HLE16: A Local KohnâSham Gradient Approximation with Good

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HLE16: A Local Kohn–Sham Gradient Approximation with Good Performance for Semiconductor Band Gaps and Molecular Excitation Energies Pragya Verma, and Donald G. Truhlar J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b02757 • Publication Date (Web): 29 Dec 2016 Downloaded from http://pubs.acs.org on January 2, 2017

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The Journal of Physical Chemistry Letters

1 revised for JPC Lett., 12/26/2016

HLE16: A Local Kohn–Sham Gradient Approximation with Good Performance for 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: Local exchange–correlation functionals have low cost and convenient portability, but are known to seriously underestimate semiconductor band gaps and the energies of molecular Rydberg states. Here we present a new local approximation to the exchange–correlation functional called HLE16 that gives good performance for semiconductor band gaps and molecular excitation energies – competitive with hybrid functionals. By simultaneously increasing the local exchange and decreasing the local correlation, electronic excitation energies were improved without excessively degrading the ground-state solid-state cohesive energies, molecular bond energies, or chemical reaction barrier heights, although the new functional is not recommended for optimizing lattice constants or molecular bond lengths. The new functional can be useful as is for calculations on semiconductors or excited states where it is essential to control the cost, and it can also be useful in establishing a starting point for developing even better new functionals that perform well for excited states.

ACS Paragon Plus Environment


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Kohn-Sham density functional theory (KS–DFT) is widely used for problems in solid-state physics and molecular science and is a basic tool for computing band structure1 and electronic excitation energies.2 The accuracy of KS–DFT depends on the accuracy of the exchange– correlation functional, which is unknown but approximated in a variety of ways. We may divide the most useful approximations into two groups, typically labeled as local and nonlocal in chemistry (and typically labeled as semilocal and nonlocal in physics – we use the chemistry language). For a local functional, the energy density at a point depends only on local properties like the spin densities, their gradients, or the local kinetic energy density. For nonlocal functionals, the energy density at a point involves an integral over all space. The most common nonlocal ingredient is Hartree–Fock (HF) exchange,3 but there are also functionals that involve nonlocal correlation.4 The main advantages of local functionals are low cost and the absence of static correlation error caused by HF exchange. The cost advantage can be significant; in our experience it is up to three orders of magnitude in plane wave codes and an order of magnitude in codes that use Gaussian basis functions. The absence of static correlation error can be a significant advantage for many kinds of systems including excited states and systems containing transition-metal elements. However, local functionals also have important disadvantages, and for the purposes of this article we single out their significant underestimation of solid-state band gaps1 (known since the early days of KS–DFT5) and their large errors for many kinds of molecular excited states.6 For these reasons, one often uses functionals with a portion of HF exchange (called hybrid functionals), despite their cost. For example, band gaps in semiconductors are often calculated with screened exchange functionals that add HF exchange at small interelectronic distances but not at large distances;7,8 in our experience that can bring the cost disadvantage in plane wave codes down from three to two orders of magnitude. For molecules it is more common to use functionals with a distance-independent percentage X of HF exchange; these are called global hybrids, and one often finds the best overall performance with X greater than or equal to 25%,9 even 35–45%.10 While high X ameliorates some problems (e.g., Rydberg states), it is not a panacea; for example, it often makes the functional less accurate for valence-excited states due to static correlation error.10 This situation often drives workers to use other more expensive methods such as GW1,11,12 or EOM-CCSD.13 In the present work we report the design of a new local exchange–correlation functional that gives band gaps of semiconductors and electronic excitation energies of molecules that are as


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accurate as those given by many hybrids. We also show that the improved performance of electronically excited states can be achieved without significant degradation of the accuracy for lattice constants or ground-state molecular energetic properties although molecular geometries become less accurate. The cost savings of the new functional can open up new vistas for simulation of large systems and for simulations that require real-time dynamics or ensemble averaging, where the number of required calculations is large. We note that the present paper follows the popular conventions by which solid-state band gaps are calculated as the difference in energy of the lowest unoccupied crystal orbital and the highest occupied crystal orbital,1,5,7 whereas molecular excitation energies are calculated by time-dependent density functional theory2,6,9 (TDDFT). Furthermore the new functional we present is a reparametrized generalized gradient approximation14 (GGA), which means it should be very easy to port to various software packages since essentially all modern DFT codes support GGA-type functionals. For these reasons, we believe the new functional can be used for new applications without invoking nonstandard methodology and with a minimum of additional coding. The first strategic decision in the present functional design project is to optimize the new functional to semiconductor band gaps rather than to ground-state data, which has been the usual method of parametrization of new functionals. The second key strategic decision is to increase the local exchange functional. This is motivated in part by work15 showing that the errors in Kohn-Sham orbital energies are due to incorrect density dependencies of exchange functionals leading to too repulsive response potentials. Increased local exchange has also been use to improve the accuracy of Rydberg-state excitation energies.16 The new functional is called High Local Exchange 2016 and abbreviated HLE16. In this letter, the performance of the newly designed HLE16 is compared to six other density functionals – five local functionals with the same ingredients, namely spin densities and their gradients, plus one hybrid functional that is widely used for semiconductor band gaps. The local functionals include two popular GGA functionals – PBE17 and BLYP,18,19 a GGA functional especially designed to perform well for solids – PBEsol,20 a recently developed nonseparable gradient approximation optimized to a broad data set in order to have good accuracy for molecules – GAM,21 and another GGA (newer than PBE and BLYP but older than PBEsol and GAM), which was developed by optimization to a different broad data set – HCTH/407.22 The hybrid functional is the screened exchange HSE06 functional23,24 that has 25% HF exchange at short interelectronic distances and 0% HF exchange at long interelectronic distances; this



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functional is very often used in plane-wave calculations, as it is known to often give good predictions for solid-state properties (for example, band gaps and lattice constants). We examined a diverse set of data, which includes molecular energies and bond lengths as well as solid-state energies and lattice constants. The 16 databases tested in this work are shown in Table 1 along with the references25,26,27,28,29,30,31,32,33,34,35,36,37,38 where the database was first developed or was developed in the form used here; the reference values we used were taken from these references. Table S1 of the Supporting Information (SI) gives the expanded form of this table with additional details of the databases. All calculations in the present letter were done using Gaussian0939 or a locally modified version40 of Gaussian09; the solid-state calculations were done using the periodic boundary conditions41 code of Gaussian09. As mentioned already, the new HLE16 exchange–correlation functional is designed to obtain good performance for semiconductor band gaps. After a series of preliminary trials, which also involved use of commonly used GGAs – BLYP and PBE – and a selection of band gap data and molecular data, we elected to make the new functional by changing two parameters in the HCTH/407 functional;22 the details of this functional form are presented in earlier papers about the HCTH/9342 and HCTH/14743 functionals. The first new parameter multiplies all the exchange terms in the HCTH/407 functional; the optimized value of this parameter is 1.25. This parameter improved the band gaps significantly; however, changing only this parameter would make the molecular properties worse. To remedy this, we multiplied all the correlation terms in HCTH/407 by 0.50, which resulted in a more balanced treatment of the various kinds of energetic data against which we tested. In particular, the performances for most of the molecular energetic properties were found to be similar to HCTH/407, as discussed below. We note that this strategy does not attempt to separately estimate exchange and correlation energies, but rather just their sum, a strategy that proved reasonably successful in designing the GAM functional21 and the earlier N1236 and N12-SX8 functionals. Notice that the parameters are only lightly optimized, with values of 5/4 and 1/2. The main point of the letter is not a statistical optimization of a parameterized functional but rather a demonstration that the dependence of band gap on the exchange functional may be used to design a local functional that is useful for band gaps. Whereas previous workers thought that it was necessary to introduce DFT+U with systemdependent parameters or hybrid functionals with expensive nonlocal exchange, this letter shows how a local functional dependent only on spin densities and their gradients can give equally accurate band gaps. In addition, we show that the resulting functional does not make unphysical predictions for other properties, so the improvement of the band gaps does not require an unphysical distortion of the overall physics of solids or molecules.


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The performance of the HLE16 functional for band gaps is illustrated for the SBG31 database,25,26 which has 31 semiconductor band gaps, in Table 2, which also compares the results to those of six other density functionals including HCTH/407. All calculations in this table are for consistently optimized lattice constants, except for the ones that are indicated by “//”; where consistently optimized means that the geometry is optimized with the same functional as used for the energy calculation. As is well known,1,5,44,45,46 local functionals tend to underestimate band gaps, and this is illustrated in Table 2 where five of the local density functionals – PBE, BLYP, PBEsol, GAM, and HCTH/407 – are seen to underestimate band gaps considerably compared to the reference values, resulting in large mean unsigned errors (MUEs) of approximately 1 eV. On the other hand, the screened exchange functional HSE06 of Scuseria and coworkers shows very good performance with an MUE of only 0.31 eV, but at a higher computational cost than the local functionals. We find that with the new local functional, HLE16, we can achieve a mean unsigned error of a similar magnitude (0.30 eV). For almost every solid in the SBG31 database, this new functional, due to the increased exchange, increases band gaps compared to HCTH/407, thereby making them closer to the experimental values. This is achieved at the computational cost of a GGA, and hence it has the advantage of being easily and affordably applied to large systems such as metal-organic frameworks, zeolites, etc., where the high cost of hybrid functionals can make them impractical for such applications. The lattice constants predicted by HLE16 for these solids are not improved, although they are also not significantly worse than in the original HCTH/407 functional (see Table S11 of the SI, which shows mean unsigned errors for 34 semiconductor lattice constants of 0.157 Å, as compared to 0.155 Å for HCTH/407). The HLE16 functional systematically underestimates lattice constant. To show that the increased HLE16 band gaps are not simply an artifact of decreased lattice constants, we did single-point calculations using HLE16 on PBEsol and HSE06 optimized geometries – these two functionals being the best performing functionals for lattice constants in Table S11. The new results have been added to Table 2 under columns HLE16//PBEsol and HLE16//HSE06. The MUEs with these two methods are 0.36 and 0.37 eV, respectively, and even though the MUEs worsen a bit compared to HLE16, the results are still better than any previous gradient approximation. This clearly shows that increased band gaps predicted by HLE16 are not an artifact of inaccurate geometries. In our earlier work26 on the SBG31 database, various local density functionals (M11-L,47 M06-L,48 TPSS,49 SOGGA11,50 PBE, revTPSS,51 LSDA,52,53,54,55 SOGGA,28 and PBEsol) were tested, and the meta-GGA functional, M11-L, with an MUE of 0.54 eV was found to be the best local functional for band gaps. The HLE16 functional introduced here improves over the best local functional tested in that work. (Three of the functionals used in this work (PBE, PBEsol,



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and HSE06) were also in Ref. 26, and the small difference between the present work and Ref. 26 for these three functionals can be attributed to small differences in the optimized structures.) The TMOBG4 database of band gaps in transition metal oxides29 was also used to test the HLE16 functional; the results are in Table S7 of the SI. We find that the band gap of MnO significantly improves compares to other local functionals, and for the remaining systems band gaps show a small improvement. However, HSE06 gives a much better band gap compared to HLE16 for these systems. It was also shown in a recent work56 that with hybrid density functionals – B3LYP,18,19,57,58 B3PW91,18,19,57 and HSE – band gaps for these systems could be significantly improved. Besides SBG31 and TMOBG4 databases we performed calculations with HCTH/407 and HLE16 on 10 compounds – BN, BSb, AlN, MgO, CaS, CaSe, CaTe, SrS, SrSe, and SrTe (they are in Table S15 of the SI) – that were not in the validation presented above or in our previous work because of a lack of experimental band gaps or very high band gaps. Even in the absence of experimental values, we can compare HCTH/407 to HLE16 for these 10 compounds; we find that for every solid except one (CaTe), the band gaps increase as one goes from HCTH/407 to HLE16. Just as the band gaps depend on the orbital energy differences, the molecular electronic excitation energies calculated using TDDFT also depend on orbital energy differences, and local density functionals are well known6 to underestimate these excitations. For accurate predictions, the use of hybrid functionals has been required, and again this raises the computational cost although the increase is smaller for molecular calculations with Gaussian basis functions than for solid-state calculations with plane waves. To test whether the new local functional presented here provides improved predictions of electronic excitation energies for molecules as well as semiconductors, we examined the performance for the EE23 database,35,37 which comprises 18 n → π* and π → π* valence excitation energies of 14 unsaturated organic molecules (14 excitations to the lowest excited singlet and 4 excitations to the lowest excited triplet), two Rydberg excitation energies (the lowest singlet and triplet of the water molecule), and three intramolecular or intermolecular charge transfer (CT) excitations. The results are in Table 3. Except for a few cases we see that the excitation energy increases when one goes from HCTH/407 to HLE16, and therefore it becomes closer to experiments. The table shows that all functionals give similar moderate errors for valence excitations; the MUEs lie between 0.30 and 0.36 eV. However the performances of the functionals vary more for Rydberg and charge transfer (CT) excitations. The HLE16 functional is the best performing local functional for the Rydberg states and gives an MUE of 0.32 eV, which is only 0.05 eV more than the computationally expensive hybrid functional, HSE06. In a previous article,35 functionals with an


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MUE of 0.31–0.36 eV were defined as “moderately successful;” by this criterion, HLE16 is the only GGA that is moderately successful for both valence and Rydberg states. Besides the Rydberg excitations of water molecule presented in Table 3, we computed Rydberg excitations of six atoms (H, He, B, Ne, Al, and Ar) using all the seven functionals. They are reported in Table S16 of the SI and HLE16 turns out to be the best performing gradient approximation in the table; the error being dominated by inert gases. The performance of HLE16 for CT states is similar to that of other local functionals, which – as is well known – are all poor for CT excitations. The behavior of the π → π* triplets of naphthalene, furan, and hexatriene with HLE16 is surprising (give values much lower than HCTH/407). These systems are highly multireference ones though, and we attribute the breakdown in part to the breakdown of TDDFT for some multireference triplet excitations, an effect studied previously by others.59 A full discussion of this issue also involves a discussion of stability and spin contamination and of partial ionic character in some of the contributing configurations in these conjugated systems; such a full discussion is beyond the scope of this letter. In past work37 on the EE23 database, it was found that both multiconfiguration pair-density functional theory (MC-PDFT)60 and complete active space perturbation theory (CASPT2)61 show good performances for valence, Rydberg, and charge transfer excitations. The PBE062 global hybrid exchange-correlation functional also provided reasonably good performance for valence and Rydberg states (with MUEs of 0.29 eV for both) but not for most CT states. The key point here is that the HLE16 functional, although being less computationally expensive for large systems, gives useful accuracy for both valence and Rydberg states. Good performance for Rydberg states is very important when one treats whole spectra, because other local functionals systematically underestimate their excitation energies; this causes them to mix with lower-energy valence states and makes the computed low-energy manifold of excited states unphysical. Since the HLE16 functional gives excellent performance for the SBG31 database (both at consistently optimized geometries and at more accurate geometries) and useful accuracy for both valence and Rydberg molecular excitation energies, it is interesting to see whether the parameterization of this functional occurs at the expense of its performance on ground-state molecular energies and geometries. We therefore compared the seven functionals shown in Tables 2 and 3 for ground-state molecular properties and other solid-state properties besides band gaps. A molecule-by-molecule comparison is provided in the SI, and MUEs are compared in Table 4. We see that for the atomization energies database of main-group molecules (the AE6 database), the MUE of the HLE16 functional is still better than two of the other local functionals, and for the transition-metal average bond energies database, TMABE10, its still within the range



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of those for the other local functionals, although all functionals perform worse for transition metals than for the main group. For the diverse barrier heights database, DBH24, which contains forward and reverse barrier heights of a representative set of 12 diverse chemical reactions, HLE16 has the second best performance of the six local functionals, and its performance is comparable with the more expensive hybrid functional, HSE06. However, the performance of HLE16 on next two molecular energy databases and the three molecular bond length databases is poor, demonstrating that the present functional is not a universal functional. For cohesive energies of solids, the performance of HLE16 is better than two of the other local functionals. The main purpose of examining the additional databases in Table 4 was to show that by adjusting the local exchange to improve band gaps, we did not make a functional that is awful for all other properties (which would indicate that the functional is unphysical). The results for bond energies, barrier heights, and cohesive energies shows that for these other energies properties the results are not out of line with the performance of some previously designed GGAs that have found practical uses in the past. For lattice constants of solids, HLE16 is comparable to HCTH/407, and only slightly worse than BLYP and GAM, but none of these can be recommended for calculating lattice constants. Keeping in mind that the accuracy of HLE16 on bond lengths of molecules and lattice constants of solids in low, it is recommended to use this functional mainly for single-point calculations to obtain band gaps and excitation energies, although, because of its low cost, it could also be useful for simulations requiring a very large number of energy calculations. It is possible that HLE16 could be improved by adding post-SCF molecular mechanics terms to improve the geometrical predictions; these would be damped dispersion terms63 to improve noncovalent interactions and short-range repulsive terms to lengthen the predicted bond lengths. However, we will not pursue this here. In summary, we have presented a local density functional named HLE16 that gives good performance for band gaps of semiconductors and valence and Rydberg excitation energies of organic molecules, without unreasonably compromising the accuracy for ground-state cohesive energies, bond energies, or barrier heights. The proton affinities and molecular geometries do not turn out to be very accurate, and therefore, this functional would need to be further improved to make it a general purpose functional for both ground-state and excited-state properties. We conclude that the HLE16 local functional is important for two main reasons: (i) it provides a practical method for calculating band gaps of semiconductors at low cost, and (ii) it opens a new strategy to design improved exchange-correlation functionals, either at the GGA level, as here, or by adding more ingredients (e.g., HF exchange, kinetic energy density, or post-SCF molecular mechanics terms) with a better gradient approximation as the starting point.


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! ASSOCIATED CONTENT Supporting Information. Computational details, tables for additional databases, and geometries. This material is available free of charge via the Internet at http://pubs.acs.org. ! AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected]; [email protected] Notes The authors declare no competing financial interest. ! ACKNOWLEDGMENT We thank Kaining Duanmu and Haoyu Yu for helpful discussions, and Soumen Ghosh and Chad Hoyer for help with the excitation energy database. 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. PV acknowledges partial funding from the Richard D. Amelar and Arthur S. Lodge Fellowship.



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Table 1. The Databases Considered in This Work Database Description Solid-state energies SBG31 Semiconductor Band Gaps SSCE8 Solid-State Cohesive Energies TMOBG4 Transition-Metal Oxide Band Gaps Solid-state lattice constants MGLC4 Main Group Lattice Constants ILC5 Ionic Lattice Constants TMLC4 Transition Metal Lattice Constants SLC34 Semiconductor Lattice Constants Molecular energies AE6 Atomization Energies TMABE10 Transition-Metal Average Bond Energies DBH24 Diverse Barrier Heights PA3 Proton Affinities NCCE31 Noncovalent Complexation Energies EE23 Excitation Energies Molecular bond lengths MGHBL9 Main Group Hydrogenic Bond Lengths MGNHBL11 Main Group Non-Hydrogenic Bond Lengths DGH4 Bond lengths for diatomic molecules (geometries) with one or more heavy atoms a

Opt indicates optimization and SP indicates single-point calculations.


Reference(s) 25, 26 27, 28 29 27, 28 27, 28 27, 28 25, 26 30 21, 38 31 32 33, 34 35, 37 28 28, 36 21

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Table 2. Semiconductor Band Gaps (eV) SBG31 C Si Ge SiC BP BAs AlP AlAs AlSb GaN β-GaN GaP GaAs GaSb InN InP InAs InSb ZnO ZnS ZnSe ZnTe CdS CdSe CdTe MgS MgSe MgTe BaS BaSe BaTe MSEb MUEc a

PBE 4.13 0.67 0.00 1.40 1.37 1.23 1.76 1.55 1.29 1.70 1.55 1.71 0.19 0.00 0.00 0.68 0.00 0.00 0.88 2.16 1.19 1.14 1.11 0.48 0.61 3.38 1.69 2.55 2.39 2.05 1.61 –1.11 1.11

PBE//PBEsol 4.17 1.37 0.01 1.38 1.34 1.20 1.71 1.49 1.30 1.93 1.76 1.75 0.51 0.13 0.00 0.92 0.00 0.01 0.96 2.36 1.39 1.38 1.23 0.58 0.78 3.50 1.74 2.68 2.31 1.96 1.49 –1.04 1.04

BLYP 4.37 0.94 –0.17 1.75 1.67 1.47 2.07 1.83 1.06 1.54 1.41 1.16 0.00 0.00 0.00 0.19 0.00 0.00 0.94 1.88 0.88 0.72 0.91 0.26 0.32 3.35 1.66 2.41 2.52 2.21 1.81 –1.15 1.15

PBEsol 4.05 0.53 0.01 1.27 1.23 1.09 1.56 1.37 1.22 1.86 1.70 1.62 0.41 0.06 0.00 0.83 0.00 0.01 0.84 2.22 1.25 1.29 1.08 0.45 0.67 3.34 1.69 2.58 2.15 1.83 1.38 –1.13 1.13

GAM 4.44 0.91 0.01 1.91 1.58 1.47 2.28 1.84 1.17 1.70 1.57 2.15 0.09 0.00 0.00 1.04 0.00 0.00 1.16 2.42 1.12 0.87 1.28 0.40 0.41 3.75 1.71 2.36 2.53 2.15 1.67 –0.99 0.99

HCTH/407 4.32 1.00 0.01 1.72 1.60 1.49 2.29 2.00 1.34 1.98 1.81 1.87 0.12 0.00 0.00 0.74 0.00 0.00 1.41 2.45 1.33 1.06 1.39 0.64 0.57 3.85 2.00 2.67 2.94 2.57 2.15 –0.89 0.89

HLE16 4.93 1.06 1.03 1.84 1.55 1.54 2.51 2.19 1.95 3.89 3.63 2.58 1.85 0.99 1.14 2.12 0.67 0.25 3.22 4.10 2.83 2.39 2.78 1.85 1.62 5.53 2.86 3.88 3.58 2.97 2.59 0.04 0.30

HLE16//PBEsol 4.57 1.43 0.01 2.08 1.94 1.82 2.90 2.51 1.81 2.69 2.50 2.46 0.80 0.20 0.55 1.31 0.00 0.01 2.98 3.80 2.53 2.12 2.57 1.68 1.46 5.11 2.72 3.62 3.68 3.08 2.63 –0.23 0.36

HLE16//HSE06

D = direct band gap; I = indirect band gap. bMSE = mean signed error. cMUE = mean unsigned error. ACS Paragon Plus Environment

4.58 1.45 0.00 2.08 1.95 1.83 2.92 2.53 1.71 2.70 2.50 2.35 0.70 0.07 0.55 1.22 0.00 0.00 2.96 3.65 2.40 1.95 2.48 1.59 1.34 5.07 2.69 3.54 3.72 3.15 2.73 –0.27 0.37

HSE06 5.42 1.22 0.54 2.32 2.12 1.89 2.44 2.16 1.85 3.15 2.97 2.42 1.18 0.70 0.66 1.61 0.36 0.28 2.55 3.37 2.27 2.16 2.10 1.36 1.49 4.48 2.58 3.49 3.21 2.80 2.22 –0.24 0.31

Expt. 5.48 1.17 0.74 2.42 2.40 1.46 2.51 2.23 1.68 3.50 3.30 2.35 1.52 0.73 0.69 1.42 0.41 0.23 3.40 3.66 2.70 2.38 2.55 1.90 1.92 5.40 2.47 3.60 3.88 3.58 3.08 0.00 0.00

Typea I I I I I I I I I D D I D D D D D D D D D D D D D D I I I I I


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Table 3. Excitation Energies (eV) for EE23 Database as Calculated Using Linear-Response TDDFT EE23 acetaldehyde acetone formaldehyde pyrazine pyridazine pyridine pyrimidine s-tetrazine ethylene butadiene benzene

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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 Naphthalene B3u, π → π* 3 B2u, π → π* 1 Furan B2, π → π* 3 B2, π → π* 1 Hexatriene Bu, π → π* 3 Bu, π → π* Water Singlet, 2px → 3s Triplet, 2px → 3s pNA Intramolecular CT,1A1, π → π* DMABN Intramolecular CT,1A1, π → π* B-TCNE Intermolecular CT,1A, π → π* a MSE valence b MUE valence a MSE Rydberg b MUE Rydberg a MSE charge transfer b MUE charge transfer a

1

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 4.02 2.79 5.88 3.90 4.42 2.27 6.36 6.01 3.55 4.36 1.35 –0.33 0.36 –1.01 1.01 –1.07 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 3.99 2.82 5.77 3.95 4.42 2.31 6.22 5.91 3.50 4.34 1.32 –0.32 0.34 –1.14 1.14 –1.10 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 4.03 2.87 5.92 4.03 4.42 2.36 6.42 6.09 3.56 4.35 1.36 –0.32 0.35 –0.95 0.95 –1.07 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 4.04 2.75 5.89 3.87 4.43 2.22 6.53 6.31 3.56 4.40 1.37 –0.29 0.33 –0.78 0.78 –1.05 1.05

MSE = Mean signed error. bMUE = Mean unsigned error.


HCTH/407 4.19 4.28 3.85 3.58 3.20 4.39 3.82 1.89 7.33 5.40 5.14 3.78 4.03 2.71 5.87 3.83 4.42 2.18 6.43 6.28 3.55 4.39 1.31 –0.32 0.35 –0.84 0.84 –1.07 1.07

HLE16 4.43 4.50 4.19 3.88 3.55 4.73 4.10 2.14 7.73 5.60 5.15 4.23 4.07 2.01 6.07 2.96 4.52 1.29 7.71 7.33 3.46 4.46 0.95 –0.25 0.35 0.32 0.32 –1.20 1.20

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 4.35 2.52 6.04 3.55 4.65 1.90 7.16 6.71 4.10 4.74 1.98 –0.18 0.30 –0.27 0.27 –0.55 0.66

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 4.00 3.11 6.06 4.17 4.93 2.69 7.40 7.00 4.30 4.57 3.59 0.00 0.00 0.00 0.00 0.00 0.00

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Table 4. MUEs for All Considered Molecular and Solid-State Databases Database PBE BLYP PBEsol GAM HCTH/407 HLE16 Molecular properties – energies AE6 4.6 1.8 9.7 4.1 2.3 4.3 TMABE10 8.3 7.8 12.7 4.9 6.4 11.0 DBH24 8.2 7.7 10.3 5.0 4.7 4.9 NCCE31 1.2 1.5 1.8 1.0 1.1 2.1 PA3 1.0 2.0 2.1 2.6 2.1 16.8 0.36 0.34 0.35 0.33 0.35 0.35 EE23 – valence EE23 – Rydberg 1.01 1.14 0.95 0.78 0.84 0.32 EE23 – CT 1.07 1.10 1.07 1.05 1.07 1.20 Molecular properties – bond lengths MGHBL9 0.011 0.010 0.014 0.004 0.003 0.053 MGNHBL11 0.009 0.016 0.006 0.007 0.005 0.072 DGH4 0.021 0.039 0.012 0.037 0.055 0.033 Solid-state properties – energies SSCE8 0.11 0.30 0.33 0.10 0.23 0.27 TMOBG4 2.9 2.9 3.1 2.8 2.8 2.1 1.11 1.15 1.13 0.99 0.89 0.30 SBG31 Solid-state properties – lattice constants MGLC4 0.037 0.069 0.023 0.037 0.110 0.140 ILC5 0.078 0.121 0.020 0.110 0.219 0.142 TMLC4 0.065 0.139 0.019 0.106 0.072 0.113 SLC34 0.095 0.180 0.031 0.158 0.155 0.157 a

HSE06 2.6 5.6 3.8 0.7 0.8 0.30 0.27 0.66

0.002 0.009 0.015 0.10 b 0.6 0.31

0.045 0.023 0.050 0.052

Molecule-by-molecule comparison for each database with each functional is provided in the SI. The reference values that were used to calculate MUEs with each functional are also in the SI. The units for various databases are: AE6, TMABE10, DBH24, NCCE31, and PA3 are in kcal/mol, MGHBL9, MGNHBL11, DGH4, ILC5, MGLC4, TMLC4, and SLC34 are in Å, and EE23, SSCE8, TMOBG4, and SBG31 are in eV. b The TMOBG4 database is using HSE03 and was reported in Ref. 64.



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