www.sciencemag.org/content/355/6320/49/suppl/DC1

Supplementary Materials for

Density functional theory is straying from the path toward the exact

functional

Michael G. Medvedev,*† Ivan S. Bushmarinov,*† Jianwei Sun, John P. Perdew,† Konstantin A. Lyssenko†

*Corresponding author. Email: medvedev.m.g@gmail.com (M.G.M.); ib@xrlab.ru (I.S.B.) †These authors contributed equally to this work.

Published 6 January 2017, Science 355, 49 (2017)

DOI: 10.1126/science.aah5975

This PDF file includes: Materials and Methods

Figs. S1 to S5

Table S1

Captions for data S1 to S4

References (30–96)

Other supplementary material for this manuscript includes the following:

Data S1 to S4 (Excel format)

2

Materials and Methods:

Calculations CCSD-full, MP4(sdq)-full, MP3-full, B3LYP*, revB3LYP, B97-1, B97-2, MN12-L,

MN12-SX, N12, N12-SX, mPW1PW91, mPW1LYP, mPWPBE, mPW1PBE, mPW3PBE, B1B95, Xα, HISS, APFD, BC, TPSSLYP1W, PBEh1PBE, PBELYP1W, PBE1KCIS, MPWLYP1W, HSE06 and O3LYP calculations were performed using Gaussian09 (16). SCAN, MS0, MS1, MS2, MVS calculations were performed using modified Gaussian03 (30). All other calculations were performed using GAMESS-US (17) on the IBM BlueGene/P supercomputer at Moscow State University’s Faculty of Computational Mathematics and Cybernetics. Big grids were used in both cases: “JANS=2” in GAMESS-US and “integral=ultrafine” in Gaussian. Analysis

Conversion of wfn files into RDF tables (50000 radial points, range: 0-10 Å) was done by Multiwfn (29) on the Lomonosov supercomputer of the Supercomputing Center of Lomonosov Moscow State University (28). Subsequent analysis was done using python and R (31). All descriptors are used as implemented in Multiwfn, and their definitions, along with definitions of RMSD and normalized errors, may be found below. Median errors in RMSDs that were used to calculate normalized errors are 0.009909618 for RHO, 0.091951402 for GRD, and 1.433784114 for LR. Sensitivity of the results to the exclusion of certain atoms:

The Ne and Ne+6 atoms demonstrate the highest errors in our study and therefore have the largest impact on the ordering of the functionals in Tables 1 and 2. The exclusion of Ne, Ne+6 and Ne+8 from consideration replaces PBE1KCIS, B97-2, B1B95, TPSSh, TPSSm, SCAN and B3LYP with revB3LYP, TPSSLYP1W, revTPSS and HCTH407 in L1, but does not remove any functional except Xα from L2. L2 is expanded with PW91VWN, PW91PZ81, BVWN, BPZ81, BVWN1RPA, ωB97, B97-K, M05-2X, BMK. Thus, HCTH407 appears to be the only GGA competing with mGGAs and hGGAs.

All conclusions of the manuscript hold up on this truncated dataset. Some additional highly empirical and LDA-correlation-based functionals are added to the upper half of L2, replacing the “best of the worst” Xα. The lower half of L1 (with Max NE > 1.95) is reordered, which is unsurprising given the close competition between the best functionals. Notably we can see that the performance of L2 functionals is not a result of some catastrophic failure in the description of Ne atom and ions. Correlation of the functional accuracy with the Jacob’s Ladder rung

If we denote the rungs of the Jacob’s Ladder by integers (1 for LDA, 2 for GGA, 3 for meta-GGA, and 4 for hybrid) as defined in (19), with ab initio methods excluded, the correlation coefficient of “Max NE” with “Rung” is -0.47 in L1 and +0.44 in L2.

3

Definitions

��� = ����� = 4 � × � ���Ȃ ���� ��� = 4 � × � �� �� ��,����� ��

�,

where ηi is the occupation number of orbital i, ϕ is an orbital wavefunction, χ is a basis function. C is the coefficient matrix: The element of the i-th row and j-th column corresponds to the expansion coefficient of orbital j with respect to basis function i.

��� = ���∇��� = 4 � × !"��"# $� + !"��"& $� + !"��"' $�

(� = ��)���� = 4 � × !"���"#� + "���"&� + "���"'� $

For a property P, atom a, and method f:

�*+�,,-,. = Ȃ /,0,1�234,0,5567�2389:3;< = , where N is the number of radial points.

The errors for different properties are on different scales, so to put them on the same one, we normalize them by the median error for a given property:

*>?@,,-,. = �*+�,,-,.median, �*+�,,-,. In the Additional Data table S1, we list for every functional its maximal median-

normalized error, defined as follows: HI#*>?@,,. = max- *>?@,,-,.

In the Additional Data table S2, we list for every functional its mean median-normalized error, defined as follows: HKIL*>?@,,. = mean- *>?@,,-,.

4

Studied functionals 1. LDAs: SLATER (32), SVWN (33), SVWN1RPA (33), SPZ81 (34), Xα (35); 2. GGAs: BECKE (36), BLYP (37), BOP, BP86, BPBE, BPW91, BPZ81, BVWN,

BVWN1RPA, EDF1 (38), GILL (39), GLYP, GOP, GP86, GPBE, GPW91, GPZ81, GVWN, GVWN1RPA, HCTH407 (8), MOHLYP (40), MPWLYP1W (41), mPWPBE, N12 (42), OLYP, OP86, OPBE (43), OPTX (44), OPW91, OPZ81, OVWN, OVWN1RPA, PBE (45), PBELYP, PBELYP1W (41), PBEOP, PBEP86, PBEPW91, PBEPZ81, PBEsol (46), PBEVWN, PBEX (45), PW91 (47, 48), PW91LYP, PW91P86, PW91PBE, PW91PZ81, PW91VWN, PW91X (47), revPBE (49), RPBE (50), SLYP, SOGGA (51), SOGGA11 (52), SOP, SP86, SPBE, SPW91;

3. mGGAs: BMK (53), M06-L (54), M11-L (55), MN12-L (56), MS0 (57, 58), MS1 (58), MS2 (58), MVS (59), PKZB (60), revTPSS (24), SCAN (5, 18), τHCTH (61), TPSS (23, 62), TPSSLYP1W (41), TPSSm (63);

4. hGGA*s: BC (22), LYP, OP (64), P86 (65), PBEC (45), PW91C (47), PZ81, VWN1RPA, VWN5;

5. hGGAs: APFD (66), B1B95 (22), B3LYP (67), B3LYP* (68), B3LYPV1R (67), B3P86 (21), B3PW91 (21), B97-1 (69), B97-2 (9), B97-3 (70), B97-K (53), B98 (71, 72), BHHLYP (73), CAM-B3LYP (74), HISS (75), HSE06 (76–78), M05 (79), M05-2X (80), M06 (81), M06-2X (81, 82), M06-HF (83), M08-HX (84), M08-SO (84), M11 (85), MN12-SX (86), mPW1LYP (87), mPW1PBE, mPW1PW91 (88), mPW3PBE, N12-SX (86), O3LYP (89), PBE0 (90), PBE1KCIS (91), PBEh1PBE (92), revB3LYP (93), SOGGA11X (94), τHCTHhyb (61), TPSSh (23), ωB97 (95), ωB97X (95), X3LYP (96).

Replacement of VWN5 (which is used as the local correlation in VWN-listed functionals) by VWN3 does not affect the electron density because the RPA correlation energy per electron of a uniform electron gas of a given density (VWN3) is about 0.5 eV lower than the beyond-RPA correlation energy (VWN5), over a wide range of high- or valence-electron densities. Then the VWN3 correlation potential will be lower by a constant compared to the VWN5 one, over most of the density. A constant shift of the correlation potential does not change the density. Thus, B3LYPV3, BVWN3, OVWN3, PBEVWN3, PW91VWN3, SVWN3 and VWN3 were also considered, but excluded from the investigation.

5

Fig. S1.

Plots of RHO and errors in it for Be (left) and Ne (right).

6

Fig. S2

Plots of GRD and errors in it for Be (left) and Ne (right).

7

Fig. S3

Plots of LR and errors in it for Be (left) and Ne (right).

8

Fig. S4

The historical trends in maximum and averaged over all systems RMSDs for all studied functionals and properties.

9

Fig. S5

Details of PBE0 analysis. Computed points are connected by linear segments. Dotted lines denote the optimal HF fraction per atom, which minimizes the maximum normalized error.

0

1

2

3

4

0.00 0.25 0.50 0.75 1.00

Hartree�Fock fraction

No

rma

lize

d e

rro

r (M

NA

E)

Descriptor RHO GRD LR

Atom Be0

F+5

Ne+6

Ne0

10

Table S1.

Source table for the barplot (Figure 2). Average Median-Normalized Absolute Errors (MNAEs) for Energy and Electron Density (ED).

11

Additional Data table S1 (separate file)

Maximum median-normalized absolute errors. Sorted by Max column. Colored from 10th to 90th percentiles.

Additional Data table S2 (separate file)

Mean median-normalized absolute errors. Sorted by Max column. Colored from 10th to 90th percentiles.

Additional Data table S3 (separate file)

Average median-normalized absolute errors over all descriptors and systems with a given number of electrons. Sorted by Max column. Each column colored separately from 10th to 90th percentiles.

Additional Data table S4 (separate file)

Median-normalized absolute errors for all studied methods, descriptors and systems.

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