Supplementary Material
to accompany
“Effect of the Orbital-overlap Dependence in the Meta Generalized Gradient
Approximation”
by
Jianwei Sun, Bing Xiao, and Adrienn Ruzsinszky
The Journal of Chemical Physics
(Dated: June 16, 2012)
In this supplementary material we give the proposed exchange enhancement factor Fx ( p,  ) and its
derivatives, the table of exchange energies of hydrogenic anions, and the figure of f ( ) and two variants vs.
,
followed by the tables that show detailed results on which the error statistics of Tables I and II of our article
are based.
The following expressions give the proposed exchange enhancement factor Fxint ( p,  ) and its derivatives
with respect to the density n , the density gradient n , and the kinetic energy density  , which are needed for self-
n
n
  w
3
2/3
w
consistent implementation. Here, p 
and
with
and  unif   3  n5/ 3 .




2 / 3 8/ 3
unif
10
8n

4  3  n
2
2
Most of the results in this supplementary material were obtained from self-consistent calculations using GAUSSIAN
and VASP codes. We obtained the self-consistent subroutines of the present MGGA by modifying the revTPSS
subroutines of these two codes based on the following expressions. For the complete self-consistent
implementation, we refer readers to Ref [2] and references therein.
The
exchange
enhancement
factor
reads: Fxint ( p,  )  Fx1 ( p)  f ( )  Fx0 ( p)  Fx1 ( p)  ,
where


(1   2 )3
0
F
(
p
)

1



,
,
and
with  GE =10/81,  =0.29,
f
(

)

x
GE
GE
3
6
(c   p )
 p
1   
1
1


and c =0.28771. Note a good feature of f ( ) for implementation in computer codes is that it doesn't have any
singularity, even for  < 0 which is unphysical but possible, e.g., in the project augmented wavefunction (PAW)
Fx1 ( p)  1   
method [2].
Let x and y stand for n , n , and  . We have
Fxint ( p,  ) Fxint ( p,  ) p Fxint ( p,  ) 
,


x
p
x

x
and
 2 Fxint ( p,  )  2 Fxint ( p,  ) p p  2 Fxint ( p,  )   Fxint ( p,  )  2 p



xy
p 2
y x
 2
y x
p
xy
Fxint ( p,  )  2  2 Fxint ( p,  )   p p  


 y x  y x 

xy
p


,
p
2 p 

5 p
 2 p 88 p
p
8p
5 
5 p

1




,
,
,
,
,
,




n
3n  n n n
3n  unif 3 n  n
  unif
n2 9 n2
3  n
where
2 p
2p
 2 40 
5 2 p
 2
5 1
 2
5 2 p
2 p
16 p


,
,
,
,
,
 2 unif 


2
2
2
2
n n
3 n n n
n n
3n n  n
n
9n 
3 n
3n  unif
n
 2
 n
2

5 2 p
p  2 p
2 p
2 p
 2
 2





 0.
,
and
 n  n   2  n   2
3  n 2
The derivatives of Fx ( p,  ) with respect to p and
 are given:
 dFx0 ( p) dFx1 ( p) 
Fxint ( p,  ) dFx1 ( p)

 f ( ) 

,
p
dp
dp 
 dp
Fxint ( p,  ) df ( ) 0
 Fx ( p)  Fx1 ( p)  ,


d
 d 2 Fx0 ( p) d 2 Fx1 ( p) 
 2 Fxint ( p,  ) d 2 Fx1 ( p)


f
(

)


,
2
p 2
dp 2
dp 2 
 dp
 2 Fxint ( p,  ) df ( )  dFx0 ( p) dFx1 ( p) 



,
p
d  dp
dp 
and
 2 Fxint ( p,  ) df ( )  dFx0 ( p) dFx1 ( p) 



,
p
d  dp
dp 
where
dFx1 ( p)
 AK

2
dp
  AK p 
1 
 

d 2 Fx0 ( p)

dp 2
2   AK 
,
2
 c   AK p 
 1 




3
,
dFx0 ( p)
 AK

2
dp
 c   AK p 
1 




2   AK 
d 2 Fx1 ( p)

3
dp 2
  AK p 
 1 
 

2
,
df ( )
3(2 + 2 -4 3   4  4 5   6  4 7   8  2 9 )
d
(1   3   6 )2
,
df ( ) 6(-1- +6 2 +7 3  8 4  3 5  24 6  3 7  27 8  2 9 +13 10 -6 11  10 12  2 13  3 14 )

d
(1   3   6 )3
.
,
and
Table SI. Exchange energies of N-noninteracting-electron hydrogenic anion with the nuclear charge Z=1 [1],
which is the coefficient of the leading term for the high-Z limit of the Hartree-Fock exchange energies of Ninteracting-electron hydrogenic cation. In these hydrogenic anion, we can obtain the electronic densities,
their gradients, and the kinetic energy densities from the hydrogenic orbitals, and therefore the exchange
energies for different functionals. MATHEMATICA has been used to do the calculations. The details of the
calculations can be found in the Appendix of Ref [1]. Unit is hartree.
N
2
HF
PBE
TPSS
revTPSS present
-0.6250
-0.6119
-0.6250
-0.6250
-0.6250
4
-0.8192
-0.8121
-0.8242
-0.8222
-0.8222
10 -1.7479 -1.7286 -1.7425
-1.7289
-1.7449
12 -1.8596 -1.8486 -1.8594
-1.8441
-1.8596
ME
0.0126
0.0002
0.0079
0.0000
MAE
0.0126
0.0026
0.0094
0.0015
Figure SI, f ( ) , f 0 ( ) , and f * ( ) (see the text for their definitions) vs.  . Note the corresponding
Fxint ( p,  ) ,
*int
x
F
Fx0int ( p,  ) ,
and
Fx*int ( p,  )
( p,  )  F ( p,  ) for all p and  .
int
x
have
the
relation:
Fx*int ( p,  )  Fx0int ( p,  )  1
and
Table SII. Static-lattice lattice constants (Å) of 20 solids calculated with VASP. The computational details
can be found in Ref [2]. The experimental values [3] in the last column are obtained by subtracting the
zero-point anharmonic expansion (ZPAE) from the experimental zero-temperature values. See the
definitions of Fx0int , Fx*int and Fxint from the text. Note Fxint denotes the present functional here. This table
shows the details of Table I about the lattice constants.
solids
LDA
PBEsol
Li
Na
Ca
Sr
Ba
Al
Cu
Rh
Pd
Ag
C
SiC
Si
Ge
GaAs
LiF
LiCl
NaF
NaCl
MgO
3.362
4.051
5.332
5.791
4.770
3.983
3.522
3.759
3.844
4.002
3.533
4.332
5.405
5.631
5.615
3.913
4.968
4.506
5.467
4.170
3.426
4.170
5.448
5.916
4.894
4.011
3.565
3.781
3.876
4.050
3.552
4.355
5.432
5.680
5.665
4.007
5.063
4.632
5.603
4.223
3.378
4.074
5.364
5.809
4.770
3.981
3.522
3.753
3.838
3.994
3.534
4.333
5.397
5.619
5.604
3.925
4.952
4.524
5.460
4.181
3.307
3.971
5.278
5.786
4.796
3.937
3.497
3.776
3.847
3.980
3.514
4.298
5.357
5.540
5.541
3.839
4.882
4.425
5.365
4.125
3.430
4.217
5.545
6.112
5.093
4.027
3.572
3.783
3.880
4.066
3.551
4.361
5.441
5.670
5.663
3.984
5.096
4.609
5.604
4.209
ME
MAE
MRE(%)
-0.081
0.081
-1.72
-0.012
0.036
-0.23
-0.079
0.079
-1.66
-0.126
0.126
-2.69
0.016
0.023
0.30
MARE(%)
1.72
1.66
2.69
0.48
0.74
Fx0int
Fx*int
Fxint
Expt
(ZPAE)
3.451
4.207
5.553
6.042
5.004
4.019
3.595
3.793
3.876
4.063
3.555
4.348
5.422
5.644
5.641
3.974
5.072
4.570
5.565
4.188
Table SIII. Atomization energies of the AE6 molecules, in kcal/mol, calculated self-consistently from
GAUSSIAN using the 6-311+G(3df,2p) basis set and standard geometries, as in Ref. [4]. See the definitions
of Fx0int , Fx*int and Fxint from the text. Note Fxint denotes the present functional here. This table shows the
details of Table I about the atomization energies.
Molecule LSDA PBEsol
SiH4
SiO
S2
C3H4
C2H2O2
C4H8
347.4
223.9
135.1
802.1
754.9
1304.4
323.7
205.0
123.5
749.9
698.0
1218.6
ME
MAE
77.4
77.4
35.9
35.9
Fx0int
Fx*int
Fxint
Expt.
325.0 333.8 326.7 322.4
214.1 190.2 178.9 192.1
132.3 108.8 103.3 101.7
774.6 729.2 703.4 704.8
729.4 668.3 633.2 633.4
1258.4 1204.8 1161.0 1149.0
55.1
55.1
22.0
22.6
0.6
5.5
Table SIV. Exchange energies of rare-gas atoms, calculated self-consistently from GAUSSIAN using the 6311+G(3df,2p) basis set. The exact exchange energies are from Ref. [5].
Exact
LDA
PBE
PBEsol TPSS
M06L revTPSS present
He -1.026 -0.865 -1.008 -0.955 -1.031 -1.039
-1.030
-1.030
Ne -12.11 -10.969 -12.028 -11.611 -12.147 -12.065 -12.073 -12.157
Ar -30.19 -27.786 -29.959 -29.086 -30.190 -30.026 -29.978 -30.223
Kr -93.89 -88.500 -93.345 -91.343 -93.641 -93.270 -92.973 -93.532
2.274
0.219
1.055
0.052
0.204
0.291
ME
0.068
2.274
0.219
1.055
0.073
0.210
0.293
MAE
0.111
Table SV. Atomization energies of the AE6 molecules, in kcal/mol, calculated self-consistently from
GAUSSIAN using the 6-311+G(3df,2p) basis set and standard geometries, as in Ref. [4].
Molecule LSDA
PBE
PBEsol
TPSS
M06L
revTPSS present
Expt.
SiH4
SiO
S2
C3H4
C2H2O2
C4H8
ME
MAE
347.4 313.2
223.9 195.7
135.1 114.8
802.1 721.2
754.9 665.1
1304.4 1167.7
77.4
77.4
12.4
15.5
323.7
205.0
123.5
749.9
698.0
1218.6
35.9
35.9
333.7 327.9
186.7 189.1
108.7 110.9
707.5 707.9
636.0 637.5
1155.5 1148.9
4.1
5.9
3.2
4.2
338.2
185.7
109.1
704.1
632.8
1153.4
326.7
178.9
103.3
703.4
633.2
1161.0
3.3
5.9
0.6
5.5
322.4
192.1
101.7
704.8
633.4
1149.0
Table SVI. Dissociation energies (kcal/mol) of the W6 water clusters [6], calculated self-consistently from
GAUSSIAN using the aug-cc-PVQZ basis set. See Ref. [6] for the details of the reference values and the
structures of water clusters.
Water clusters
LSDA PBE PBEsol TPSS M06L revTPSS present Reference
Non-planar open Cs dimer 7.947 5.049
5.852
4.511 4.716
4.549
4.99
5.012
Cyclic C2h dimer
6.047 3.336
3.919
2.656 3.240
2.804
4
3.827
C3 local min trimer
24.771 15.058 18.033 13.415 15.568 13.667
14.99
15.353
Ice VIII 50 GPa dimer
5.754 -0.697 1.765 -1.624 -1.851 -2.259
-1.3
-1.446
NVT dimer
8.250 4.731
5.815
4.138 4.590
4.138
4.6
4.656
NVT trimer
12.847 6.780
8.434
5.505 6.906
5.623
7.05
7.141
5.214 -0.012 1.581 -0.955 -0.194 -0.968
ME
0.036
5.214 0.299
1.608
0.955 0.386
0.968
MAE
0.142
Table SVII. Errors for the enthalpies of formation of the 223 molecules of the G3 test set in kcal/mol,
calculated self-consistently from GAUSSIAN using the 6-311+G(3df,2p) basis set and standard geometries.
For M06L, a larger basis set aug-cc-PVQZ has been used. 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. Except the values
of M06L and the present functional, all the other values are from Ref. [7].
Molecule
LiH
BeH
CH
CH2(3B1)
TPSS
-1.1
-10.2
-3.4
-8.2
M06L revTPSS present
-2.17
-2.96
3.40
-9.94
-10.84
-10.13
1.09
-3.68
-3.15
-2.45
-7.24
-3.75
Expt.
33.30
81.70
142.80
93.70
CH2(1A1)
-0.5
1.38
0.26
-1.33
102.30
CH3
Methane(CH4)
NH
NH2
Ammonia(NH3)
OH
Water(H2O)
Hydrogenfluoride(HF)
SiH2(1A1)
-6.3
-4.6
-6.7
-6.1
-1.7
-0.6
3.5
1.3
-5.3
0.14
1.62
1.11
1.07
6.67
2.19
8.62
6.10
-3.32
-4.59
-3.01
-6.64
-4.11
1.25
2.27
7.94
3.58
-7.71
-4.88
-1.69
-7.52
-10.53
-2.41
-2.33
8.90
12.60
-1.20
35.00
-17.83
85.20
45.10
-11.00
9.40
-57.80
-65.10
65.20
SiH2(3B1)
-10.7
-5.01
-12.11
-7.26
86.20
SiH3
Silane (SiH4)
PH2
PH3
Hydrogen sulfide (H2S)
Hidrogen chloride (HCl)
Li2
LiF
Acetylene(C2H2)
Ethylene(H2C=CH2)
Ethane(H3C-CH3)
CN
Hydrogencyanide(HCN)
CO
HCO
-11.8
-11.6
-8.8
-8.0
-3.0
-1.1
1.2
1.2
0.7
-4.3
-6.1
-1.6
-1.0
4.2
-4.9
-5.99
-5.72
-0.31
1.74
1.13
-0.12
-0.72
0.25
-1.02
0.55
1.60
-3.61
-0.18
1.11
-2.52
-14.82
-16.31
-10.03
-9.85
-2.10
-0.08
0.42
2.11
2.87
-2.08
-3.64
-0.97
-0.17
5.04
-3.08
-6.94
-4.84
-8.62
-6.25
0.10
3.94
4.41
13.38
5.84
-2.25
-5.25
0.81
0.63
7.81
-0.32
47.90
8.20
33.10
1.30
-4.90
-22.10
51.60
-80.10
54.35
12.52
-20.10
104.90
31.50
-26.40
10.00
Formaldehyde(H2C=O)
Methanol (CH3-OH)
N2
Hydrazine(H2N-NH2)
NO
O2
Hydrogenperoxide(HO-OH)
F2
Carbon dioxide (CO2)
Na2
Si2
P2
S2
Cl2
NaCl
Silicon monoxide (SiO)
CS
SO
ClO
Chlorine monofluoride (FCl)
Si2H6
Methyl chloride (CH3Cl)
Methanethiol (H3CSH)
Hypochlorous acid (HOCl)
Sulfur dioxide (SO2)
BF3
BCl3
AlF3
AlCl3
Carbon tetrafuoride (CF4)
Carbon tetrachloride (CCl4)
-3.1
-2.6
0.9
-5.0
-4.1
-6.8
-0.2
-6.7
-1.8
-2.4
-3.8
0.0
-7.1
-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
-2.5
-0.18
5.91
3.80
9.12
0.86
-4.72
9.48
1.70
-9.38
-3.61
-4.17
3.96
-7.67
-3.18
-11.43
4.79
-1.08
-1.81
-2.56
0.61
-11.00
-1.55
1.11
3.26
5.00
3.82
-13.65
10.02
-13.47
1.47
-8.63
-1.17
0.96
-0.22
-1.85
-4.16
-5.62
4.21
-8.06
0.79
-2.57
-4.83
-2.59
-7.56
-4.53
0.66
6.59
1.20
-4.06
-7.36
-6.66
-24.59
-4.75
-3.96
-0.80
2.49
6.53
-1.51
11.28
0.10
-6.42
-6.67
1.09
0.96
-2.89
-10.88
-4.30
-3.04
3.27
3.28
7.09
1.23
-0.52
3.53
-1.65
3.44
5.59
13.42
6.49
0.24
-3.52
4.16
-11.30
-1.54
-3.19
3.05
12.12
25.69
0.60
36.54
6.39
22.42
5.67
-26.00
-48.00
0.00
22.75
21.60
0.00
-32.50
0.00
-94.05
34.00
141.00
34.30
30.70
0.00
-43.60
-24.60
66.90
1.20
24.20
-13.20
19.10
-19.60
-5.50
-17.80
-71.00
-271.40
-96.30
-289.00
-139.70
-223.00
-22.90
Carbon oxide sulfide (COS)
Carbon bisulfide (CS2)
-6.0
-8.5
-13.16
-15.55
-4.72
-8.49
2.47
0.33
-33.10
28.00
Carbonic difluoride (COF2)
Silicon tertrafluoride (SiF4)
-0.3
16.2
-0.10
16.11
0.14
16.39
17.53
50.96
-149.10
-386.00
Silicon tetrachloride (SiCl4)
3.4
-14.00
0.77
11.88
-158.40
Dinitrogen monoxide (N2O)
Nitrogen chloride oxide (ClNO)
Nitrogen trifluoride (NF3)
PF3
O3
F2O
Chlorine trifluoride (ClF3)
-12.0
-13.4
-19.4
1.8
-9.0
-14.3
-22.9
-11.12
-9.76
-2.79
12.22
-0.17
0.01
-2.03
-12.19
-13.89
-23.26
-0.35
-8.54
-15.77
-25.71
-7.02
-6.36
-0.97
26.46
1.02
-1.47
6.56
19.60
12.40
-31.60
-229.10
34.10
5.90
-38.00
Ethene, tetrafluoro- (F2C=CF2)
Ethene, tetrachloro- (C2Cl4)
Acetonitrile, trifluoro- (CF3CN)
-14.8
-6.5
-5.5
-9.91
-17.93
-3.52
-15.52
-9.73
-6.59
11.70
2.21
11.98
-157.40
-3.00
-118.40
Propyne(C3H4)
Allene(C3H4)
Cyclopropene(C3H4)
Propylene(C3H6)
Cyclopropane(C3H6)
Propane(C3H8)
Trans-1,3-butadiene (C4H6)
Dimethylacetylene (C4H6)
Methylenecyclopropane (C4H6)
Bicyclo[1.1.0]butane (C4H6)
Cyclobutene (C4H6)
Cyclobutane (C4H8)
Isobutene(C4H8)
Trans-butane(C4H10)
Isobutane(C4H10)
Spiropentane(C5H8)
Benzene(C6H6)
Difluoromethane(CH2F2)
Trifluoromethane(CHF3)
CH2Cl2
CHCl3
Methylamine(H3C-NH2)
Acetonitrile(CH3-CN)
Nitromethane(CH3-NO2)
Methyl nitrite(CH3-O-N=O)
Methyl silane (CH3SiH3)
Formic acid (HCOOH)
Methyl formate (HCOOCH3)
Acetamide (CH3CONH2)
Aziridine (C2H4NH)
Cyanogen (NCCN)
Dimethylamine ((CH3)2NH )
Trans ethylamine (CH3CH2NH2)
-2.6
-7.0
-5.2
-5.6
-7.4
-6.6
-5.8
-4.8
-10.7
-7.7
-5.1
-7.1
-5.6
-6.8
-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
-6.6
-3.88
-5.92
-7.43
-0.12
-4.62
2.21
-2.35
-5.24
-10.60
-11.03
-3.32
-1.32
0.25
3.02
3.87
-11.89
-9.53
2.91
2.84
-4.22
-6.35
6.07
-2.77
-3.23
-1.95
-2.89
2.34
-0.31
2.23
-2.93
-8.03
5.93
5.75
0.69
-3.66
-3.26
-2.50
-5.41
-3.50
-2.14
-0.49
-7.84
-6.56
-2.95
-4.79
-1.90
-3.09
-2.18
-8.50
-1.57
-7.22
-7.05
-5.48
-5.84
-2.29
-2.12
-12.08
-12.68
-11.93
3.16
-3.49
0.20
-7.93
-2.40
-5.22
-3.07
1.31
-3.82
-0.47
-5.67
-5.90
-8.20
-6.91
-2.04
-9.35
-4.87
-8.22
-12.26
-8.26
-10.87
-10.08
-10.89
-12.68
8.16
15.32
0.42
3.07
-7.37
-3.72
-9.76
-9.85
-5.06
5.38
-1.96
-6.70
-10.44
-3.79
-11.71
-11.19
44.20
45.50
66.20
4.80
12.70
-25.00
26.30
34.80
47.90
51.90
37.40
6.80
-4.00
-30.00
-32.10
44.30
19.70
-108.10
-166.60
-22.80
-24.70
-5.50
18.00
-17.80
-15.90
-7.00
-90.50
-85.00
-57.00
30.20
73.30
-4.40
-11.30
Ketene (CH2CO)
Oxirane (C2H4O)
Acetaldehyde (CH3CHO)
Glyoxal (HCOCOH)
Ethanol (CH3CH2OH)
Dimethylether (CH3OCH3)
Thiirane (C2H4S)
Dimethyl sulfoxide ((CH3)2SO)
Ethanethiol (C2H5SH)
Dimethyl sulfide (CH3SCH3)
Vinyl fluoride (CH2=CHF)
Ethyl chloride (C2H5Cl)
Vinyl chloride (CH2=CHCl)
-6.0
-8.2
-4.4
-2.8
-2.9
-7.1
-8.4
-5.6
-5.5
-7.1
-7.3
-6.2
-8.8
-8.79
-3.48
-0.70
-2.74
6.52
4.37
-7.38
5.22
2.19
1.19
-1.60
-1.08
-7.67
-2.72
-6.92
-1.27
0.61
1.51
-4.74
-8.04
-3.33
-3.53
-5.41
-5.53
-4.66
-7.45
0.15
-4.96
-2.12
0.32
-1.75
-5.52
-5.16
-2.18
-5.76
-6.15
0.95
-4.51
-5.11
-11.40
-12.60
-39.70
-50.70
-56.20
-44.00
19.60
-36.20
-11.10
-8.90
-33.2
-26.80
8.90
Acrylonitrile(CH2=CHCN)
Acetone (CH3COCH3)
Acetic acid (CH3COOH)
Acetyl fluoride (CH3COF)
CH3COCl (acetyl chloride)
CH3CH2CH2Cl (propyl chloride)
Isopropanol (CH3)2CHOH)
Methyl ethyl ether (C2H5OCH3)
Trimethylamine ((CH3)3N)
Furan (C4H4O)
C4H4S (thiophene)
Pyrrole (C4H5N)
Pyridine (C5H5N)
H2
HS
CCH
C2H3 (2A')
-2.7
-4.7
-0.9
-4.8
-5.9
-6.8
-2.5
-7.8
-9.6
-5.8
-5.1
-7.3
-8.7
-3.2
-2.8
0.1
-7.5
-3.05
-0.65
2.30
-1.67
-7.01
-0.67
7.48
4.47
5.89
-5.50
-7.57
-4.41
-9.09
5.31
0.14
-6.72
-4.75
-0.24
-0.69
4.09
-2.37
-3.79
-4.64
2.36
-4.61
-8.08
-3.28
-3.71
-4.58
-6.26
-5.01
-2.12
2.33
-5.00
-3.37
-4.82
2.68
4.33
-0.72
-7.50
-4.17
-8.64
-15.87
-5.58
-5.81
-11.74
-17.96
3.65
-2.34
3.77
-5.95
43.2
-51.9
-103.4
-105.7
-58.00
-31.50
-65.2
-51.7
-5.7
-8.3
27.50
25.9
33.6
0.00
34.20
135.10
71.60
CH3CO (2A')
-7.1
-3.66
-4.07
-4.17
-2.40
2
-5.0
3.63
-1.16
-1.61
-4.10
2
CH3O Cs( A')
-8.1
-1.67
-5.74
-9.47
4.10
CH3CH2O (2A'')
H2COH ( A)
-9.9
-2.27
-6.56
-13.37
-3.70
2
-6.9
-1.80
-5.70
-6.63
29.80
2
C2H5 ( A')
-8.6
-1.67
-5.74
-8.54
28.90
(CH3)2CH (2A')
-10.3
-2.98
-6.50
-12.01
21.50
(CH3)3C (t-butyl radical)
NO2
Methyl allene (C4H6)
Isoprene (C5H8)
Cyclopentane (C5H10)
n-Pentane (C5H12)
Neo pentane (C5H12)
1,3 Cyclohexadiene (C6H8)
1,4 Cyclohexadiene (C6H8)
Cyclohexane (C6H12)
n-Hexane (C6H14)
3-Methyl pentane (C6H14)
Toluene (C6H5CH3 )
n-Heptane (C7H16)
Cyclooctatetraene (C8H8)
n-Octane (C8H18)
Naphthalene (C10H8)
Azulene (C10H8)
Acetic acid methyl ester (CH3COOCH3)
t-Butanol (CH3)3COH
Aniline (C6H5NH2)
-10.1
-14.3
-8.0
-5.3
-4.9
-6.9
-3.7
-3.1
-2.2
-2.7
-7.3
-5.2
-5.6
-7.5
-2.0
-7.7
-5.6
-8.3
-5.4
-0.9
-6.5
-2.57
-9.59
-5.62
-2.05
2.47
3.91
5.68
-2.68
-2.38
4.27
4.55
5.42
-9.21
5.34
-7.72
6.16
-18.96
-20.24
0.70
8.99
-7.53
-5.66
-13.74
-3.90
-1.15
-1.76
-2.58
-0.48
1.16
2.29
1.05
-2.39
-0.68
-1.03
-1.97
3.71
-1.59
-0.24
-2.81
-1.55
4.04
-1.24
-13.92
-8.56
-6.83
-9.10
-13.78
-13.43
-11.43
-12.02
-11.37
-15.84
-16.30
-14.70
-15.18
-18.96
-13.47
-21.63
-23.14
-24.10
-3.58
-5.78
-18.53
12.30
7.90
38.8
18
-18.3
-35.1
-40.2
25.4
25
-29.5
-39.9
-41.1
12
-44.9
70.7
-49.9
35.9
69.10
-98.40
-74.70
20.80
CH3S ( A')
Phenol (C6H5OH)
Divinyl ether (C4H6O)
Tetrahydrofuran (C4H8O)
Cyclopentanone(C5H8O)
Benzoquinone(C6H4O2)
Pyrimidine(C4H4N2)
Dimethyl sulphone (C2H6O2S)
Chlorobenzene (C6H5Cl)
Butanedinitrile(NºC-CH2-CH2-CºN)
Pyrazine(C4H4N2)
Acetyl acetylene (CH3-C(=O)-CºCH)
Crotonaldehyde (CH3-CH=CH-CHO)
Acetic anhydride (CH3-C(=O)-O-C(=O)-CH3)
2,5-Dihydrothiophene (C4H6S)
Isobutane nitrile((CH3)2CH-CN)
Methyl ethyl ketone(CH3-CO-CH2-CH3)
Isobutanal((CH3)2CH-CHO)
1,4-Dioxane(C4H8O2)
Tetrahydrothiophene (C4H8S)
t-Butyl chloride ((CH3)3C-Cl)
n-Butyl chloride (CH3-CH2-CH2-CH2-Cl)
Tetrahydropyrrole(C4H8NH)
Nitro-s-butane (CH3-CH2-CH(CH3)-NO2)
Diethyl ether(CH3-CH2-O-CH2-CH3)
Dimethyl acetal(CH3-CH(OCH3)2)
t-Butanethiol ((CH3)3C-SH)
Diethyl disulfide (CH3-CH2-S-S-CH2-CH3)
t-Butylamine ((CH3)3C-NH2)
Tetramethylsilane (Si(CH3)4)
2-Methyl thiophene (C5H6S)
N-methyl pyrrole (cyc-C4H4N-CH3)
Tetrahydropyran(C5H10O)
Diethyl ketone (CH3-CH2-CO-CH2-CH3)
Isopropyl acetate (CH3-C(=O)-O-CH(CH3)2)
Tetrahydrothiopyran (C5H10S)
Piperidine(cyc-C5H10NH)
t-Butyl methyl ether((CH3)3C-O-CH3)
1,3-Difluorobenzene(C6H4F2)
1,4-Difluorobenzene(C6H4F2)
Fluorobenzene (C6H5F)
Di-isopropyl ether ((CH3)2CH-O-CH(CH3)2)
PF5
SF6
P4
SO3
SCl2
-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
-4.9
-3.4
-5.0
-4.3
-4.7
-6.5
-6.3
-12.2
-7.6
-6.8
-3.6
-8.1
-3.3
-3.0
-5.7
-9.1
-3.7
-5.8
-4.9
-3.3
-4.1
-4.6
-9.4
-9.3
-7.2
-5.6
7.5
-3.8
-9.9
1.9
-5.6
-6.62
-1.50
4.56
-1.65
-9.46
-9.56
10.21
-13.79
-4.65
-5.80
-3.95
-4.75
-4.60
-2.43
1.35
-0.27
2.64
7.02
2.23
0.87
0.72
5.47
-0.48
5.45
7.95
4.76
1.95
9.38
7.18
-7.47
-4.01
5.77
-0.55
1.98
3.22
7.59
6.95
-13.07
-12.75
-10.94
7.77
19.87
12.89
-15.19
2.46
-7.00
3.32
-2.71
-2.24
0.19
4.02
-11.85
3.02
-3.12
0.26
-8.15
4.74
-2.82
0.48
-3.19
0.38
-0.33
0.67
-2.10
-2.66
-2.81
-3.73
-3.76
-9.24
-3.59
-3.19
-1.54
-6.18
0.63
3.38
-3.63
-6.72
-0.38
-0.66
0.33
-1.04
-0.91
-0.85
-6.59
-6.46
-3.82
-0.70
4.40
-10.70
-18.21
3.23
-7.75
-8.84
-5.69
-10.52
-11.30
-9.19
-24.20
3.52
-11.65
-7.22
-20.50
2.86
-8.30
-2.87
-9.73
-7.37
-7.55
-6.12
-11.13
-10.74
-9.05
-9.64
-17.67
-6.29
-3.20
-0.80
-0.25
1.24
-5.84
5.02
1.16
-6.93
-4.06
-1.67
1.62
-2.62
-9.54
-2.12
4.64
4.79
0.44
-3.47
51.38
45.13
2.59
19.74
10.01
-23.00
-3.30
-44.00
-45.90
-29.40
46.80
-89.20
12.40
50.10
46.90
15.60
-24.00
-136.80
20.80
5.60
-57.10
-51.60
-75.50
-8.20
-43.50
-37.00
-0.80
-39.10
-60.30
-93.10
-26.20
-17.90
-28.90
-55.70
20.00
24.60
-53.40
-61.60
-115.10
-15.20
-11.30
-67.80
-73.90
-73.30
-27.70
-76.30
-381.10
-291.70
14.10
-94.60
-4.20
POCl3
PCl5
Cl2O2S
PCl3
Cl2S2
SiCl2 singlet
CF3Cl
Ethane,-hexafluoro- (C2F6)
CF3
C6H5 (phenyl radical)
0.8
-7.3
-0.6
-4.7
-12.8
-0.9
-5.9
-6.6
-10.8
-9.6
-4.71
-14.46
-2.40
-9.02
-15.41
-8.93
-2.81
0.18
-3.10
-16.01
-2.08
-14.01
-2.10
-8.32
-15.54
-2.15
-8.45
-10.02
-11.46
-5.59
19.07
16.88
21.30
12.57
8.44
13.49
21.66
35.15
14.72
-8.64
ME
MAE
Max
Min
-5.07
5.70
16.20
-22.93
-1.58
5.20
19.87
-20.24
-3.60
4.81
16.39
-25.71
-1.63
8.25
51.38
-24.20
-133.80
-86.10
-84.80
-69.00
-4.00
-40.30
-169.50
-321.30
-111.30
81.20
Table SVIII. Jellium surface exchange energies in erg/cm2, for bulk density parameter , which are
evaluated on LSDA densities as in Ref. [7]. Except for the values of M06L and the present functional, all the
other values are from Ref. [7].
σx
rs exact LSDA PBE PBEsol TPSS M06L revTPSS present
2 2624 3036 2436 2666
2553 2088
2657
2661
3 526
669
465
540
498
316
532
525
4 157
224
128
162
141
52
157
150
6
22
43.5 11.8
22.9
15.5 -16.8
20.6
16.3
MRE (%)
MARE(%)
45.8
45.8
-20.9
20.9
2.9
2.9
-11.9
11.9
-75.9
75.9
-1.0
2.2
-7.3
8.0
Table SIX. Jellium surface exchange-correlation energies in erg/cm2, for bulk density parameter
,
evaluated on LSDA densities as in Ref. [7]. The diffusion Monte Carlo (DMC) jellium surface exchangecorrelation energies [8, 9] have been taken as the reference values. The reason to use the DMC results as
references is to show that the jellium surface exchange-correlation energies of the present functional are
in a safe range [8, 9], since the exact values are not known. Except the values of M06L and the present
functional, all the other functionals’ values are from Ref. [7].
LSDA PBE PBEsol TPSS M06L revTPSS present
2
3
3354
764
3265
741
3374
774
3380
772
3525
867
3428
783
3434.0
776.1
DMC
3392±50
768±10
4
6
MRE (%)
MARE (%)
261
53
-0.4
0.4
252
52
-3.1
3.1
267
56.3
2.2
2.5
266
55.5
1.7
1.9
326
82.8
24.5
24.5
268
55.4
2.6
2.6
261.6
51.0
-0.3
1.6
261±8
53
Table SX. Static-lattice lattice constants (Å) of 20 solids calculated with VASP. The computational details
can be found in Ref [2]. The experimental values [3] in the last column are obtained by subtracting the
zero-point anharmonic expansion (ZPAE) from the experimental zero-temperature values. The values of
the column M06L are from Ref [10].
Solids
LDA
PBE
PBEsol
TPSS
M06L
revTPSS
present
Li
Na
Ca
Sr
Ba
Al
Cu
Rh
Pd
Ag
C
SiC
Si
Ge
GaAs
LiF
LiCl
NaF
NaCl
MgO
3.362
4.051
5.332
5.791
4.770
3.983
3.522
3.759
3.844
4.002
3.533
4.332
5.405
5.631
5.615
3.913
4.968
4.506
5.467
4.170
3.431
4.198
5.518
6.027
5.030
4.035
3.633
3.831
3.942
4.145
3.569
4.378
5.468
5.768
5.752
4.070
5.151
4.705
5.695
4.261
3.426
4.170
5.448
5.916
4.894
4.011
3.565
3.781
3.876
4.050
3.552
4.355
5.432
5.680
5.665
4.007
5.063
4.632
5.603
4.223
3.448
4.239
5.522
6.028
5.009
4.008
3.580
3.805
3.904
4.086
3.568
4.366
5.453
5.729
5.718
4.050
5.121
4.710
5.701
4.244
3.533
4.002
3.971
3.590
3.896
3.956
4.099
3.562
4.348
5.431
5.772
5.744
4.023
5.174
4.606
5.682
4.181
3.440
4.215
5.504
6.007
4.986
4.005
3.558
3.785
3.884
4.052
3.558
4.357
5.439
5.682
5.680
4.029
5.109
4.680
5.667
4.240
3.430
4.217
5.545
6.112
5.093
4.027
3.572
3.783
3.880
4.066
3.551
4.361
5.441
5.670
5.663
3.984
5.096
4.609
5.604
4.209
ME
-0.081
0.051
MAE
MRE(%
)
MARE(
%)
0.081
-1.72
0.059
1.10
0.012
0.036
-0.23
0.035
0.015
0.015
0.016
0.043
0.74
0.071
0.033
0.29
0.023
0.30
1.72
1.27
0.74
0.90
0.70
0.48
Expt
(ZPAE)
3.451
4.207
5.553
6.042
5.004
4.019
3.595
3.793
3.876
4.063
3.555
4.348
5.422
5.644
5.641
3.974
5.072
4.570
5.565
4.188
Table SXI. Bulk moduli (GPa) in GPa of 20 solids calculated with VASP. The computational details can be
found in Ref [2]. All the values other than those of the present functional are from Ref [2].
solids
LDA
PBE
PBEsol TPSS revTPSS present Expt.
Li
15.1
13.8
13.7
13.3
13.4
14.1
13.9
Na
9.2
7.8
7.8
7.3
7.5
7.6
7.7
Ca
19.4
17.5
17.9
17.6
17.9
14.2
18.7
Sr
14.5
11.1
12.9
10.7
10.9
11.3
12.5
Ba
10.6
8.8
9.4
8.4
8.7
7.8
9.4
Al
83.7
77.3
81.9
85.6
85.7
76.4
82
Cu
187.4
138.0
166.0
162.4
173.8
159.3
145
Rh
315.6
256.4
295.0
281.9
296.1
295.4
272.1
Pd
226.3
169.4
205.2
195.4
209.7
202.4
198.1
Ag
138.5
90.9
118.9
110.0
120.5
111.3
110.8
C
465.8
433.2
450.2
430.3
439.5
465.2
454.7
SiC
229.5
212.8
221.9
217.2
221.5
227.1
229.1
Si
97.0
90.0
92.8
92.0
93.0
101.7
100.8
Ge
70.5
59.4
65.8
60.2
65.0
72.9
77.3
GaAs
75.1
60.5
69.9
64.8
66.8
70.6
76.7
LiF
86.7
66.9
72.2
66.2
68.9
79.1
76.3
LiCl
41.5
31.7
35.4
33.4
34.0
35.7
38.7
NaF
61.5
45.2
48.8
42.9
44.0
53.7
53.1
NaCl
31.2
23.6
26.0
22.4
24.1
28.0
27.6
MgO
172.1
149.5
157.6
155.0
155.5
164.6
169.8
5
8.84
10.514
10.0
68
10.5
21
10.52
ME
MAE
MRE(%)
9.30
10.7
MARE(%)
7
0.257
4.867
-0.890
1.207
6.209
7.9
42
8.664
4.567
-1.93
7.65
-4.26
-2.61
9.04
5.79
10.6
1
9.6
5.69
5
Table SXII. Static-lattice cohesive energies in eV/atom of 20 solids calculated with VASP. The
computational details can be found in Ref [2]. All the values other than those of the present functional are
from Ref [2].
solids
Li
Na
Ca
Sr
Ba
Al
Cu
Rh
Pd
Ag
C
SiC
Si
Ge
GaAs
LiF
LiCl
NaF
NaCl
MgO
LDA
1.810
1.256
2.220
1.893
2.246
4.038
4.545
7.563
5.016
3.642
9.011
7.457
5.348
4.628
4.095
4.945
3.835
4.384
3.503
5.863
PBE
1.605
1.082
1.917
1.609
1.871
3.438
3.474
5.688
3.714
2.516
7.714
6.401
4.559
3.716
3.148
4.322
3.364
3.826
3.097
4.973
PBEsol
1.677
1.154
2.117
1.808
2.109
3.817
4.027
6.642
4.435
3.078
8.275
6.876
4.940
4.144
3.555
4.474
3.518
3.959
3.223
5.299
TPSS
1.631
1.135
2.027
1.750
2.017
3.478
3.787
5.776
3.981
2.733
7.246
6.189
4.435
3.642
3.120
4.223
3.362
3.736
3.104
4.941
revTPSS
1.637
1.155
2.068
1.822
2.094
3.570
4.121
6.155
4.379
3.034
7.312
6.255
4.504
3.783
3.259
4.228
3.391
3.740
3.137
4.930
present
1.536
1.063
1.886
1.781
1.999
3.409
3.838
5.137
4.162
2.817
7.499
6.323
4.684
3.903
3.346
4.119
3.337
3.651
3.125
4.922
ME
MAE
MRE(%)
MARE(%)
0.642
0.642
16.50
16.50
-0.121
0.144
-3.68
4.23
0.234
0.253
5.97
6.52
-0.107
0.173
-1.99
4.70
0.006
0.206
1.22
5.73
-0.096
0.167
-2.18
4.57
Expt.
1.658
1.119
1.86
1.73
1.91
3.431
3.524
5.783
3.938
2.985
7.545
6.478
4.685
3.918
3.337
4.457
3.586
3.970
3.337
5.203
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