Nuclear orbital plus molecular orbital approach for positron annihilation spectra of atoms and molecules Yasuhiro Ikabata,1 Risa Aiba,2 Toru Iwanade,2 Hiroaki Nishizawa,3 Feng Wang,4 Hiromi Nakai1,2,5,6,*
1 Waseda
Research Institute for Science and Engineering, Waseda University, 3-4-1
Okubo, Shinjuku-ku, Tokyo 169-8555, Japan 2 Department
of Chemistry and Biochemistry, School of Advanced Science and
Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan 3 Department
of Theoretical and Computational Molecular Science, Division of
Computational Molecular Science, Institute for Molecular Science, Myodaiji, Okazaki, Japan 4 Department
of Chemistry and Biotechnology, Faculty of Science, Engineering and
Technology, Swinburne University of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia 5 CREST,
Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama
332-0012, Japan 6 Elements
Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University,
Katsura, Kyoto 615-8520, Japan _____________________________________________________________________ *Corresponding author. E-mail address:
[email protected] (H. Nakai).
S1
Table SI. The full width of half maximum (FWHM), Δε, and percentage contribution of each molecular orbital (MO) at ε = 0, p, of positron annihilation spectra of methane and its fluoro derivatives. The FWHMs are given in keV. PWP/HF CH4
CH3F
CH2F2
CHF3
CF4
1a1 2a1 1t2 Total 1a1 2a1 3a1 4a1 1e1 5a1 2e1 Total 1b2 1a1 2a1 3a1 2b2 4a1 1b1 5a1 3b2 1a2 4b2 6a1 2b1 Total 1e1 1a1 2a1 3a1 2e1 4a1 5a1 3e1 4e1 5e1 1a2 6a1 Total 1t2 1a1 2a1 3a1 2t2 4a1 3t2 1e1 4t2 1t1 Total
Δε 10.40 1.97 3.32 2.94 15.90 10.36 2.35 2.78 3.47 5.18 5.33 3.72 16.00 15.91 10.41 2.40 3.72 3.27 3.68 4.05 5.44 5.36 5.86 4.61 4.70 4.21 16.0 15.86 10.42 2.13 3.73 3.93 3.81 5.06 5.20 5.83 5.82 4.72 4.52 15.68 15.81 10.43 1.98 3.73 4.55 4.89 4.89 5.81 6.05 4.74
p 6.1 33.0 60.9 2.9 4.5 18.9 17.5 28.7 9.9 17.7 2.3 2.3 3.5 14.1 10.1 11.4 9.7 9.3 7.1 6.8 6.6 8.7 8.1 3.7 1.9 2.9 12.6 16.5 7.7 7.9 12.0 11.5 11.1 5.0 7.0 4.7 1.6 2.4 11.5 21.0 5.7 15.6 10.0 14.8 12.7
NOMO/HF Δε 8.51 1.56 2.63 2.32 14.49 8.84 2.45 2.88 3.26 2.56 3.92 3.15 14.56 14.38 9.12 1.98 3.14 3.33 2.97 3.52 3.46 4.19 4.66 2.16 4.08 3.30 14.52 14.18 9.36 1.72 3.08 3.38 3.18 3.70 3.47 4.20 4.79 1.94 3.30 15.31 15.11 10.03 1.56 3.06 3.39 3.70 3.74 2.60 4.79 3.35
p 0.3 29.4 70.2 0.6 0.2 17.6 11.2 30.4 20.2 20.0 0.3 0.3 0.1 11.8 8.9 8.3 9.6 9.1 12.3 9.3 8.6 15.2 6.3 0.4 0.2 0.1 8.9 12.9 7.8 6.9 15.3 17.9 13.9 5.9 9.8 1.0 0.4 0.1 6.9 15.9 8.4 11.9 17.7 23.3 14.4
LDA Δε 8.36 1.55 2.56 2.27 14.27 8.64 2.32 2.77 2.85 2.29 3.94 2.99 14.15 14.33 8.89 1.89 3.07 3.19 3.12 2.73 3.27 3.98 4.38 2.21 3.90 3.11 14.13 14.26 9.11 1.67 3.02 3.20 2.96 3.50 3.05 4.11 4.68 1.79 3.15 14.48 15.53 9.76 1.54 3.01 3.05 3.54 3.60 2.40 4.69 3.31
S2
p 0.3 29.1 70.5 0.5 0.2 15.2 11.5 27.9 20.8 23.9 0.3 0.3 0.1 10.5 7.7 8.6 9.4 9.5 12.0 9.3 8.5 16.4 7.4 0.4 0.2 0.1 8.1 11.3 8.1 6.8 15.6 18.9 13.3 5.6 11.7 0.6 0.7 0.1 6.2 14.0 9.0 12.5 18.4 24.7 13.9
NOMO/DFT BLYP B3LYP Δε p Δε p 8.44 0.3 8.45 0.3 1.55 28.9 1.56 29.1 2.55 70.8 2.58 70.6 2.27 2.29 14.36 0.5 14.38 0.5 8.73 0.2 8.74 0.2 2.33 15.0 2.36 15.7 2.72 11.3 2.77 11.3 2.81 27.5 2.91 28.1 2.27 21.1 2.35 20.9 3.91 24.4 3.95 23.4 2.97 3.02 14.28 0.3 14.44 0.3 14.38 0.3 14.27 0.3 8.98 0.1 9.00 0.1 1.89 10.4 1.91 10.8 3.07 7.7 3.09 8.0 3.17 8.2 3.21 8.3 3.07 9.6 3.23 9.3 2.70 9.3 2.76 9.4 3.24 12.0 3.29 12.1 3.95 9.3 4.02 9.3 4.34 8.6 4.44 8.6 2.20 16.6 2.21 16.3 3.86 7.6 3.94 7.2 3.08 3.15 15.14 0.8 14.37 0.4 15.20 0.4 14.12 0.2 7.38 0.0 9.23 0.1 1.70 7.3 1.69 8.2 3.09 10.8 3.04 11.7 3.11 7.8 3.23 7.9 2.95 6.4 3.01 6.8 3.44 15.7 3.53 15.5 3.02 19.5 3.12 18.7 4.18 13.1 4.13 13.4 4.72 5.6 4.69 5.7 1.79 12.4 1.84 11.4 3.16 3.19 15.19 1.0 15.22 1.0 15.10 0.3 15.06 0.4 9.87 0.1 9.89 0.1 1.54 6.1 1.54 6.3 3.02 13.9 3.03 14.4 3.04 8.9 3.16 8.8 3.52 12.4 3.57 12.3 3.56 18.4 3.61 18.3 2.36 25.0 2.42 24.5 4.67 13.9 4.71 14.0 3.28 3.34
LC-BLYP Δε p 8.41 0.3 1.52 30.0 2.56 69.7 2.24 14.34 0.5 8.71 0.2 2.28 15.6 2.75 11.3 2.86 27.9 2.28 21.3 3.92 23.2 2.96 14.40 0.3 14.22 0.3 8.96 0.1 1.86 10.7 3.03 7.8 3.13 8.5 3.19 9.3 2.72 9.3 3.23 12.2 3.96 9.3 4.38 8.4 2.14 16.6 3.89 7.1 3.08 14.33 0.4 14.08 0.2 9.19 0.1 1.65 8.1 2.98 11.4 3.07 8.1 2.97 6.8 3.48 15.5 3.04 19.0 4.06 13.2 4.65 5.6 1.79 11.5 3.11 15.19 0.9 15.04 0.3 9.86 0.1 1.51 6.2 2.97 14.1 2.78 9.3 3.52 12.3 3.55 18.3 2.37 24.7 4.66 13.8 3.26
Table SII. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for benzene. The FWHMs are given in keV.
NOMO/HF 1a1g 1e1u 1e2g 1b1u 2a1g 2e1u 2e2g 3a1g 2b1u 1b2u 3e1u 1a2u 3e2g 1e1g Total
Δε 9.29 9.59 9.72 9.93 1.21 2.03 2.72 1.25 3.17 3.88 1.90 2.03 3.26 2.62 2.36
p 0.2 0.4 0.4 0.2 6.4 10.6 10.0 9.5 6.2 2.3 13.3 11.5 9.0 20.0
LDA Δε 9.05 9.37 9.47 9.53 1.20 2.00 2.67 1.20 3.13 3.79 1.82 1.98 3.11 2.61 2.31
p 0.2 0.4 0.4 0.2 6.5 11.2 10.3 9.8 6.3 2.4 13.7 11.0 9.5 18.0
NOMO/DFT BLYP B3LYP Δε p Δε p 9.21 0.2 9.22 0.2 9.50 0.4 9.51 0.4 9.59 0.4 9.61 0.4 9.73 0.2 9.76 0.2 1.19 6.3 1.20 6.4 1.99 11.0 2.01 10.9 2.66 10.3 2.68 10.3 1.19 10.0 1.21 9.9 3.12 6.4 3.14 6.4 3.79 2.4 3.82 2.4 1.80 14.1 1.84 13.9 1.95 10.7 1.98 10.9 3.08 9.7 3.14 9.5 2.56 17.8 2.59 18.3 2.28 2.31
S3
LC-BLYP Δε p 9.14 0.2 9.45 0.4 9.55 0.4 9.67 0.2 1.17 6.6 1.97 11.4 2.65 10.4 1.18 10.1 3.10 6.3 3.83 2.3 1.82 13.5 1.96 10.9 3.12 9.2 2.58 18.0 2.36
Table SIII. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for fluorobenzene. The FWHMs are given in keV.
NOMO/HF a1 a1 a1 b2 b2 a1 a1 a1 a1 b2 a1 b2 a1 a1 a1 b2 b1
a1 b2 b2 a1 b1
b2 a2 b1 Total
Δε 15.26 9.39 9.63 9.48 9.89 9.81 9.89 2.52 1.72 2.08 3.07 2.71 3.22 3.15 4.03 3.66 1.83 3.05 2.10 4.51 1.68 2.51 2.66 2.66 3.24 2.99
p 0.8 0.1 0.1 0.1 0.1 0.1 0.0 13.5 2.4 2.5 2.9 3.6 6.6 8.6 7.4 13.8 7.6 3.0 3.9 4.2 3.0 4.5 3.0 4.0 4.0
LDA Δε p 14.94 0.6 9.17 0.1 9.29 0.2 9.41 0.2 9.49 0.1 9.57 0.1 9.58 0.1 2.37 10.6 1.57 2.8 2.04 3.2 2.93 3.1 2.65 3.7 3.03 7.2 2.88 7.7 1.65 8.3 3.67 3.8 3.05 12.4 2.92 3.1 1.87 4.3 4.25 6.2 1.61 3.4 2.72 5.3 2.69 4.8 2.63 4.5 3.44 4.4 2.77
NOMO/DFT BLYP B3LYP Δε p Δε p 15.06 0.6 15.10 0.7 9.29 0.1 9.31 0.1 9.43 0.2 9.44 0.1 9.53 0.1 9.54 0.1 9.64 0.1 9.69 0.1 9.69 0.1 9.72 0.1 9.72 0.1 9.75 0.0 2.39 10.7 2.42 11.4 1.58 2.7 1.62 2.7 2.03 3.0 2.05 2.9 2.92 2.9 2.97 3.0 2.63 3.6 2.65 3.6 2.99 6.7 3.06 6.7 2.89 8.1 2.95 8.2 1.67 8.6 1.72 8.4 3.66 3.9 3.73 4.5 3.06 12.5 3.21 12.9 2.91 3.2 2.95 3.1 1.86 4.4 1.89 4.2 4.22 6.1 4.33 6.2 1.58 3.6 1.60 3.4 2.71 5.1 2.68 4.9 2.71 5.1 2.66 4.3 2.58 4.4 2.62 4.3 3.42 4.3 3.39 4.2 2.76 2.83
S4
LC-BLYP Δε p 15.07 0.7 9.24 0.1 9.48 0.1 9.38 0.1 9.61 0.1 9.67 0.1 9.69 0.0 2.34 11.4 1.59 2.6 2.01 2.9 2.94 3.0 2.62 3.6 2.95 6.8 2.89 8.1 1.72 9.0 3.68 4.3 3.08 12.8 2.93 3.0 1.87 4.1 4.26 6.4 2.85 5.0 1.59 3.2 2.62 4.2 2.61 4.1 3.44 4.1 2.78
Table SIV. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for o-difluorobenzene. The FWHMs are given in keV.
NOMO/HF b2 a1 a1 b2 a1 b2 a1 b2 a1 b2 a1 b2 a1 a1 b2 a1 b2 a1 b1
a1 b2 a2
b2 a1 b2 b1
a1 a2 b1 Total
Δε 15.38 15.21 9.27 9.51 9.84 9.70 9.91 10.20 2.09 2.99 2.51 2.89 3.41 3.15 3.59 2.85 3.37 3.09 3.34 4.09 4.69 2.31 2.86 3.71 3.57 2.78 3.23 3.09 3.72 3.34
p 0.6 0.6 0.1 0.1 0.0 0.0 0.0 0.0 10.4 8.2 1.3 1.4 2.1 2.7 4.7 10.4 4.8 7.7 9.5 8.8 6.8 3.3 1.8 4.4 1.5 1.9 2.6 1.9 2.3
LDA Δε 15.04 14.92 9.06 9.27 9.57 9.46 9.59 9.82 1.94 2.89 2.18 2.72 3.27 3.01 3.44 2.75 2.72 2.89 2.77 1.70 2.60 3.75 4.46 3.08 4.28 3.16 3.00 3.18 3.94 3.09
p 0.5 0.5 0.1 0.1 0.1 0.1 0.0 0.0 8.9 6.7 1.5 1.8 2.2 2.8 5.2 9.5 7.5 4.1 9.0 3.2 1.9 8.4 5.4 5.1 2.7 3.3 4.3 2.3 3.0
NOMO/DFT BLYP B3LYP Δε p Δε p 15.15 0.5 15.20 0.5 15.03 0.5 15.07 0.5 9.20 0.1 9.20 0.1 9.39 0.1 9.41 0.1 9.71 0.1 9.73 0.1 9.58 0.1 9.60 0.1 9.71 0.0 9.75 0.0 9.96 0.0 10.01 0.0 1.96 8.9 1.99 9.3 2.90 6.8 2.93 7.1 2.21 1.4 2.30 1.4 2.70 1.7 2.76 1.6 3.27 2.1 3.31 2.1 2.97 2.7 3.03 2.7 3.44 4.8 3.48 4.8 2.74 9.3 2.79 9.5 2.72 7.8 2.79 7.8 2.83 4.3 2.95 4.4 2.77 9.0 2.90 9.1 1.79 3.4 1.91 3.4 2.66 2.0 3.83 8.5 3.74 8.4 2.80 1.8 4.41 5.2 4.49 6.0 3.00 4.9 3.20 5.2 4.28 3.1 4.13 2.4 3.17 3.2 3.00 3.7 3.02 4.7 3.26 3.0 3.15 2.3 3.15 2.1 3.94 2.9 3.91 2.7 3.08 3.15
S5
LC-BLYP Δε p 15.17 0.5 15.03 0.5 9.12 0.1 9.34 0.1 9.66 0.1 9.54 0.1 9.70 0.0 9.95 0.0 1.92 9.2 2.88 6.9 2.27 1.4 2.73 1.6 3.26 2.2 2.98 2.7 3.41 4.8 2.69 9.4 2.72 8.0 2.85 4.5 2.79 8.9 1.88 3.6 3.76 8.5 2.68 1.8 4.47 5.8 3.11 5.4 4.12 2.4 3.33 3.2 2.95 3.6 3.20 2.1 3.93 2.7 3.09
Table SV. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for m-difluorobenzene. The FWHMs are given in keV.
NOMO/HF a1 b2 a1 b2 a1 a1 b2 a1 a1 b2 a1 a1 b2 a1 b2 a1 b2 a1 b2 b1 a2
a1 a1 b2 b2 a1 b1 b1 a2 Total
Δε 15.19 15.17 9.36 9.50 9.87 9.66 9.89 9.87 2.45 2.50 1.84 2.31 3.00 2.59 3.41 3.31 3.48 1.67 3.43 3.67 3.07 4.16 2.46 4.73 2.00 2.06 2.61 2.81 3.22 2.98
p 0.4 0.4 0.1 0.1 0.0 0.2 0.1 0.1 7.2 6.6 2.2 2.7 2.3 4.1 4.2 6.7 5.1 6.7 7.4 7.2 5.0 5.3 3.9 3.1 2.8 4.2 4.3 4.5 3.5 100.0
LDA Δε 14.86 14.86 9.12 9.27 9.41 9.58 9.56 9.62 2.27 2.38 1.71 2.25 2.88 2.55 3.29 3.09 3.00 1.39 2.98 2.68 3.20 2.13 3.98 2.36 4.10 2.96 2.16 2.90 3.40 2.80
p 0.3 0.3 0.1 0.1 0.2 0.1 0.1 0.1 6.0 5.4 2.4 3.1 2.5 4.3 4.4 6.1 4.9 7.5 3.7 7.3 6.9 4.2 4.7 3.1 4.5 4.4 5.1 4.7 3.5 100.0
NOMO/DFT BLYP B3LYP Δε p Δε p 14.97 0.3 15.02 0.3 14.97 0.3 15.02 0.3 9.25 0.1 9.27 0.1 9.39 0.1 9.41 0.1 9.54 0.2 9.55 0.2 9.71 0.1 9.73 0.1 9.71 0.1 9.77 0.1 9.75 0.1 9.73 0.1 2.30 6.0 2.33 6.3 2.39 5.4 2.41 5.7 1.72 2.3 1.76 2.3 2.24 3.0 2.26 2.9 2.87 2.4 2.91 2.4 2.51 4.2 2.53 4.2 3.28 4.2 3.32 4.2 3.07 6.2 3.13 6.3 2.99 4.8 3.12 4.7 1.40 7.6 1.45 7.3 2.95 4.0 2.95 4.3 2.69 7.2 2.87 7.3 3.19 7.0 3.31 7.1 2.14 4.2 2.18 3.9 3.96 4.7 3.99 5.1 2.14 3.1 3.25 3.0 4.12 4.6 3.87 4.2 2.16 5.4 2.10 5.0 2.96 4.3 2.91 4.3 2.86 4.6 2.85 4.6 3.39 3.4 3.36 3.4 2.79 100.0 2.84 100.0
S6
LC-BLYP Δε p 14.98 0.3 14.97 0.3 9.18 0.1 9.35 0.1 9.49 0.2 9.69 0.1 9.69 0.1 9.70 0.1 2.23 6.3 2.35 5.6 1.74 2.3 2.23 3.0 2.87 2.4 2.46 4.2 3.27 4.3 3.05 6.3 3.10 4.5 1.40 7.7 2.83 4.6 2.70 7.3 3.23 7.1 2.15 4.0 3.95 5.0 3.17 3.0 3.77 4.3 3.07 4.3 2.09 4.9 2.87 4.4 3.40 3.3 2.79 100.0
Table SVI. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for p-difluorobenzene. The FWHMs are given in keV.
NOMO/HF b1u ag b1u ag b2u ag b1u b3g ag b1u ag b2u b1u b3g ag ag b1u b2u b3u b1u b2g b2u b3g ag b2u b3u b3g b1g b2g Total
Δε 15.23 15.08 9.53 9.44 9.74 9.59 10.00 9.87 2.44 2.49 1.65 2.02 3.09 2.66 3.16 2.84 2.92 3.66 3.41 3.60 4.19 2.93 1.83 1.45 4.73 2.57 2.46 2.63 3.29 2.91
p 0.3 0.3 0.1 0.1 0.1 0.1 0.1 0.1 6.5 6.1 2.5 2.8 2.7 3.3 5.2 5.1 8.1 3.9 6.8 6.7 5.0 3.6 4.1 5.8 3.0 3.3 5.1 5.0 4.2 100.0
LDA Δε 14.91 14.75 9.27 9.21 9.49 9.33 9.65 9.60 2.30 2.33 1.57 1.99 3.02 2.62 2.99 2.61 2.63 3.45 2.62 2.81 3.10 1.77 1.26 3.92 4.41 2.85 2.50 2.61 3.55 2.74
p 0.3 0.3 0.1 0.1 0.1 0.2 0.1 0.1 5.3 4.9 2.8 3.3 2.8 3.5 5.4 5.0 7.6 2.8 6.7 4.0 6.4 4.6 6.3 4.3 3.9 5.3 4.4 5.2 3.9 100.0
NOMO/DFT BLYP B3LYP Δε p Δε p 15.02 0.3 15.06 0.3 14.88 0.3 14.92 0.3 9.41 0.1 9.43 0.1 9.33 0.1 9.35 0.1 9.61 0.1 9.64 0.1 9.47 0.1 9.48 0.1 9.81 0.1 9.84 0.1 9.72 0.1 9.75 0.1 2.32 5.3 2.35 5.6 2.35 5.0 2.38 5.2 1.57 2.7 1.59 2.7 1.98 3.2 1.99 3.1 3.02 2.7 3.04 2.7 2.61 3.5 2.62 3.4 2.89 5.3 2.97 5.3 2.65 5.2 2.70 5.1 2.58 7.9 2.67 7.9 3.44 2.8 3.48 3.0 2.83 4.0 2.82 6.7 2.63 6.7 2.85 3.9 3.09 6.5 3.21 6.5 1.76 4.7 1.78 4.5 1.27 6.5 3.97 4.5 3.89 4.3 1.32 6.3 4.37 4.0 4.48 3.8 2.83 5.2 2.74 5.2 2.49 4.5 2.51 4.2 2.56 5.1 2.59 5.1 3.54 3.8 3.49 3.9 2.72 100.0 2.78 100.0
S7
LC-BLYP Δε p 15.02 0.3 14.88 0.3 9.35 0.1 9.29 0.1 9.58 0.1 9.42 0.1 9.77 0.1 9.69 0.1 2.26 5.6 2.30 5.2 1.57 2.7 1.95 3.2 3.02 2.8 2.59 3.4 2.80 5.4 2.68 5.3 2.51 8.2 3.44 2.9 2.64 6.8 2.83 3.8 3.13 6.6 1.77 4.4 3.93 4.5 1.31 6.2 4.39 3.9 2.90 5.1 2.47 4.1 2.59 4.9 3.52 3.8 2.73 100.0
Table SVII. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for 1,2,4-trifluorobenzene. The FWHMs are given in keV.
NOMO/HF a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a' a'' a'' a' a' a'' a' a' a' a' a'' a'' a'' Total
Δε p 15.22 0.6 15.23 0.5 15.07 0.0 9.41 0.1 9.48 0.1 9.59 0.0 9.88 0.0 9.86 0.1 10.18 0.0 2.11 10.1 2.95 7.7 2.36 0.5 2.51 1.3 2.83 1.5 3.40 2.0 3.11 2.6 3.59 4.1 2.81 8.4 3.43 3.9 3.21 6.4 3.31 9.0 2.71 5.6 4.03 6.9 3.93 2.4 4.33 5.6 4.34 3.5 2.79 2.2 3.31 2.7 3.99 3.3 2.59 2.1 3.23 2.6 3.11 2.1 3.69 2.2 3.30 100.0
LDA Δε p 14.89 0.4 14.90 0.4 14.75 0.0 9.19 0.1 9.24 0.1 9.31 0.0 9.61 0.1 9.58 0.1 9.81 0.0 1.95 8.4 2.85 6.1 2.24 0.8 2.19 1.4 2.64 1.8 3.29 2.1 2.94 2.9 3.47 4.3 2.80 7.6 2.78 6.9 3.27 2.6 2.25 5.3 2.70 8.0 2.56 2.6 2.97 2.1 2.47 3.9 3.72 7.7 3.47 3.2 4.41 5.3 3.37 4.3 2.76 2.9 3.33 3.2 3.23 2.7 3.91 2.5 3.04 100.0
NOMO/DFT BLYP B3LYP Δε p Δε p 15.01 0.4 15.06 0.5 15.02 0.4 15.07 0.4 14.87 0.0 14.91 0.0 9.32 0.1 9.33 0.1 9.37 0.1 9.38 0.1 9.43 0.0 9.46 0.0 9.74 0.1 9.76 0.1 9.71 0.1 9.73 0.1 9.94 0.0 9.99 0.0 1.97 8.4 2.00 8.8 2.86 6.2 2.88 6.6 2.26 0.8 2.28 0.7 2.22 1.4 2.30 1.3 2.63 1.8 2.69 1.7 3.29 2.0 3.32 2.0 2.90 2.8 2.96 2.7 3.46 4.1 3.49 4.1 2.79 7.5 2.82 7.7 2.76 7.1 2.82 6.9 3.21 2.6 3.28 3.0 2.23 5.5 2.29 5.4 2.70 8.0 2.84 8.3 2.57 2.6 3.12 1.8 2.97 2.0 2.73 2.5 2.42 4.0 3.80 7.8 3.71 7.8 3.21 4.3 3.39 3.2 3.69 3.4 4.39 5.5 4.33 4.9 3.32 4.6 3.46 4.0 2.73 2.9 2.74 2.9 3.33 3.2 3.36 3.1 3.19 2.6 3.19 2.5 3.91 2.4 3.88 2.3 3.02 100.0 3.10 100.0
S8
LC-BLYP Δε p 15.02 0.5 15.03 0.4 14.87 0.0 9.26 0.1 9.31 0.1 9.40 0.0 9.71 0.1 9.68 0.1 9.95 0.0 1.93 8.7 2.83 6.4 2.20 0.7 2.29 1.3 2.66 1.7 3.28 2.1 2.91 2.7 3.43 4.1 2.72 7.8 2.88 5.0 2.98 5.1 2.20 5.7 2.72 7.9 3.02 2.0 2.60 2.2 3.73 8.0 3.21 4.5 3.61 3.4 4.29 4.9 3.34 4.1 2.74 2.8 3.44 3.1 3.24 2.4 3.90 2.3 3.03 100.0
Table SVIII. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for 1,2,4,5-tetrafluorobenzene. The FWHMs are given in keV. NOMO/HF a1 b2 a1 b2 a1 a1 a1 b2 b2 a1 a1 a1 b2 a1 b2 a1 a1 a1 b2 a1 b2 a1 b2 a1 b2 a1 b1
b2 a2 b1
a1 a2 b1
a1 b2 b2 a1 b2 b1 b1 a2 Total
Δε 15.11 15.14 15.13 15.08 15.09 9.55 9.61 9.50 9.74 9.84 10.39 1.87 2.21 2.74 2.85 3.09 3.20 3.40 3.51 3.60 3.66 3.07 3.04 3.26 3.02 2.64 2.80 3.58 3.43 3.79 3.99 4.15 3.89 4.81 4.32 3.87 3.53 3.51 3.60 3.70 3.77 3.40
p 0.4 0.3 0.3 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 7.4 3.7 3.4 2.3 3.1 0.9 1.5 1.3 2.5 2.3 2.8 5.6 5.6 5.4 6.3 5.6 2.7 3.6 4.1 2.6 4.1 2.8 4.7 2.1 2.6 2.7 1.9 2.1 1.8 1.5 100.0
LDA Δε 14.77 14.81 14.80 14.75 14.76 9.33 9.34 9.39 9.37 9.41 10.02 1.69 2.09 2.60 2.77 3.05 2.78 3.26 3.43 3.51 3.62 2.75 2.99 3.04 2.66 2.46 2.43 2.46 2.92 3.24 2.75 3.67 3.92 4.58 4.12 3.69 3.45 3.06 3.52 3.89 4.00 3.18
p 0.3 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.1 6.2 3.4 3.0 2.0 2.6 0.9 1.6 1.4 2.6 2.4 5.9 2.4 5.4 4.8 5.9 4.9 3.0 3.7 4.1 2.0 2.9 4.4 4.8 2.9 3.2 2.7 3.2 2.6 2.2 1.8 100.0
NOMO/DFT BLYP B3LYP Δε p Δε p 14.90 0.3 14.94 0.3 14.94 0.2 14.98 0.2 14.92 0.2 14.96 0.2 14.87 0.1 14.92 0.1 14.89 0.1 14.93 0.1 9.46 0.1 9.47 0.1 9.47 0.1 9.53 0.0 9.52 0.1 9.48 0.1 9.51 0.0 9.59 0.0 9.54 0.0 9.58 0.0 10.14 0.1 10.19 0.0 1.71 6.3 1.75 6.6 2.11 3.4 2.13 3.5 2.62 3.1 2.65 3.2 2.78 2.0 2.79 2.0 3.05 2.6 3.06 2.7 2.85 0.9 2.94 0.9 3.26 1.5 3.30 1.5 3.43 1.4 3.45 1.3 3.50 2.5 3.52 2.5 3.61 2.3 3.63 2.3 2.70 5.9 2.78 5.9 2.98 2.4 3.00 2.5 3.02 5.4 3.08 5.5 2.63 4.8 2.71 4.8 2.44 6.0 2.49 6.1 2.42 4.9 2.54 4.9 2.44 3.1 2.52 3.1 2.91 3.7 3.02 3.6 2.64 2.0 3.36 4.1 3.23 4.1 3.04 2.0 3.65 2.9 3.73 2.8 3.90 4.4 3.97 4.4 4.56 4.9 4.64 4.9 4.10 3.0 4.18 2.8 3.66 3.3 3.71 3.1 3.42 2.7 3.48 2.5 3.02 3.2 3.11 3.1 3.53 2.5 3.59 2.4 3.88 2.1 3.87 2.0 4.00 1.7 3.97 1.6 3.17 100.0 3.23 100.0
S9
LC-BLYP Δε p 14.90 0.3 14.94 0.2 14.92 0.2 14.87 0.1 14.88 0.1 9.40 0.1 9.47 0.0 9.41 0.0 9.49 0.0 9.52 0.0 10.16 0.0 1.66 6.4 2.07 3.4 2.58 3.0 2.74 2.0 3.03 2.7 2.90 0.9 3.25 1.5 3.41 1.4 3.48 2.5 3.58 2.3 2.95 2.4 2.61 6.1 2.99 5.6 2.59 4.9 2.40 6.2 2.44 4.7 2.47 3.2 2.94 3.6 3.26 4.1 2.99 2.1 3.67 2.8 3.91 4.5 4.59 4.9 4.11 2.8 3.64 3.1 3.43 2.5 3.03 3.1 3.58 2.5 3.88 2.0 3.97 1.6 3.16 100.0
Table SIX. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for pentafluorobenzene. The FWHMs are given in keV. NOMO/HF a1 b2 a1 b2 a1 a1 a1 b2 b2 a1 a1 a1 b2 a1 b2 a1 a1 a1 b2 a1 b2 a1 b2 a1 b2 a1 b1
b2 a2 b1
a1 a2 b1
a1 b2 b2 a1 b2 b1 b1 a2 Total
Δε 15.11 15.14 15.13 15.08 15.09 9.55 9.61 9.50 9.74 9.84 10.39 1.87 2.21 2.74 2.85 3.09 3.20 3.40 3.51 3.60 3.66 3.07 3.04 3.26 3.02 2.64 2.80 3.58 3.43 3.79 3.99 4.15 3.89 4.81 4.32 3.87 3.53 3.51 3.60 3.70 3.77 3.40
p 0.4 0.3 0.3 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 7.4 3.7 3.4 2.3 3.1 0.9 1.5 1.3 2.5 2.3 2.8 5.6 5.6 5.4 6.3 5.6 2.7 3.6 4.1 2.6 4.1 2.8 4.7 2.1 2.6 2.7 1.9 2.1 1.8 1.5 100.0
LDA Δε 14.77 14.81 14.80 14.75 14.76 9.33 9.34 9.39 9.37 9.41 10.02 1.69 2.09 2.60 2.77 3.05 2.78 3.26 3.43 3.51 3.62 2.75 2.99 3.04 2.66 2.46 2.43 2.46 2.92 3.24 2.75 3.67 3.92 4.58 4.12 3.69 3.45 3.06 3.52 3.89 4.00 3.18
p 0.3 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.1 6.2 3.4 3.0 2.0 2.6 0.9 1.6 1.4 2.6 2.4 5.9 2.4 5.4 4.8 5.9 4.9 3.0 3.7 4.1 2.0 2.9 4.4 4.8 2.9 3.2 2.7 3.2 2.6 2.2 1.8 100.0
NOMO/DFT BLYP B3LYP Δε p Δε p 14.90 0.3 14.94 0.3 14.94 0.2 14.98 0.2 14.92 0.2 14.96 0.2 14.87 0.1 14.92 0.1 14.89 0.1 14.93 0.1 9.46 0.1 9.47 0.1 9.47 0.1 9.53 0.0 9.52 0.1 9.48 0.1 9.51 0.0 9.59 0.0 9.54 0.0 9.58 0.0 10.14 0.1 10.19 0.0 1.71 6.3 1.75 6.6 2.11 3.4 2.13 3.5 2.62 3.1 2.65 3.2 2.78 2.0 2.79 2.0 3.05 2.6 3.06 2.7 2.85 0.9 2.94 0.9 3.26 1.5 3.30 1.5 3.43 1.4 3.45 1.3 3.50 2.5 3.52 2.5 3.61 2.3 3.63 2.3 2.70 5.9 2.78 5.9 2.98 2.4 3.00 2.5 3.02 5.4 3.08 5.5 2.63 4.8 2.71 4.8 2.44 6.0 2.49 6.1 2.42 4.9 2.54 4.9 2.44 3.1 2.52 3.1 2.91 3.7 3.02 3.6 2.64 2.0 3.36 4.1 3.23 4.1 3.04 2.0 3.65 2.9 3.73 2.8 3.90 4.4 3.97 4.4 4.56 4.9 4.64 4.9 4.10 3.0 4.18 2.8 3.66 3.3 3.71 3.1 3.42 2.7 3.48 2.5 3.02 3.2 3.11 3.1 3.53 2.5 3.59 2.4 3.88 2.1 3.87 2.0 4.00 1.7 3.97 1.6 3.17 100.0 3.23 100.0
S10
LC-BLYP Δε p 14.90 0.3 14.94 0.2 14.92 0.2 14.87 0.1 14.88 0.1 9.40 0.1 9.47 0.0 9.41 0.0 9.49 0.0 9.52 0.0 10.16 0.0 1.66 6.4 2.07 3.4 2.58 3.0 2.74 2.0 3.03 2.7 2.90 0.9 3.25 1.5 3.41 1.4 3.48 2.5 3.58 2.3 2.95 2.4 2.61 6.1 2.99 5.6 2.59 4.9 2.40 6.2 2.44 4.7 2.47 3.2 2.94 3.6 3.26 4.1 2.99 2.1 3.67 2.8 3.91 4.5 4.59 4.9 4.11 2.8 3.64 3.1 3.43 2.5 3.03 3.1 3.58 2.5 3.88 2.0 3.97 1.6 3.16 100.0
Table SX. The FWHM, Δε, and percentage contribution of each MO, p, of positron annihilation spectra for hexafluorobenzene. The FWHMs are given in keV.
NOMO/HF b1u e2g e1u a1g a1g e1u e2g b1u a1g e1u e2g b1u a1g e1u e2g b2u a1g b1u e1u e2g a2u e1g e2u b2g e1u a2g e2g b2u a2u e1g Total
Δε 15.14 15.07 15.01 14.84 9.51 9.76 9.90 10.28 1.36 2.05 2.80 3.10 3.80 3.48 3.69 3.18 3.00 3.32 2.72 2.98 2.74 3.29 3.95 4.13 4.47 4.92 3.76 3.47 3.57 3.73 3.40
p 0.2 0.3 0.3 0.2 0.0 0.1 0.1 0.0 4.4 7.0 5.9 2.7 0.8 2.7 4.4 2.3 4.8 4.6 7.1 8.8 4.5 7.0 6.4 3.0 5.9 2.6 5.2 2.5 2.2 3.5
LDA Δε 14.77 14.79 14.56 14.52 9.23 9.47 9.59 9.86 1.26 1.96 2.73 3.08 3.20 3.41 3.65 3.16 2.69 3.11 2.75 2.46 2.22 2.84 3.63 3.94 4.22 4.86 3.55 3.02 3.55 3.97 3.19
p 0.1 0.3 0.3 0.1 0.1 0.1 0.1 0.0 4.0 6.2 5.0 2.3 0.8 2.8 4.5 1.9 4.8 4.5 8.8 8.5 3.7 7.3 6.5 3.0 6.1 2.4 6.0 3.1 2.7 3.8
NOMO/DFT BLYP B3LYP Δε p Δε p 14.92 0.1 14.96 0.1 14.91 0.3 14.96 0.3 14.69 0.3 14.72 0.3 14.65 0.1 14.69 0.1 9.37 0.1 9.40 0.0 9.60 0.1 9.63 0.1 9.71 0.1 9.74 0.1 10.03 0.0 10.08 0.0 1.27 4.0 1.29 4.1 1.97 6.2 1.98 6.4 2.74 5.0 2.75 5.2 3.08 2.3 3.08 2.4 3.38 0.8 3.55 0.8 3.41 2.7 3.43 2.7 3.64 4.3 3.65 4.4 3.15 1.9 3.15 2.0 2.64 4.9 2.71 4.9 3.08 4.6 3.14 4.6 2.72 9.0 2.80 8.9 2.43 8.7 2.31 7.2 2.21 3.7 2.48 4.4 2.82 7.3 2.92 7.2 3.61 6.5 3.70 6.5 3.92 3.0 3.97 3.0 4.21 6.1 4.28 6.1 4.85 2.4 4.87 2.5 3.53 6.1 3.60 5.8 2.99 3.2 3.07 3.0 3.55 2.6 3.60 2.5 3.97 3.7 3.93 3.6 3.18 3.24
S11
LC-BLYP Δε p 14.90 0.1 14.92 0.3 14.67 0.3 14.63 0.1 9.33 0.0 9.56 0.1 9.69 0.1 10.01 0.0 1.23 4.1 1.94 6.2 2.70 5.1 3.05 2.3 3.46 0.8 3.39 2.8 3.61 4.4 3.12 2.0 2.55 5.1 3.05 4.6 2.69 9.1 2.22 7.0 2.43 4.4 2.85 7.1 3.63 6.5 3.91 3.0 4.22 6.0 4.83 2.4 3.53 5.8 3.00 3.1 3.58 2.6 3.94 3.6 3.16
Difference from total energy w/o a positron [Hartree]
1.0E+00 1.0E-01 1.0E-02 1.0E-03 1.0E-04 1.0E-05 1s1d 2s2d 3s3d 4s4d 5s5d 6s6d 7s7d 8s8d Positron basis function
FIG. S1. Convergence of the NOMO/HF total energy of a neon atom with a positron with respect to the positronic basis set. The (nsnd) basis functions have the exponents
m 1.0 10 m / 2 from m = 1 to m = n.
S12
Radial positron density [bohr−1]
1.0E+02 1.0E+01 1.0E+00 1.0E-01 1.0E-02 1.0E-03 1.0E-04 1.0E-05 1.0E-06 1.0E-07 1.0E-08 1.0E-09 1.0E-10
1s1d 2s2d 3s3d 4s4d 5s5d 6s6d 7s7d 8s8d 0.0
1.0
2.0
3.0
4.0 5.0 6.0 r [bohr]
7.0
8.0
9.0 10.0
1.0E+02 Radial contact density [bohr−1]
PWP 1.0E+00
1s1d 2s2d
1.0E-02
3s3d 4s4d
1.0E-04
5s5d 1.0E-06
6s6d
1.0E-08
7s7d 8s8d
1.0E-10 0.0
1.0
2.0 3.0 r [bohr]
4.0
5.0
FIG. S2. Positronic basis-set dependence of the radial distribution of positron density (upper) and contact density (lower) of a neon atom obtained by the NOMO/HF calculation. The (nsnd) basis functions have the exponents m 1.0 10 m / 2 from m = 1 to m = n.
S13
Integrated ratio of contact density
1.0 0.9
PWP
0.8
1s1d
0.7
2s2d
0.6
3s3d
0.5
4s4d
0.4
5s5d
0.3
6s6d
0.2
7s7d
0.1
8s8d
0.0 0.0
1.0
2.0 r [bohr]
3.0
4.0
FIG. S3. Positronic basis-set dependence of the integrated ratio of contact density, which is defined as
r
0
r' 2 ep ( r' )dr'
0
r'' 2 ep ( r'' )dr'' with ρep given by Eq. (25) in the main
text. The (nsnd) basis functions have the exponents m 1.0 10 m / 2 from m = 1 to m = n.
S14