Ali Malek Degao Peng Weitao Yang Robert Balawender Andrzej Holas
Supplemental Material for
Testing exchange-correlation functionals at fractional electron numbers.
Ali Malek · Degao Peng · Weitao Yang · Robert Balawender · Andrzej Holas
A. Malek · R. Balawender · A. Holas
Institute of Physical Chemistry of Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, Poland
D. Peng · W. Yang (*)
Department of Chemistry, Duke University, Durham, North Carolina 27708, USA
E-mail:
W. Yang
Department of Physics, Duke University, Durham, North Carolina 27708, USA
Table S1. The average (over the four atoms set) of the test indicator for the global constancy condition . All values are per thousand. 2
Table S2. The average (over the four atoms set) of the test indicator for the local linearity condition . All values are per thousand. 3
Table S3. The average (over the four atoms set) of the test indicator for the local constancy condition . All values are per thousand. 4
Table S4. The test indicator for the global linearity condition, . All values are per thousand. 5
Table S5 The test indicator for the global constancy condition . All values are per thousand. 6
Table S6. The test indicator for the local linearity condition . All values are per thousand. 7
Table S7. The test indicator for the local constancy condition . All values are per thousand. 8
Table S1. The average (over the four atoms set) of the test indicator for the global constancy condition . All values are per thousand.
0.0 / 0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99 / 1.0HF / 22.50 / 21.86 / 16.51 / 6.51 / -1.51 / -8.13 / -13.71 / -15.96 / 6144.86
SVWN5 / -402.44 / -394.39 / -321.38 / -156.49 / 9.14 / 169.22 / 307.13 / 327.09 / 4218.16
PBE / -403.80 / -395.78 / -323.18 / -159.88 / 3.96 / 165.56 / 322.59 / 338.76 / 4201.96
B3LYP / -332.79 / -326.24 / -266.96 / -133.22 / 2.11 / 136.83 / 264.59 / 301.98 / 4491.85
CAM-B3LYP / -165.29 / -162.09 / -133.10 / -67.80 / -1.49 / 65.99 / 136.26 / 167.40 / 4783.63
rCAM-B3LYP / 14.27 / 13.72 / 8.90 / -0.45 / -7.19 / -8.79 / 3.71 / 27.69 / 5061.80
Table S2. The average (over the four atoms set) of the test indicator for the local linearity condition . All values are per thousand.
0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99HF / 0.38 / 3.09 / 6.02 / 6.08 / 4.39 / 1.63 / 0.17
SVWN5 / 3.00 / 26.92 / 60.75 / 69.74 / 56.32 / 23.19 / 2.56
PBE / 2.88 / 25.81 / 58.67 / 68.38 / 57.01 / 25.31 / 3.01
B3LYP / 2.41 / 21.66 / 49.28 / 57.29 / 47.30 / 20.43 / 2.38
CAM-B3LYP / 1.53 / 13.77 / 31.54 / 36.98 / 30.89 / 13.61 / 1.61
rCAM-B3LYP / 0.63 / 5.58 / 12.64 / 14.86 / 12.80 / 6.17 / 0.79
Table S3. The average (over the four atoms set) of the test indicator for the local constancy condition . All values are per thousand.
0.0 / 0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99 / 1.0HF / 660.41 / 36.29 / 24.57 / 6.34 / 4.89 / 11.57 / 15.61 / 16.83 / 1070.05
SVWN5 / 873.29 / 296.99 / 235.20 / 106.25 / 23.74 / 119.98 / 210.43 / 250.43 / 713.60
PBE / 812.10 / 284.20 / 226.00 / 106.00 / 25.29 / 108.85 / 215.73 / 285.86 / 834.95
B3LYP / 756.33 / 238.11 / 189.84 / 88.52 / 17.20 / 93.68 / 177.34 / 230.73 / 765.28
CAM-B3LYP / 680.56 / 151.06 / 121.09 / 57.60 / 8.22 / 58.63 / 116.10 / 155.61 / 705.29
rCAM-B3LYP / 906.66 / 61.77 / 48.63 / 22.66 / 3.79 / 20.93 / 48.18 / 76.14 / 656.10
Table S4. The test indicator for the global linearity condition, . All values are per thousand.
Fun. / Atom /0.0 / 0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99 / 1.0
HF / He / 0.00 / -0.19 / -1.73 / -4.02 / -4.80 / -4.10 / -1.82 / -0.24 / 0.00
Li / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00
Be / 0.00 / -0.01 / -0.08 / -0.18 / -0.20 / -0.16 / -0.06 / -0.01 / 0.00
Na / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00
Mg / 0.00 / 0.00 / 0.00 / -0.01 / -0.01 / -0.01 / 0.00 / 0.00 / 0.00
SVWN5 / He / 0.00 / 1.14 / 10.62 / 26.30 / 33.05 / 28.96 / 12.33 / 1.10 / 0.00
Li / 0.00 / 0.11 / 1.04 / 2.40 / 2.79 / 2.22 / 0.82 / 0.07 / 0.00
Be / 0.00 / 0.09 / 0.79 / 1.85 / 2.20 / 1.84 / 0.78 / 0.08 / 0.00
Na / 0.00 / 0.01 / 0.05 / 0.11 / 0.13 / 0.10 / 0.04 / 0.00 / 0.00
Mg / 0.00 / 0.01 / 0.05 / 0.11 / 0.14 / 0.11 / 0.05 / 0.01 / 0.00
PBE / He / 0.00 / 1.10 / 10.26 / 25.13 / 31.04 / 26.21 / 9.67 / 0.07 / 0.00
Li / 0.00 / 0.12 / 1.05 / 2.43 / 2.85 / 2.32 / 0.93 / 0.08 / 0.00
Be / 0.00 / 0.08 / 0.77 / 1.79 / 2.13 / 1.77 / 0.73 / 0.06 / 0.00
Na / 0.00 / 0.01 / 0.05 / 0.11 / 0.13 / 0.11 / 0.05 / 0.00 / 0.00
Mg / 0.00 / 0.01 / 0.05 / 0.11 / 0.14 / 0.11 / 0.05 / 0.00 / 0.00
B3LYP / He / 0.00 / 0.87 / 8.16 / 20.27 / 25.66 / 22.82 / 9.92 / 0.90 / 0.00
Li / 0.00 / 0.10 / 0.87 / 2.04 / 2.40 / 1.97 / 0.79 / 0.08 / 0.00
Be / 0.00 / 0.07 / 0.63 / 1.49 / 1.78 / 1.50 / 0.63 / 0.06 / 0.00
Na / 0.00 / 0.00 / 0.04 / 0.09 / 0.11 / 0.09 / 0.04 / 0.00 / 0.00
Mg / 0.00 / 0.00 / 0.04 / 0.10 / 0.11 / 0.10 / 0.04 / 0.00 / 0.00
CAM-B3LYP / He / 0.00 / 0.61 / 5.73 / 14.17 / 17.85 / 15.73 / 6.69 / 0.54 / 0.00
Li / 0.00 / 0.05 / 0.42 / 0.98 / 1.15 / 0.95 / 0.39 / 0.04 / 0.00
Be / 0.00 / 0.04 / 0.33 / 0.78 / 0.95 / 0.82 / 0.37 / 0.04 / 0.00
Na / 0.00 / 0.00 / 0.02 / 0.04 / 0.05 / 0.04 / 0.02 / 0.00 / 0.00
Mg / 0.00 / 0.00 / 0.02 / 0.05 / 0.06 / 0.05 / 0.02 / 0.00 / 0.00
rCAM-B3LYP / He / 0.00 / 0.32 / 3.00 / 7.33 / 9.06 / 7.74 / 2.96 / 0.16 / 0.00
Li / 0.00 / -0.01 / -0.06 / -0.13 / -0.13 / -0.08 / 0.00 / 0.00 / 0.00
Be / 0.00 / 0.00 / 0.00 / 0.02 / 0.06 / 0.09 / 0.07 / 0.01 / 0.00
Na / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00
Mg / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00 / 0.00
Table S5 The test indicator for the global constancy condition . All values are per thousand.
Fun. / Atom /0.0 / 0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99 / 1.0
HF / He / 64.42 / 62.63 / 47.51 / 18.99 / -4.11 / -23.28 / -39.51 / -46.06 / -46.75
Li / 0.16 / 0.15 / 0.12 / 0.06 / 0.00 / -0.06 / -0.13 / -0.15 / 13223.17
Be / 46.32 / 44.99 / 33.91 / 13.31 / -3.18 / -16.71 / -28.09 / -32.65 / 1254.13
Na / 1.09 / 1.07 / 0.86 / 0.42 / -0.01 / -0.43 / -0.84 / -1.02 / 8871.04
Mg / 42.43 / 41.25 / 31.14 / 12.27 / -2.86 / -15.31 / -25.80 / -30.01 / 1231.08
SVWN5 / He / -363.41 / -356.06 / -289.49 / -139.34 / 10.91 / 154.47 / 274.46 / 270.56 / 201.27
Li / -422.09 / -413.39 / -334.45 / -156.52 / 20.41 / 185.49 / 302.29 / 261.22 / 9884.74
Be / -379.82 / -372.34 / -304.51 / -151.44 / 2.86 / 154.85 / 300.83 / 351.10 / 584.26
Na / -426.17 / -417.60 / -339.87 / -163.98 / 12.64 / 181.93 / 319.21 / 326.95 / 5851.40
Mg / -381.69 / -374.24 / -306.69 / -154.02 / 0.65 / 154.61 / 306.17 / 369.09 / 552.21
PBE / He / -357.36 / -349.92 / -282.57 / -130.95 / 19.90 / 161.13 / 260.75 / 176.78 / 22.53
Li / -422.89 / -414.22 / -335.70 / -159.80 / 13.46 / 175.72 / 318.01 / 316.34 / 9806.28
Be / -376.61 / -369.09 / -301.03 / -148.13 / 4.96 / 154.83 / 298.81 / 307.02 / 608.41
Na / -432.96 / -424.53 / -348.21 / -176.11 / -1.68 / 175.70 / 357.99 / 390.66 / 5830.37
Mg / -382.74 / -375.29 / -307.79 / -155.50 / -0.91 / 155.98 / 315.54 / 341.02 / 562.79
B3LYP / He / -279.68 / -274.04 / -223.11 / -109.06 / 4.81 / 116.62 / 219.36 / 216.14 / 784.75
Li / -351.67 / -344.53 / -280.02 / -135.22 / 9.07 / 147.41 / 265.14 / 283.88 / 10380.88
Be / -306.39 / -300.42 / -246.30 / -124.21 / -0.11 / 125.42 / 248.71 / 282.11 / 673.52
Na / -358.27 / -351.22 / -287.39 / -143.23 / 2.52 / 146.90 / 283.64 / 330.51 / 6279.94
Mg / -314.83 / -308.80 / -254.13 / -130.21 / -3.04 / 127.58 / 260.88 / 311.42 / 633.06
CAM-B3LYP / He / -196.78 / -192.77 / -156.49 / -75.41 / 4.93 / 82.64 / 152.06 / 142.39 / 101.22
Li / -170.23 / -166.69 / -134.92 / -64.64 / 4.22 / 69.65 / 127.56 / 144.70 / 10782.35
Be / -157.39 / -154.47 / -127.95 / -67.49 / -4.70 / 61.86 / 137.35 / 173.48 / 836.03
Na / -175.37 / -171.88 / -140.30 / -69.85 / 0.56 / 70.32 / 139.45 / 169.50 / 6707.04
Mg / -158.18 / -155.32 / -129.22 / -69.24 / -6.03 / 62.11 / 140.69 / 181.94 / 809.11
rCAM-B3LYP / He / -103.87 / -101.67 / -81.74 / -37.52 / 5.31 / 45.19 / 76.07 / 51.62 / 39.99
Li / 23.37 / 22.81 / 17.74 / 6.49 / -4.27 / -12.90 / -11.24 / 12.27 / 11133.92
Be / 2.96 / 2.52 / -1.15 / -7.18 / -9.33 / -4.30 / 18.09 / 45.12 / 1010.32
Na / 20.67 / 20.10 / 15.01 / 4.18 / -5.33 / -11.50 / -6.01 / 15.19 / 7107.54
Mg / 10.08 / 9.43 / 4.01 / -5.31 / -9.84 / -6.45 / 13.99 / 38.17 / 995.44
Table S6. The test indicator for the local linearity condition . All values are per thousand.
Fun. / Atom /0.0 / 0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99 / 1.0
HF / He / 0.00 / 0.68 / 5.67 / 10.95 / 10.91 / 7.74 / 2.82 / 0.27 / 0.00
Li / 0.00 / 0.01 / 0.06 / 0.14 / 0.17 / 0.14 / 0.06 / 0.01 / 0.00
Be / 0.00 / 0.73 / 6.03 / 11.64 / 11.63 / 8.30 / 3.06 / 0.32 / 0.00
Na / 0.00 / 0.05 / 0.39 / 0.88 / 1.04 / 0.86 / 0.37 / 0.04 / 0.00
Mg / 0.00 / 0.72 / 5.89 / 11.42 / 11.48 / 8.23 / 3.05 / 0.32 / 0.00
SVWN5 / He / 0.00 / 2.42 / 21.76 / 49.35 / 56.45 / 44.74 / 17.21 / 1.54 / 0.00
Li / 0.00 / 3.36 / 30.02 / 67.22 / 76.25 / 60.37 / 23.79 / 2.52 / 0.00
Be / 0.00 / 2.62 / 23.46 / 52.90 / 60.56 / 48.61 / 19.62 / 2.05 / 0.00
Na / 0.00 / 3.39 / 30.59 / 69.50 / 80.67 / 66.47 / 28.81 / 3.45 / 0.00
Mg / 0.00 / 2.64 / 23.62 / 53.40 / 61.48 / 49.84 / 20.54 / 2.25 / 0.00
PBE / He / 0.00 / 2.38 / 21.40 / 48.42 / 55.25 / 43.55 / 16.29 / 1.16 / 0.00
Li / 0.00 / 3.13 / 27.91 / 62.12 / 70.58 / 57.45 / 25.47 / 2.85 / 0.00
Be / 0.00 / 2.54 / 22.82 / 51.67 / 59.72 / 49.10 / 21.53 / 2.52 / 0.00
Na / 0.00 / 3.25 / 29.31 / 67.70 / 80.65 / 68.83 / 30.82 / 3.78 / 0.00
Mg / 0.00 / 2.58 / 23.22 / 53.18 / 62.55 / 52.68 / 23.40 / 2.89 / 0.00
B3LYP / He / 0.00 / 1.96 / 17.65 / 39.92 / 45.68 / 36.45 / 14.20 / 1.22 / 0.00
Li / 0.00 / 2.68 / 24.06 / 54.47 / 62.83 / 51.29 / 21.99 / 2.62 / 0.00
Be / 0.00 / 2.07 / 18.56 / 41.96 / 48.33 / 39.32 / 16.39 / 1.80 / 0.00
Na / 0.00 / 2.76 / 24.87 / 57.09 / 67.19 / 56.52 / 25.25 / 3.04 / 0.00
Mg / 0.00 / 2.13 / 19.13 / 43.61 / 50.82 / 42.06 / 18.10 / 2.07 / 0.00
CAM-B3LYP / He / 0.00 / 1.88 / 16.87 / 38.08 / 43.41 / 34.40 / 13.22 / 1.12 / 0.00
Li / 0.00 / 1.52 / 13.72 / 31.43 / 36.73 / 30.52 / 13.58 / 1.69 / 0.00
Be / 0.00 / 1.53 / 13.76 / 31.09 / 35.88 / 29.37 / 12.41 / 1.37 / 0.00
Na / 0.00 / 1.58 / 14.23 / 33.16 / 39.70 / 34.06 / 15.65 / 1.92 / 0.00
Mg / 0.00 / 1.49 / 13.36 / 30.48 / 35.61 / 29.61 / 12.78 / 1.45 / 0.00
rCAM-B3LYP / He / 0.00 / 1.76 / 15.77 / 35.38 / 39.99 / 31.23 / 11.50 / 0.85 / 0.00
Li / 0.00 / 0.34 / 3.06 / 7.14 / 8.72 / 8.03 / 4.68 / 0.77 / 0.00
Na / 0.00 / 0.97 / 8.62 / 19.05 / 21.68 / 17.78 / 7.72 / 0.87 / 0.00
Be / 0.00 / 0.36 / 3.23 / 7.83 / 9.97 / 9.47 / 5.28 / 0.73 / 0.00
Mg / 0.00 / 0.85 / 7.42 / 16.53 / 19.07 / 15.90 / 7.02 / 0.79 / 0.00
Table S7. The test indicator for the local constancy condition . All values are per thousand.
Fun. / Atom /0.0 / 0.01 / 0.1 / 0.3 / 0.5 / 0.7 / 0.9 / 0.99 / 1.0
HF / He / 1347.97 / 66.82 / 44.89 / 10.93 / 9.62 / 21.16 / 27.49 / 29.17 / 29.32
Li / 0.66 / 0.64 / 0.52 / 0.26 / 0.01 / 0.28 / 0.54 / 0.66 / 1865.43
Be / 1004.90 / 71.12 / 47.67 / 11.68 / 9.95 / 22.37 / 29.41 / 31.39 / 31.57
Na / 4.30 / 4.21 / 3.39 / 1.64 / 0.06 / 1.68 / 3.25 / 3.94 / 3392.36
Mg / 944.23 / 69.18 / 46.68 / 11.79 / 9.51 / 21.98 / 29.25 / 31.35 / 31.55
SVWN5 / He / 1385.15 / 239.01 / 190.83 / 85.92 / 14.78 / 101.43 / 168.67 / 167.33 / 132.76
Li / 339.76 / 331.78 / 261.25 / 114.64 / 30.37 / 136.98 / 224.83 / 246.83 / 1820.27
Be / 970.98 / 258.56 / 204.80 / 91.70 / 16.87 / 105.09 / 182.15 / 204.81 / 207.12
Na / 344.98 / 337.15 / 268.22 / 124.89 / 30.55 / 133.17 / 248.11 / 329.21 / 1174.88
Mg / 1325.56 / 260.46 / 206.52 / 93.76 / 17.17 / 104.66 / 186.64 / 220.86 / 232.97
PBE / He / 1360.57 / 235.15 / 187.38 / 83.79 / 15.20 / 100.77 / 166.23 / 155.89 / 106.67
Li / 317.06 / 309.35 / 241.83 / 105.50 / 37.34 / 117.16 / 216.64 / 282.11 / 1823.77
Be / 1047.49 / 251.18 / 199.50 / 91.20 / 16.58 / 96.93 / 183.24 / 242.78 / 580.07
Na / 328.67 / 321.50 / 258.67 / 129.61 / 31.13 / 123.86 / 263.08 / 350.48 / 1273.18
Mg / 1006.71 / 254.77 / 203.99 / 97.71 / 16.11 / 97.46 / 199.97 / 268.08 / 391.04
B3LYP / He / 1343.84 / 194.20 / 154.50 / 69.25 / 11.90 / 80.69 / 139.44 / 143.36 / 124.59