How Well Can the M06 Suite of Functionals Describe the Electron

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How Well Can the M06 Suite of Functionals Describe the Electron Densities of Ne, Ne6+ and Ne8+? Ying Wang, Xianwei Wang, Donald G. Truhlar, and Xiao He J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00865 • Publication Date (Web): 07 Nov 2017 Downloaded from http://pubs.acs.org on November 7, 2017

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Journal of Chemical Theory and Computation

How Well Can the M06 Suite of Functionals Describe the Electron Densities of Ne, Ne6+ and Ne8+? Ying Wang1, Xianwei Wang1,2, Donald G. Truhlar,3* and Xiao He1,4* 1

2

State Key Laboratory of Precision Spectroscopy, School of Chemistry and Molecular Engineering, East China Normal University, Shanghai, 200062, China

Center for Optics & Optoelectronics Research, College of Science, Zhejiang University of Technology, Hangzhou, Zhejiang, 310023, China

3

Department of Chemistry, Chemical Theory Center, Nanoporous Materials Genome Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota, 55455-0431, USA 4

NYU-ECNU Center for Computational Chemistry at NYU Shanghai, Shanghai, 200062, China

*Corresponding authors. Email: [email protected], [email protected]

Abstract The development of better approximations to the exact exchange-correlation functional is essential to the accuracy of density functionals. A recent study suggested that functionals with few parameters provide more accurate electron densities than recently developed many-parameter functionals for light closed-shell atomic systems. In this study, we calculated electron densities, their gradients, and Laplacians of Ne, Ne6+, and Ne8+ using 19 electronic structure methods, with reference to the CCSD results. Two basis sets, namely aug-cc-pωCV5Z and aug-cc-pV5Z, are utilized in the calculations. We found that the choice of basis set has a significant impact on the errors and rankings of some of the selected methods. The errors of electron densities, their gradients, and Laplacians calculated with the aug-cc-pV5Z basis set are substantially reduced, especially for Minnesota density functionals, as compared to the results using the aug-cc-pωCV5Z basis set (a larger basis set utilized in earlier work (Medvedev et al. Science 2017, 355, 49-52)). The rankings of the M06 suite of functionals among the 19 methods are greatly improved with the aug-cc-pV5Z basis set. In addition, the performances of the HSE06, BMK, MN12-L, and MN12-SX functionals are also improved with the aug-cc-pV5Z basis set. The M06 suite of functionals is capable of providing accurate electron densities, gradients, and Laplacians using the aug-cc-pV5Z basis set, and thus it is suitable for a wide range of applications in chemistry and physics.

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1. Introduction Kohn-Sham density functional theory (DFT) has become the most popular computational method for quantum mechanical modeling of electronic structure in chemistry, physics, and biology. A foundation of this approach is the theorem1 showing the existence of an exact exchange-correlation (XC) functional that can give the exact energy and one-electron density of any system. However, the exact functional is essentially unknowable,2 hence it is essential to develop good density functional approximations (DFAs), and much effort has been devoted to this task. Approaches have been developed using a combination of fundamental principles and constraints on the one hand,3 and adjustment of parameters to experimental data and results of high-level wave function calculations on the other,4-8 with strategies varying in terms of how much they rely on each of these kinds of sources of knowledge of the functional. A recent study by Medvedev et al.9 suggested that recent developments of improved DFAs are leading to more accurate energies only at the expense of predicting inaccurate densities. They calculated 14 closed-shell atoms and atomic cations with 2, 4, or 10 electrons (including such chemically unusual systems as Ne8+) by 128 DFT functionals, with all-electron coupled-cluster singles and doubles (CCSD-full)10,

11

results as the benchmarks. Three descriptors, the local electron

density (RHO), the gradient norm of RHO (GRD), and the Laplacian of RHO (LR) were calculated to demonstrate how accurately the functionals can describe the atomic electron

density.

The

authors

ranked

the

functionals

by

the

average

maximum-normalized errors over the atoms and cations, and they concluded that older functionals with less parameters provide more accurate electron densities than newer ones that are more highly parameterized to energies. In addition, the accuracy in describing the atomic electron density increases as one adds more independent variables to the DFA, moving along from the local density approximation (LDA)12 to generalized gradient approximations (GGAs)13, 14 to meta-GGAs15-17 to hybrid18, 19 functionals. In their evaluation, most of the best-performing functionals are few-parameter hybrid functionals. They particularly criticized several Minnesota


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Journal of Chemical Theory and Computation 3

functionals, e. g., M06-L20, M06-2X21, M0621, M1122, M11-L23, MN12-SX24, and MN12-L25. In followup work, Kepp26 questioned whether the energies and densities really behave differently, and he found a linear correlation between errors in energies (E) and densities (ρ) for most functionals, excluding some of the Minnesota functionals. He concluded, based on studying the very same systems as in the previous work, “M06-2X is very ‘exact’ when put on actual E[ρ] paths and much more exact than MP2, PBE0, or TPSSh. In table 2 in [ref. 9], M06-2X was ranked very low mainly because of the Laplacian of ρ and thus claimed to be off-path, despite E[ρ] and ρ being excellently on-path.” The original Minnesota functionals did not enforce smoothness restraints, and that may be partly responsible for the lower accuracy on Laplacians; this may be remedied the smoothness restraints5-8 in newer Minnesota functionals that are not the subject of the present investigation. In a more recent study, Gould27 calculated the density changes of open-shell atoms Li, C, and F, and pointed out that many Minnesota functionals give much better results than do some other functionals; he emphasized two critical points: (i) there is "a delicate balancing act" in evaluating densities, to strike a balance between chemically relevant density differences and "near-core behavior." (ii) He supports the view that a method that "sacrifices unimportant core densities but captures chemically important core density differences" does not constitute an example of "straying from the path toward the exact functional." Gould also concluded “Many ‘Minnesota functionals’ fare much better here than in a recent analysis of electron densities.” In another related study, Brorsen et al.28 investigated the accuracy of the electron densities in the bonding regions and the atomization energies for small molecules with atomic numbers 1, 3, 6–9, 11, and 14–17. They found that the errors in densities and atomization energies are decoupled for hybrid-GGAs, but not for GGAs and meta-GGAs. In table 1 of ref. 28, they report the best functionals, and the recent well parameterized (to energies) M06-2X and SOGGA11-X are in this table. Table 2 of ref. 28 has functionals with largest errors and some of the Minnesota functionals are in that table as well. They concluded, "Although the debate over the optimal strategy for current functional development is not likely to end soon, the ultimate goal is still to



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obtain both accurate densities and accurate energies," and we agree with that. A related but independent work found that various kinds of density functionals give similar accuracy in predicting dipole moments (which are the first moments of the electron density) for a variety of neutral organic and inorganic molecules.29 Unlike most of the other work on densities, this study included transition-metal-containing molecules and other inherently multiconfigurational systems; the test set included molecules with atomic numbers 1, 3, 5–9, 11–17, 19, 21–24, 26, 29, 32–35, 37–40, 47, 53, 57, 72, 81, and 82. Dipole moments were also tested (for 53 exchange-correlation functionals) in a paper on BeF radical, BeO, BeO+ radical cation, HF, KF, KO−, KOH, LiF, LiO radical, LiO−, MgO, MgS, NaO radical, and NaO− and CaO; the more recent functionals were neither best nor worst, but rather somewhere in the middle of the pack.30 Examples of the findings are that the well parametrized GAM5 functional gives slightly more accurate dipole moments than the “lightly parameterized” revTPSSh31 and PBE032 functionals; these are gradient and hybrid gradient functionals. Similarly the recent and well parametrized M06-L20, a meta-GGA, does better than the older TPSS meta-GGA. Comparing range-separated functionals, the newer more highly parameterized N12-SX24 gives more accurate dipole moments than the older, less highly parameterized CAM-B3LYP33. One may conclude that there is a danger in overgeneralizing. Most recently, Hait and Head-Gordon developed a database of 200 benchmark dipole moments of molecules based on CCSD(T) calculations34, and they assessed the performance of 76 popular density functionals for molecular dipole moment prediction. As demonstrated in their study, the Minnesota functionals developed with smoothness restraints clearly reduced the errors on the dipole moments as compared to their predecessors that were obtained without imposing smoothness restraints. For instance, the revM06-L8, MN15-L6, and MN157 functionals give much lower errors than the M06-L20, MN12-L25, and MN12-SX24 functionals, respectively. In addition, Minnesota functionals N1235, SOGGA11-X36, and PW6B9537 are the top performers among GGA, hybrid GGA and hybrid meta-GGA functionals, respectively, which


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give the lowest regularized root-mean-square errors (RMSE) of dipole moments in each category. Among 76 tested functionals, PW6B95 gives the lowest RMSE of 4.9%, which is very close to the error of 4.4% calculated by CCSD. The results from the study of Hait and Head-Gordon show that the development of Minnesota functionals is on the right path. Although some of the above studies give arguments why a poor performance for core densities need not be a serious detriment to the usefulness of modern density functionals and although the studies of dipole moments show reasonable results for the first moments of densities predicted by modern density functionals, it is still interesting, especially in light of the findings of Kepp, to further examine the question of the performance of various density functionals for the systems in Ref. 9. Are the densities of modern functionals really inaccurate? Note that the basis set used in Medvedev et al.’s and Kepp’s work is aug-cc-pωCV5Z, but in Gould’s study the aug-cc-pV5Z basis set was utilized. The impact of different basis sets on the accuracy of electron density has not been discussed in previous studies. In the present work, we investigate the influence of basis sets on the accuracy of electron densities by different electronic structure methods. Two basis sets, namely aug-cc-pωCV5Z and aug-cc-pV5Z, are utilized in the present calculations. The previous study also demonstrated that the Ne and Ne6+ atoms cause the largest errors in electron densities, which have the most significant impact on the ranking of the density functionals. In addition, it was suggested that most of the Minnesota functionals give relatively poor performance on these systems. Therefore, in this work, we performed calculations on the densities of Ne, Ne6+ and Ne8+ using a variety of density functionals, and focused on the performance of Minnesota functionals on describing the electron densities with the two different basis sets. This means that the present study examines the validity of the previous conclusions for the systems where their evidence was strongest, even though those systems are least relevant to chemically interesting questions.

2. Computational approaches

The radial distribution functions (RDFs) of electron density (ρ, where r is a ACS Paragon Plus Environment


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point

in

real

(|∇| = (∇ =

space),

+

+

+

the +

gradient

),

of

the

electron

density

and the Laplacian of the electron density

) of Ne, Ne6+, and Ne8+, were calculated by 19

electronic structure methods. The CCSD-full10, 11 result is taken as the reference. We define RHO, GRD and LR as RDFs of electron density, its gradient, and its

Laplacian, respectively. For spherically symmetric atoms, RHO is 4π , GRD

is 4π |∇|, and LR is 4π ∇ . For a property P (which may be RHO, GRD,

or LR), the RMSD of a given method F with reference to the CCSD result is calculated by, RMSD =

∑& !'( r! − ##$% r! )

where N is the number of radial points, and r! denotes the property P at radial

point r! calculated by the method F. In this study, 800 radial points evenly spaced between 0 to 4 Å are employed. Because the RMSDs for these three properties are on different scales, we normalize the RMSDs by three median errors from the previous study,9 which are 0.009909618 e/Å for RHO, 0.091951402 e/Å2 for GRD, and 1.433784114 e/Å3 for LR, respectively. Furthermore, the RMSD calculated in the previous study9 was over the radius ranging from 0 to 10 Å. The electron density of neon atom is nearly zero after 4 Å. Therefore, we rescale our results by dividing by 1.58 ( ≈

10/ 4 ) to match the results calculated by the previous study.9

We selected 17 density functionals and 2 wave function methods to compare the performance of electronic structure methods for describing the electron density with two different basis sets (namely, aug-cc-pωCV5Z and aug-cc-pV5Z). The aug-cc-pωCV5Z basis set includes tight functions (whose presence is denoted by “ωC”) for capturing core-core and core-valence correlations. We included in our tests some representative Minnesota functionals, namely M06-2X21, M06-L20, M0621, MN12-L25, M1122, and MN12-SX24, as well as the nonseparable gradient approximation N1235. In addition, based on the previous results9, we selected for comparison several methods that give diverse performances for the accuracies of the


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electron densities of neon atom and its cations; this includes the good-performing methods MP238, PBE032, B97-139, HF40, TPSS41, HSE0642, and B3LYP14, 43, 44, the medium-performing

functionals

ωB97X45,

PBE46,

and

ωB9745,

and

the

poor-performing functionals BLYP14, 43 and BMK47. All the calculations were carried out using the Gaussian 09 program48. The grid with 99 radial shells and 590 angular points per shell was utilized for all DFT calculations. We emphasize that we use the same normalized error measure as previous authors (in order to make comparisons to their conclusions) even though this is not necessarily the best measure of accuracy of density functionals. The normalized error of the density, gradient, and Laplacian for aug-cc-pωCV5Z is based on the average error of many functionals; it is not based on the weights of their contributions to the total energy. The main goal of the study by Medvedev et al. was to study how well the electron densities obtained with various functionals reproduce the density, gradient and Laplacian from CCSD calculations. The Laplacian gives the largest error; therefore, their rankings of functionals (especially for Minnesota functionals) are largely determined by the errors of Laplacian. We shall see in the discussion below not only that the conclusions drawn using previously used error measures depend on the basis set, but also that other possible ways to measure errors in densities also lead to quite different conclusions. This was also pointed out by Medvedev et al.49 who commented that “if one neglects its errors in the derivatives of the density, M06-2X would be one of the best-performing ... functionals in our study.”

3. Results and discussion Table 1 shows the maximum-normalized errors among electron density, its gradient, and its Laplacian after averaging over Ne, Ne6+, and Ne8+ from calculations using 19 selected methods, which are listed in the order of increasing error with the aug-cc-pωCV5Z basis set. As shown in Table 1, when the aug-cc-pV5Z basis set is utilized, most of the functionals, especially the Minnesota functionals, give much smaller errors than with the other basis set. The maximum normalized-errors of



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M06-L, BMK, M06, M06-2X, M11, N12, MN12-L, and MN12-SX are all reduced by more than 40%, as compared to the results calculated using the aug-cc-pωCV5Z basis set. The errors of PBE0, HSE06, B3LYP, ωB97X, PBE and BLYP decreased, in those cases by 13–25%. The average maximum normalized error for these 19 methods decreased from 3.4 to 1.7. The errors between these two basis sets are very close for MP2, HF, TPSS, and ωB97. We note that we can reproduce the results calculated by Medvedev et al.9 with the aug-cc-pωCV5Z basis set for 15 methods presented in Table 1. However, we obtained much lower errors for M06-L, M06 and M11 using aug-cc-pωCV5Z, such that the error of M06-L is smaller than ωB97, BLYP, and BMK in our calculations. Meanwhile, the error of M06-2X is slightly increased with this basis set. Table S1 of the Supplementary Information and Figure 1 give the rankings of the 19 selected methods based on their maximum-normalized errors given in Table 1. MP2 and PBE0 are consistently ranked as the top two methods using both of the basis sets. The rankings of M06 suite of functionals are much improved when the aug-cc-pV5Z basis set is employed. The M06, M06-L, and M06-2X functionals are ranked in fourth, fifth, and sixth places with aug-cc-pV5Z, respectively, as compared to the fourteenth, tenth, and fifteenth places with aug-cc-pωCV5Z. Instead of being poor-performing functionals in describing the electron densities of Ne atom and its cations using the aug-cc-pωCV5Z basis set, the M06 suite of functionals are among the top-performing functionals when the aug-cc-pV5Z basis set is utilized. The performance of the M06 suite of functionals is completely overturned, outperforming other functionals which were top-performing functionals when the aug-cc-pωCV5Z basis set was adopted, including B97-1, HF, TPSS, and B3LYP. As shown in Fig. 1, the rankings of B97-1, TPSS, B3LYP, ωB97X, PBE, ωB97, BLYP, M11 and N12 functionals all became worse, among which the changes of the rankings are most remarkable for TPSS and ωB97. On the contrary, the rankings of HSE06, M06-L, BMK, M06, M06-2X, MN12-L, and MN12-SX functionals are all improved. Among them, M06, M06-2X, M06-L and BMK improved most significantly in the rankings. HSE06 now becomes the third best-performing method,


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and both MN12-L and MN12-SX outperforms BLYP and ωB97 when the aug-cc-pV5Z basis set is utilized. The maximum-normalized errors for most of the methods originate from the deviations of LR. To give a full spectrum of errors from electron density, its gradient, and its Laplacian, we list the average normalized errors over Ne, Ne6+, and Ne8+ for these three properties in Table S2 of the Supplementary Information. In addition, Table 2 shows the rankings of 19 selected methods based on the errors in Table S2. For RHO, the errors with the aug-cc-pV5Z basis set decreased, as compared to those using the aug-cc-pωCV5Z basis, by more than 9.9% for all Minnesota functionals (see Table S2). The differences between the two basis sets for other functionals are relatively smaller (below 5.8%). As a result, the rankings of the M06 suite of functionals are improved when the aug-cc-pV5Z basis set is used (see Table 2). We find that M06-2X, M06, and M06-L are ranked in second, fourth, and eighth places among 19 selected methods with aug-cc-pV5Z. Therefore, as also noted by Kepp26 and Medvedev et al.49, M06-2X and M06 are among the best-performing functionals in the description of electron densities with both of the two basis sets. Although the other four functionals from Minnesota (namely, the MN12-SX, MN12-L, and M11 Minnesota functionals and the nonseparable gradient approximation N12) also give smaller errors for RHO, they still ranked at the bottom of the list. For GRD, the errors of seven Minnesota functionals and BMK decreased by more than 27 percent when smaller basis set is used (see Table S2). As shown in Table 2, the rankings of BMK, M06-2X, M06 and M06-L are improved to the fourth, fifth, sixth and ninth places (with aug-cc-pV5Z), respectively, from the tenth, fifteenth, fourteenth and eleventh places with the aug-cc-pωCV5Z basis set. As shown in Table S2, the lowest error of GRD is 1.04 given by PBE0 among all functionals using aug-cc-pV5Z. The errors calculated by M06, M06-2X and M06-L are between 1.11-1.20. Hence, instead of being the medium to poor-performing functionals in description of GRD with the aug-cc-pωCV5Z basis set, the M06 suite of functionals become the top-performing functionals using the aug-cc-pV5Z basis set. Nevertheless, the other three Minnesota functionals of M11, MN12-L, and MN12-SX and the



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nonseparable gradient approximation N12, still in the bottom four even with much reduced errors in GRD. For LR, the errors using aug-cc-pV5Z are significantly decreased, as compared to those with aug-cc-pωCV5Z calculations. As can be seen from Table S2, the average normalized error of the Laplacian over 19 selected methods using aug-cc-pV5Z is reduced by 59% from the result of aug-cc-pωCV5Z. In particular, the errors of BMK, the nonseparable gradient approximation N12, and six Minnesota density functionals decreased by more than 52% when the smaller basis set is used. As a result, the rankings of M06-L, BMK, M06, M06-2X and MN12-L are substantially improved for LR with the aug-cc-pV5Z basis set. It is especially interesting that the M06-L functional gives the third-lowest error of 0.944 for LR among 19 selected methods (see Table S2), with MP2 and HF being the top two with errors of 0.404 and 0.700, respectively. The performance of the BMK, M06, and M06-2X functionals are also above the average with errors of 1.12-1.20, which are lower than the mean error of 1.32. Hence, the M06 suite of functionals performs significantly better on LR with the aug-cc-pV5Z basis set. Furthermore, the MN12-L functional outperforms three non-Minnesota functionals, including BLYP, ωB97X, and ωB97. The M11, MN12-SX, ωB97, and N12 functionals are the lowest ranked functionals for calculating LR with aug-cc-pV5Z. Overall, six Minnesota density functionals, the nonseparable gradient approximation N12, and the BMK functional give much smaller errors for RHO, GRD and LR, when the basis set is changed from aug-cc-pωCV5Z to aug-cc-pV5Z. The errors for these three properties calculated by M06, M06-2X, and M06-L are all decreased significantly, and thus the rankings are improved in all three categories among 19 selected methods. For M11, MN12-L, MN12-SX, and N12, although the errors are also decreased substantially, the rankings for these four functionals do not have dramatic changes, because the errors of those are still larger than most of the other functionals. The errors of B3LYP, B97-1, BLYP, PBE, TPSS, ωB97, and ωB97X have smaller decreases than those of Minnesota functionals when the basis set is changed from aug-cc-pωCV5Z to aug-cc-pV5Z, but their rankings mostly got worse


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due to the large improvement of the M06 suite of functionals and BMK in rankings. Table S3 of the Supplementary Information gives single-point energies of Ne, Ne6+ and Ne8+ using HF and CCSD-full with aug-cc-pV5Z and aug-cc-pωCV5Z, respectively. As shown in Table S3, the HF energies using the two basis sets are very close (within 0.0001 a.u.), while the CCSD-full energies differ by 0.02-0.03 a.u.. Therefore, adding the tight basis functions in the core region will mainly lower the correlation energy. Figures S1-S3 of the Supplementary Information shows the plots of RHO, GRD, and LR of Ne atom, Ne6+ and Ne8+ cations calculated by CCSD-full method with the two basis sets. The black and red lines (in the left panels) represent the results for aug-cc-pωCV5Z and aug-cc-pV5Z, respectively. The panels on the right (Figs. S1-S3) show the differences between the results of these two basis sets. As one can see from Figs. S1-S3, the results from these two basis sets overlap with each other, and the difference between them is about three orders of magnitude smaller than the absolute value of the curve. Therefore, those three properties calculated by CCSD with aug-cc-pV5Z and aug-cc-pωCV5Z are very close to each other for Ne atom and its cations. Figure 2 show the deviations of RHO, GRD, and LR of Ne atom (with reference to the CCSD-full results) calculated by PBE0, M06-L, M06, M06-2X, and BLYP using the two basis sets. As shown in Fig. 2, the errors of RHO between 0.3-0.5 Å (around the n=2 shell) changed more notably when different basis sets were used. The errors of the M06 suite of density functionals are all decreased upon changing from aug-cc-pωCV5Z to aug-cc-pV5Z, and this contributes the most to the reduction of errors given in Table S2, while the error of PBE0 and BLYP did not change much between these two basis sets. As for GRD in Fig. 2, the errors for the M06 suite of functionals also clearly decreased, especially for distances 0.03-0.1 Å from the nucleus (corresponding to the inner shell of 1s core electrons). The largest absolute errors of M06-2X and M06 near the nucleus (< 0.1 Å) are around 2 e/Å2 when calculated with aug-cc-pωCV5Z, while the errors become much smaller (~0.8 e/Å2) when aug-cc-pV5Z was applied. On the other hand, the deviations for PBE0 and



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BLYP are also reduced, but less than for the M06 suite of density functionals. Furthermore, Fig. 2 shows that the errors of LR are all decreased substantially, especially for the M06 suite of functionals. The largest deviation of LR is over 100 e/Å3 around 0.05 Å from M06-2X /aug-cc-pωCV5Z calculation, which is reduced to ~ 13 e/Å3 when using the aug-cc-pV5Z basis set. Such a deviation is even smaller than the counterpart from BLYP/aug-cc-pV5Z calculation (~ 20 e/Å3), and also very close to the result given by PBE0. The decreases of deviations of LR for BLYP and PBE0 are much less than those for the M06 suite of functionals. Consequently, the errors of GRD and LR for the M06 suite of functionals become smaller than those for BLYP when the aug-cc-pV5Z basis set is used. Especially noteworthy is that the largest LR errors occurs at the same radius (around 0.05 Å) for different functionals, while the radial distances where the largest GRD errors are observed vary among the methods (see Figure 2). For instance, the largest GRD errors occur around 0.05 Å from the nucleus for M06 and M06-2X, but for BLYP, the largest GRD error shows up at r = 0.25 Å. Figures S4 and S5 of the Supplementary Information show the deviations of RHO, GRD, and LR for Ne6+ and Ne8+ cations calculated by PBE0, M06-L, M06, M06-2X, and BLYP using the two basis sets, respectively. As shown in Fig. S4, the errors of RHO of Ne6+ changed very slightly for all five methods, and the curves with the two basis sets are very close for each functional. Nevertheless, the errors of GRD of Ne6+ are clearly decreased for M06 and M06-2X when the aug-cc-pV5Z basis set is used, especially for distance around 0.03 Å from the nucleus (1s shell). Moreover, similar to Ne atom, the changes of LR of Ne6+ (see Fig. S4) are the largest when different basis sets are used, especially for the M06 suite of functionals. The deviations of M06 and M06-2X are all reduced by ~ 50 e/Å3 at 0.08 Å when using the aug-cc-pV5Z basis set. Fig. S5 shows the error of RHO for Ne8+ cation calculated by M06-2X decreased significantly at around 0.005 Å. The deviations of GRD of Ne8+ are also significantly reduced at 0.03 and 0.07 Å for the M06 suite of functionals. Similar to the results for Ne atom, the errors of GRD of Ne8+ also decreased for PBE0 and BLYP, but not as


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much as those for the M06 suite of density functionals. As shown in Fig. S5, the error changes of LR for Ne8+ cation between the two basis sets are also consistent with the results for Ne and Ne6+. Figures S6-7 (in the Supplementary Information) and Figure 3 compare the errors of RHO, GRD, and LR, respectively, for Ne atom calculated by M06-L, M06, M06-2X, and MN12-L using the aug-cc-pωCV5Z and aug-cc-pV5Z basis sets. Figures S8-10 and S11-13 of the Supplementary Information give the same comparison for Ne6+ and Ne8+ cations, respectively. As shown in those figures, the differences occurred more dramatically for the 1s shell electrons (< 0.1 Å from the nucleus). The changes of errors for GRD and LR are much larger than that for RHO between the two basis sets. As for the error changes of RHO for Ne atom (see Fig. S6), the deviations of M06-L and MN12-L changed more significantly than M06-2X and M06 between the two basis sets. However, Fig. S8 shows that the error changes of RHO of Ne6+ are very slight for those four functionals, while the error changes of RHO of Ne8+ are more significant for M06, M06-2X, and MN12-L than that for M06-L when different basis sets are used (see Fig. S11). Moreover, Fig. S7, Fig. 3, Figs. S9-10, and Figs. S12-13 clearly show that the errors of GRD and LR calculated by all four Minnesota functionals decreased substantially for Ne, Ne6+ and Ne8+, respectively, when the aug-cc-pV5Z basis set was applied. The trends of error changes in RHO, GRD and LR between the two basis sets are similar for Ne, Ne6+ and Ne8+. Owing to lack of smoothness restraints, the exchange enhancement factors of some Minnesota functionals show some oscillations, and these can affect the derivatives in the kinetic energy density.23, 49, 50 This lack of smoothness has been recognized for a long time and is ameliorated by smoothness constraints in some later functionals.5-8 It was suggested49 that it may cause oscillations of electron density; the present results are consistent with nonsmoothness causing inaccuracy close to the core region when tight basis functions are employed. This is the main reason that the Minnesota functionals ranked low in describing the electron density among functionals investigated in the study of Medvedev et al.9. However, in this study we show that, when the tight basis functions are removed as in aug-cc-pV5Z, even



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though one retains a very extended basis in the valence and diffuse regions, the errors of electron density and its derivatives are significantly reduced with Minnesota functionals. The finding that one must use care in selecting basis sets that well match to the method, especially for core electrons, is not unique to density functional theory; it has also been observed in wave function theory, where one can obtain spurious results if one does not appropriately match the basis set to the quantity calculated.51 From one point of view a true evaluation of a method can only be accomplished by using a complete basis set. But if methods perform well with generally useful and widely used basis sets, that is also an important consideration and is probably more important for practical work. In optimizing a density functional, one can try to make the functional as universal as possible, and in our experience this involves tradeoffs. For example, if one optimizes the functional to give more accurate transition metal bond energies, one may lose some accuracy in calculating barrier heights in organic chemistry. Nevertheless there has been good progress in optimizing density functionals to give accurate results for a range of properties.4-8 One could include core densities in one's objective function, and in this way could no doubt improve them, just as recent density functionals give much more accurate absolute atomic energies than older less parameterized ones.37, 52-54 One should not judge a density functional on any one attribute, especially absolute atomic energies or core densities, which cancel out for many chemical properties. Nevertheless, if one does evaluate core densities, it is important to realize that the conclusions may be basis set dependent. In a very recent study, Mayer et al. pointed out that when a finite basis set is used, there does not exist a DFT functional that could be used with a single Kohn-Sham determinant to strictly reproduce the electron density from a full configuration interaction calculation using the same finite basis set.55 This shows that one cannot rigorously test DFT functionals by comparing the density to that from a high-level wave function calculation with the same basis set, and the results of all such tests should therefore be considered with appropriate mathematical caution.


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Another discussion point we want to mention for completeness is that it would be interesting to check the accuracy of the CCSD results used as a benchmark. A recent study by Mezei et al.56 demonstrated that when the composite CCSDTQ densities of many-electron systems are taken as reference, the M06-L functional provides clearly better atomic electron densities than the LDA method, which is contradictory to the conclusion given by Medvedev et al.9. In

addition,

Mezei

et

al.

also

investigated

the

density-driven

exchange-correlation energy (DDXCs) errors of various density functionals.56 They found that the rankings of DDXC errors do not always follow the same order as the rankings of density errors for different functionals. This is partially because the locally positive or negative DDXC errors arising from the deviations of electron densities cancel each other in the exchange-correlation energy integral. Furthermore, the DDXC errors are smaller with meta-GGA electron densities than those with GGA densities, and the DDXC errors of the PBE exchange-correlation energy are generally smaller for molecules than those for atoms. The role of the percentage of HF exchange in mean absolute DDXC errors for various density functionals was also investigated by Mezei et al.56. For both atoms and molecules, the mean absolute DDXC errors in the PBE exchange-correlation energy is the lowest when one includes about 40% of HF exchange. Therefore, they concluded that mixing with 40% of HF exchange strikes a compromise between the overall density errors and the DDXC errors.

4. Conclusion The electron density (RHO), its gradient (RHO), and its Laplacian of RHO (LR) of Ne, Ne6+, and Ne8+ are calculated by 19 electronic structure methods in this study. The CCSD-full results were taken as the references. Two different basis sets of aug-cc-pωCV5Z and aug-cc-pV5Z are utilized in the calculations. The choice of the basis sets has a great impact on the rankings of various electronic structure methods on the accuracy of describing atomic electron densities. In particular, we conclude that Minnesota functionals show much better performance for RHO, GRD, and LR



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when the basis set is changed from aug-cc-pωCV5Z to aug-cc-pV5Z. The M06 suite of functionals is capable of providing accurate electron density, its gradient, and its Laplacian for Ne, Ne6+ and Ne8+ with the aug-cc-pV5Z basis set even though these properties are very far from any data or goals used for parameterization. The errors significantly decreased for the Minnesota functionals, when the basis set is changed from aug-cc-pωCV5Z to aug-cc-pV5Z. As a result, the rankings of the M06 suite of functionals on RHO, GRD, and LR are all improved substantially. Overall, the M06-2X and M06 functionals are among the best-performing functionals in the description of atomic electron densities for both of the two basis sets. As for the description of the gradients of electron densities, instead of being medium to poor-performing functionals, as was found with the aug-cc-pωCV5Z basis set, the M06 suite of functionals become the top-performing functionals using the aug-cc-pV5Z basis set. Furthermore, the M06 suite of functionals performs significantly better on LR with the aug-cc-pV5Z basis set. M06-L functional with aug-cc-pV5Z is among the top-performing functionals for LR, and M06 and M06-2X functionals are among the medium-performing functionals for LR. The MN12-L functional also outperforms three non-Minnesota functionals, including BLYP, ωB97X, and ωB97, for LR calculations using the aug-cc-pV5Z basis set. The reduction of errors between the two basis sets mainly occurs for the 1s shell electrons (arising from the tight basis functions employed in the aug-cc-pωCV5Z basis set), which have less impact on the chemically relevant bonding regions. The M06 suite of functionals is capable of providing accurate electron densities, gradients, and Laplacians using the aug-cc-pV5Z basis set. Especially noteworthy is that the M06 suite of functionals was not fit to chemical properties that depend explicitly on the electron density. However, the M06 suite of functionals naturally gives a good description of electron densities for atoms studied in this work. Therefore, we conclude that the development of the M06 suite of functionals is a step in the right direction towards a density functional that gives good energies based on a realistic prediction of the density. We agree with previous assessments that it is important to obtain both accurate energies and accurate densities. Evaluation of density functionals


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can be very subtle, though, and a full evaluation cannot be based on any one property, such as densities of light closed-shell atoms, especially when one attempts to evaluate density functionals that were designed to be broadly accurate for chemical and physical properties, including transition metals, which are missing in some of the discussions of energies and densities. We agree with previous comments that the electron densities in the core are not the key quantities for chemistry, but it is still encouraging that we confirm the conclusion of Kepp that recent functionals are not as bad for core densities as may have been concluded from previous work. ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ The rankings of 19 selected methods based on their maximum-normalized errors of RHO, GRD and LR, average normalized errors, the RDFs of RHO, GRD and LR of Ne, Ne6+, and Ne8+ calculated by CCSD-full with the aug-cc-pωCV5Z and aug-cc-pV5Z basis sets, the deviations of RDFs for RHO, GRD and LR (with respect to CCSD-full) calculated by various functionals, are provided. (PDF)

AUTHOR INFORMATION Corresponding Authors Email: [email protected], [email protected]

ORCID Ying Wang: 0000-0002-4359-3753 Xianwei Wang: 0000-0003-4471-426X Donald G. Truhlar: 0000-0002-7742-7294 Xiao He: 0000-0002-4199-8175 Notes The authors declare no competing financial interest.



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ACKNOWLEDGMENTS This work was supported by the Ministry of Science and Technology of China (No. 2016YFA0501700), National Natural Science Foundation of China (No. 21673074), Youth Top-Notch Talent Support Program of Shanghai, NYU-ECNU Center for Computational Chemistry at NYU Shanghai, and U.S. Department of Energy, Basic Energy Sciences (Award DE-SC0012702 to the Inorganometallic Catalysis Design Center). This work was also supported in part by the National Science Foundation of the U.S.A. under grant no. CHE–1464536. The authors are grateful to Pragya Verma, Roberto Peverati, Haoyu Yu, and Yan Zhao for many helpful discussions.

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Density-Driven Exchange-Correlation Energy Errors in Approximate Density Functional Calculations. J Chem Theory Comp 2017, 13, 4753-4764.


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Table 1. Maximum-normalized errors among electron density, its gradient, and its Laplacian after averaging over Ne, Ne6+, and Ne8+ from calculations with the aug-cc-pωCV5Z and aug-cc-pωCV5Z basis sets. Method

aug-cc-pωCV5Z aug-cc-pV5Z Differencea

MP2 0.76 0.76 0.3% 1.15 -19% PBE0 1.43 B97-1 1.46 1.35 -7.9% 1.9% HF 1.56 1.59 1.8% TPSS 1.56 1.59 -25% HSE06 1.62 1.22 -13% B3LYP 1.78 1.53 ωB97X 1.99 1.65 -17% -17% PBE 2.10 1.75 -41% M06-L 2.14 1.27 -5.4% ωB97 2.36 2.24 -16% BLYP 2.59 2.18 BMK 2.95 1.37 -54% -60% M06 3.09 1.25 -65% M06-2X 3.82 1.33 -49% M11 5.25 2.70 -41% N12 6.53 3.84 MN12-L 8.95 2.16 -76% -82% MN12-SX 12.0 2.16 b -48% ME 3.36 1.74 a The percentage change between the errors from calculations using the aug-cc-pωCV5Z and aug-cc-pV5Z basis sets. Difference = ./0123324567 − ./01233248#567 /./01233248#567 , where ./0123324567 and ./01233248#567 denote the errors calculated using the basis sets of aug-cc-pV5Z and aug-cc-pωCV5Z, respectively. b ME: mean error. “Difference” in this row is based on the ME values.



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Table 2. The rankings of two wave function theories (namely, HF and MP2) and 17 selected density functionals based on their average normalized errors over Ne, Ne6+, and Ne8+ for electron density, its gradient, and its Laplacian. RHO

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

GRD

LR

aug-cc-pωCV5Z

aug-cc-pV5Z

aug-cc-pωCV5Z

aug-cc-pV5Z

aug-cc-pωCV5Z

aug-cc-pV5Z

MP2 PBE0 HSE06 B97-1 M06-2X M06 BMK ωB97X TPSS B3LYP M06-L PBE HF ωB97 BLYP MN12-SX M11 MN12-L N12

MP2 M06-2X PBE0 M06 HSE06 B97-1 BMK M06-L ωB97X B3LYP TPSS PBE HF ωB97 BLYP MN12-SX MN12-L M11 N12

MP2 HF TPSS PBE0 B97-1 HSE06 B3LYP PBE ωB97X BMK M06-L BLYP ωB97 M06 M06-2X M11 MN12-SX MN12-L N12

MP2 PBE0 HSE06 BMK M06-2X M06 HF B97-1 M06-L B3LYP TPSS PBE ωB97X BLYP ωB97 M11 MN12-L MN12-SX N12

MP2 HF TPSS PBE0 B97-1 B3LYP HSE06 PBE ωB97X BLYP M06-L ωB97 BMK M06 M06-2X M11 N12 MN12-L MN12-SX

MP2 HF M06-L PBE0 HSE06 TPSS B3LYP BMK B97-1 M06 M06-2X PBE MN12-L BLYP ωB97X M11 MN12-SX ωB97 N12


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Figure 1. The rankings of 17 functionals based on their maximum-normalized errors among electron density, its gradient, and its Laplacian after averaging over Ne, Ne6+, and Ne8+.



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Figure 2. The deviations of RDFs for electron density (top), its gradient (middle) and Laplacian (bottom) of Ne atom calculated by PBE0, M06-L, M06, M06-2X, and BLYP functionals using the aug-cc-pωCV5Z (left panels) and aug-cc-pV5Z (right panels) basis sets, respectively, with reference to the CCSD-full results.


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Figure 3. The deviations of LR of Ne atom calculated by M06-L, M06, M06-2X, and MN12-L functionals using the aug-cc-pωCV5Z and aug-cc-pV5Z basis sets, respectively, with reference to the CCSD-full results.



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TOC graphic


How Well Can the M06 Suite of Functionals Describe the Electron

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