.79 1.90 7.35 5.33 5.09 3.94 4.31 3.99 2.82 2.97 5.77 3.95 4.08 4.42 2.31 2.50 6.22 5.91 7.00 3.50 4.34 1.32 –0.32 0.34 –0.28 0.32 –1.14 1.14 –0.59 0.59 –1.10

PBEsol 4.07 4.19 3.73 3.47 3.06 4.27 3.71 1.79 7.39 5.43 5.16 4.04 4.39 4.03 2.87 3.00 5.92 4.03 4.15 4.42 2.36 2.52 6.42 6.09 7.10 3.56 4.35 1.36 –0.32 0.35 –0.28 0.34 –0.95 0.95 –0.44 0.54 –1.07

GAM 4.21 4.31 3.90 3.59 3.22 4.41 3.83 1.90 7.36 5.42 5.15 3.84 4.26 4.04 2.75 2.94 5.89 3.87 4.01 4.43 2.22 2.43 6.53 6.31 7.05 3.56 4.40 1.37 –0.29 0.33 –0.24 0.29 –0.78 0.78 –0.41 0.46 –1.05

HCTH/407 HLE16 4.19 4.43 4.28 4.50 3.85 4.19 3.58 3.88 3.20 3.55 4.39 4.73 3.82 4.10 1.89 2.14 7.33 7.73 5.40 5.60 5.14 5.15 3.78 4.23 4.25 3.70 4.03 4.07 2.71 2.01 2.92 2.67 5.87 6.07 3.83 2.96 4.00 3.49 4.42 4.52 2.18 1.29 2.41 2.06 6.43 7.71 6.28 7.33 7.29 7.76 3.55 3.46 4.39 4.46 1.31 0.95 –0.32 –0.25 0.35 0.35 –0.26 –0.17 0.30 0.25 –0.84 0.32 0.84 0.32 –0.34 0.54 0.63 0.54 –1.07 –1.20


TPSS 4.29 4.36 4.01 3.72 3.34 4.53 3.96 2.04 7.42 5.50 5.23 3.75 4.23 4.10 2.70 2.95 5.95 3.78 4.01 4.50 2.14 2.43 6.56 6.24 6.97 3.64 4.46 1.41 –0.24 0.29 –0.17 0.23 –0.80 0.80 –0.44 0.44 –0.98

HLE17 4.61 4.62 4.43 4.09 3.77 4.94 4.33 2.38 7.69 5.64 5.20 4.23 3.77 4.11 1.66 2.63 6.05 2.63 3.38 4.57 0.73 2.00 7.82 7.31 7.28 3.52 4.54 1.09 –0.22 0.47 –0.08 0.32 0.37 0.37 0.35 0.35 –1.10

HSE06 4.24 4.39 3.90 3.95 3.61 4.81 4.30 2.27 7.46 5.62 5.36 3.46 4.83 4.35 2.52 3.05 6.04 3.55 3.94 4.65 1.90 2.46 7.16 6.71 6.90 4.10 4.74 1.98 –0.18 0.30 –0.02 0.22 –0.27 0.27 –0.17 0.17 –0.55

Ref. 4.28 4.43 4.00 3.97 3.60 4.74 4.18 2.25 8.02 6.21 4.90 4.12 4.12 4.00 3.11 3.11 6.06 4.17 4.17 4.93 2.69 2.69 7.40 7.00 7.00 4.30 4.57 3.59 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00


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14 MUE charge transfer (LR)

1.07

1.10

1.07

1.05

1.07

1.20

0.98

1.10

0.66

0.00

aSome of the results presented in Table 7 were part of our work in Ref. 18, and they are presented here again to facilitate comparison and because ΔSCF was

not done in ref. 18. Rows not marked as ΔSCF are calculated by LR-TDDFT. bLR denotes that the MUEs and MSEs are calculated using only LR-TDDFT calculations. cLR/Δ denotes that the MUEs and MSEs are calculated using LR-TDDFT calculations for singlet states and the ΔSCF method for the triplet states.

Table 8. HOMO, LUMO, LUMO – HOMO, and LR-TDDFT Energies (in eV) of the EE23 Database. TPSS Molecule acetaldehyde acetone formaldehyde pyrazine pyridazine pyridine pyrimidine s-tetrazine ethylene butadiene benzene

” naphthalene

” furan

” hexatriene

” water

” pNA DMABN B-TCNE

Transition 1

A″ n → π* A2 n → π* 1 A2 n → π* 1 B3u n → π* 1 B1 n → π* 1 B1 n → π* 1 B1 n → π* 1 B3u n → π* 1 B1u π → π* 1 Bu π → π* 1 B2u, π → π* 3 B1u, π → π* 1 B3u, π → π* 3 B2u, π → π* 1 B2, π → π* 3 B2, π → π* 1 Bu, π → π* 3 Bu, π → π* Singlet, 2px → 3s Triplet, 2px → 3s Intramolecular CT,1A1, π → π* Intramolecular CT,1A1, π → π* Intermolecular CT,1A, π → π* 1

HOMO

LUMO

-6.05 -5.75 -6.39 -6.03 -5.57 -6.06 -6.08 -5.95 -6.71 -5.79 -6.27 -6.27 -5.45 -5.45 -5.64 -5.64 -5.36 -5.36 -7.31 -7.31 -5.88 -5.08 -6.77

-1.88 -1.52 -2.50 -2.53 -2.47 -1.74 -2.24 -4.13 -0.90 -1.74 -1.09 -1.09 -1.95 -1.95 -0.73 -0.73 -2.19 -2.19 -0.81 -0.81 -2.89 -1.47 -5.37

LUMO– HOMO 4.17 4.23 3.89 3.50 3.10 4.33 3.84 1.82 5.82 4.06 5.18 5.18 3.51 3.51 4.91 4.91 3.17 3.17 6.51 6.51 2.99 3.61 1.40

LRTDDFT 4.29 4.36 4.01 3.72 3.34 4.53 3.96 2.04 7.42 5.50 5.23 3.75 4.10 2.70 5.95 3.78 4.50 2.14 6.56 6.24 3.64 4.46 1.41

Average

diff.a

HOMO

LUMO

0.12 0.13 0.13 0.21 0.24 0.21 0.12 0.22 1.60 1.44 0.05 1.43 0.60 0.80 1.04 1.13 1.33 1.02 0.05 0.26 0.65 0.84 0.01 0.59

-7.90 -7.56 -8.30 -7.87 -7.45 -7.83 -7.84 -7.95 -7.92 -7.03 -7.50 -7.50 -6.71 -6.71 -6.97 -6.97 -6.61 -6.61 -9.11 -9.11 -7.43 -6.54 -8.02

-3.27 -2.94 -3.84 -3.90 -3.84 -3.01 -3.56 -5.70 -1.95 -2.88 -2.28 -2.28 -3.16 -3.16 -2.07 -2.07 -3.37 -3.37 -1.32 -1.32 -4.56 -2.83 -6.94

MSE (singlet valence and Rydberg) -0.73 -0.23 MUE (singlet valence and Rydberg) 0.77 0.29 a The diff. columns give the absolute value of the difference between the LR-TDDFT and LUMO–HOMO columns. ACS Paragon Plus Environment

HLE17 LUMO– HOMO 4.62 4.63 4.46 3.98 3.61 4.82 4.28 2.25 5.97 4.14 5.21 5.21 3.54 3.54 4.91 4.91 3.24 3.24 7.79 7.79 2.86 3.72 1.08

LRTDDFT 4.61 4.62 4.43 4.09 3.77 4.94 4.33 2.38 7.69 5.64 5.20 4.23 4.11 1.66 6.05 2.63 4.57 0.73 7.82 7.31 3.52 4.54 1.09

-0.39 0.62

0.09 0.25

diff.a 0.02 0.01 0.03 0.12 0.16 0.12 0.05 0.13 1.72 1.49 0.02 0.98 0.57 1.88 1.14 2.28 1.33 2.51 0.03 0.48 0.66 0.82 0.01 0.72

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15 a

Table 9. Excitation Energies (eV) of Atoms in Database EEA11. Atom H He Li Be B Ne Na Mg Al Ar K

a

states type 2 S−2S 1s−2s 1 S−1S 1s2−1s2s 2 S−2P 2s−2p 1 S−1P 2s2−2s2p 2 P−2S 2p−3s 1 S−1P 2p6−2p53s 2 S−2P 3s−3p 1 S−1P 3s2−3s3p 2 P−2S 3p−4s 1 S−1P 3p6−3p54s 2 S−2P 4s−4p MSE valence MUE valence MSE Rydberg MUE Rydberg MSE overall MUE overall

PBE 8.05 17.61 2.00 4.99 4.14 15.85 2.13 4.24 2.73 11.04 1.57 –0.05 0.12 –1.31 1.31 –0.74 0.77

BLYP 7.96 17.63 2.01 4.92 4.12 15.67 2.25 4.26 2.56 10.81 1.72 –0.01 0.17 –1.43 1.43 –0.78 0.86

PBEsol 7.93 17.43 2.00 4.96 4.14 16.00 2.14 4.25 2.74 11.15 1.60 –0.05 0.12 –1.32 1.32 –0.74 0.78

GAM 7.86 18.08 2.11 5.03 4.60 16.36 2.09 4.11 2.90 11.37 1.70 –0.03 0.17 –1.02 1.02 –0.57 0.64

HCTH/407 7.85 17.93 2.07 4.97 4.33 15.92 1.95 4.08 2.92 11.00 1.41 –0.14 0.23 –1.22 1.22 –0.73 0.77

HLE16 9.48 19.77 2.14 5.28 5.40 18.47 1.92 4.34 3.74 12.76 1.21 –0.06 0.18 0.38 0.91 0.18 0.58

Some of the results presented in Table 9 were reported in the SI of ref. 18.


TPSS 8.25 18.13 2.02 5.12 4.52 16.21 2.05 4.24 3.02 11.33 1.46 –0.06 0.13 –0.97 0.97 –0.56 0.59

HLE17 10.36 20.83 1.99 5.33 5.82 18.34 2.13 4.49 3.88 12.71 1.45 0.04 0.10 0.77 0.77 0.44 0.47

HSE06 8.75 18.87 1.97 4.98 4.59 17.03 2.11 4.26 2.95 11.71 1.57 –0.06 0.11 –0.57 0.68 –0.34 0.42

Ref. 10.20 20.62 1.85 5.28 4.96 16.71 2.10 4.35 3.14 11.68 1.61 0.00 0.00 0.00 0.00 0.00 0.00


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16

Next we consider the excitation energies of atoms to see if the increased excitation energies seen for the case of molecules with HLE16 and HLE17 functionals in Table 7 are also found for singlet and doublet main-group atoms. We limit this examination to systems with closed subshells or with at most one open-shell electron. Table 9 summarizes the results for valence excitations of Li, Be, Na, Mg, and K and Rydberg excitations of H, He, B, Ne, Al, and Ar. The “MUE valence” value shows that all functionals perform well for valence excitations with HLE17 being the best performing functional with an MUE of 0.10 eV. The “MUE Rydberg” value shows that none of the functionals perform well for Rydberg excitations with the hybrid functional, HSE06, having the smallest MUE of 0.68 eV. Encouragingly, HLE17 has the lowest MUE of any local functional with an MUE of 0.77 eV, and lags behind HSE06 by only 0.09 eV. The performance of the newly developed functional was also tested on other databases besides the four presented so far. The objective is to see whether the good performance of HLE17 on band gaps and excitation energies (properties that depend directly on orbital energies) occurs at the cost of its poor performance on other properties. All the databases that were tested with HLE16 in ref. 18 are tested here, and four more have been added, namely EA13, IP21, EEA11, and DM20 (see Table 1 for expansions of database acronyms). The MUEs of the newly added databases along with all the results reported in ref. 18 are all summarized in Table 10, and the detailed molecule-by-molecule comparison for each of these databases is presented in the SI. Besides the SLC34 database, lattice constants were calculated for three other databases (MGLC4, ILC5, and TMLC4). For three of the four lattice constant databases (SLC34, ILC5, and TMLC4), HLE17 gives significant improvement over HLE16 as shown in Table 10, and it gives performance similar to HLE16 for MGLC4. For solid-state cohesive energies (database SSCE8), the HLE17 functional is better than HLE16, but one very disappointing result in Table 10 is that all local functionals, even our new ones, perform poorly for the band gaps of transition metal oxides (TMOBG4 database). The TMOBG4 database contains the band gaps of MnO, FeO, CoO, and NiO. The strong correlation effects in these oxides are discussed elsewhere,76,77,78,79,80,81,82,83,84,85 and it is well known that they provide a strong challenge for approximation methods that represent the density by a single Slater determinant with only local exchange. However, we find that HLE17 does significantly better than the widely used PBE+U method with U = 4 eV for this database (MUE = 1.5 eV vs. 2.2 eV), and the improvement with respect to TPSS (MUE = 2.7 eV) and PBE (MUE = 2.9 eV) is even greater.


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17

Next we consider molecular energetic properties. Table 10 shows that HLE17 performs reasonably well for atomization energies of main-group molecules (AE6 database), forward and reverse barriers heights of 12 reactions (DBH24 database), and noncovalent complexation energies (NCCE31 database), and it is better than HLE16 for these databases. The performance of HLE17 for barrier heights is particularly noteworthy. Whereas PBE+U hardly improves the barrier heights with respect to PBE, HLE17 greatly improves them as compared to similar functionals with lower local exchange and even with respect to HSE06. However, the performance of HLE17 on proton affinity (PA3 database), electron affinity (EA13 database), and ionization potential (IP21 database), all of which involve charged species, is not good. Also for molecular bond lengths, HLE17 is not as good as most of the local functionals but is slightly better than HLE16. Finally we look at molecular densities, as judged by dipole moments. The dipole moment is a leading indicator of the quality of the density because it is the first nonzero moment of the charge distribution for neutral molecules. The DM20 database contains dipole moments of 20 molecules (of which 11 molecules contain transition metals and the remainder contain only main-group elements); the overall MUEs for HLE16 and HLE17 are larger on this database than those for other functionals. To understand this source of error in HLE16 and HLE17, we calculated MUEs for the transition-metal-containing molecules and main-group elements containing molecules separately (see Table S19 of the SI). We find that the MUEs for the maingroup containing molecules are close to each other for all the functionals studied than are those for transition-metal-containing molecules, and they are reasonably small (~ 0.1 D); in contrast, the errors in dipole moments are large for the transition-metal-containing molecules. One possible reason for this could be multireference nature of these molecules; to estimate the extent of their multireference character we calculated the B1 diagnostic52 (see Table S20 of the SI for the resulting B1 values). Only two molecules in Table S20 have a B1 diagnostic value less than 10 kcal/mol, and the rest are clearly highly multireference cases. It is well known86,87 that KS-DFT with currently available functionals does more poorly for multireference cases than for singlereference ones. Because the MUEs of dipole moments of molecules with transition metals are higher for HLE16 and HLE17 than other functionals for molecules, we conclude that increasing local exchange can give poor performance for multireference systems. This prompts one final comparison. In particular, the TMABE10 database considered in Table 10 contains open-shell molecules, which often have multireference character, and we look at the spin densities for these molecules using TPSS, HLE17, and HSE06. A hybrid functional with nonlocal exchange is known to localize electron density more than a local functional, and it would be interesting to see



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18

if the newly developed functional, HLE17, with enhanced local exchange, shows a similar trend. Table 11 gives Hirshfeld spin densities on transition metals of the molecules in the TMABE10 database, and we see that HLE17 shows the same trend as HSE06 in that for most of the cases the spin densities increase as one goes from TPSS to HLE17 or to HSE06. The present work is also relevant to the difference between Kohn-Sham theory and generalized Kohn-Sham theory,3,88,89 but a discussion of this issue is beyond the scope of the present article. Finally we provide a comment on the physical interpretation of the parameterized exchangecorrelation potential, which is the functional derivative of the exchange-correlation energy. The starting exchange potential for the present investigation reduces for zero density gradient to the Gáspár-Kohn-Sham exchange potential of a uniform electron gas at the Fermi level.90,91 The earlier exchange potential of Slater92 corresponds to averaging the uniform electron gas exchange potential over all occupied states of same-spin electrons, and it is higher by a factor of 1.5. Our exchange potential is scaled up by the intermediate factor of 1.25, and so it corresponds (in the homogeneous-gas limit) to a weighted average over the Fermi sea in which the states near the Fermi level have a higher weight than the lower energy states. Thus it is a physical exchange potential. We also note the work of Handy and Cohen,93 who found that they needed to increase the amount of local exchange (in the small density-gradient region) as compared to the GáspárKohn-Sham approximation in order to approximate Hartree-Fock exchange with a local functional. The exchange energy is much larger in magnitude than the correlation energy, and so it is not unreasonable that if exchange is scaled up by 25%, the correlation potential that was empirically found to give reasonable results consistent with the Gáspár-Kohn-Sham exchange potential has been empirically scaled down by a larger percentage (50%) to compensate in the total energy. We also note that Becke94 found it reasonable to scale down the correlation energy (by 19-28%) when he added Hartree-Fock exchange in the B3PW91 functional, and this is consistent with our increased local exchange simulating in some ways the effect of Hartree-Fock exchange.


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19 a

b

Table 10. MUEs for All Molecular and Solid-State Databases. Databasec

PBE

PBE+Ud

BLYP

PBEsol GAM HCTH/407 HLE16 TPSS HLE17 Molecular properties AE6 4.6 4.5 1.8 9.7 4.1 2.3 4.3 2.3 3.4 TMABE10 8.3 6.1 7.8 12.7 4.9 6.4 11.0 8.1 10.6 DBH24 8.2 7.7 10.3 5.0 4.7 4.9 8.2 3.7 e 8.6 8.2 7.8 11.3 5.4 5.3 5.0 8.3 3.5 DBH18 NCCE31 1.2 1.1 1.5 1.8 1.0 1.1 2.1 1.2 1.0 PA3 1.0 2.0 2.1 2.6 2.1 16.8 1.8 15.4 EA13 2.3 2.7 2.2 4.5 3.6 18.1 2.4 19.3 IP21 6.2 6.6 5.7 4.1 6.4 32.0 4.1 33.1 EE23 – valence 0.36 0.34 0.35 0.33 0.35 0.35 0.29 0.47 EE23 – Rydberg 1.01 1.14 0.95 0.78 0.84 0.32 0.80 0.37 EE23 – CT 0.98 1.10 1.07 1.10 1.07 1.05 1.07 1.20 EEA11 0.77 0.86 0.78 0.64 0.77 0.58 0.59 0.47 DM20 0.294 0.305 0.305 0.268 0.262 0.486 0.262 0.462 MGHBL9 0.011 0.014 0.010 0.014 0.004 0.003 0.053 0.007 0.050 MGNHBL11 0.009 0.016 0.016 0.006 0.007 0.005 0.072 0.007 0.052 DGH4 0.021 0.021 0.039 0.012 0.037 0.055 0.015 0.039 0.033 Solid-state properties SSCE8 0.11 0.49 0.30 0.33 0.10 0.23 0.27 0.14 0.22 TMOBG4 2.2 2.9 2.9 3.1 2.8 2.8 2.1 2.7 1.5 SBG31 1.11 1.19 1.15 1.13 0.99 0.89 0.30 0.96 0.32 MGLC4 0.037 0.076 0.069 0.023 0.037 0.110 0.140 0.051 0.145 ILC5 0.078 0.081 0.121 0.020 0.110 0.219 0.142 0.062 0.083 TMLC4 0.065 0.038 0.139 0.019 0.106 0.072 0.113 0.029 0.028 SLC34 0.095 0.184 0.180 0.031 0.158 0.155 0.157 0.080 0.077 a The reference values used to calculate MUEs along with molecule-by-molecule comparison with each functional is in the SI. b

HSE06 2.6 5.6 3.8 4.2 0.7 0.8 2.8 3.7 0.30 0.27 0.66 0.42 0.275 0.002 0.009 0.015 0.10 f 0.6 0.31 0.045 0.023 0.050 0.052

Some of the results presented in Table 10 were part of our work in ref. 18 and they are presented here again to facilitate easy comparison with HLE17. c The units for various databases are: AE6, TMABE10, DBH24, DBH18, NCCE31, PA3, EA13, and IP21 are in kcal/mol; MGHBL9, MGNHBL11, DGH4, ILC5, MGLC4, TMLC4, and SLC34 are in Å; EE23, EEA11, SSCE8, TMOBG4, and SBG31 are in eV; and DM20 is in D. d The PBE+U (U = 4.0 eV) values were computed using VASP and the rest of the table is calculated using Gaussian 09. e A subset DBH18 of DBH24 database that contains only reactions of neutral species. f The TMOBG4 database uses HSE03 and the values are from Ref. 95. ACS Paragon Plus Environment


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20

Table 11. Hirshfeld spin densities (atomic units) on transition metal ions of molecules in TMABE10 database. Molecule TPSS HLE17 HSE06 AgH CoCl2 CoH CrCl CrCl2 FeCl2 FeH MnF2 TiCl VF5

0.00 2.52 2.00 4.93 4.01 3.58 3.00 4.74 2.95 0.00

0.00 2.68 2.00 4.97 3.91 3.63 3.00 4.80 2.99 0.00

0.00 2.78 2.00 4.95 4.00 3.71 3.00 4.81 2.95 0.00

6. Conclusions A new density functional, HLE17, was calibrated and validated against a diverse set of data that includes ground-state and excited-state properties for both molecules and solids. It is a meta-GGA and has the advantage of giving band gaps and excitation energies comparable in accuracy to hybrid GGAs at a lower computational cost. Furthermore, for atomization energies, barrier heights, noncovalent complexation energies, cohesive energies, and lattice constants, it improves over the recently developed high local exchange functional, HLE16, which is a GGA. However its prediction of molecular bond lengths, proton affinities, electron affinities, and ionization potentials is not good and is similar to HLE16 in performance. We also compared HLE17 with PBE+U (with a fixed value of U equal to 4 eV), and we find that PBE+U does better than HLE17 on only 3 out of 14 databases for which this comparison is made. We anticipate that HLE17 will be especially useful for solid-state calculations on non-oxide semiconductors, where it provides greatly enhanced performance compared to other functionals with comparable cost and even provides performance comparable to the hybrid HSE06 functional for our largest semiconductor band gap database. In addition to its practical utility, the success of HLE17 exposes a new strategic direction for making more useful exchangecorrelation functionals for Kohn-Sham theory.


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21

! ASSOCIATED CONTENT Supporting Information. Additional tables. This material is available free of charge via the Internet at http://pubs.acs.org. ! AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] and [email protected] ORCID Pragya Verma: 0000-0002-5722-0894 Donald G. Truhlar: 0000-0002-7742-7294

Notes The authors declare no competing financial interest. ! ACKNOWLEDGMENTS The authors thank Soumen Ghosh and Chad Hoyer for help with the excitation energy database. PV acknowledges a Richard D. Amelar and Arthur S. Lodge Fellowship. This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under award number DE-SC0015997.. ! REFERENCES 1 Suhai, S. Quasiparticle Energy-Band Structures in Semiconducting Polymers: Correlation Effects on the Band Gap in Polyacetylene. Phys. Rev. B 1983, 27, 3506−3518. 2 Heybertsen, J.; Louie, S. G. Electron Correlation in Semiconductors and Insulators: Band Gaps and Quasiparticle Energies. Phys. Rev. B 1986, 34, 5390−5413. 3 Seidl, A.; Görling, A.; Vogl, P.; Majewski, J. A.; Levy, M. Generalized Kohn-Sham Schemes and the Band-Gap Problem. Phys. Rev. B 1996, 53, 3764−3774. 4 Muscat, J. Wander, A.; Harrison, N. M. On the Prediction of Band Gaps from Hybrid Functional Theory. Chem. Phys. Lett. 2001, 342, 397−401. 5 de P. R. Moreira, I.; Illas, F.; Martin, R. L. Effect of Fock Exchange on the Electronic Structure and Magnetic Coupling in NiO. Phys. Rev. B 2002, 65, 155102. 6 Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A Long-Range-Corrected TimeDependent Density Functional Theory. J. Chem. Phys. 2004, 120, 8425−8433. 7 Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid Exchange-Correlation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51−57. 8 Grüning, M.; Marini, M.; Rubio, A. Density Functionals from Many-Body Perturbation Theory: The Band Gap for Semiconductors and Insulators. J. Chem. Phys. 2006, 124, 154108. 9 Paier, J.; Marsman, M.; Hummer, K.; Kresse, G.; Gerber, I. C.; Ángyán, J. G. Screened Hybrid Density Functionals Applied to Solids. J. Chem. Phys. 2006, 124, 154709.



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