the ranking in the tables remains mostly unchanged upon removal of Ne-based systems from
the comparison (see supplementary materials for
details). Full tables of normalized errors are
available in the supplementary materials. Another
way to ensure the fairness of the ordering is to
average the normalized errors over all descriptors
and atoms for a given number of electrons (ne = 2,
4, or 10). The resulting table, sorted by maximum
error among these three classes, is available in
the supplementary materials; its top is populated
by L1 functionals and its bottom by L2 ones, with
L2 methods showing the worst performance for
Two conclusions are immediately apparent.
First, functionals constructed with little or no
empiricism tend to produce more accurate electron densities than highly empirical ones. Second, at the level of little or no empiricism, the
accuracy of the density tends to increase along
with the complexity of the density functional
approximation as we ascend the Jacob’s Ladder
(a hierarchy of approximate functionals classified
by their inputs) from LDA to GGA to meta-GGA
to hybrids ( 19).
The list of best methods (Table 1) includes
four post-HF methods, 20 hybrid functionals,
and three meta-GGAs. No DFT functional outperformed the MP2 method. The best hybrid
functionals are mainly three- or one-parameter,
as proposed by Becke ( 21, 22). They often include
PBE, mPW, or B88 exchange along with PBE,
PW91, or LYP correlation. All the best meta-GGAs
(TPSS, TPSSm, and SCAN) are the result of thorough constraint-satisfaction work ( 5, 18, 23, 24).
They demonstrate that good accuracy for electron densities (at least for atoms and atomic
ions) can be achieved within the meta-GGA form,
as can accurate energetics ( 5, 18, 23, 24).
Hybrid functionals with 100% exact exchange
are formally hGGAs but differ in that their exchange and correlation components are strongly
misbalanced, so we denote them as hGGA*s.
They excellently reproduce LR and are able to
describe the two-electron ions relatively well.
However, they expectedly fail for systems with
higher correlation contribution.
Three functionals (HCTH407, MOHLYP, and
revB3LYP) demonstrated good if uneven performance: Their maximum normalized errors for
RHO were relatively large (>2.1), whereas the GRD
and LR ones were among the lowest for studied
functionals. If we were to average the maximum
errors over descriptors, these methods would appear among the best, which is notable because
MOHLYP and HCTH407 are GGAs.
In the second list (L2, Table 2), the functionals
published before 1985 (including all tested LDAs)
and the modern highly parameterized methods
tend to produce density distributions that visi-
bly deviate from the exact ones. Unlike L1, the
maximum normalized error of the L2 function-
als for electron densities positively correlates
with their rung of the Jacob’s Ladder (see sup-
plementary materials). Despite their excellent per-
formance for energies and geometries, we must
suspect that modern highly parameterized func-
tionals need further guidance from exact con-
straints, or exact density, or both. For energies
and energy differences, the density-driven error
( 25) could be compensated by the functional er-
ror [the error that the approximate functional
would make when applied to the exact density,
as defined in ( 25)]. Although their deviation from
the exact density is considerable, relative errors in
density distributions are small enough to make
the trends in resulting integral properties (e.g.,
atoms-in-molecules charges) similar to trends from
MP2 even for a complicated case of a supra-
molecular stereoelectronic effect ( 26).
We also tested our approach to error estima-
tion for electron densities on the well-studied
hybrid functional PBE0. The optimal fraction of
HF in it was estimated theoretically to be 25% for
molecules and their constituent atoms ( 27). We
found that the dependence of atomic normalized
errors on HF fraction in the four systems demon-
strating the largest errors in our study (Be, Ne, F5+,
Ne6+) closely follows the underlying physical
model ( 27) (Fig. 2 and fig. S5): For Be, F5+, and
Ne6+, optimum values are close to 25% as pro-
posed for chemically correlated cases, and for
weakly correlated Ne the optimum value is close
to 50% (although 25% is still reasonable; see Fig.
2). Be, F5+, and Ne6+ have relatively small energy
gaps between their occupied 2s and unoccupied
2p orbitals, whereas Ne has a huge gap between
its occupied 2p and unoccupied 3s orbitals. As
the gap increases, the correlation energy becomes
less important compared to the exchange energy,
and the optimum fraction of exact exchange in-
creases toward 100%. The overall optimum value
is 26.3%, which is very close to the one typically
used, and we obtain it without any energy fitting.
We conclude that the latest trend of devel-
oping functionals using unconstrained forms
leads to unphysical electron densities despite the
excellent energy-related performance of these
methods. The meta-GGA functionals constructed
by the constraint-satisfaction approach produce
much better electron densities, and hybrid func-
tionals with physically sound formulations show
the best performance. Our findings suggest that
the long-neglected electron density will play a
crucial role in the future of DFT development.
Minimization of deviations from the exact elec-
tron density, along with constraint satisfaction
and controlled energy fitting, may result in more
accurate approximations to the exact functional,
providing new computational methods for mo-
lecular and solid-state physics. Overall, DFT is in
need of new strategies for functional development.
REFERENCES AND NOTES
1. R. O. Jones, Rev. Mod. Phys. 87, 897–923 (2015).
2. P. Hohenberg, W. Kohn, Phys. Rev. 136, B864–B871 (1964).
3. M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062–6065 (1979).
4. R. Peverati, D. G. Truhlar, Philos. Trans. R. Soc. A 372,
5. J. Sun et al., Nat. Chem. 8, 831–836 (2016).
6. J. Poater, M. Duran, M. Solà, J. Comput. Chem. 22, 1666–1678
7. G. I. Csonka, N. A. Nguyen, I. Kolossváry, J. Comput. Chem. 18,
8. A. D. Boese, N. C. Handy, J. Chem. Phys. 114, 5497–5503 (2001).
9. P. J. Wilson, T. J. Bradley, D. J. Tozer, J. Chem. Phys. 115,
10. R. J. Boyd, J. Wang, L. A. Eriksson, Recent Adv. Density Funct.
Methods 1, 369–401 (1995).
11. V. Polo, E. Kraka, D. Cremer, Theor. Chem. Acc. 107, 291–303
12. A. D. Bochevarov, R. A. Friesner, J. Chem. Phys. 128, 034102
13. A. A. Rykounov et al., Acta Crystallogr. B 67, 425–436 (2011).
14. V. Tognetti, L. Joubert, J. Phys. Chem. A 115, 5505–5515 (2011).
15. I. Grabowski et al., Mol. Phys. 112, 700–710 (2014).
16. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria,
M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone,
B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato,
X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng,
J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda,
J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao,
H. Nakai, T. Vreven, J. A. Montgomery Jr., J. E. Peralta,
F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin,
V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari,
A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi,
N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken,
C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev,
J. B. Foresman, J. V. Ortiz, J. Cioslowski, D. J. Fox, Gaussian 09,