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: [email protected]

W. Yang Department of Physics, Duke University, Durham, North Carolina 27708, USA

1 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 6144.86

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

2 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 0.17

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

3 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 1.0

HF 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

4 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 0.00

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 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 0.00

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 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 0.00

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 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 0.00

He 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 0.00

He 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 0.00

5 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 1231.08

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 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 0.65 154.61 306.17 369.09 552.21

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 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 -0.91 155.98 315.54 341.02 562.79

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 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 -3.04 127.58 260.88 311.42 633.06

He -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 62.11 140.69 181.94 809.11

He -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 38.17 995.44

6 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 0.00

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 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 0.00

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 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 0.00

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 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 0.00

He 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 0.00

He 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 0.00

7 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 31.55

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 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 17.17 104.66 186.64 220.86 232.97

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 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 97.46 199.97 268.08 391.04

He 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 82.65 158.63 201.74 222.21

He 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 57.18 112.09 141.24 151.69

He 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 77.76

8 9