Ali Malek Degao Peng Weitao Yang Robert Balawender Andrzej Holas

Supplemental Material forTesting exchange-correlation functionals at fractional electron numbers. Ali Malek · Degao Peng · Weitao Yang · Robert Balawender · Andrzej HolasA. Malek · R. Balawender · A. Holas Institute of Physical Chemistry of Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw, PolandD. Peng · W. Yang () Department of Chemistry, Duke University, Durham, North Carolina 27708, USA E-mail: [email protected]W. Yang Department of Physics, Duke University, Durham, North Carolina 27708, USA1 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 1.0 HF 22.50 21.86 16.51 6.51 -1.51 -8.13 -13.71 -15.96 6144.86SVWN5 -402.44 -394.39 -321.38 -156.49 9.14 169.22 307.13 327.09 4218.16PBE -403.80 -395.78 -323.18 -159.88 3.96 165.56 322.59 338.76 4201.96B3LYP -332.79 -326.24 -266.96 -133.22 2.11 136.83 264.59 301.98 4491.85CAM-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 5061.802 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 0.99 HF 0.38 3.09 6.02 6.08 4.39 1.63 0.17SVWN5 3.00 26.92 60.75 69.74 56.32 23.19 2.56PBE 2.88 25.81 58.67 68.38 57.01 25.31 3.01B3LYP 2.41 21.66 49.28 57.29 47.30 20.43 2.38CAM-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 0.793 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 1070.05SVWN5 873.29 296.99 235.20 106.25 23.74 119.98 210.43 250.43 713.60PBE 812.10 284.20 226.00 106.00 25.29 108.85 215.73 285.86 834.95B3LYP 756.33 238.11 189.84 88.52 17.20 93.68 177.34 230.73 765.28CAM-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 76.14 656.104 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 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 HF 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 0.00He 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 SV Be 0.00 0.09 0.79 1.85 2.20 1.84 0.78 0.08 0.00 WN5 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 0.00He 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 PBE 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 0.00He 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 B3L Be 0.00 0.07 0.63 1.49 1.78 1.50 0.63 0.06 0.00 YP 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 0.00He 0.00 0.61 5.73 14.17 17.85 15.73 6.69 0.54 0.00 CA Li 0.00 0.05 0.42 0.98 1.15 0.95 0.39 0.04 0.00 M- Be 0.00 0.04 0.33 0.78 0.95 0.82 0.37 0.04 0.00 B3L Na 0.00 0.00 0.02 0.04 0.05 0.04 0.02 0.00 0.00 YP Mg 0.00 0.00 0.02 0.05 0.06 0.05 0.02 0.00 0.00He 0.00 0.32 3.00 7.33 9.06 7.74 2.96 0.16 0.00 rCA Li 0.00 -0.01 -0.06 -0.13 -0.13 -0.08 0.00 0.00 0.00 M- Be 0.00 0.00 0.00 0.02 0.06 0.09 0.07 0.01 0.00 B3L Na 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 YP Mg 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.005 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 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 HF 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 1231.08He -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 SV Be -379.82 -372.34 -304.51 -151.44 2.86 154.85 300.83 351.10 584.26 WN Na -426.17 -417.60 -339.87 -163.98 12.64 181.93 319.21 326.95 5851.40 5 Mg -381.69 -374.24 -306.69 -154.02 0.65 154.61 306.17 369.09 552.21He -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 PBE 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 -0.91 155.98 315.54 341.02 562.79He -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 B3L Be -306.39 -300.42 -246.30 -124.21 -0.11 125.42 248.71 282.11 673.52 YP 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 -3.04 127.58 260.88 311.42 633.06He -196.78 -192.77 -156.49 -75.41 4.93 82.64 152.06 142.39 101.22 CA Li -170.23 -166.69 -134.92 -64.64 4.22 69.65 127.56 144.70 10782.35 M- Be -157.39 -154.47 -127.95 -67.49 -4.70 61.86 137.35 173.48 836.03 B3L Na -175.37 -171.88 -140.30 -69.85 0.56 70.32 139.45 169.50 6707.04 YP Mg -158.18 -155.32 -129.22 -69.24 -6.03 62.11 140.69 181.94 809.11He -103.87 -101.67 -81.74 -37.52 5.31 45.19 76.07 51.62 39.99 rCA Li 23.37 22.81 17.74 6.49 -4.27 -12.90 -11.24 12.27 11133.92 M- Be 2.96 2.52 -1.15 -7.18 -9.33 -4.30 18.09 45.12 1010.32 B3L Na 20.67 20.10 15.01 4.18 -5.33 -11.50 -6.01 15.19 7107.54 YP Mg 10.08 9.43 4.01 -5.31 -9.84 -6.45 13.99 38.17 995.446 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 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 HF 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 0.00He 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 SV Be 0.00 2.62 23.46 52.90 60.56 48.61 19.62 2.05 0.00 WN5 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 0.00He 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 PBE 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 0.00He 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 B3L Be 0.00 2.07 18.56 41.96 48.33 39.32 16.39 1.80 0.00 YP 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 0.00He 0.00 1.88 16.87 38.08 43.41 34.40 13.22 1.12 0.00 CA Li 0.00 1.52 13.72 31.43 36.73 30.52 13.58 1.69 0.00 M- Be 0.00 1.53 13.76 31.09 35.88 29.37 12.41 1.37 0.00 B3L Na 0.00 1.58 14.23 33.16 39.70 34.06 15.65 1.92 0.00 YP Mg 0.00 1.49 13.36 30.48 35.61 29.61 12.78 1.45 0.00He 0.00 1.76 15.77 35.38 39.99 31.23 11.50 0.85 0.00 rCA Li 0.00 0.34 3.06 7.14 8.72 8.03 4.68 0.77 0.00 M- Na 0.00 0.97 8.62 19.05 21.68 17.78 7.72 0.87 0.00 B3L Be 0.00 0.36 3.23 7.83 9.97 9.47 5.28 0.73 0.00 YP Mg 0.00 0.85 7.42 16.53 19.07 15.90 7.02 0.79 0.007 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 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 HF 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 31.55He 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 SV Be 970.98 258.56 204.80 91.70 16.87 105.09 182.15 204.81 207.12 WN5 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 17.17 104.66 186.64 220.86 232.97He 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 PBE 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 97.46 199.97 268.08 391.04He 1343.84 194.20 154.50 69.25 11.90 80.69 139.44 143.36 124.59 Li 271.26 265.08 210.45 96.24 21.87 105.10 190.07 252.32 1825.25 B3L Be 1054.29 204.45 162.16 73.49 12.66 81.30 147.72 176.92 369.81 YP Na 278.66 272.59 218.88 105.76 21.42 105.68 212.96 291.94 1284.54 Mg 833.62 210.32 167.85 78.58 12.83 82.65 158.63 201.74 222.21He 1324.34 185.77 147.58 65.47 11.73 77.62 130.92 124.55 97.28 CA Li 153.91 150.57 120.79 57.15 9.20 58.85 112.80 161.86 1822.79 M- Be 759.95 151.66 120.11 54.47 7.85 59.20 110.68 134.93 173.94 B3L Na 158.34 155.11 126.26 63.68 8.83 59.29 128.83 184.43 1280.76 YP Mg 1006.26 146.89 117.20 55.09 7.01 57.18 112.09 141.24 151.69He 1293.99 173.93 137.54 59.57 12.60 73.61 119.66 99.14 77.27 rCA Li 34.36 33.63 27.18 13.93 2.26 9.04 28.46 70.81 1819.29 M- Na 976.36 96.08 74.16 31.53 4.89 34.66 67.23 85.45 87.39 B3L Be 35.42 34.79 29.18 16.89 4.51 10.39 36.42 70.36 1218.78 YP Mg 2193.16 82.60 64.01 28.30 3.51 29.63 60.63 77.95 77.768 9

Ali Malek Degao Peng Weitao Yang Robert Balawender Andrzej Holas

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