Table 4.1: The unrestricted singlet energy(ground state energy), singlet-triplet energy gap, ionization potentail, electron affinity, fundamental gap of fullerenes Cn(n=20 to n=46), calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.The unrestricted singlet energy are in Hartrees, the singlet-triplet energy gap are in kcal/mol, ionization potentail, electron affinity, fundamental gap are given in eVs.
US ST gap IP EA FG
c20
isomer1 0.00 4.56 6.86 1.76 5.11 c24
isomer1 0.00 3.82 7.22 2.46 4.76 c26
isomer1 0.00 1.18 7.31 2.79 4.52 c28
isomer1 0.00 1.21 7.39 2.95 4.45 isomer2 16.27 5.68 7.42 2.79 4.62
c30
isomer1 0.00 6.24 7.47 2.88 4.59 isomer2 8.59 13.09 7.41 2.51 4.90 isomer3 42.94 4.69 6.92 2.50 4.42
c32
isomer1 0.00 19.11 7.52 2.42 5.10 isomer2 23.60 12.37 7.36 2.60 4.77 isomer3 44.71 5.16 7.01 2.62 4.38 isomer4 57.30 6.42 7.03 2.57 4.46 isomer5 59.93 5.49 7.07 2.66 4.41 isomer6 78.77 14.22 7.31 2.47 4.84
c34
isomer1 0.00 7.37 7.06 2.62 4.43 Continued on next page…
US ST gap IP EA FG isomer2 16.13 6.56 7.06 2.68 4.38 isomer3 22.02 3.33 6.98 2.75 4.22 isomer4 29.09 5.97 7.09 2.75 4.35 isomer5 31.05 6.93 7.08 2.69 4.39 isomer6 70.98 4.07 7.01 2.79 4.23
c36
isomer1 0.00 4.80 6.81 2.62 4.19 isomer2 2.04 5.48 6.95 2.71 4.24 isomer3 9.23 5.56 6.95 2.70 4.25 isomer4 9.32 4.66 6.89 2.69 4.21 isomer5 14.90 6.10 7.04 2.76 4.28 isomer6 26.46 6.15 7.03 2.76 4.27 isomer7 33.83 5.80 7.03 2.80 4.23 isomer8 36.75 5.07 7.03 2.82 4.21 isomer9 38.01 5.60 6.98 2.74 4.24 isomer10 44.21 4.72 7.03 2.83 4.19 isomer11 54.90 3.70 7.04 2.90 4.13 isomer12 70.94 4.42 7.06 2.89 4.16 isomer13 76.92 3.69 7.01 2.89 4.11 isomer14 90.55 10.79 7.17 2.71 4.46 isomer15 106.41 7.03 6.95 2.69 4.26
c38
isomer1 0.00 7.47 6.92 2.69 4.23 isomer2 13.17 5.18 7.04 2.91 4.13 isomer3 16.34 4.83 7.11 2.99 4.12 isomer4 18.06 4.21 6.92 2.84 4.08 isomer5 18.53 4.57 6.95 2.85 4.10 isomer6 29.45 5.00 7.00 2.87 4.12 isomer7 32.99 5.79 7.04 2.88 4.16 isomer8 37.07 4.67 6.98 2.88 4.11 isomer9 39.92 3.29 7.00 2.95 4.04 isomer10 48.12 8.47 7.02 2.75 4.28 isomer11 51.83 4.72 7.10 3.00 4.10 isomer12 57.23 6.08 7.01 2.84 4.17 isomer13 59.15 5.98 6.92 2.77 4.15 isomer14 64.56 5.56 7.08 2.95 4.14 isomer15 69.25 6.45 7.03 2.85 4.18 isomer16 88.02 4.68 7.02 2.94 4.08 isomer17 107.73 5.07 6.99 2.92 4.08
c40
isomer1 0.00 7.56 6.97 2.80 4.17 isomer2 8.71 5.47 7.03 2.96 4.07 isomer3 8.72 9.23 7.06 2.84 4.22 isomer4 9.27 5.35 7.01 2.94 4.07 isomer5 14.35 5.96 7.01 2.91 4.10 isomer6 15.63 4.65 7.00 2.97 4.03 isomer7 22.96 2.81 6.97 3.03 3.94 isomer8 24.07 4.23 6.94 2.93 4.01 isomer9 24.23 4.92 7.06 3.02 4.04 isomer10 25.35 4.85 6.96 2.91 4.04 isomer11 25.44 6.23 7.00 2.90 4.10 isomer12 26.38 4.04 6.95 2.95 4.00 isomer13 30.69 5.68 7.05 2.97 4.08 isomer14 31.08 7.05 7.09 2.95 4.15 isomer15 31.19 7.02 7.03 2.88 4.14 isomer16 31.23 4.89 7.02 2.98 4.04 isomer17 36.69 4.51 6.94 2.92 4.02 isomer18 37.64 2.73 6.99 3.04 3.95 isomer19 38.82 5.96 6.98 2.89 4.09 isomer20 39.29 6.89 7.10 2.97 4.13 isomer21 40.98 5.04 6.94 2.90 4.04 isomer22 41.35 5.81 7.08 3.01 4.08 isomer23 44.65 4.99 6.90 2.86 4.04 isomer24 45.07 3.76 6.90 2.91 3.99 isomer25 45.46 8.36 7.06 2.86 4.20 isomer26 48.90 5.23 6.95 2.90 4.05 isomer27 49.36 6.20 7.00 2.90 4.10 isomer28 60.60 7.93 7.09 2.92 4.18 isomer29 62.57 6.44 7.04 2.92 4.11 isomer30 67.36 5.92 7.10 3.02 4.08
Continued on next page…
US ST gap IP EA FG isomer31 67.42 4.01 7.04 3.04 4.00 isomer32 71.52 6.96 6.96 2.83 4.13 isomer33 75.80 6.66 6.96 2.88 4.08 isomer34 76.95 6.00 7.09 3.02 4.07 isomer35 88.42 5.02 6.97 2.93 4.04 isomer36 89.14 7.33 7.03 2.88 4.14 isomer37 103.17 6.85 6.90 2.78 4.12 isomer38 105.64 5.63 6.98 2.92 4.05 isomer39 131.87 5.16 6.75 2.76 3.99 isomer40 164.94 2.70 6.72 2.87 3.86
c42
isomer1 0.00 9.11 7.04 2.87 4.17 isomer2 16.77 5.36 7.06 3.07 3.99 isomer3 17.43 5.54 7.01 3.01 4.00 isomer4 17.59 5.90 7.01 2.99 4.02 isomer5 18.32 5.77 6.99 2.98 4.02 isomer6 24.38 4.83 7.00 3.03 3.97 isomer7 24.49 5.04 7.00 3.01 3.98 isomer8 29.91 5.20 7.06 3.08 3.98 isomer9 30.26 6.95 6.99 2.93 4.06 isomer10 31.07 7.80 6.97 2.88 4.10 isomer11 31.65 6.31 7.00 2.96 4.04 isomer12 33.87 5.92 7.09 3.07 4.02 isomer13 34.05 5.73 7.02 3.00 4.02 isomer14 34.30 4.48 6.99 3.04 3.95 isomer15 37.27 3.17 6.96 3.08 3.88 isomer16 38.92 4.07 6.95 3.02 3.93 isomer17 39.67 5.52 6.99 2.99 4.00 isomer18 44.83 5.25 6.97 2.99 3.98 isomer19 45.50 7.64 7.02 2.92 4.10 isomer20 47.17 7.35 7.02 2.94 4.08 isomer21 48.36 5.11 6.96 2.98 3.98 isomer22 48.44 6.39 7.02 2.99 4.03 isomer23 48.61 5.93 7.03 3.02 4.01 isomer24 50.52 3.78 6.91 3.00 3.91 isomer25 51.98 5.64 6.97 2.98 3.99 isomer26 52.51 7.23 7.02 2.95 4.07 isomer27 55.85 7.03 6.97 2.90 4.07 isomer28 56.07 7.84 7.03 2.92 4.11 isomer29 58.81 5.83 7.00 2.98 4.01 isomer30 60.33 5.78 6.95 2.95 4.00 isomer31 66.58 5.03 6.94 2.98 3.96 isomer32 66.62 8.72 7.01 2.87 4.14 isomer33 67.74 7.32 6.94 2.87 4.07 isomer34 70.69 5.04 7.06 3.09 3.97 isomer35 70.94 6.58 6.98 2.94 4.04 isomer36 77.26 5.26 6.97 2.96 4.01 isomer37 77.86 5.28 6.91 2.93 3.99 isomer38 78.69 7.16 6.93 2.87 4.06 isomer39 79.92 7.66 6.98 2.90 4.08 isomer40 87.34 6.20 6.90 2.90 4.00 isomer41 92.99 7.19 6.83 2.77 4.06 isomer42 93.40 6.23 6.95 2.93 4.02 isomer43 93.44 7.41 6.96 2.89 4.07 isomer44 102.79 6.57 6.89 2.87 4.02 isomer45 116.11 6.35 6.90 2.89 4.01
c44
isomer1 0.00 8.57 7.09 3.01 4.08 isomer2 0.71 8.38 7.10 3.03 4.07 isomer3 5.75 12.95 7.11 2.85 4.27 isomer4 11.77 12.90 7.12 2.84 4.28 isomer5 15.19 7.14 7.07 3.05 4.02 isomer6 16.57 7.84 7.09 3.04 4.05 isomer7 20.37 6.10 7.03 3.06 3.97 isomer8 21.55 7.07 6.99 2.98 4.01 isomer9 22.16 5.95 7.00 3.03 3.96 isomer10 22.82 5.69 7.01 3.06 3.95 isomer11 23.00 5.94 7.02 3.06 3.96 isomer12 31.30 5.96 7.06 3.10 3.96 isomer13 34.16 8.57 7.04 2.96 4.08
Continued on next page…
US ST gap IP EA FG isomer14 34.43 11.47 7.10 2.88 4.22 isomer15 34.74 11.50 7.02 2.82 4.20 isomer16 37.21 4.79 6.98 3.08 3.90 isomer17 37.84 5.72 7.01 3.06 3.95 isomer18 38.01 6.98 7.00 3.00 4.01 isomer19 39.19 5.62 7.05 3.11 3.94 isomer20 40.35 5.96 7.04 3.08 3.96 isomer21 42.47 3.61 6.97 3.12 3.84 isomer22 44.88 5.70 7.05 3.11 3.94 isomer23 45.36 6.36 7.00 3.02 3.98 isomer24 47.06 2.80 6.94 3.13 3.81 isomer25 47.07 5.63 7.05 3.11 3.94 isomer26 47.57 5.96 7.02 3.07 3.95 isomer27 49.60 6.56 6.98 3.00 3.98 isomer28 50.18 10.02 7.03 2.89 4.14 isomer29 50.50 5.68 6.96 3.02 3.94 isomer30 51.34 10.70 6.98 2.81 4.16 isomer31 52.41 5.22 7.01 3.09 3.91 isomer32 52.52 4.95 6.91 2.99 3.91 isomer33 52.77 7.78 6.92 2.88 4.04 isomer34 52.87 6.85 6.95 2.94 4.00 isomer35 53.55 7.53 6.93 2.90 4.03 isomer36 54.73 7.20 6.98 2.98 4.00 isomer37 55.84 6.09 6.94 2.98 3.95 isomer38 56.06 5.88 6.95 3.00 3.95 isomer39 56.88 6.43 6.90 2.93 3.97 isomer40 57.39 4.63 6.89 3.01 3.89 isomer41 58.52 8.67 7.02 2.97 4.05 isomer42 59.97 7.09 6.96 2.96 4.00 isomer43 60.96 7.10 6.98 2.97 4.00 isomer44 62.01 5.18 6.89 2.96 3.92 isomer45 63.21 6.20 6.95 2.99 3.96 isomer46 68.18 5.96 6.93 2.98 3.95 isomer47 68.71 7.50 7.01 2.99 4.02 isomer48 69.33 5.38 6.91 2.99 3.92 isomer49 70.04 5.70 7.00 3.06 3.94 isomer50 70.46 6.20 6.87 2.92 3.95 isomer51 71.17 6.63 6.89 2.91 3.98 isomer52 71.71 5.33 6.89 2.98 3.92 isomer53 71.92 6.06 6.90 2.95 3.95 isomer54 72.52 5.14 6.92 3.01 3.91 isomer55 72.62 8.69 7.05 2.97 4.08 isomer56 73.29 12.23 7.00 2.77 4.24 isomer57 75.74 9.40 6.91 2.80 4.11 isomer58 75.79 6.85 7.02 3.03 3.99 isomer59 76.88 5.44 6.88 2.96 3.93 isomer60 77.47 7.59 7.00 2.97 4.03 isomer61 77.96 5.45 6.90 2.98 3.92 isomer62 78.96 5.15 6.86 2.97 3.89 isomer63 79.09 4.79 6.82 2.93 3.89 isomer64 79.35 7.02 6.93 2.93 3.99 isomer65 80.30 7.15 6.94 2.94 4.00 isomer66 80.42 5.86 6.96 3.02 3.94 isomer67 82.65 6.08 6.80 2.86 3.94 isomer68 85.10 4.72 6.89 3.01 3.88 isomer69 88.44 5.38 6.84 2.93 3.91 isomer70 95.12 4.86 6.83 2.94 3.89 isomer71 99.38 4.40 6.89 3.01 3.87 isomer72 100.95 9.55 6.94 2.84 4.10 isomer73 102.11 5.37 6.81 2.90 3.91 isomer74 103.59 5.06 6.84 2.95 3.89 isomer75 104.52 6.15 6.91 2.97 3.94 isomer76 109.21 7.65 6.97 2.95 4.02 isomer77 110.39 6.08 6.86 2.91 3.94 isomer78 111.18 5.41 6.87 2.97 3.90 isomer79 112.90 6.09 6.84 2.91 3.94 isomer80 120.73 5.46 6.82 2.92 3.91 isomer81 125.43 6.34 6.92 2.97 3.96 isomer82 127.58 8.54 6.95 2.89 4.05 isomer83 127.64 6.09 6.90 2.97 3.94
Continued on next page…
US ST gap IP EA FG isomer84 129.03 6.41 6.91 2.95 3.96 isomer85 142.00 4.86 6.78 2.90 3.87 isomer86 145.77 5.66 6.81 2.90 3.91 isomer87 150.24 4.38 6.84 3.00 3.84 isomer88 176.04 6.78 6.77 2.86 3.92 isomer89 178.65 3.67 6.80 3.01 3.78
c46
isomer1 0.00 7.23 7.10 3.15 3.96 isomer2 1.57 6.72 7.05 3.10 3.94 isomer3 1.88 6.80 7.06 3.11 3.95 isomer4 2.43 7.72 7.04 3.05 3.99 isomer5 2.59 7.86 7.12 3.13 3.99 isomer6 5.23 9.01 6.99 2.93 4.06 isomer7 7.32 10.51 7.05 2.93 4.11 isomer8 16.34 9.78 7.02 2.95 4.07 isomer9 16.59 7.79 7.06 3.07 3.98 isomer10 16.95 9.11 7.03 2.99 4.04 isomer11 17.63 6.49 7.03 3.11 3.92 isomer12 18.74 9.91 7.04 2.96 4.08 isomer13 18.75 6.93 7.04 3.09 3.95 isomer14 22.07 4.27 6.96 3.15 3.82 isomer15 23.50 9.56 7.01 2.95 4.06 isomer16 24.18 3.53 6.95 3.16 3.79 isomer17 24.90 6.62 6.97 3.05 3.93 isomer18 25.11 5.39 6.95 3.08 3.87 isomer19 25.36 6.48 6.88 2.96 3.93 isomer20 25.48 5.72 6.99 3.11 3.88 isomer21 25.74 9.31 7.02 2.96 4.06 isomer22 25.82 9.73 6.96 2.90 4.06 isomer23 25.92 8.15 6.96 2.96 4.00 isomer24 26.99 7.32 6.99 3.03 3.96 isomer25 27.11 10.85 6.99 2.88 4.11 isomer26 29.53 5.80 6.92 3.04 3.88 isomer27 32.65 6.44 7.10 3.19 3.90 isomer28 33.19 6.55 6.98 3.06 3.92 isomer29 34.49 5.30 6.97 3.11 3.86 isomer30 35.96 9.01 6.98 2.94 4.03 isomer31 36.93 6.22 7.05 3.14 3.91 isomer32 37.27 5.33 6.92 3.06 3.86 isomer33 37.30 6.45 6.86 2.94 3.92 isomer34 37.49 4.10 6.95 3.15 3.80 isomer35 37.99 6.37 6.90 3.00 3.91 isomer36 39.29 5.16 6.97 3.12 3.85 isomer37 39.87 8.14 6.94 2.95 3.99 isomer38 40.52 5.24 6.96 3.09 3.86 isomer39 42.53 6.51 6.89 2.98 3.91 isomer40 43.22 7.68 6.95 2.99 3.96 isomer41 43.66 8.33 6.93 2.93 4.00 isomer42 43.74 6.57 6.90 2.98 3.92 isomer43 43.85 4.78 6.91 3.08 3.83 isomer44 44.22 6.59 6.91 2.98 3.93 isomer45 45.01 4.26 6.90 3.10 3.81 isomer46 45.51 6.10 6.89 2.99 3.90 isomer47 45.64 6.46 6.88 2.97 3.91 isomer48 45.79 4.76 6.83 3.00 3.83 isomer49 47.55 5.08 6.93 3.09 3.85 isomer50 48.18 5.28 6.90 3.04 3.86 isomer51 49.12 5.82 6.84 2.96 3.88 isomer52 49.89 6.17 6.88 2.98 3.90 isomer53 49.90 9.88 6.99 2.92 4.07 isomer54 50.19 5.14 6.93 3.07 3.85 isomer55 51.27 5.72 6.96 3.07 3.89 isomer56 51.33 5.30 6.84 2.98 3.86 isomer57 51.79 5.41 6.87 3.01 3.87 isomer58 52.06 6.49 6.85 2.94 3.91 isomer59 52.24 6.25 6.89 2.99 3.89 isomer60 52.32 7.00 6.93 3.00 3.93 isomer61 53.73 5.52 6.84 2.97 3.87 isomer62 54.51 5.40 6.93 3.06 3.86 isomer63 54.89 6.38 6.88 2.97 3.91
Continued on next page…
US ST gap IP EA FG isomer64 55.16 8.25 6.84 2.85 3.99 isomer65 57.01 4.95 6.88 3.04 3.84 isomer66 60.74 4.71 6.88 3.06 3.82 isomer67 61.35 4.28 6.82 3.02 3.80 isomer68 61.47 9.30 6.91 2.87 4.04 isomer69 61.91 6.33 6.94 3.04 3.90 isomer70 62.57 6.71 6.92 3.00 3.92 isomer71 64.43 7.06 6.89 2.94 3.95 isomer72 64.51 5.73 6.87 3.00 3.87 isomer73 65.82 5.59 6.83 2.97 3.86 isomer74 67.02 5.64 6.91 3.04 3.87 isomer75 67.38 5.76 6.81 2.93 3.87 isomer76 67.91 7.46 6.93 2.98 3.95 isomer77 68.96 5.35 6.87 3.01 3.85 isomer78 69.24 9.80 6.92 2.87 4.05 isomer79 71.75 7.08 6.78 2.85 3.92 isomer80 73.95 4.57 6.83 3.02 3.81 isomer81 73.96 7.45 6.97 3.02 3.95 isomer82 74.43 5.38 6.83 2.98 3.85 isomer83 74.64 5.47 6.83 2.95 3.87 isomer84 75.13 3.99 6.84 3.05 3.79 isomer85 76.08 5.83 6.83 2.95 3.88 isomer86 76.66 6.84 6.81 2.89 3.92 isomer87 79.01 4.19 6.77 2.99 3.78 isomer88 79.18 5.19 6.90 3.06 3.84 isomer89 79.74 4.61 6.84 3.03 3.81 isomer90 82.38 3.80 6.85 3.07 3.78 isomer91 82.80 5.36 6.81 2.96 3.85 isomer92 83.09 7.63 6.89 2.94 3.95 isomer93 83.57 5.64 6.83 2.96 3.87 isomer94 84.00 5.62 6.75 2.90 3.85 isomer95 86.07 5.26 6.91 3.06 3.85 isomer96 86.91 5.42 6.77 2.93 3.85 isomer97 89.98 4.98 6.82 3.00 3.82 isomer98 90.17 3.88 6.78 2.99 3.79 isomer99 92.01 4.52 6.86 3.06 3.80 isomer100 95.17 4.32 6.83 3.03 3.80 isomer101 95.39 5.29 6.76 2.93 3.83 isomer102 95.82 4.56 6.81 3.00 3.81 isomer103 100.35 4.46 6.83 3.03 3.79 isomer104 100.56 4.06 6.81 3.02 3.78 isomer105 100.93 6.81 6.94 3.02 3.92 isomer106 102.78 4.54 6.84 3.05 3.80 isomer107 105.07 3.94 6.79 3.01 3.77 isomer108 106.34 8.20 6.89 2.93 3.96 isomer109 106.44 8.07 6.87 2.89 3.97 isomer110 114.09 5.33 6.77 2.94 3.83 isomer111 114.16 4.49 6.86 3.05 3.81 isomer112 120.19 4.12 6.79 3.02 3.76 isomer113 124.63 5.96 6.80 2.93 3.87 isomer114 127.98 5.58 6.82 2.96 3.85 isomer115 142.84 4.65 6.73 2.92 3.81 isomer116 157.57 4.80 6.82 3.04 3.78
Table 4.2: The comparison of singlet-triplet gaps, ionization potentials, electron affinities and fundamental gaps to O2(Optimized second-order opposite-spin) and experiment data of some fullerenes Cn.
TAO-LDA /6-31G* O2
/6-31G*
O2/cc-pVDZ
Experiment Singlet-triplet gap (kcal/mol)C20 4.56 -0.63
C36(isomer2) 5.48 21.66 [7] 22.07
∼8 [7]
Ionization potential (eV)
C20 6.86 6.03
Electron affinity (eV)
C20 1.76 1.37 2.25
± 0.03 [4]
C30(isomer1) 2.88 2.79
∼3.5 [46]
C32(isomer1) 2.42 1.88
∼2.8 [46]
C34(isomer1) 2.62 2.25
C36(isomer1) 2.62 2.19
∼3.0 [46]
C40(isomer1) 2.80
∼2.8 [46]
C42(isomer1) 2.87
∼3.0 [46]
C44(isomer1) 3.01
∼3.3 [46]
Fundamental gap (eV)
C20 5.11 4.67
Figure 4.1: Singlet-triplet energy gap of fullerenes C20 to C26, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.2: Singlet-triplet energy gap of fullerenes C28, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.3: Singlet-triplet energy gap of fullerenes C30, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.4: Singlet-triplet energy gap of fullerenes C32, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.5: Singlet-triplet energy gap of fullerenes C34, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.6: Singlet-triplet energy gap of fullerenes C36, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.7: Singlet-triplet energy gap of fullerenes C38, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.8: Singlet-triplet energy gap of fullerenes C40, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.9: Singlet-triplet energy gap of fullerenes C42, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.10: Singlet-triplet energy gap of fullerenes C44, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.11: Singlet-triplet energy gap of fullerenes C46, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.12: Ionization potential of fullerenes C20to C26, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.13: Ionization potential of fullerenes C28, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.14: Ionization potential of fullerenes C30, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.15: Ionization potential of fullerenes C32, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.16: Ionization potential of fullerenes C34, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.17: Ionization potential of fullerenes C36, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.18: Ionization potential of fullerenes C38, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.19: Ionization potential of fullerenes C40, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.20: Ionization potential of fullerenes C42, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.21: Ionization potential of fullerenes C44, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.22: Ionization potential of fullerenes C46, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.23: Electron affinity of fullerenes C20 to C26, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.24: Electron affinity of fullerenes C28, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.25: Electron affinity of fullerenes C30, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.26: Electron affinity of fullerenes C32, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.27: Electron affinity of fullerenes C34, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.28: Electron affinity of fullerenes C36, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.29: Electron affinity of fullerenes C38, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.30: Electron affinity of fullerenes C40, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.31: Electron affinity of fullerenes C42, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.32: Electron affinity of fullerenes C44, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.33: Electron affinity of fullerenes C46, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.34: Fundamental gap of fullerenes C20 to C26, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.35: Fundamental gap of fullerenes C28, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.36: Fundamental gap of fullerenes C30, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.37: Fundamental gap of fullerenes C32, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.38: Fundamental gap of fullerenes C34, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.39: Fundamental gap of fullerenes C36, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.40: Fundamental gap of fullerenes C38, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.41: Fundamental gap of fullerenes C40, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.42: Fundamental gap of fullerenes C42, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.43: Fundamental gap of fullerenes C44, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Figure 4.44: Fundamental gap of fullerenes C46, calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.
Table 4.3: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO+1, ..., and LUMO+9) of fullerenes C20-C30, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C20 C24 C26 C28-1 C28-2 C30-1 C30-2 C30-3 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-5 2.0000 2.0000 2.0000 2.0000 2.0000 1.9964 1.9997 2.0000 H-4 2.0000 2.0000 2.0000 1.9528 1.9929 1.9922 1.9978 1.9996 H-3 2.0000 1.9999 1.9648 1.9526 1.9890 1.9855 1.9975 1.9996 H-2 2.0000 1.7325 1.4202 1.9523 1.8467 1.9261 1.9804 1.9925 H-1 2.0000 1.7325 1.4071 1.1882 1.7312 1.8623 1.9443 1.9925 H 0.5090 1.2280 1.1015 1.1827 1.5738 1.5748 1.9218 0.5514 L 0.5048 1.2279 1.0652 1.1800 0.8631 0.6592 0.1084 0.5465 L+1 0.4960 0.0398 1.0409 0.5913 0.0017 0.0033 0.0483 0.4607 L+2 0.4885 0.0395 0.0004 0.0000 0.0016 0.0001 0.0017 0.4558 L+3 0.0006 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0014 L+4 0.0006 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0006 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.4: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C32, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C32-1 C32-2 C32-3 C32-4 C32-5 C32-6 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 1.9997 1.9999 2.0000 2.0000 2.0000 2.0000 H-5 1.9997 1.9998 1.9999 2.0000 2.0000 2.0000 H-4 1.9989 1.9977 1.9996 1.9999 2.0000 1.9953 H-3 1.9933 1.9964 1.9985 1.9968 1.9868 1.9952 H-2 1.9933 1.9900 1.9962 1.9921 1.9866 1.9945 H-1 1.9688 1.9855 1.9854 1.9752 1.9566 1.9437 H 1.9678 1.8195 0.9787 1.3826 1.3990 1.9435 L 0.0717 0.1724 0.5870 0.5593 0.3393 0.0576 L+1 0.0034 0.0380 0.4459 0.0731 0.3308 0.0570 L+2 0.0033 0.0008 0.0087 0.0211 0.0007 0.0133 L+3 0.0000 0.0000 0.0000 0.0001 0.0002 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.5: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C34, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C34-1 C34-2 C34-3 C34-4 C34-5 C34-6 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-5 1.9998 1.9999 2.0000 1.9991 1.9999 1.9999 H-4 1.9997 1.9995 1.9996 1.9990 1.9979 1.9973 H-3 1.9996 1.9995 1.9984 1.9952 1.9976 1.9972 H-2 1.9964 1.9937 1.9973 1.9952 1.9934 1.9753 H-1 1.9917 1.9341 1.6131 1.9907 1.9697 1.6365 H 1.3697 1.4829 1.2403 0.9551 1.3459 1.2062 L 0.5857 0.4736 0.9006 0.9494 0.6501 1.0502 L+1 0.0517 0.1102 0.2467 0.1161 0.0409 0.1283 L+2 0.0059 0.0067 0.0041 0.0001 0.0046 0.0091 L+3 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.6: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C36, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C36-1 C36-2 C36-3 C36-4 C36-5 C36-6 C36-7 C36-8 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 1.9999 H-5 2.0000 2.0000 2.0000 2.0000 1.9999 1.9999 1.9999 1.9996 H-4 2.0000 2.0000 2.0000 2.0000 1.9995 1.9997 1.9991 1.9995 H-3 1.9995 1.9999 1.9990 1.9996 1.9978 1.9975 1.9991 1.9966 H-2 1.9995 1.8681 1.9984 1.9980 1.9952 1.9897 1.9675 1.9853 H-1 1.9332 1.8657 1.8094 1.7686 1.6878 1.8150 1.6120 1.6755 H 1.1174 1.6155 1.3571 1.2004 1.6229 1.6045 1.6107 1.3271 L 0.4282 0.5597 0.7848 0.9058 0.6702 0.4899 0.7701 0.9790 L+1 0.4282 0.0457 0.0383 0.1065 0.0230 0.1023 0.0209 0.0318 L+2 0.0939 0.0456 0.0132 0.0212 0.0039 0.0015 0.0207 0.0057 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C36-9 C36-10 C36-11 C36-12 C36-13 C36-14 C36-15 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 1.9999 1.9999 2.0000 2.0000 2.0000 2.0000 H-5 1.9999 1.9995 1.9999 1.9990 1.9998 1.9971 1.9999 H-4 1.9989 1.9994 1.9988 1.9990 1.9981 1.9965 1.9971 H-3 1.9926 1.9949 1.9946 1.9928 1.9756 1.9887 1.9769 H-2 1.9926 1.9927 1.8854 1.8863 1.9098 1.9860 1.9726 H-1 1.9502 1.8372 1.7191 1.4953 1.5954 1.9310 1.9724 H 1.3560 1.3206 1.3686 1.4258 1.2224 1.8152 1.5360 L 0.3530 0.6594 0.7586 1.1613 1.1116 0.2593 0.4230 L+1 0.3530 0.1952 0.2740 0.0299 0.1828 0.0218 0.0824 L+2 0.0035 0.0012 0.0010 0.0106 0.0044 0.0043 0.0395 L+3 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.7: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C38, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C38-1 C38-2 C38-3 C38-4 C38-5 C38-6 C38-7 C38-8 C38-9 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 2.0000 1.9999 2.0000 1.9999 1.9999 2.0000 1.9999 1.9999 H-5 2.0000 1.9999 1.9997 2.0000 1.9999 1.9999 1.9997 1.9999 1.9999 H-4 2.0000 1.9985 1.9952 1.9999 1.9999 1.9997 1.9976 1.9985 1.9998 H-3 1.9996 1.9970 1.9917 1.9999 1.9988 1.9948 1.9925 1.9979 1.9591 H-2 1.9319 1.8391 1.8140 1.6424 1.9065 1.8908 1.9303 1.9327 1.8648 H-1 1.8939 1.7534 1.5811 1.6416 1.6851 1.7597 1.8092 1.8571 1.6806 H 1.7412 1.5433 1.3029 1.5156 1.4544 1.4818 1.6218 1.3846 1.4722 L 0.2438 0.8202 1.3023 1.1239 0.8370 0.7914 0.5380 0.5700 0.6613 L+1 0.1260 0.0468 0.0124 0.0385 0.1123 0.0769 0.1095 0.2583 0.3606 L+2 0.0636 0.0019 0.0009 0.0382 0.0062 0.0051 0.0012 0.0011 0.0019 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C38-10 C38-11 C38-12 C38-13 C38-14 C38-15 C38-16 C38-17 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 1.9998 2.0000 2.0000 2.0000 H-6 1.9999 1.9999 1.9999 2.0000 1.9998 1.9999 1.9999 1.9998 H-5 1.9998 1.9995 1.9996 1.9999 1.9998 1.9994 1.9996 1.9998 H-4 1.9978 1.9903 1.9980 1.9984 1.9962 1.9969 1.9813 1.9951 H-3 1.9930 1.9265 1.9942 1.9984 1.9920 1.9744 1.9742 1.9104 H-2 1.9603 1.8923 1.9603 1.9148 1.9274 1.9290 1.9103 1.8777 H-1 1.9152 1.7326 1.9230 1.9146 1.7399 1.8992 1.6968 1.8773 H 1.7667 1.6015 1.3411 1.6021 1.2147 1.6481 1.3549 1.1545 L 0.2625 0.7680 0.7231 0.1968 1.1168 0.4661 1.0222 1.1526 L+1 0.0534 0.0887 0.0570 0.1882 0.0121 0.0724 0.0556 0.0252 L+2 0.0513 0.0006 0.0038 0.1868 0.0015 0.0145 0.0051 0.0077 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.8: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C40, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C40-1 C40-2 C40-3 C40-4 C40-5 C40-6 C40-7 C40-8 C40-9 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 1.9999 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 1.9999 1.9999 2.0000 1.9999 1.9999 1.9999 2.0000 1.9999 H-5 2.0000 1.9999 1.9999 1.9999 1.9999 1.9998 1.9999 1.9999 1.9998 H-4 1.9999 1.9999 1.9999 1.9999 1.9988 1.9986 1.9999 1.9999 1.9972 H-3 1.9617 1.8943 1.9553 1.9301 1.9641 1.9737 1.8253 1.9510 1.9020 H-2 1.9397 1.8839 1.9550 1.8627 1.9201 1.9198 1.8245 1.9290 1.8436 H-1 1.8811 1.7261 1.8706 1.7401 1.7563 1.6282 1.8237 1.7552 1.7913 H 1.7959 1.6147 1.8691 1.6518 1.6395 1.5487 0.8467 1.2279 1.3717 L 0.2996 0.8584 0.1750 0.7690 0.6790 0.8312 0.8417 0.9770 1.0662 L+1 0.0954 0.0215 0.1741 0.0420 0.0376 0.0993 0.8376 0.1549 0.0275 L+2 0.0268 0.0015 0.0013 0.0047 0.0048 0.0008 0.0004 0.0053 0.0006 L+3 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C40-10 C40-11 C40-12 C40-13 C40-14 C40-15 C40-16 C40-17 C40-18 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 1.9999 1.9999 1.9999 2.0000 1.9999 1.9999 2.0000 1.9999 2.0000 H-5 1.9999 1.9999 1.9999 1.9984 1.9972 1.9990 1.9998 1.9998 1.9997 H-4 1.9999 1.9999 1.9992 1.9980 1.9960 1.9983 1.9980 1.9986 1.9786 H-3 1.9509 1.9622 1.9669 1.9682 1.9727 1.9803 1.9524 1.9796 1.9784 H-2 1.9508 1.9005 1.9667 1.8980 1.9340 1.9621 1.9154 1.9494 1.8888 H-1 1.9013 1.8167 1.4866 1.8511 1.7907 1.8376 1.6922 1.7139 1.7629 H 1.0412 1.7537 1.2466 1.6302 1.7588 1.6782 1.5078 1.4939 0.8523 L 1.0400 0.3695 1.1998 0.5360 0.5204 0.4955 0.8545 0.6548 0.8487 L+1 0.1152 0.1961 0.1270 0.1198 0.0297 0.0436 0.0786 0.1988 0.6904 L+2 0.0004 0.0017 0.0074 0.0002 0.0005 0.0053 0.0013 0.0114 0.0001 L+3 0.0004 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0001 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C40-19 C40-20 C40-21 C40-22 C40-23 C40-24 C40-25 C40-26 C40-27 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 1.9999 2.0000 1.9999 2.0000 1.9999 2.0000 2.0000 2.0000 H-6 1.9999 1.9998 2.0000 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 H-5 1.9999 1.9998 1.9999 1.9982 1.9999 1.9999 1.9980 1.9992 1.9982 H-4 1.9989 1.9963 1.9990 1.9915 1.9996 1.9999 1.9962 1.9984 1.9957 H-3 1.9826 1.9581 1.9630 1.9634 1.9655 1.9817 1.9831 1.9867 1.9891 H-2 1.9446 1.9518 1.9491 1.8717 1.9593 1.9569 1.9465 1.9670 1.9718 H-1 1.7707 1.7390 1.8441 1.8509 1.7897 1.5212 1.9162 1.8673 1.8890 H 1.6074 1.5993 1.4537 1.2827 1.4531 1.2874 1.7448 1.3749 1.5785 L 0.6274 0.7503 0.6019 1.0345 0.6921 1.0487 0.3626 0.6454 0.3126 L+1 0.0633 0.0030 0.1773 0.0046 0.0901 0.1999 0.0502 0.1559 0.2617 L+2 0.0052 0.0028 0.0120 0.0025 0.0508 0.0045 0.0025 0.0053 0.0033 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
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C40-28 C40-29 C40-30 C40-31 C40-32 C40-33 C40-34 C40-35 C40-36 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 1.9999 2.0000 1.9999 2.0000 2.0000 2.0000 1.9999 2.0000 2.0000 H-6 1.9999 1.9999 1.9999 2.0000 1.9999 1.9999 1.9998 1.9999 1.9998 H-5 1.9982 1.9971 1.9894 1.9995 1.9997 1.9979 1.9995 1.9980 1.9964 H-4 1.9960 1.9918 1.9862 1.9800 1.9979 1.9957 1.9869 1.9931 1.9905 H-3 1.9815 1.9802 1.9683 1.9342 1.9886 1.9683 1.9791 1.9897 1.9843 H-2 1.9664 1.9526 1.9349 1.9265 1.9879 1.9650 1.9020 1.9806 1.9815 H-1 1.8421 1.8335 1.8737 1.6629 1.7754 1.9434 1.7826 1.8096 1.8944 H 1.6349 1.6984 1.3096 1.2032 1.7558 1.5765 1.4607 1.4865 1.5942 L 0.5748 0.4623 0.9252 1.2020 0.3120 0.2970 0.8718 0.4796 0.5240 L+1 0.0046 0.0781 0.0126 0.0903 0.1770 0.2035 0.0173 0.2611 0.0277 L+2 0.0017 0.0061 0.0003 0.0014 0.0058 0.0528 0.0003 0.0017 0.0071 L+3 0.0001 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 L+4 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C40-37 C40-38 C40-39 C40-40
H-9 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 2.0000 2.0000
H-6 1.9999 1.9999 2.0000 2.0000
H-5 1.9992 1.9985 1.9996 2.0000
H-4 1.9988 1.9853 1.9981 1.9955
H-3 1.9945 1.9788 1.9925 1.9954
H-2 1.9921 1.9721 1.9867 1.8185
H-1 1.9423 1.8402 1.9567 1.8184
H 1.4690 1.4263 1.1128 0.5775
L 0.4410 0.7336 0.4789 0.5723
L+1 0.1519 0.0625 0.3606 0.5445
L+2 0.0113 0.0025 0.1128 0.5422
L+3 0.0001 0.0003 0.0012 0.1358
L+4 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000
Table 4.9: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C42, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C42-1 C42-2 C42-3 C42-4 C42-5 C42-6 C42-7 C42-8 C42-9 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 2.0000 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 2.0000 H-5 1.9999 1.9998 1.9999 1.9999 1.9999 1.9994 1.9999 1.9986 1.9999 H-4 1.9725 1.9641 1.9686 1.9807 1.9799 1.9880 1.9862 1.9791 1.9793 H-3 1.9567 1.9472 1.9621 1.9677 1.9717 1.9696 1.9498 1.9609 1.9539 H-2 1.9566 1.8901 1.8870 1.8727 1.8970 1.9187 1.9099 1.8581 1.8844 H-1 1.8750 1.7747 1.8844 1.8376 1.7588 1.6674 1.7596 1.7322 1.8765 H 1.8745 1.2797 1.5832 1.6884 1.6730 1.3379 1.4183 1.2716 1.8272 L 0.3254 1.1271 0.5920 0.5513 0.6552 1.0737 0.9197 1.1799 0.2516 L+1 0.0198 0.0162 0.1221 0.1008 0.0575 0.0387 0.0521 0.0183 0.2174 L+2 0.0196 0.0011 0.0007 0.0009 0.0071 0.0067 0.0046 0.0014 0.0098 L+3 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C42-10 C42-11 C42-12 C42-13 C42-14 C42-15 C42-16 C42-17 C42-18 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 1.9999 1.9999 1.9996 1.9999 1.9999 2.0000 1.9999 1.9999 1.9999 H-5 1.9999 1.9995 1.9976 1.9990 1.9999 1.9999 1.9994 1.9999 1.9989 H-4 1.9940 1.9883 1.9853 1.9912 1.9819 1.9773 1.9893 1.9908 1.9862 H-3 1.9610 1.9660 1.9638 1.9687 1.9680 1.9317 1.9722 1.9274 1.9719 H-2 1.9208 1.9183 1.9263 1.8929 1.8930 1.9079 1.9108 1.8991 1.9661 H-1 1.8789 1.8662 1.6636 1.8832 1.6555 1.7409 1.9023 1.8360 1.8233 H 1.8333 1.6008 1.4916 1.4583 1.5552 1.0137 1.1635 1.5444 1.1198 L 0.2556 0.6035 0.9667 0.7645 0.7952 0.7581 0.6616 0.7440 1.0647 L+1 0.1398 0.0474 0.0049 0.0418 0.1498 0.6687 0.3994 0.0556 0.0587 L+2 0.0168 0.0101 0.0005 0.0005 0.0017 0.0018 0.0016 0.0029 0.0106 L+3 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C42-19 C42-20 C42-21 C42-22 C42-23 C42-24 C42-25 C42-26 C42-27 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 1.9990 1.9994 1.9999 1.9998 1.9999 1.9999 1.9986 1.9989 1.9999 H-5 1.9985 1.9988 1.9994 1.9988 1.9998 1.9999 1.9975 1.9983 1.9993 H-4 1.9940 1.9918 1.9899 1.9915 1.9919 1.9889 1.9974 1.9973 1.9948 H-3 1.9823 1.9906 1.9817 1.9701 1.8950 1.9596 1.9905 1.9889 1.9845 H-2 1.9782 1.9333 1.9116 1.9202 1.8054 1.9337 1.9767 1.9006 1.9587 H-1 1.9121 1.8892 1.8355 1.8030 1.8045 1.7734 1.8740 1.8695 1.9005 H 1.6720 1.7384 1.5399 1.5521 1.7307 1.2299 1.0814 1.7028 1.5763 L 0.3874 0.3346 0.5293 0.7509 0.7429 0.5990 1.0032 0.4965 0.5298 L+1 0.0747 0.1224 0.2069 0.0082 0.0276 0.5038 0.0800 0.0461 0.0477 L+2 0.0017 0.0016 0.0060 0.0054 0.0023 0.0118 0.0005 0.0010 0.0084 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0001 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C42-28 C42-29 C42-30 C42-31 C42-32 C42-33 C42-34 C42-35 C42-36 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 1.9999 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 1.9984 1.9998 1.9993 1.9999 1.9994 1.9999 1.9994 1.9999 1.9997 H-5 1.9982 1.9998 1.9991 1.9991 1.9980 1.9991 1.9963 1.9992 1.9956 H-4 1.9980 1.9942 1.9978 1.9950 1.9949 1.9980 1.9833 1.9957 1.9947 H-3 1.9842 1.9857 1.9931 1.9859 1.9939 1.9937 1.9748 1.9920 1.9803 H-2 1.9781 1.8773 1.9450 1.9057 1.9544 1.9071 1.8841 1.8885 1.9743 H-1 1.8934 1.8407 1.8655 1.7513 1.9131 1.8621 1.6916 1.7875 1.8554 H 1.6039 1.2241 1.4260 1.5914 1.7178 1.7542 1.2831 1.6866 1.3846 L 0.5245 1.0614 0.6890 0.5694 0.3972 0.3983 1.1691 0.6173 0.7700 L+1 0.0199 0.0095 0.0790 0.1974 0.0238 0.0745 0.0178 0.0270 0.0438 L+2 0.0015 0.0075 0.0062 0.0050 0.0075 0.0130 0.0006 0.0065 0.0016 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0001 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C42-37 C42-38 C42-39 C42-40 C42-41 C42-42 C42-43 C42-44 C42-45 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 2.0000 2.0000 H-6 1.9999 1.9990 1.9991 1.9990 1.9997 1.9993 1.9999 1.9995 1.9997 H-5 1.9996 1.9989 1.9979 1.9986 1.9994 1.9982 1.9992 1.9988 1.9975 H-4 1.9954 1.9988 1.9939 1.9981 1.9985 1.9972 1.9975 1.9983 1.9931 H-3 1.9884 1.9942 1.9929 1.9918 1.9926 1.9917 1.9968 1.9938 1.9924 H-2 1.9273 1.9807 1.9658 1.9807 1.9922 1.9691 1.9388 1.9397 1.9444 H-1 1.8179 1.9102 1.8690 1.8785 1.9841 1.8549 1.8653 1.9167 1.9178 H 1.5046 1.5613 1.7218 1.4837 1.4577 1.5876 1.6939 1.5850 1.5312 L 0.6331 0.4571 0.3618 0.5193 0.3380 0.4440 0.4156 0.4266 0.5233 L+1 0.1222 0.0939 0.0865 0.1366 0.1694 0.1562 0.0910 0.0875 0.0593 L+2 0.0115 0.0057 0.0110 0.0136 0.0684 0.0018 0.0020 0.0540 0.0405 L+3 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0000 0.0001 0.0008 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.10: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C44, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C44-1 C44-2 C44-3 C44-4 C44-5 C44-6 C44-7 C44-8 C44-9 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 1.9999 2.0000 1.9999 1.9999 2.0000 2.0000 H-6 1.9999 2.0000 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 H-5 1.9784 1.9744 1.9826 1.9969 1.9823 1.9917 1.9838 1.9960 1.9932 H-4 1.9752 1.9730 1.9765 1.9842 1.9684 1.9698 1.9837 1.9895 1.9871 H-3 1.9702 1.9332 1.9649 1.9792 1.9549 1.9594 1.9700 1.9677 1.9666 H-2 1.9409 1.9087 1.9648 1.9483 1.9256 1.9427 1.9161 1.9549 1.9364 H-1 1.8899 1.8959 1.9586 1.9482 1.9162 1.9006 1.9160 1.9072 1.8993 H 1.7941 1.8810 1.9585 1.9338 1.6522 1.6714 1.3658 1.6802 1.5066 L 0.4339 0.4061 0.1533 0.1911 0.5768 0.5539 0.8459 0.3917 0.6266 L+1 0.0162 0.0272 0.0206 0.0179 0.0211 0.0099 0.0094 0.1091 0.0823 L+2 0.0013 0.0005 0.0204 0.0003 0.0026 0.0006 0.0093 0.0038 0.0020 L+3 0.0000 0.0001 0.0000 0.0003 0.0000 0.0001 0.0000 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C44-10 C44-11 C44-12 C44-13 C44-14 C44-15 C44-16 C44-17 C44-18 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 1.9999 1.9997 2.0000 2.0000 2.0000 1.9999 H-6 1.9999 1.9999 2.0000 1.9999 1.9991 1.9999 1.9999 1.9999 1.9995 H-5 1.9935 1.9909 1.9714 1.9970 1.9986 1.9968 1.9889 1.9924 1.9973 H-4 1.9771 1.9811 1.9701 1.9847 1.9933 1.9965 1.9843 1.9854 1.9934 H-3 1.9458 1.9502 1.9408 1.9580 1.9881 1.9859 1.9694 1.9585 1.9738 H-2 1.9310 1.9386 1.9039 1.9458 1.9465 1.9545 1.8997 1.9138 1.9341 H-1 1.8740 1.8486 1.8524 1.9285 1.8988 1.9447 1.8906 1.7965 1.9153 H 1.5715 1.5552 1.6186 1.7295 1.8850 1.8658 1.4146 1.6166 1.5463 L 0.6067 0.6816 0.6986 0.4388 0.2880 0.2141 0.6085 0.6685 0.6107 L+1 0.0988 0.0521 0.0418 0.0164 0.0016 0.0209 0.2421 0.0677 0.0255 L+2 0.0016 0.0019 0.0023 0.0014 0.0011 0.0208 0.0020 0.0006 0.0041 L+3 0.0000 0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0001 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C44-19 C44-20 C44-21 C44-22 C44-23 C44-24 C44-25 C44-26 C44-27 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 1.9999 2.0000 2.0000 2.0000 1.9999 2.0000 1.9999 2.0000 1.9999 H-6 1.9999 1.9995 2.0000 1.9999 1.9999 2.0000 1.9999 1.9999 1.9994 H-5 1.9932 1.9955 1.9921 1.9949 1.9960 1.9877 1.9935 1.9936 1.9982 H-4 1.9757 1.9726 1.9766 1.9489 1.9922 1.9877 1.9885 1.9703 1.9851 H-3 1.9558 1.9450 1.9300 1.9197 1.9694 1.9789 1.9589 1.9568 1.9801 H-2 1.9271 1.9056 1.9142 1.8955 1.9245 1.9341 1.8531 1.9475 1.9797 H-1 1.6629 1.7925 1.8696 1.8005 1.8827 1.5095 1.6900 1.7470 1.9400 H 1.3308 1.6026 1.0128 1.5635 1.4244 0.9544 1.4271 1.6624 1.1462 L 1.1462 0.7639 0.8325 0.8558 0.7917 0.9544 1.0807 0.6550 0.9459 L+1 0.0066 0.0193 0.4712 0.0187 0.0174 0.6925 0.0076 0.0666 0.0235 L+2 0.0018 0.0035 0.0012 0.0028 0.0018 0.0008 0.0007 0.0008 0.0020 L+3 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0001 0.0001 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C44-28 C44-29 C44-30 C44-31 C44-32 C44-33 C44-34 C44-35 C44-36 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 1.9999 2.0000 2.0000 1.9999 2.0000 2.0000 2.0000 2.0000 2.0000 H-6 1.9993 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 1.9997 1.9999 H-5 1.9990 1.9931 1.9979 1.9936 1.9989 1.9992 1.9962 1.9991 1.9956 H-4 1.9926 1.9915 1.9976 1.9853 1.9845 1.9958 1.9949 1.9969 1.9932 H-3 1.9841 1.9854 1.9831 1.9528 1.9845 1.9808 1.9830 1.9863 1.9676 H-2 1.9533 1.9258 1.9640 1.8477 1.9773 1.9560 1.9407 1.9806 1.9244 H-1 1.9425 1.8421 1.9175 1.7182 1.6364 1.8589 1.9275 1.9398 1.8297 H 1.7720 1.5877 1.8783 1.5972 1.3174 1.7715 1.5708 1.5898 1.6876 L 0.3374 0.5104 0.1419 0.8495 1.0298 0.3295 0.5099 0.4080 0.5703 L+1 0.0178 0.1627 0.0856 0.0541 0.0387 0.0710 0.0615 0.0950 0.0214 L+2 0.0019 0.0011 0.0341 0.0019 0.0328 0.0373 0.0156 0.0048 0.0104 L+3 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C44-37 C44-38 C44-39 C44-40 C44-41 C44-42 C44-43 C44-44 C44-45 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 2.0000 1.9999 1.9999 H-6 1.9999 1.9999 1.9997 1.9999 2.0000 1.9999 1.9996 1.9999 1.9994 H-5 1.9975 1.9978 1.9988 1.9976 1.9862 1.9985 1.9973 1.9992 1.9987 H-4 1.9948 1.9893 1.9971 1.9949 1.9408 1.9961 1.9945 1.9903 1.9963 H-3 1.9703 1.9666 1.9801 1.9502 1.9406 1.9756 1.9685 1.9861 1.9814 H-2 1.9537 1.9497 1.9577 1.9173 1.9276 1.9557 1.9638 1.9611 1.9368 H-1 1.8646 1.9120 1.9284 1.9083 1.9274 1.8150 1.8726 1.8372 1.8472 H 1.4551 1.1599 1.5256 1.1504 1.9272 1.6684 1.6061 1.3281 1.5733 L 0.7031 0.9971 0.4773 0.8899 0.1172 0.5562 0.5558 0.7791 0.5846 L+1 0.0434 0.0176 0.0825 0.1520 0.1169 0.0292 0.0335 0.1091 0.0805 L+2 0.0176 0.0101 0.0528 0.0394 0.1160 0.0054 0.0084 0.0099 0.0018 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C44-46 C44-47 C44-48 C44-49 C44-50 C44-51 C44-52 C44-53 C44-54 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 2.0000 1.9995 2.0000 2.0000 1.9997 1.9998 2.0000 1.9996 1.9999 H-6 1.9998 1.9992 1.9999 1.9998 1.9996 1.9994 1.9999 1.9993 1.9999 H-5 1.9959 1.9992 1.9957 1.9969 1.9992 1.9989 1.9979 1.9993 1.9970 H-4 1.9940 1.9956 1.9939 1.9871 1.9974 1.9977 1.9932 1.9972 1.9952 H-3 1.9843 1.9491 1.9868 1.9519 1.9933 1.9937 1.9777 1.9948 1.9619 H-2 1.9670 1.8608 1.9570 1.9185 1.9815 1.9866 1.9280 1.9527 1.9504 H-1 1.9006 1.8591 1.7161 1.7913 1.9574 1.9094 1.9278 1.9358 1.8962 H 1.3647 1.8065 1.6098 1.3620 1.0665 1.5465 1.1581 1.0417 1.1160 L 0.7070 0.5112 0.6051 0.9676 0.9352 0.3656 0.9226 1.0339 0.9634 L+1 0.0805 0.0099 0.1240 0.0222 0.0438 0.1994 0.0672 0.0284 0.1179 L+2 0.0061 0.0099 0.0117 0.0028 0.0264 0.0028 0.0275 0.0172 0.0021 L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.0001 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C44-55 C44-56 C44-57 C44-58 C44-59 C44-60 C44-61 C44-62 C44-63 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 1.9994 1.9999 2.0000 1.9999 1.9999 1.9999 1.9996 2.0000 2.0000 H-6 1.9991 1.9997 1.9997 1.9993 1.9998 1.9999 1.9995 1.9994 1.9999 H-5 1.9991 1.9993 1.9991 1.9963 1.9990 1.9989 1.9994 1.9991 1.9995 H-4 1.9931 1.9980 1.9985 1.9926 1.9964 1.9896 1.9982 1.9991 1.9967 H-3 1.9392 1.9931 1.9928 1.9694 1.9826 1.9477 1.9832 1.9806 1.9832 H-2 1.9239 1.9614 1.9841 1.8920 1.9566 1.9417 1.9706 1.9306 1.9777 H-1 1.9230 1.9204 1.9029 1.7734 1.9457 1.8862 1.9530 1.9014 1.9092 H 1.7564 1.9179 1.7973 1.7155 1.2323 1.6686 1.1688 1.2084 1.0983 L 0.4635 0.1448 0.2197 0.6464 0.7332 0.5487 0.7419 0.8225 0.7640 L+1 0.0016 0.0602 0.0892 0.0091 0.1507 0.0111 0.1842 0.1311 0.2526 L+2 0.0016 0.0051 0.0167 0.0061 0.0036 0.0077 0.0015 0.0276 0.0189 L+3 0.0001 0.0001 0.0001 0.0000 0.0001 0.0000 0.0001 0.0000 0.0000 L+4 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C44-64 C44-65 C44-66 C44-67 C44-68 C44-69 C44-70 C44-71 C44-72 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 1.9997 1.9999 1.9999 1.9999 1.9998 1.9999 1.9998 2.0000 1.9997 H-6 1.9993 1.9996 1.9999 1.9991 1.9994 1.9995 1.9996 1.9999 1.9990 H-5 1.9992 1.9984 1.9976 1.9989 1.9992 1.9988 1.9990 1.9972 1.9989 H-4 1.9986 1.9973 1.9971 1.9986 1.9977 1.9984 1.9965 1.9972 1.9977 H-3 1.9907 1.9881 1.9717 1.9985 1.9825 1.9953 1.9960 1.9554 1.9915 H-2 1.9700 1.9434 1.9006 1.9895 1.9588 1.9697 1.9903 1.9500 1.9813 H-1 1.9300 1.9070 1.7983 1.9422 1.9042 1.9318 1.8822 1.7405 1.9642 H 1.4507 1.6034 1.6574 1.3208 1.0192 1.2867 1.1439 1.3359 1.7257 L 0.6157 0.4808 0.5689 0.5303 0.9221 0.5374 0.6884 0.8434 0.1977 L+1 0.0449 0.0800 0.1075 0.1732 0.2143 0.2724 0.3006 0.1769 0.1207 L+2 0.0010 0.0019 0.0011 0.0487 0.0028 0.0101 0.0036 0.0036 0.0233 L+3 0.0002 0.0001 0.0001 0.0001 0.0000 0.0001 0.0001 0.0000 0.0001 L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 C44-73 C44-74 C44-75 C44-76 C44-77 C44-78 C44-79 C44-80 C44-81 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 H-7 1.9999 2.0000 1.9996 1.9999 1.9999 2.0000 2.0000 1.9998 1.9999 H-6 1.9998 1.9990 1.9994 1.9999 1.9998 1.9994 1.9990 1.9995 1.9999 H-5 1.9988 1.9984 1.9987 1.9987 1.9990 1.9983 1.9972 1.9990 1.9995 H-4 1.9980 1.9955 1.9972 1.9971 1.9980 1.9921 1.9965 1.9980 1.9995 H-3 1.9974 1.9944 1.9925 1.9684 1.9787 1.9912 1.9897 1.9967 1.9406 H-2 1.9750 1.9762 1.9224 1.9237 1.9555 1.9307 1.9839 1.9750 1.9320 H-1 1.9098 1.9372 1.8527 1.8540 1.9297 1.8956 1.9752 1.8705 1.7904 H 1.1739 1.1593 1.5552 1.8237 1.4082 1.4259 1.2490 1.1561 1.7888 L 0.8039 0.5973 0.6077 0.2350 0.6296 0.6175 0.6467 0.9042 0.2749 L+1 0.1189 0.3396 0.0657 0.1979 0.0715 0.1398 0.1466 0.0878 0.2730 L+2 0.0242 0.0017 0.0089 0.0017 0.0299 0.0092 0.0148 0.0134 0.0015 L+3 0.0003 0.0013 0.0001 0.0000 0.0001 0.0003 0.0015 0.0000 0.0000 L+4 0.0001 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C44-82 C44-83 C44-84 C44-85 C44-86 C44-87 C44-88 C44-89 H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 H-7 1.9997 1.9999 1.9999 1.9999 2.0000 2.0000 2.0000 2.0000 H-6 1.9997 1.9997 1.9994 1.9998 2.0000 1.9998 2.0000 1.9999 H-5 1.9997 1.9987 1.9994 1.9989 1.9985 1.9985 1.9998 1.9999 H-4 1.9935 1.9977 1.9991 1.9969 1.9985 1.9930 1.9848 1.9415 H-3 1.9882 1.9828 1.9611 1.9964 1.9475 1.9678 1.9807 1.8736 H-2 1.9713 1.9342 1.9599 1.9778 1.8764 1.9525 1.9188 1.8578 H-1 1.9187 1.8803 1.9083 1.8773 1.8755 1.8050 1.8157 1.8553 H 1.7465 1.5100 1.3178 1.1371 1.6958 1.2839 1.7740 1.1533 L 0.2507 0.6037 0.8215 0.7608 0.2307 0.7251 0.3684 1.1446 L+1 0.1298 0.0902 0.0174 0.2171 0.2293 0.2696 0.0975 0.1689 L+2 0.0019 0.0027 0.0161 0.0380 0.1477 0.0041 0.0593 0.0049 L+3 0.0002 0.0001 0.0000 0.0001 0.0000 0.0005 0.0010 0.0001 L+4 0.0001 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.11: Active orbital occupation numbers (HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO +1, ..., and LUMO +9) of fullerenes C46, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C46-1 C46-2 C46-3 C46-4 C46-5 C46-6 C46-7 C46-8 C46-9
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 2.0000 2.0000 2.0000
H-6 1.9937 1.9910 1.9918 1.9967 1.9948 1.9985 1.9983 1.9986 1.9973
H-5 1.9842 1.9896 1.9821 1.9912 1.9872 1.9968 1.9976 1.9955 1.9909
H-4 1.9621 1.9799 1.9819 1.9903 1.9357 1.9922 1.9860 1.9881 1.9738
H-3 1.9450 1.9603 1.9636 1.9589 1.9339 1.9908 1.9683 1.9724 1.9463
H-2 1.9118 1.9390 1.9340 1.9484 1.9127 1.9787 1.9334 1.9666 1.9319
H-1 1.8821 1.9288 1.9315 1.9247 1.8503 1.9062 1.9283 1.9521 1.8671
H 1.6535 1.5862 1.5851 1.5950 1.8080 1.7266 1.8919 1.8062 1.7451
L 0.6632 0.5829 0.5963 0.5837 0.5750 0.3861 0.2634 0.2232 0.5337
L+1 0.0035 0.0418 0.0330 0.0090 0.0016 0.0205 0.0310 0.0932 0.0108
L+2 0.0009 0.0006 0.0006 0.0019 0.0008 0.0037 0.0018 0.0043 0.0030
L+3 0.0000 0.0001 0.0001 0.0000 0.0001 0.0001 0.0002 0.0000 0.0000
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-10 C46-11 C46-12 C46-13 C46-14 C46-15 C46-16 C46-17 C46-18
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 2.0000 2.0000 2.0000
H-6 1.9981 1.9942 1.9980 1.9951 1.9900 1.9995 1.9908 1.9984 1.9987
H-5 1.9947 1.9914 1.9943 1.9860 1.9899 1.9956 1.9908 1.9962 1.9941
H-4 1.9873 1.9852 1.9876 1.9811 1.9896 1.9918 1.9848 1.9952 1.9902
H-3 1.9791 1.9521 1.9726 1.9670 1.9592 1.9767 1.9642 1.9535 1.9783
H-2 1.9462 1.9321 1.9552 1.9399 1.9592 1.9712 1.9641 1.9429 1.9559
H-1 1.9031 1.8446 1.9436 1.8922 1.9202 1.9348 1.8646 1.8351 1.8627
H 1.8120 1.6043 1.7933 1.6277 0.9132 1.7568 0.8297 1.5832 1.2020
L 0.3345 0.6621 0.3339 0.5812 0.9112 0.3392 0.7059 0.6631 0.9411
L+1 0.0439 0.0312 0.0169 0.0288 0.3670 0.0332 0.7049 0.0267 0.0748
L+2 0.0010 0.0029 0.0045 0.0010 0.0002 0.0010 0.0002 0.0057 0.0020
L+3 0.0001 0.0000 0.0000 0.0000 0.0002 0.0002 0.0002 0.0000 0.0001
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-19 C46-20 C46-21 C46-22 C46-23 C46-24 C46-25 C46-26 C46-27
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 1.9999 2.0000 1.9999 2.0000 2.0000 2.0000 2.0000
H-6 1.9992 1.9977 1.9990 1.9992 1.9986 1.9965 1.9990 1.9991 1.9972
H-5 1.9989 1.9966 1.9954 1.9978 1.9978 1.9960 1.9984 1.9973 1.9699
H-4 1.9940 1.9860 1.9927 1.9934 1.9939 1.9912 1.9910 1.9865 1.9497
H-3 1.9902 1.9722 1.9604 1.9774 1.9870 1.9752 1.9827 1.9628 1.9170
H-2 1.9824 1.8607 1.9579 1.9669 1.9651 1.9695 1.9706 1.9393 1.8358
H-1 1.9583 1.7826 1.9449 1.9496 1.9172 1.8537 1.9375 1.9391 1.7801
H 1.3526 1.4813 1.7572 1.7897 1.6647 1.5889 1.8503 1.3502 1.7684
L 0.6277 0.8955 0.3692 0.2347 0.4292 0.6102 0.1996 0.7225 0.7713
L+1 0.0944 0.0232 0.0203 0.0633 0.0422 0.0103 0.0585 0.0908 0.0103
L+2 0.0022 0.0042 0.0032 0.0280 0.0044 0.0084 0.0124 0.0124 0.0004
L+3 0.0002 0.0000 0.0001 0.0000 0.0001 0.0000 0.0000 0.0000 0.0001
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C46-28 C46-29 C46-30 C46-31 C46-32 C46-33 C46-34 C46-35 C46-36
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9998 2.0000
H-6 1.9970 1.9942 1.9994 1.9967 1.9972 1.9993 1.9957 1.9997 1.9986
H-5 1.9960 1.9920 1.9957 1.9850 1.9959 1.9987 1.9897 1.9981 1.9946
H-4 1.9916 1.9864 1.9905 1.9713 1.9908 1.9973 1.9816 1.9940 1.9855
H-3 1.9683 1.9695 1.9836 1.9521 1.9812 1.9957 1.9755 1.9915 1.9573
H-2 1.9228 1.9386 1.9647 1.9290 1.9471 1.9425 1.9120 1.9829 1.9371
H-1 1.9114 1.8847 1.9401 1.8230 1.8812 1.9101 1.8424 1.6909 1.8250
H 1.5147 1.3868 1.7329 1.4916 1.1640 1.5602 1.2336 1.6510 1.3738
L 0.6572 0.7177 0.3281 0.8406 0.9360 0.4434 0.5858 0.6391 0.8337
L+1 0.0403 0.1243 0.0601 0.0085 0.0839 0.1148 0.4820 0.0498 0.0938
L+2 0.0005 0.0058 0.0048 0.0022 0.0227 0.0381 0.0016 0.0032 0.0003
L+3 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-37 C46-38 C46-39 C46-40 C46-41 C46-42 C46-43 C46-44 C46-45
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 2.0000 2.0000 1.9999 1.9998 2.0000 2.0000 2.0000
H-6 1.9995 1.9980 1.9995 1.9992 1.9994 1.9997 1.9994 1.9990 1.9989
H-5 1.9979 1.9967 1.9983 1.9967 1.9983 1.9984 1.9968 1.9986 1.9969
H-4 1.9938 1.9873 1.9963 1.9920 1.9948 1.9940 1.9908 1.9915 1.9906
H-3 1.9838 1.9791 1.9905 1.9672 1.9853 1.9904 1.9558 1.9792 1.9765
H-2 1.9687 1.9616 1.9509 1.9488 1.9698 1.9787 1.9238 1.9686 1.9133
H-1 1.8566 1.7678 1.7967 1.9221 1.9507 1.9028 1.8807 1.9256 1.9038
H 1.7549 1.2017 1.6262 1.6996 1.6848 1.4130 1.1161 1.4600 1.0686
L 0.3801 1.0565 0.5597 0.3568 0.2838 0.6675 0.9897 0.6227 0.7919
L+1 0.0590 0.0509 0.0601 0.1131 0.1196 0.0493 0.1382 0.0447 0.3586
L+2 0.0057 0.0003 0.0219 0.0046 0.0137 0.0063 0.0088 0.0100 0.0005
L+3 0.0000 0.0002 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0003
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-46 C46-47 C46-48 C46-49 C46-50 C46-51 C46-52 C46-53 C46-54
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 1.9998 2.0000 2.0000 2.0000 2.0000 1.9999 1.9999 1.9999 2.0000
H-6 1.9996 1.9995 1.9996 1.9991 1.9995 1.9997 1.9996 1.9997 1.9983
H-5 1.9992 1.9990 1.9990 1.9987 1.9977 1.9993 1.9980 1.9977 1.9968
H-4 1.9949 1.9930 1.9977 1.9805 1.9909 1.9954 1.9964 1.9892 1.9952
H-3 1.9868 1.9856 1.9819 1.9431 1.9867 1.9932 1.9834 1.9764 1.9550
H-2 1.9753 1.9580 1.9710 1.9189 1.9543 1.9661 1.9680 1.9751 1.9074
H-1 1.9224 1.9479 1.8566 1.9162 1.8252 1.9370 1.8840 1.8911 1.8605
H 1.1305 1.5063 1.1512 1.4670 1.3767 1.3337 1.4428 1.8324 1.4510
L 0.9545 0.3917 0.8122 0.4832 0.7460 0.6140 0.6579 0.3014 0.7052
L+1 0.0336 0.2066 0.2142 0.2919 0.1207 0.1548 0.0600 0.0333 0.1277
L+2 0.0033 0.0123 0.0167 0.0012 0.0024 0.0068 0.0099 0.0038 0.0028
L+3 0.0001 0.0000 0.0000 0.0002 0.0001 0.0001 0.0000 0.0000 0.0001
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C46-55 C46-56 C46-57 C46-58 C46-59 C46-60 C46-61 C46-62 C46-63
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 1.9999 1.9999 2.0000 2.0000 2.0000 2.0000
H-7 2.0000 2.0000 2.0000 1.9998 1.9998 2.0000 1.9998 2.0000 2.0000
H-6 1.9989 1.9991 1.9995 1.9998 1.9996 1.9994 1.9998 1.9981 1.9992
H-5 1.9954 1.9986 1.9988 1.9995 1.9988 1.9990 1.9990 1.9937 1.9975
H-4 1.9823 1.9971 1.9891 1.9976 1.9949 1.9900 1.9974 1.9881 1.9922
H-3 1.9786 1.9909 1.9819 1.9840 1.9881 1.9760 1.9895 1.9807 1.9880
H-2 1.9512 1.9847 1.9648 1.9800 1.9821 1.9560 1.9839 1.9417 1.9772
H-1 1.8509 1.7696 1.9185 1.9498 1.8334 1.8817 1.7372 1.8944 1.9395
H 1.2564 1.5268 1.2775 1.0934 1.3910 1.6552 1.6242 1.3216 1.4121
L 0.9536 0.4852 0.7260 0.9581 0.7654 0.4415 0.3724 0.7765 0.5629
L+1 0.0298 0.2377 0.1388 0.0237 0.0397 0.0987 0.2935 0.0987 0.1111
L+2 0.0030 0.0103 0.0051 0.0143 0.0071 0.0026 0.0032 0.0066 0.0201
L+3 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0000 0.0000
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-64 C46-65 C46-66 C46-67 C46-68 C46-69 C46-70 C46-71 C46-72
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 1.9999 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 1.9999 2.0000 1.9999 1.9999 1.9999 1.9999 2.0000 2.0000 1.9999
H-6 1.9999 1.9991 1.9993 1.9998 1.9998 1.9993 1.9995 1.9992 1.9993
H-5 1.9996 1.9976 1.9983 1.9992 1.9983 1.9973 1.9974 1.9991 1.9985
H-4 1.9993 1.9943 1.9939 1.9971 1.9967 1.9852 1.9923 1.9910 1.9934
H-3 1.9853 1.9823 1.9778 1.9897 1.9919 1.9700 1.9746 1.9777 1.9855
H-2 1.9849 1.9512 1.9457 1.9409 1.9785 1.9523 1.9575 1.9615 1.9659
H-1 1.9762 1.7376 1.8374 1.7794 1.9587 1.9146 1.9015 1.9096 1.8982
H 1.5802 1.5157 1.3036 1.2167 1.7176 1.5695 1.6213 1.5982 1.4039
L 0.3487 0.6493 0.6986 0.7968 0.2363 0.4560 0.3814 0.5041 0.5742
L+1 0.1219 0.1705 0.2388 0.2709 0.1074 0.1551 0.1687 0.0362 0.1774
L+2 0.0038 0.0022 0.0066 0.0094 0.0149 0.0005 0.0058 0.0233 0.0037
L+3 0.0004 0.0001 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 0.0001
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-73 C46-74 C46-75 C46-76 C46-77 C46-78 C46-79 C46-80 C46-81
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 1.9999
H-7 1.9999 1.9999 1.9999 1.9997 2.0000 1.9999 1.9999 1.9998 1.9999
H-6 1.9996 1.9988 1.9998 1.9996 1.9996 1.9992 1.9998 1.9995 1.9993
H-5 1.9994 1.9975 1.9990 1.9994 1.9980 1.9984 1.9994 1.9991 1.9982
H-4 1.9938 1.9957 1.9960 1.9916 1.9955 1.9920 1.9987 1.9975 1.9812
H-3 1.9912 1.9671 1.9912 1.9824 1.9810 1.9906 1.9927 1.9935 1.9741
H-2 1.9636 1.9353 1.9773 1.9690 1.9543 1.9902 1.9876 1.9738 1.9307
H-1 1.8691 1.8569 1.9578 1.8197 1.7847 1.9605 1.9662 1.5729 1.8573
H 1.3565 1.4499 1.1607 1.6785 1.5503 1.7213 1.4714 1.4055 1.7038
L 0.6927 0.7155 0.7389 0.5226 0.5094 0.2596 0.3634 0.9315 0.5214
L+1 0.1011 0.0790 0.1569 0.0330 0.2221 0.0532 0.1629 0.1219 0.0325
L+2 0.0329 0.0043 0.0225 0.0044 0.0050 0.0352 0.0577 0.0049 0.0017
L+3 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0001 0.0000 0.0000
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C46-82 C46-83 C46-84 C46-85 C46-86 C46-87 C46-88 C46-89 C46-90
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 1.9999 2.0000 2.0000 2.0000 1.9999 2.0000 1.9997 1.9999 2.0000
H-7 1.9998 1.9999 1.9999 2.0000 1.9999 1.9999 1.9997 1.9998 2.0000
H-6 1.9996 1.9997 1.9996 1.9999 1.9997 1.9998 1.9995 1.9996 1.9994
H-5 1.9993 1.9995 1.9972 1.9995 1.9995 1.9991 1.9989 1.9993 1.9983
H-4 1.9978 1.9957 1.9965 1.9918 1.9994 1.9985 1.9888 1.9948 1.9856
H-3 1.9905 1.9830 1.9861 1.9856 1.9903 1.9920 1.9884 1.9924 1.9780
H-2 1.9821 1.9716 1.9309 1.9629 1.9802 1.9081 1.9742 1.9849 1.9587
H-1 1.6635 1.9094 1.6531 1.9349 1.7884 1.7538 1.5597 1.6180 1.5793
H 1.5523 1.3536 1.4348 1.4305 1.7286 1.0792 1.3252 1.2648 1.3142
L 0.7029 0.6037 0.6556 0.4280 0.3663 1.0303 1.1272 1.0293 0.9144
L+1 0.0988 0.1389 0.3442 0.2562 0.1368 0.1213 0.0303 0.1144 0.2651
L+2 0.0133 0.0449 0.0018 0.0106 0.0107 0.1177 0.0083 0.0027 0.0071
L+3 0.0001 0.0000 0.0001 0.0001 0.0003 0.0001 0.0000 0.0001 0.0000
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-91 C46-92 C46-93 C46-94 C46-95 C46-96 C46-97 C46-98 C46-99
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 1.9999 2.0000 2.0000 1.9999 1.9999 2.0000 2.0000 2.0000 1.9998
H-7 1.9998 2.0000 1.9998 1.9999 1.9999 1.9999 1.9997 1.9998 1.9998
H-6 1.9998 1.9996 1.9997 1.9999 1.9995 1.9998 1.9996 1.9995 1.9997
H-5 1.9993 1.9970 1.9992 1.9996 1.9994 1.9987 1.9991 1.9992 1.9991
H-4 1.9977 1.9966 1.9982 1.9994 1.9809 1.9986 1.9978 1.9969 1.9909
H-3 1.9912 1.9815 1.9880 1.9945 1.9796 1.9895 1.9897 1.9880 1.9898
H-2 1.9860 1.9502 1.9565 1.9525 1.9188 1.9790 1.9725 1.9843 1.9788
H-1 1.7262 1.8981 1.9161 1.9483 1.7856 1.8704 1.6628 1.6955 1.6761
H 1.5738 1.6848 1.1930 1.3238 1.5810 1.3437 1.5842 1.1677 1.1978
L 0.5040 0.3966 0.8736 0.5232 0.5875 0.6257 0.5637 0.9942 1.0194
L+1 0.2175 0.0677 0.0661 0.1847 0.1673 0.1292 0.2263 0.1657 0.1480
L+2 0.0046 0.0281 0.0099 0.0740 0.0003 0.0654 0.0042 0.0082 0.0005
L+3 0.0001 0.0000 0.0001 0.0004 0.0003 0.0001 0.0002 0.0011 0.0002
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
C46-100 C46-101 C46-102 C46-103 C46-104 C46-105 C46-106 C46-107 C46-108
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 1.9999 1.9999 1.9999 1.9999
H-7 2.0000 1.9999 1.9999 1.9999 2.0000 1.9999 1.9998 1.9999 1.9998
H-6 1.9997 1.9998 1.9997 1.9995 1.9997 1.9995 1.9997 1.9996 1.9998
H-5 1.9994 1.9996 1.9996 1.9982 1.9990 1.9993 1.9988 1.9990 1.9997
H-4 1.9815 1.9933 1.9956 1.9972 1.9918 1.9880 1.9972 1.9974 1.9902
H-3 1.9666 1.9922 1.9869 1.9846 1.9781 1.9759 1.9762 1.9944 1.9826
H-2 1.9605 1.9875 1.9583 1.9523 1.8334 1.9436 1.8960 1.9724 1.9701
H-1 1.8025 1.6722 1.8340 1.6250 1.7999 1.8690 1.6293 1.6690 1.9012
H 1.2848 1.6337 1.2544 1.4137 1.2912 1.4978 1.5262 1.2596 1.7031
L 0.7430 0.4484 0.6450 0.8610 0.8793 0.7074 0.8317 0.7146 0.3620
L+1 0.2443 0.2175 0.3175 0.1650 0.1711 0.0171 0.1434 0.3897 0.0878
L+2 0.0177 0.0552 0.0089 0.0035 0.0564 0.0027 0.0011 0.0046 0.0036
L+3 0.0000 0.0006 0.0001 0.0001 0.0000 0.0000 0.0004 0.0000 0.0001
L+4 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0001
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Continued on next page
C46-109 C46-110 C46-111 C46-112 C46-113 C46-114 C46-115 C46-116
H-9 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-8 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000
H-7 1.9999 1.9999 1.9999 1.9999 1.9999 1.9999 2.0000 2.0000
H-6 1.9999 1.9998 1.9996 1.9999 1.9999 1.9996 1.9994 1.9997
H-5 1.9996 1.9986 1.9995 1.9985 1.9994 1.9990 1.9989 1.9969
H-4 1.9902 1.9974 1.9808 1.9957 1.9963 1.9969 1.9980 1.9824
H-3 1.9866 1.9942 1.9667 1.9624 1.9860 1.9885 1.9965 1.9705
H-2 1.9636 1.9444 1.9301 1.9175 1.9614 1.9662 1.9842 1.8595
H-1 1.9284 1.6940 1.8827 1.7943 1.9484 1.8458 1.6805 1.8189
H 1.7071 1.6335 1.3483 1.2327 1.2872 1.4071 1.5277 1.2777
L 0.2301 0.5765 0.4474 0.7966 0.6902 0.6892 0.5211 0.9946
L+1 0.1752 0.1207 0.4399 0.2924 0.1228 0.0959 0.2271 0.0772
L+2 0.0194 0.0408 0.0051 0.0098 0.0085 0.0117 0.0666 0.0220
L+3 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000 0.0003
L+4 0.0000 0.0000 0.0000 0.0001 0.0000 0.0001 0.0000 0.0002
L+5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+6 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+7 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+8 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
L+9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Table 4.12: The symmetrized von Neumann entropy of fullerene C20to C38, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C20 2.2562
C24 2.3162
C26 3.3717
C28-1 C28-2 2.9721 1.9383
C30-1 C30-2 C30-3 1.6546 0.6984 2.3111
C32-1 C32-2 C32-3 C32-4 C32-5 C32-6 0.3942 0.7924 1.9223 1.5322 1.7044 0.6099
C34-1 C34-2 C34-3 C34-4 C34-5 C34-6 1.4131 1.5265 2.2510 1.6789 1.4979 2.1930
C36-1 C36-2 C36-3 C36-4 C36-5 C36-6 C36-7 C36-8 C36-9 2.0656 1.7905 1.7588 1.9973 1.6609 1.6150 1.8588 1.9366 1.7443 C36-10 C36-11 C36-12 C36-13 C36-14 C36-15
1.9293 2.3418 2.2063 2.4389 1.0202 1.5481
C38-1 C38-2 C38-3 C38-4 C38-5 C38-6 C38-7 C38-8 C38-9 1.4907 2.0043 2.2036 2.3702 2.1339 2.0241 1.7868 2.0245 2.4800 C38-10 C38-11 C38-12 C38-13 C38-14 C38-15 C38-16 C38-17
1.2979 2.1439 1.7224 1.8091 1.9847 1.6431 2.1996 2.1195
Table 4.13: The symmetrized von Neumann entropy of fullerene C40 and C42, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C40-1 C40-2 C40-3 C40-4 C40-5 C40-6 C40-7 C40-8 C40-9 1.4710 2.0690 1.2962 2.0369 1.8572 2.1398 2.9390 2.2923 2.2058 C40-10 C40-11 C40-12 C40-13 C40-14 C40-15 C40-16 C40-17 C40-18 2.0380 1.7792 2.3400 1.8512 1.5996 1.5667 2.1384 2.1501 2.7088 C40-19 C40-20 C40-21 C40-22 C40-23 C40-24 C40-25 C40-26 C40-27 1.8145 1.8037 2.0239 2.0014 2.0596 2.3926 1.3507 1.9208 1.6941 C40-28 C40-29 C40-30 C40-31 C40-32 C40-33 C40-34 C40-35 C40-36 1.5352 1.6431 1.9113 2.3494 1.5568 1.7098 1.9599 1.9483 1.5262 C40-37 C40-38 C40-39 C40-40
1.5955 1.8677 2.1101 3.2597
C42-1 C42-2 C42-3 C42-4 C42-5 C42-6 C42-7 C42-8 C42-9 1.3056 2.1692 1.9666 1.8825 1.9345 2.1822 2.1407 2.2014 1.6682 C42-10 C42-11 C42-12 C42-13 C42-14 C42-15 C42-16 C42-17 C42-18 1.4838 1.7970 2.0359 1.8989 2.2786 2.7854 2.3084 2.0044 2.0478 C42-19 C42-20 C42-21 C42-22 C42-23 C42-24 C42-25 C42-26 C42-27 1.4263 1.5012 2.0232 1.8400 2.0102 2.5107 1.9089 1.5904 1.6002 C42-28 C42-29 C42-30 C42-31 C42-32 C42-33 C42-34 C42-35 C42-36 1.4716 1.9872 1.8422 2.0732 1.3321 1.5438 2.1687 1.7455 1.8157 C42-37 C42-38 C42-39 C42-40 C42-41 C42-42 C42-43 C42-44 C42-45 1.9655 1.5445 1.4736 1.7624 1.5858 1.7086 1.5419 1.6772 1.7130
Table 4.14: The symmetrized von Neumann entropy of fullerene C44, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C44-1 C44-2 C44-3 C44-4 C44-5 C44-6 C44-7 C44-8 C44-9 1.4578 1.4818 0.8780 0.8713 1.7042 1.6040 1.8866 1.5860 1.8479 C44-10 C44-11 C44-12 C44-13 C44-14 C44-15 C44-16 C44-17 C44-18 1.9316 1.9048 1.9871 1.4137 1.0378 1.0052 2.1670 1.9534 1.6589 C44-19 C44-20 C44-21 C44-22 C44-23 C44-24 C44-25 C44-26 C44-27 2.1553 1.9715 2.5831 2.1067 1.8332 2.8689 2.1700 1.9414 1.7481 C44-28 C44-29 C44-30 C44-31 C44-32 C44-33 C44-34 C44-35 C44-36 1.1875 1.8926 1.0772 2.1698 2.1464 1.4833 1.6416 1.4701 1.6972 C44-37 C44-38 C44-39 C44-40 C44-41 C44-42 C44-43 C44-44 C44-45 1.8528 1.8814 1.7214 2.2371 1.4439 1.6371 1.6396 2.0045 1.7833 C44-46 C44-47 C44-48 C44-49 C44-50 C44-51 C44-52 C44-53 C44-54 1.8285 1.6002 1.9635 2.0605 1.7526 1.6100 1.9938 1.7996 2.0553 C44-55 C44-56 C44-57 C44-58 C44-59 C44-60 C44-61 C44-62 C44-63 1.4203 0.8840 1.1821 1.7729 1.9020 1.6058 1.9033 2.0758 2.0963 C44-64 C44-65 C44-66 C44-67 C44-68 C44-69 C44-70 C44-71 C44-72 1.5918 1.5976 1.8905 1.8136 2.0935 1.9227 2.0532 2.2603 1.2042 C44-73 C44-74 C44-75 C44-76 C44-77 C44-78 C44-79 C44-80 C44-81 1.9206 2.0094 1.7909 1.5122 1.7945 1.9216 1.7774 1.9251 1.7666 C44-82 C44-83 C44-84 C44-85 C44-86 C44-87 C44-88 C44-89
1.3142 1.8016 1.8065 2.1090 2.0028 2.2654 1.7414 2.5570
Table 4.15: The symmetrized von Neumann entropy of fullerene C46, calculated by spin-restricted TAO-LDA(θ = 7 mHartree)/6-31G*.
C46-1 C46-2 C46-3 C46-4 C46-5 C46-6 C46-7 C46-8 C46-9 1.8054 1.7234 1.7217 1.5961 1.7164 1.2825 1.1278 1.2015 1.6323 C46-10 C46-11 C46-12 C46-13 C46-14 C46-15 C46-16 C46-17 C46-18 1.3140 1.8499 1.2219 1.7044 2.3203 1.2428 2.5097 1.7997 2.0046 C46-19 C46-20 C46-21 C46-22 C46-23 C46-24 C46-25 C46-26 C46-27 1.6635 2.0722 1.2879 1.2101 1.4254 1.6460 1.0659 1.9268 2.0697 C46-28 C46-29 C46-30 C46-31 C46-32 C46-33 C46-34 C46-35 C46-36 1.7688 2.0449 1.3107 1.9809 2.0626 1.7187 2.4517 1.7567 2.1005 C46-37 C46-38 C46-39 C46-40 C46-41 C46-42 C46-43 C46-44 C46-45 1.4285 2.0563 1.7671 1.5355 1.3774 1.6923 2.1973 1.6774 2.3179 C46-46 C46-47 C46-48 C46-49 C46-50 C46-51 C46-52 C46-53 C46-54 1.7693 1.7191 2.1475 2.0921 2.0048 1.8185 1.7667 1.1980 2.0688 C46-55 C46-56 C46-57 C46-58 C46-59 C46-60 C46-61 C46-62 C46-63 1.9519 1.9507 1.9313 1.7220 1.8017 1.6291 1.8786 1.9859 1.7443 C46-64 C46-65 C46-66 C46-67 C46-68 C46-69 C46-70 C46-71 C46-72 1.3820 2.0716 2.1802 2.2974 1.2314 1.7588 1.6872 1.5972 1.8844 C46-73 C46-74 C46-75 C46-76 C46-77 C46-78 C46-79 C46-80 C46-81 1.9447 1.9374 1.8890 1.5825 1.9969 1.1962 1.6226 2.1739 1.6249 C46-82 C46-83 C46-84 C46-85 C46-86 C46-87 C46-88 C46-89 C46-90 1.9669 1.9294 2.3739 1.8413 1.5891 2.4327 2.1060 2.1622 2.4758 C46-91 C46-92 C46-93 C46-94 C46-95 C46-96 C46-97 C46-98 C46-99 1.9277 1.5525 1.8659 1.9410 2.0364 1.9828 2.0503 2.2171 2.2036 C46-100 C46-101 C46-102 C46-103 C46-104 C46-105 C46-106 C46-107 C46-108
2.2952 2.0173 2.2025 2.2482 2.4620 1.7521 2.2580 2.3792 1.4471 C46-109 C46-110 C46-111 C46-112 C46-113 C46-114 C46-115 C46-116
1.4445 1.9974 2.2266 2.4055 1.8294 1.8924 2.1346 2.2725
Table 4.16: The difference(in Kcal/mol) between unrestricted singlet and restricted singlet energy of fullerenes (from n=20 to n=46), calculated by TAO-LDA(θ = 7 mHartree)/6-31G*.The energy difference between unrestricted singlet and restricted singlet energy of fullerenes (from n=36 to n=46) are simply all zeros and omitted to avoid repeatness.
fullerenes energy difference c20
0.00 c24
0.00 c26
0.00 c28
1 0.00
2 0.00
c30
1 0.00
2 0.00
3 0.00
c32
1 0.00
2 0.00
3 0.00
4 0.00
5 0.00
6 0.00
c34
1 0.00
2 0.00
3 0.00
4 0.00
5 0.00
6 0.00
4.2 Discussions
In table 4.2, the comparison of singlet-triplet gaps, ionization potentials, electron affin-ity and fundamental gaps to O2 (Optimized second-order opposite-spin) and experiment data of some fullerenes Cnhas showed. For the singlet-triplet gap of C36, cost-expensive O2[12] results overestimates experiment values nearly three times, while TAO-LDA give much acceptable results. Though O2improve scaled second-order Møller-Plesset methods for open-shell systems, this poor result of singlet-triplet gap of C36indicates O2 can not well deal multi-reference systems [7, 12]. The electron affinities calculated by TAO-LDA are compared to O2in cc-pVDZ basis set since O2
/cc-pVDZ give almost the same results
in singlet-triplet gap compared to O2/6-31G* result. TAO-LDA give better result than
O2 in C20, C30, C32 and C36 compared to experiment result [4]. Ionization potential and fundamental gap calculted by TAO-LDA also are reasonable compare to O2result. Since vertical electron affinities are usually smaller than adiabatic electron affinities, TAO-LDA predicted results may be closer to experiment data.Biradical character or even polyradical character of molecules can be judged quanti-tively by natural orbital occupation numbers (NOONs) [47, 48].The natural orbital occu-pation numbers can offer the evidence of polyradical character of molecules, since in the single-reference system the natural orbital occupation numbers are either 0 (fully unoccu-pied) or 2 (fully occuunoccu-pied), natural orbital occupation numbers between 2 and 0 may be interpreted as fractionally occupied electrons . From (Table 4.8) to (Table 4.12), the ac-tive orbital occupation numbers(HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO+1, ..., and LUMO+9) of fullerenes C20to C46calculated by restricted TAO-LDA are provided to study the biradical or even polyradical character of fullerenes C20to C46. For convenience, HOMO and LUMO are denoted as H and L such that HOMO-9, ..., HOMO-1, HOMO, LUMO, LUMO+1, ..., and LUMO+9 are denoted as H-9, ...,H-1, H, L, L+1, ...,and L+9.
As can be seen from (Table 4.8) to (Table 4.12), the orbital occupation numbers of each fullerene all include occupation numbers between 0 and 2, and most fullerenes have one or more orbital occupation numbers between 0.5 and 1.5, which suggests that fullerene sys-tem have biradical character or even polyradical character, and results in static correlation
error(SCE) can not be handled well by traditional KS-DFT.
A quantive criterion to measure the static correlation is the symmetrized von Neumann entropy [49]
S = −
1 2∑
i
[filn(fi) + (1
− f
i) ln(1− f
i)], (4.1) which is the symmetrized form of the von Neumann entropy , S =− ∑
if
iln(fi) [18,50].The von Neumann entropy, a scalar extracted from the occupation numbers array, is a wavefunction based criterion to measure the static correlation error [18, 49, 50]. Because the symmetrized von Neumann entropy are 0 for single-reference system, the magnitude can be refered to as an index to judge the static correlation in the systems. Although sym-metrized von Neumann entropy lost some detailed information in the process of extrac-tion, the one-dimensional number character of symmetrized von Neumann also provide a simple way to measure the degree of poly-radical or severeness of static correlation.
Another metric used to evaluate the ability of a theoretical approach handling sys-tems with severe static correlation is comparing the difference between unrestricted singlet ground state energy and restricted singlet ground state energy. For the simplest illustra-tive example of static correlation error, dissociation of H2, the energy difference between unrestricted singlet state and restricted singlet state used as quantitive measure of static correlation error(SCE) of specific theoretical approach at dissociation limit, where the multireference character dominates [11]. In table 4.16 , the unrestricted singlet energy is subtracted by the restricted singlet energy, and the differences are all zero among all fullerenes from C20 to C46, which provide evidence of excellent ability handling multi-reference system of TAO-LDA.
In table 4.3, the natural orbital occupation numbers of HOMO-1, HOMO, LUMO and LUMO+1 in C20 are close to 0.5 implies that C20 is a near-tetraradical. The active orbital occupation number of C24and C26also show significant polyradical character. The singlet-triplet gap of C20, C24and C26are all positive, which indicate the ground state is singlet state. The significant polyradical character and small singlet-triplet energy gap of C20both explain the highly lability in experiment [4]. The strongly static correlation in C26lead to different ground state including triplet state or quintet state both in B3LYP
cal-culations [46, 51], and the positive singlet-triplet energy gap also supported by the CCSD computational results [52].
The lowest energy isomer in C28 is with Td symmetry and the D2 isomer are 16.27 kcal/mol higher than isomer1, as can be seen in table 4.1. The relative energy order are the same as HF and B3LYP calculation [52,53]. However, as the static correlation effect is included in TAO-DFT method, the ground state of isomer one is singlet instead of quintet in HF and B3LYP level [52, 53] or the triplet in CCSD level [52]. In tables 4.3 and 4.12 , both isomers of C28show polyradical character, and interesting isomer 1 has more polyradical character than isomer 2.
The symmery of C30 isomers are respectively C2v, C2v and D5h and all are singlet ground state. The relative energy order are the same as B3LYP, MP2 calculations and tight-binding molecular dynamics [46, 54]. Table 4.3 and table 4.12 shows the order of polyradical character is isomer3 > isomer2 > isomer1. The orbital occupation number of isomer3 are similar to tetraradical, and it's relatively higher symmetrized von Neumann entropy, which may both explain the divergence in HF and B3LYP level by Jie Song et al [54].
The relative energy order of C32isomers are the same as B3LYP results [55], where the respective symmetry group are D3, C2, C2, D2, D3dand D3h. Compared to HF and MP2 results, the lower three results are the same [55]. The occupation numbers of C32 shows no significant polyradical character, especially for isomer1 which has almost closed-shell occupation numbers. Combined with the fact that isomer1 has largest singlet-triplet gap, ionization potential, fundamental gap and lowest electron affinity, isomer1 is most stable among all six isomers of C32.
The relative energy order of C34isomers are different from B3LYP and semi-empirical(
MNDO, AM1 and PM3) results ,which is isomer1< isomer2 < isomer5 < isomer3 < iso-mer4 < isomer6, isomer2< isomer1< isomer3< isomer5< isoiso-mer4< isomer6, isomer2<
isomer1< isomer3< isomer4< isomer5< isomer6, and isomer2< isomer1< isomer3< iso-mer5< isomer4< isomer6, respectively [56]. Ths small singlet-triplet gaps ranges from 7.37 to 3.33 kcal/mol and occupation numbers shows multireference character of C34,
which emphasize the including of static correlation is important to predict the electronic properties of C34isomers.
The symmetry of first four C36 isomers are D2d, D6h, C2 and C2v. D2d singlet state is the lowest energgy state of C36. On the other hand, calculations of HF, B3LYP and CCSD predict D6h triplet state to be the lowest energy state in C36, while calculation in AM1 level predict C2v triplet state to be the ground state [52, 57]. In table 4.6, isomer1 and isomer2 both show polyradical character, which has been mentioned in previous in-vestigation [10]. Isomer1 with higher symmetrized entropy and lower singlet-triplet gap than isomer2 indicate Isomer1 with lowest energy may be more chemically unstable than isomer2.
The lowest energy isomer of C38 of C2 symmetry calculated by TAO-DAT is in con-sistent with B3LYP and tight-binding results [9, 58]. Unlike B3LYP calculation shows no isomer has relative energy lower than 25 kcal/mol [58], relative energy until isomer5 is only 18.53 kcal/mol in TAO-DFT calculation. Isomer1 has largest EA, second larget singlet-triplet gap and fundamental gap and second smallest symmetrized entropy(in table 4.12), which provide evidence of the relatively stable character of isomer1. All C38 iso-mers shows significant diradical character in table 4.7 and the highest singlet-triplet gap is only 8.47 kcal/mol reveal the general unstable trend in 17 C38isomers.
Several theoretical calculations including semiempirical, tight-binding and DFT meth-ods show isomer1 with D2 symmetry and isomer3 with D5dare lowest and second lowest energy state of C40[9, 31, 58]. The previous investigations in relative energies among C40 isomers mostly based on single-reference closed-shell methods. The table 4.8 shows the polyradical character of C40, and the small singlet-triplet gap of C40(isomer3 has largers singlet-triplet gap 9.23 kcal/mol in C40 isomers), which shows the strongly static corre-lation in C40isomers. TAO-DFT calculation, which includes static correlation contribu-tion, predicts isomer2 with eleven pentagon adjacency edges lower than isomer3 with ten pentagon adjacency edges. There is still relation between relative energy and number of pentagon adjacency edges(The pentagon adjacency edges in order is 10, 11, 10, 11, 11, 11, 12, 11, 12,12,11,11,12,12,12,12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14,
14, 15, 14, 15, 15, 15, 16, 18, 20 ), but not as strongly as the result proposed in previous researches [31, 59].
TAO-LDA calculations shows singlet isomer1 of D3 symmetry is the lowest energy state of C42, which is in agreement with tight-binding and B3LYP calculations [9, 58]. D3
isomer1 has largest singlet-triplet gap 9.11 kcal/mol and smallest symmetrized entropy in tables 4.1 and 4.13 all provide evidence that singlet state of isomer1 is the most stable isomer in C42. In table 4.1, singlet state energy difference from isomer2 to isomer5 is only 1.55 kcal/mol. Unlike TAO-DFT calculation, B3LYP predict isomer3 with C1 symmetry is the second lowest energy state, and the relative energy order of tight-binding calculation is isomer1< isomer3< isomer4< isomer5< isomer2. Considering the general trend of small singlet-triplet gap and the fractionally occupied numbers, the C42isomers are not stable.
The first five lowest energy isomers of C44 have small energy difference below 15.2 kcal/mol, and their relative energy order is a controversial issue, for example, B3LYP calculations shows relative energy order isomer1<isomer2<isomer3<isomer5 while MP2 calculations show isomer3 < isomer4 < isomer2 < isomer1 [46, 58, 60]. Isomer3 and iso-mer4 have nearly closed-shell occupation number as can be seen in table 4.10, and largest singlet-triplet gaps about 12.9 kcal/mol and fundamental gaps ,which all claim that iso-mer3 and isomer4 are relatively stable isomers.
The lowest seven enegy states of C46have narrow relative energy difference that iso-mer7 is only 7.32 kcal/mol higher than isomer1. From orbital occupation numbers in table 4.11, most isomers shows significant polyradical character and isomer7 is the scarce exception with minor polyradical character in lower energy states. B3LYP calculations on 32 of C46isomers shows relative energy order to be isomer1< isomer5< isomer4< iso-mer7<isomer2< isomer3 [58].
In all, singlet triplet energy gaps have strongly positive correlation with fundamental gaps, which can be interpreted by the interaction between quasi-electron and quasi-hole in real world [61--64]. Singlet triplet energy gaps have strongly negative correlation with symmetrized von Neumann entropies. There is negative correlation between singlet triplet energy gaps and static correlation error because symmetrized von Neumann entropy is
an indicator of magnitude of static correlation. There are no clear dependence between relative energies, ionization potentials and electron affinities
Chapter 5 Conclusions
TAO-LDA, a method which has shown static correlation included, is used to explore electronic properties of small fullerenes Cn(n is even number from 20 to 46 except for 22 ) since their strongly-correlated character makes an obstacle for other approaches. We predict the relative isomers of each Cn which provide information for further theoretical and experimental research. All 342 isomers of fullerenes Cn (n is even number from 20 to 46 except for 22 ) has shown to be singlet ground state instead of triplet ground state.
Most isomers have small singlet-triplet energy difference and fundamental gap, which
Most isomers have small singlet-triplet energy difference and fundamental gap, which