Supplementary Information to accompany
“Workhorse Semilocal Density Functional for Condensed Matter Physics and
Quantum Chemistry”
by
John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Lucian A. Constantin, Jianwei Sun
Physical Review Letters
(Dated: June 25, 2009)
The figures show revTPSS exchange- and exchange-correlation enhancement factors:
E xc
[ n

, n

]
  d
3 rn
 x unif
( n ) F xc
( s , r s
,

,

).
(1)
Here n

3 /( 4
 r s
3
) is the electron density, s is the reduced density gradient,

is the relative spin polarization, and
 
(
  
W
) /
 unif
, where

is the positive orbital kinetic energy density. Note that the exchange enhancement factor F x
( s ,

,

) is the high-density or r s

0 limit of the exchange-correlation enhancement factor F xc
( s , r s
,

,

).
The tables show detailed results on which the error statistics of Table I of our article are based. The results shown here are based on a formula
C(

, 0 )

0 .
59

0 .
9284

2 
0 .
5734

4 
2 .
2384

6
, (2) which differs slightly from the recommended formula of Eq. (4) of our article. There is no difference at all for spin-unpolarized systems (
 
0 ) , and very little difference for any

.
In particular, the AE6 atomization energies of Table SIV are the same, to the number of digits displayed there, for both formulas.
1

Fig. S1. revTPSS exchange enhancement factor versus reduced density gradient, for a spinunpolarized (
 
0 ) region. One- and two-electron regions have
 
0 , and regions of slowly-varying density have
 
1 .
For the corresponding TPSS figure, see Fig. 1 of Ref. S1.
2

Fig. S2. revTPSS exchange enhancement factor versus reduced density gradient, for a spinunpolarized region.

can be large in a density tail region. For the corresponding TPSS figure, see Fig. 1 of Ref. S1.
3

Fig. S3. revTPSS exchange-correlation enhancement factor versus reduced density gradient, for a spin-unpolarized (
 
0 ) region with density parameter r s
.
One- and two-electron regions have
 
0 .
For the corresponding TPSS figure, see Fig. 3 of Ref. S1.
4

Fig. S4. revTPSS exchange-correlation enhancement factor, for a spin-unpolarized (
 
0 ) region with density parameter r s
. Regions of slowly-varying density have
 
1 .
For the corresponding TPSS figure, see Fig. 5 of Ref. S1.
5

Fig. S5. revTPSS exchange-correlation enhancement factor, in a fully spin-polarized (
 region, for any density parameter r s
.
One-electron densities have

1 )
 
0 .
The TPSS curve is shown for comparison.
6

Fig. S6. revTPSS exchange-correlation enhancement factor, for a fully spin-polarized (
 region with density parameter r s
.
Regions of slowly-varying density have

1 )
 
1 .
For the corresponding TPSS figure, see Fig. 6 of Ref. S1.
7

Ag
C
Si
SiC
Ge
GaAs
LiF
LiCl
NaF
NaCl
MgO
ME
MAE
MRE (%)
MARE (%)
Ca
Sr
Ba
Al
Pb
Cu
Rh
Pd
Table SI. Errors for the equilibrium lattice constants (Å) of our 21 test solids, calculated with
BAND using LSDA densities as in Ref. [24]. The zero-point anharmonic expansion (ZPAE) was subtracted from the experimental zero-temperature values to yield the static-lattice
“experimental” values (as in Ref. [24]) in the last column.
Solid LSDA PBE TPSS AM05 PBEsol revTPSS Expt.
Band Band Band VASP Band Band ZPAE
Li
Na
-0.086 -0.020 -0.004
-0.156 -0.013 0.029
0.006 -0.021
0.002 -0.043
-0.024 3.449
0.003 4.210
-0.225 -0.031 -0.029 -0.079 -0.107
-0.263 -0.040 -0.057 -0.079 -0.144
-0.249 0.020 -0.022 -0.038 -0.129
-0.043 5.553
-0.074 6.045
-0.039 4.995
-0.035 0.017 -0.013 -0.016 -0.008 -0.015 4.020
-0.028 0.133 0.082 0.037 0.024 0.034 4.902
-0.078 0.032 -0.020 -0.030 -0.034 -0.047 3.595
-0.038 0.036 0.010 -0.020 -0.013 -0.011 3.793
-0.039 0.068 0.028 -0.003 0.001 0.001 3.875
-0.054 0.092 0.030 -0.002 -0.003 -0.006 4.056
-0.011 0.027 0.025 0.008 0.010 0.016 3.543
-0.013 0.050 0.035 0.015 0.015 0.021 5.416
-0.013 0.036 0.024 0.008 0.014 0.016 4.342
-0.018 0.119 0.081 0.038 0.035 0.039 5.640
-0.033 0.108 0.075 0.034 0.023 0.035 5.638
-0.051 0.099 0.063 0.075 0.039 0.047 3.964
-0.088 0.087 0.038 0.063 0.002 0.031 5.056
-0.077 0.126 0.126 0.107 0.050 0.095 4.579
-0.101 0.134 0.137 0.120 0.036 0.105 5.566
-0.016 0.071 0.053 0.048 0.039 0.049 4.184
-0.079 0.055
1.67 1.39
0.033
0.98
0.014 -0.010
0.81 0.77
0.011
0.079 0.065 0.047 0.039 0.038 0.036
-1.67 1.19 0.70 0.30 -0.19 0.22
0.75
8

Table SII. Jellium surface exchange (
 x
) and exchange-correlation (
 xc
) energies in erg/cm
2
, for bulk density parameter r s
, evaluated on LSDA densities as in Ref. [12]. The AM05 xc values for r s
= 2, 3, 4 are from Ref. [11]. The RGE2 (regularized second-order gradient expansion) is a GGA defined in Ref. [S2], where its predicted lattice constants are displayed.
The exact value of
 xc
is imperfectly known. We have taken it to be the revTPSS value, but it could instead be the TPSS value, as used in Ref. [12]. r s
LSDA PBE TPSS AM05 PBEsol RGE2 revTPSS exact
 x
2 3036 2436 2553 2934 2666 2622 2657 2624
3 669
4 224
465 498 627 540 523
128 141 201 162 153
532 526
157 157
6 43.5 11.8 15.5 35.4 22.9 19.6
45.8
45.8
-20.9
20.9
-11.9
11.9
28.8
28.8
2.9
2.9
-3.1
3.1
20.6 22
-1.0 MRE(%)
MARE(%) 2.2
 xc
2 3354 3265 3380 3414 3374 3379 3428
3 764
4 261
741
252
772
266
782
270
774
267
773
266
783
268
6 53.0 52.0
-5.6
55.5
-0.8
56.7
0.6
56.3
-0.4
55.7
-0.7
55.4
MRE(%) -2.9 0.0
MARE(%) 2.9 5.6 0.9 0.9 1.2 1.0 0.0
Table SIII. Interpolation and extrapolation constants for jellium surface exchange-correlation
(
 xc
) energies in erg/cm
2
for the formula of Ref. [S3]. The TPSS parameters are newly optimized for better agreement; the residual sum of squares for the 4 points in Table SII is
0.21. The AM05 parameters were obtained by fitting to the 8 data points in Ref. [11]; the residual sum of squares is 0.97.
Method A B C D
RGE2
PBEsol
TPSS
49000 0.6168 -0.4675 0.2060
46525 0.4523 -0.3260 0.1590
53940 0.9733 -0.7929 0.3134 revTPSS 49000 0.5757 -0.4523 0.2150
AM05 49400 0.6197 -0.4820 0.2130
9

Table SIV. Atomization energies of the AE6 molecules, in kcal/mol, calculated selfconsistently from GAUSSIAN using the 6-311+G(3df,2p) basis set and standard geometries, as in Ref. [12].
Molecule LSDA PBE TPSS PBEsol AM05 RGE2 revTPSS Expt.
SiH
SiO
4
S
2
C
3
H
4
C
2
H
2
O
2
C
4
H
8
ME
347.4 313.2 333.7 323.7 330.0 317.3
223.9 195.7 186.7 205.0 205.5 196.3
135.1 114.8 108.7 123.5 123.3 117.9
802.1 721.2 707.5 749.9 752.9 726.9
338.2
185.7
109.1
704.1
754.9 665.1 636.0 698.0 700.0 671.7 632.8 633.4
1304.4 1167.7 1155.5 1218.6 1224.0 1182.1 1153.4 1149.0
77.4 12.4 4.1 35.9 38.7 18.1 3.3
322.4
192.1
101.7
704.8
MAE
MRE(%)
MARE(%)
77.4
17.3
17.3
15.5
3.2
4.2
5.9
1.5
2.4
35.9
8.5
8.5
38.7
9.1
9.1
19.9
4.8
5.3
5.9
1.5
2.7
10

Table SV. Errors for the enthalpies of formation of the 223 molecules of the G3 test set, calculated selfconsistently from GAUSSIAN using the 6-311+G(3df,2p) basis set and standard geometries. For other functionals, see Ref. [7]. By construction, the error of the enthalpy of formation is nearly equal in magnitude and opposite in sign to the error of the atomization energy.
Molecule TPSS revTPSS Expt. kcal/mol
LiH -1.1 -2.96 33.30
BeH -10.2 -10.84 81.70
CH
CH
2
( 3 B
1
)
-3.4
-8.2
-3.68 142.80
-7.24 93.70
CH
2
( 1 A
1
)
CH
3
Methane(CH
4
)
NH
NH
2
Ammonia(NH
3
)
OH
Water(H
2
O)
Hydrogenfluoride(HF)
SiH
2
( 1 A
1
)
SiH
2
( 3 B
1
)
-0.5
-6.3
-4.6
-6.7
-6.1
-1.7
-0.6
3.5
1.3
-5.3
-10.7
0.26 102.30
-4.59 35.00
-3.01 -17.83
-6.64 85.20
-4.11 45.10
1.25 -11.00
2.27 9.40
7.94 -57.80
3.58 -65.10
-7.71 65.20
-12.11 86.20
SiH
3
Silane (SiH
4
)
PH
2
PH
3
Hydrogen sulfide (H
2
S)
Hidrogen chloride (HCl)
Li
2
LiF
Acetylene(C
2
H
2
)
Ethylene(H
2
C=CH
2
)
Ethane(H
3
C-CH
3
)
CN
Hydrogencyanide(HCN)
CO
HCO
Formaldehyde(H
2
C=O)
Methanol (CH
3
-OH)
N
2
Hydrazine(H
2
N-NH
2
)
-11.8
-11.6
-8.8
-8.0
-4.3
-6.1
-1.6
-1.0
4.2
-4.9
-3.1
-3.0
-1.1
1.2
1.2
0.7
-2.6
0.9
-5.0
-14.82
-16.31
-10.03
-9.85
47.90
8.20
33.10
1.30
-2.10 -4.90
-0.08 -22.10
0.42 51.60
2.11 -80.10
2.87 54.35
-2.08 12.52
-3.64 -20.10
-0.97 104.90
-0.17 31.50
5.04 -26.40
-3.08 10.00
-1.17 -26.00
0.96 -48.00
-0.22 0.00
-1.85 22.75
11

NO
O
2
Hydrogenperoxide(HO-OH)
F
2
Carbon dioxide (CO
2
)
Na
2
Si
2
P
2
S
2
Cl
2
NaCl
Silicon monoxide (SiO)
CS
SO
ClO
Chlorine monofluoride (FCl)
Si
2
H
6
Methyl chloride (CH
3
Cl)
Methanethiol (H
3
CSH)
Hypochlorous acid (HOCl)
Sulfur dioxide (SO
2
)
BF
3
BCl
3
AlF
3
AlCl
3
Carbon tetrafuoride (CF
4
)
Carbon tetrachloride (CCl
4
)
Carbon oxide sulfide (COS)
Carbon bisulfide (CS
2
)
Carbonic difluoride (COF
2
)
Silicon tertrafluoride (SiF
4
)
Silicon tetrachloride (SiCl
4
)
Dinitrogen monoxide (N
2
O)
Nitrogen chloride oxide (ClNO)
Nitrogen trifluoride (NF
3
)
PF
3
O
3
F
2
O
Chlorine trifluoride (ClF
3
)
Ethene, tetrafluoro- (F
2
C=CF
2
)
Ethene, tetrachloro- (C
2
Cl
4
)
12
-3.2
0.8
5.6
1.5
-4.9
-7.5
-5.5
-17.9
-5.5
-5.3
-2.5
1.5
3.9
-2.1
9.3
-0.1
-4.2
-4.1
-6.8
-0.2
-6.7
-1.8
-2.4
-3.8
0.0
-7.1
-2.5
-6.0
-8.5
-0.3
16.2
3.4
-12.0
-13.4
-19.4
1.8
-9.0
-14.3
-22.9
-14.8
-6.5
-4.16
-5.62
21.60
0.00
4.21 -32.50
-8.06 0.00
0.79 -94.05
-2.57 34.00
-4.83 141.00
-2.59
-7.56
34.30
30.70
-4.53 0.00
0.66 -43.60
6.59 -24.60
1.20
-4.06
66.90
1.20
-7.36 24.20
-6.66 -13.20
-24.59 19.10
-4.75 -19.60
-3.96 -5.50
-0.80 -17.80
2.49 -71.00
6.53 -271.40
-1.51 -96.30
11.28 -289.00
0.10 -139.70
-6.42 -223.00
-6.67 -22.90
-4.72 -33.10
-8.49 28.00
0.14 -149.10
16.39 -386.00
0.77 -158.40
-12.19
-13.89
19.60
12.40
-23.26 -31.60
-0.35 -229.10
-8.54
-15.77
34.10
5.90
-25.71 -38.00
-15.52 -157.40
-9.73 -3.00

Acetonitrile, trifluoro- (CF
3
CN)
Propyne(C
3
H
4
)
Allene(C
3
H
4
)
Cyclopropene(C
3
H
4
)
Propylene(C
3
H
6
)
Cyclopropane(C
3
H
6
)
Propane(C
3
H
8
)
Trans-1,3-butadiene (C
4
H
6
)
Dimethylacetylene (C
4
H
6
)
Methylenecyclopropane (C
4
H
6
)
Bicyclo[1.1.0]butane (C
4
H
6
)
Cyclobutene (C
4
H
6
)
Cyclobutane (C
4
H
8
)
Isobutene(C
4
H
8
)
Trans-butane(C
4
H
10
)
Isobutane(C
4
H
10
)
Spiropentane(C
5
H
8
)
Benzene(C
6
H
6
)
Difluoromethane(CH
2
F
2
)
Trifluoromethane(CHF
3
)
CH
2
Cl
2
CHCl
3
Methylamine(H
3
C-NH
2
)
Acetonitrile(CH
3
-CN)
Nitromethane(CH
3
-NO
2
)
Methyl nitrite(CH
3
-O-N=O)
Methyl silane (CH
3
SiH
3
)
Formic acid (HCOOH)
Methyl formate (HCOOCH
3
)
Acetamide (CH
3
CONH
2
)
Aziridine (C
2
H
4
NH)
Cyanogen (NCCN)
Dimethylamine ((CH
3
)
2
NH )
Trans ethylamine (CH
3
CH
2
NH
2
)
Ketene (CH
2
CO)
Oxirane (C
2
H
4
O)
Acetaldehyde (CH
3
CHO)
Glyoxal (HCOCOH)
Ethanol (CH
3
CH
2
OH)
Dimethylether (CH
3
OCH
3
)
Thiirane (C
2
H
4
S)
13
-5.6
-10.8
-5.5
-7.3
-6.1
-5.2
-4.0
-5.1
-4.1
-13.3
-13.3
-10.0
-0.8
-6.4
-4.3
-9.1
-3.5
-7.6
-5.8
-4.8
-10.7
-7.7
-5.1
-7.1
-5.6
-6.8
-5.5
-2.6
-7.0
-5.2
-5.6
-7.4
-6.6
-6.6
-6.0
-8.2
-4.4
-2.8
-2.9
-7.1
-8.4
-6.59 -118.40
0.69 44.20
-3.66 45.50
-3.26
-2.50
66.20
4.80
-5.41 12.70
-3.50 -25.00
-2.14
-0.49
-7.84
-6.56
26.30
34.80
47.90
51.90
-2.95
-4.79
37.40
6.80
-1.90 -4.00
-3.09 -30.00
-2.18 -32.10
-8.50 44.30
-1.57 19.70
-7.22 -108.10
-7.05 -166.60
-5.48 -22.80
-5.84 -24.70
-2.29 -5.50
-2.12 18.00
-12.08 -17.80
-12.68 -15.90
-11.93 -7.00
3.16 -90.50
-3.49 -85.00
0.20 -57.00
-7.93
-2.40
-5.22
30.20
73.30
-4.40
-3.07 -11.30
-2.72 -11.40
-6.92 -12.60
-1.27 -39.70
0.61 -50.70
1.51 -56.20
-4.74 -44.00
-8.04 19.60

Dimethyl sulfoxide ((CH
3
)
2
SO)
Ethanethiol (C
2
H
5
SH)
Dimethyl sulfide (CH
3
SCH
3
)
Vinyl fluoride (CH
2
=CHF)
Ethyl chloride (C
2
H
5
Cl)
Vinyl chloride (CH2=CHCl)
Acrylonitrile(CH
2
=CHCN)
Acetone (CH
3
COCH
3
)
Acetic acid (CH
3
COOH)
Acetyl fluoride (CH
3
COF)
CH
3
COCl (acetyl chloride)
CH
3
CH
2
CH
2
Cl (propyl chloride)
Isopropanol (CH
3
)
2
CHOH)
Methyl ethyl ether (C
2
H
5
OCH
3
)
Trimethylamine ((CH
3
)
3
N)
Furan (C
4
H
4
O)
C
4
H
4
S (thiophene)
Pyrrole (C
4
H
5
N)
Pyridine (C
5
H
5
N)
H
2
HS
CCH
C
2
H
3
( 2 A')
CH
3
CO ( 2 A')
H
2
COH ( 2 A)
CH
3
O Cs( 2 A')
CH
3
CH
2
O ( 2 A'')
CH
3
S ( 2 A')
C
2
H
5
( 2 A')
(CH
3
)
2
CH ( 2 A')
(CH
3
)
3
C (t-butyl radical)
NO
2
Methyl allene (C
4
H
6
)
Isoprene (C
5
H
8
)
Cyclopentane (C
5
H
10
) n-Pentane (C
5
H
12
)
Neo pentane (C
5
H
12
)
1,3 Cyclohexadiene (C
6
H
8
)
1,4 Cyclohexadiene (C
6
H
8
)
Cyclohexane (C
6
H
12
)
14
-8.1
-9.9
-6.9
-8.6
-10.3
-10.1
-14.3
-8.0
-5.3
-4.9
-6.9
-3.7
-3.1
-2.2
-2.7
-4.7
-0.9
-4.8
-5.9
-6.8
-2.5
-7.8
-5.6
-5.5
-7.1
-7.3
-6.2
-8.8
-2.7
-9.6
-5.8
-5.1
-7.3
-8.7
-3.2
-2.8
0.1
-7.5
-7.1
-5.0
-5.74
-6.56
-5.70
-5.74
-6.50
-5.66
-13.74
-3.90
-1.15
-1.76
-2.58
-0.48
1.16
2.29
1.05
-3.33 -36.20
-3.53 -11.10
-5.41
-5.53
-8.90
-33.2
-4.66 -26.80
-7.45 8.90
-0.24 43.2
-0.69 -51.9
4.09 -103.4
-2.37 -105.7
-3.79 -58.00
-4.64 -31.50
2.36 -65.2
-4.61 -51.7
-8.08
-3.28
-3.71
-5.7
-8.3
27.50
-4.58
-6.26
25.9
33.6
-5.01 0.00
-2.12 34.20
2.33 135.10
-5.00 71.60
-4.07
-1.16
-2.40
-4.10
4.10
-3.70
29.80
28.90
21.50
12.30
7.90
38.8
18
-18.3
-35.1
-40.2
25.4
25
-29.5

n-Hexane (C
6
H
14
)
3-Methyl pentane (C
6
H
14
)
Toluene (C
6
H
5
CH
3
) n-Heptane (C
7
H
16
)
Cyclooctatetraene (C
8
H
8
) n-Octane (C
8
H
18
)
Naphthalene (C
10
H
8
)
Azulene (C
10
H
8
)
Acetic acid methyl ester (CH
3
COOCH
3
) t-Butanol (CH
3
)
3
COH
Aniline (C
6
H
5
NH
2
)
Phenol (C
6
H
5
OH)
Divinyl ether (C
4
H
6
O)
Tetrahydrofuran (C
4
H
8
O)
Cyclopentanone(C
5
H
8
O)
Benzoquinone(C
6
H
4
O
2
)
Pyrimidine(C
4
H
4
N
2
)
Dimethyl sulphone (C
2
H
6
O
2
S)
Chlorobenzene (C
6
H
5
Cl)
Butanedinitrile(NºC-CH
2
-CH
2
-CºN)
Pyrazine(C
4
H
4
N
2
)
Acetyl acetylene (CH
3
-C(=O)-CºCH)
Crotonaldehyde (CH
3
-CH=CH-CHO)
Acetic anhydride (CH
3
-C(=O)-O-C(=O)-CH
3
)
2,5-Dihydrothiophene (C
4
H
6
S)
Isobutane nitrile((CH
3
)
2
CH-CN)
Methyl ethyl ketone(CH
3
-CO-CH
2
-CH
3
)
Isobutanal((CH
3
)
2
CH-CHO)
1,4-Dioxane(C
4
H
8
O
2
)
Tetrahydrothiophene (C
4
H
8
S) t-Butyl chloride ((CH
3
)
3
C-Cl) n-Butyl chloride (CH
3
-CH
2
-CH
2
-CH
2
-Cl)
Tetrahydropyrrole(C
4
H
8
NH)
Nitro-s-butane (CH
3
-CH
2
-CH(CH
3
)-NO
2
)
Diethyl ether(CH
3
-CH
2
-O-CH
2
-CH
3
)
Dimethyl acetal(CH
3
-CH(OCH
3
)
2
) t-Butanethiol ((CH
3
)
3
C-SH)
Diethyl disulfide (CH
3
-CH
2
-S-S-CH
2
-CH
3
) t-Butylamine ((CH
3
)
3
C-NH
2
)
Tetramethylsilane (Si(CH
3
)
4
)
2-Methyl thiophene (C
5
H
6
S)
15
-2.6
-7.0
-5.0
-4.0
-1.3
-12.7
0.2
-5.8
-2.6
-9.1
0.2
-7.2
-5.0
-5.1
-2.7
-7.3
-5.2
-5.6
-7.5
-2.0
-7.7
-5.6
-8.3
-5.4
-0.9
-6.5
-7.6
-6.8
-3.6
-8.1
-3.3
-3.0
-5.7
-4.9
-3.4
-5.0
-4.3
-4.7
-6.5
-6.3
-12.2
-2.39
-0.68
-1.03
-1.97
-39.9
-41.1
12
-44.9
3.71
-1.59
70.7
-49.9
-0.24 35.9
-2.81 69.10
-1.55 -98.40
4.04 -74.70
-1.24 20.80
3.32 -23.00
-2.71 -3.30
-2.24 -44.00
0.19 -45.90
4.02 -29.40
-11.85 46.80
3.02 -89.20
-3.12 12.40
0.26
-8.15
50.10
46.90
4.74 15.60
-2.82 -24.00
0.48 -136.80
-3.19 20.80
0.38 5.60
-0.33 -57.10
0.67 -51.60
-2.10 -75.50
-2.66 -8.20
-2.81 -43.50
-3.73 -37.00
-3.76 -0.80
-9.24 -39.10
-3.59 -60.30
-3.19 -93.10
-1.54 -26.20
-6.18 -17.90
0.63 -28.90
3.38 -55.70
-3.63 20.00

N-methyl pyrrole (cyc-C
4
H
4
N-CH
3
)
Tetrahydropyran(C
5
H
10
O)
Diethyl ketone (CH
3
-CH
2
-CO-CH
2
-CH
3
)
Isopropyl acetate (CH
3
-C(=O)-O-CH(CH
3
)
2
)
Tetrahydrothiopyran (C
5
H
10
S)
Piperidine(cyc-C
5
H
10
NH) t-Butyl methyl ether((CH
3
)
3
C-O-CH
3
)
1,3-Difluorobenzene(C
6
H
4
F
2
)
1,4-Difluorobenzene(C
6
H
4
F
2
)
Fluorobenzene (C
6
H
5
F)
Di-isopropyl ether ((CH
3
)
2
CH-O-CH(CH
3
)
2
)
PF
5
SF
6
P
4
SO
3
SCl
2
POCl
3
PCl
5
Cl
2
O
2
S
PCl
3
Cl
2
S
2
SiCl
2 singlet
CF
3
Cl
Ethane,-hexafluoro- (C
2
F
6
)
CF
3
C
6
H
5
(phenyl radical)
ME
MAE
Max
Min
-9.3
-7.2
-5.6
7.5
-3.8
-9.9
1.9
-5.6
-9.1
-3.7
-5.8
-4.9
-3.3
-4.1
-4.6
-9.4
0.8
-7.3
-0.6
-4.7
-12.8
-0.9
-5.9
-6.6
-10.8
-9.6
-5.07
5.70
16.20
-22.93
16
-6.72 24.60
-0.38 -53.40
-0.66 -61.60
0.33 -115.10
-1.04 -15.20
-0.91 -11.30
-0.85 -67.80
-6.59 -73.90
-6.46 -73.30
-3.82 -27.70
-0.70 -76.30
4.40 -381.10
-10.70 -291.70
-18.21 14.10
3.23 -94.60
-7.75 -4.20
-2.08 -133.80
-14.01 -86.10
-2.10 -84.80
-8.32 -69.00
-15.54 -4.00
-2.15 -40.30
-8.45 -169.50
-10.02 -321.30
-11.46 -111.30
-5.59 81.20
-3.60
4.81
16.39
-25.71

Al
Ca
Sr
Ba
Si
SiC
Ge
Solid
Li
Na
K
Rb
Cs
LiCl
NaCl
MgO
ME
MAE
Table SVI. Total energies of the AE6 atoms, computed self-consistently from GAUSSIAN with the 6-311+G(3df,2p) basis set. The “experimental” values are accurate nonrelativistic total energies from Ref. [S4]. (hartrees)
Atom LSDA PBE TPSS PBEsol AM05 revTPSS Expt.
H
C
O
Si
S
MRE(%)
-0.478 -0.500 -0.500 -0.488 -0.485 -0.500 -0.500
-37.465 -37.794 -37.863 -37.622 -37.533 -37.853 -37.845
-74.520 -75.005 -75.099 -74.749 -74.601 -75.073 -75.067
-288.211 -289.225 -289.391 -288.688 -288.325 -289.275 -289.360
-396.731 -397.940 -398.137 -397.302 -396.856 -397.986 -398.111
-1.38 -0.06 0.02 -0.77 -1.02 -0.01
MARE(%) 1.38 0.06 0.02 0.77 1.02 0.03
Table SVII. Cohesive energies of 9 non-transition metals and 6 insulators. Many results are from Ref. [24]. The new LSDA and revTPSS values for the metals (and all values for Rb, Cs,
Ca, Sr, and Ba) are computed from BAND, and the new LSDA and revTPSS values for the insulators are computed from VASP, as in Ref. [24]. LiCl, NaCl, and MgO were also calculated with BAND, and their revTPSS values shown here are from BAND. Converged
BAND and VASP calculations agree rather well [24], but there are different technical difficulties for each code. For technical reasons, the nonselfconsistent revTPSS values from
VASP use PBE orbitals, while those from BAND use LSDA orbitals. Most experimental values are from Ref. [S6]. Experimental values for Sr and Ba from Ref. [S7]. The experimental values have been corrected to static-lattice values by addition of the zero-point vibrational energy. (eV/atom)
LSDA
1.79
1.24
1.01
0.92
0.87
3.91
2.21
1.90
2.24
5.35
7.45
4.62
3.83
3.49
5.84
3.43
1.91
1.62
1.88
4.56
6.40
3.73
3.36
3.10
5.01
PBE PBEsol revTPSS Expt.
1.61 1.67 1.61 1.668 ±
Error
0.010
1.08
0.86
0.77
0.71
1.15
0.93
0.83
0.78
1.14 1.132 ± 0.007
0.93 0.940 ± 0.008
0.83 0.857
0.76 0.813
0.32 -0.10
0.32 0.11
3.81
2.12
1.82
2.12
4.93
6.87
4.22
3.53
3.23
5.31
0.11
0.14
3.72
2.07
1.83
2.11
4.62 4.681 ± 0.083
6.45 6.490
3.95 3.886 ± 0.031
3.52
3.25
5.23
0.03
0.09
3.437 ±
1.860 ±
1.734
1.911
3.591
3.341
5.271 ±
0.042
0.008
0.050
17

Table SVIII. Cohesive energies of 3 transition metals, computed from BAND. Note that the
4 d noble metal Ag is well-described by the semilocal revTPSS, but the 3 d noble metal Cu probably needs a fully nonlocal functional as in Ref. [5]. We suggest that the highly-localized
3 d regions in metallic Cu behave somewhat like isolated open systems of fluctuating electron number [S5], in which semilocal functionals make the energy density too negative. This interpretation is supported not only by the cohesive energies in this table, but also by the lattice constants in Table SI. Experimental values are from Ref. [S6]. The experimental values have been corrected to static-lattice values by addition of the zero-point vibrational energy.
(eV/atom)
Solid LSDA PBE PBEsol revTPSS Expt.
Cu 4.54 3.50 4.05 4.13 3.518 ± 0.012
Pd 5.05 3.69 4.43 4.34 3.918
Ag 3.62 2.51 3.08 3.03 2.970 ± 0.008
___________________________________________________________________________
References
[S1] J.P. Perdew, J. Tao, V.N. Staroverov, and G.E. Scuseria, J. Chem. Phys. 120 , 6898 (2004).
[S2] A. Ruzsinszky, G.I. Csonka, and G.E. Scuseria, J. Chem. Theory. Comput. 5, 763 (2009).
[S3] L.M. Almeida, J.P. Perdew, and C. Fiolhais, Phys. Rev. B 66 , 075115 (2002).
[S4] S.J. Chakraborty and E.R. Davidson, J. Phys. Chem. 100 , 6167 (1996).
[S5] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E.Scuseria, V.N. Staroverov, and J. Tao, Phys. Rev. A 76, 040501 (2007).
[S6] J.D. Cox, D.D. Wagman, and V.A. Medvedev, CODATA Key Values for
Thermodynamics (Hemisphere Publishing, NY, 1989). http://www.codata.org/ressources/databases/key1.html
(last accessed 2009.05.20).
[S7] C. Kittel, Introduction to Solid State Physics (Wiley, NY, 1996).
18