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Supplemental tables


Table 6.3: Uniform electron gas energy versus density for closed-shell packings, with $ \mathrm {r_{s}}$ in bohr and the energy per atom in hartrees. We are comparing eFF with exchange to Hartree-Fock energies.
$ r_{s}$   fcc bcc hcp diamond   HF
1   0.5115 0.5219 0.4903 0.5870   0.6468
2   0.0359 0.0369 0.0320 0.0589   0.0472
2.5   -0.0086 -0.0082 -0.0107 0.0087   -0.0065
3   -0.0286 -0.0285 -0.0299 -0.0145   -0.0299
3.5   -0.0382 -0.0381 -0.0390 -0.0259   -0.0407
4   -0.0426 -0.0427 -0.0432 -0.0317   -0.0455
5   -0.0449 -0.0450 -0.0452 -0.0356   -0.0474
6   -0.0437 -0.0438 -0.0439 -0.0356   -0.0457
7   -0.0415 -0.0416 -0.0416 -0.0342   -0.0429
8   -0.0390 -0.0391 -0.0391 -0.0324   -0.0400
9   -0.0367 -0.0367 -0.0367 -0.0305   -0.0373
10   -0.0344 -0.0345 -0.0345 -0.0287   -0.0348



Table 6.4: Uniform electron gas energy versus density for open-shell packings, with $ \mathrm {r_{s}}$ in bohr and the energy per atom in hartrees. We are comparing eFF with exchange to Hartree-Fock energies.
$ r_{s}$   NaCl CsCl sphalerite wurtzite   HF
1   0.5109 0.5346 0.5112 0.4901   0.6468
2   0.0344 0.0390 0.0350 0.0314   0.0472
2.5   -0.0106 -0.0079 -0.0097 -0.0116   -0.0065
3   -0.0313 -0.0296 -0.0301 -0.0311   -0.0299
3.5   -0.0415 -0.0404 -0.0400 -0.0405   -0.0407
4   -0.0465 -0.0460 -0.0448 -0.0451   -0.0455
5   -0.0498 -0.0500 -0.0476 -0.0477   -0.0474
6   -0.0494 -0.0501 -0.0469 -0.0469   -0.0457
7   -0.0478 -0.0487 -0.0451 -0.0450   -0.0429
8   -0.0457 -0.0467 -0.0429 -0.0428   -0.0400
9   -0.0436 -0.0447 -0.0408 -0.0406   -0.0373
10   -0.0415 -0.0426 -0.0387 -0.0386   -0.0348



Table 6.5: Uniform electron gas energy versus density for closed-shell packings, with $ \mathrm {r_{s}}$ in bohr and the energy per atom in hartrees. We are comparing eFF with exchange and correlation to quantum Monte Carlo energies.
$ r_{s}$   fcc bcc hcp diamond   QMC
1   0.4508 0.4454 0.4305 0.5018   0.5870
2   -0.0110 -0.0233 -0.0139 -0.0119   0.0024
2.5   -0.0507 -0.0628 -0.0517 -0.0572   -0.0468
3   -0.0668 -0.0786 -0.0670 -0.0764   -0.0669
3.5   -0.0732 -0.0846 -0.0729 -0.0844   -0.0749
4   -0.0750 -0.0860 -0.0744 -0.0873   -0.0773
4.5   -0.0746 -0.0853 -0.0738 -0.0875   -0.0772
5   -0.0731 -0.0835 -0.0722 -0.0863   -0.0757
6   -0.0688 -0.0786 -0.0678 -0.0822   -0.0711
7   -0.0641 -0.0734 -0.0631 -0.0772   -0.0661
8   -0.0596 -0.0684 -0.0587 -0.0723   -0.0614
9   -0.0556 -0.0639 -0.0547 -0.0678   -0.0571
10   -0.0520 -0.0599 -0.0511 -0.0636   -0.0533



Table 6.6: Uniform electron gas energy versus density for open-shell packings, with $ \mathrm {r_{s}}$ in bohr and the energy per atom in hartrees. We are comparing eFF with exchange and correlation to quantum Monte Carlo energies.
$ r_{s}$   NaCl CsCl sphalerite wurtzite   QMC
1   0.4504 0.4836 0.4446 0.4290   0.5870
2   -0.0115 -0.0003 -0.0162 -0.0152   0.0024
2.5   -0.0512 -0.0428 -0.0555 -0.0529   -0.0468
3   -0.0674 -0.0607 -0.0714 -0.0681   -0.0669
3.5   -0.0738 -0.0682 -0.0774 -0.0740   -0.0749
4   -0.0757 -0.0708 -0.0790 -0.0754   -0.0773
4.5   -0.0753 -0.0711 -0.0784 -0.0748   -0.0772
5   -0.0738 -0.0701 -0.0767 -0.0732   -0.0757
6   -0.0695 -0.0667 -0.0720 -0.0687   -0.0711
7   -0.0648 -0.0627 -0.0671 -0.0640   -0.0661
8   -0.0604 -0.0588 -0.0624 -0.0595   -0.0614
9   -0.0564 -0.0552 -0.0582 -0.0555   -0.0571
10   -0.0528 -0.0520 -0.0544 -0.0519   -0.0533



Table 6.7: Comparison of dissociation energies (kcal/mol) of s-like geometries.
      no correlation     with correlation
energy of relative to   eFF HF     eFF B3LYP exact
$ \mathrm {H_{2}}$ $ \mathrm{H+H}$   -67.2 -83.4     -84.7 -110.0 -104.2
LiH $ \mathrm{Li+H}$   -13.6 -33.9     -36.4 -58.2 -56.6
BeH $ \mathrm{Be+H}$   -12.9 -49.9     -29.8 -57.5 -52.8
$ \mathrm{BeH_{2}}$ $ \mathrm{BeH + H}$   -66.7 -75.1     -92.3 -97.6 -98.9
$ \mathrm{H_{3}}$ (linear) $ \mathrm{H_{2} + H}$   20.3 24.3     9.1 6.0 9.7
$ \mathrm{H_{4}}$ (square) $ \mathrm{H_{2} + H_{2}}$   101.5 121.4     82.4 101.2 147.0
$ \mathrm{H_{3}^{+}}$ (triangle) $ \mathrm{2 H + H^{+}}$   -177.5 -187.6     -195.3 -214.6 -224.0
$ \mathrm{H_{4}^{2+}}$ (tetrahedron) $ \mathrm{2 H + 2 H^{+}}$   -8.3 -1.7     -25.6 -31.0  
$ \mathrm{Li_{2}}$ $ \mathrm{2 Li}$   0.9 -4.0     -22.4 -20.8 -24.5
$ \mathrm{Li_{2}^{+}}$ $ \mathrm{Li + Li^{+}}$   -31.4 -29.0     -36.9 -29.3  
$ \mathrm{Li_{3}^{+}}$ (triangle) $ \mathrm{2 Li + Li^{+}}$   -46.6 -46.0     -80.7 -65.6  
$ \mathrm{Li_{4}^{2+}}$ (tetrahedron) $ \mathrm{2 Li + 2 Li^{+}}$   0.4 1.2     -38.4 -17.6  



Table 6.8: Comparison of correlation energies (kcal/mol) of s-like geometries.
energy of relative to   eFF corr B3LYP-HF exact-HF
$ \mathrm {H_{2}}$ $ \mathrm{H+H}$   -17.5 -26.6 -20.8
LiH $ \mathrm{Li+H}$   -22.8 -24.4 -22.7
BeH $ \mathrm{Be+H}$   -16.9 -7.6 -2.9
$ \mathrm{BeH_{2}}$ $ \mathrm{BeH + H}$   -25.6 -22.6 -23.8
$ \mathrm{H_{3}}$ (linear) $ \mathrm{H_{2} + H}$   -11.1 -18.3 -14.6
$ \mathrm{H_{4}}$ (square) $ \mathrm{H_{2} + H_{2}}$   -19.1 -20.2  
$ \mathrm{H_{3}^{+}}$ (triangle) $ \mathrm{2 H + H^{+}}$   -17.8 -27.0 -36.4
$ \mathrm{H_{4}^{2+}}$ (tetrahedron) $ \mathrm{2 H + 2 H^{+}}$   -17.4 -29.2  
$ \mathrm{Li_{2}}$ $ \mathrm{2 Li}$   -23.3 -16.9 -20.5
$ \mathrm{Li_{2}^{+}}$ $ \mathrm{Li + Li^{+}}$   -5.5 -0.3  
$ \mathrm{Li_{3}^{+}}$ (triangle) $ \mathrm{2 Li + Li^{+}}$   -34.1 -19.6  
$ \mathrm{Li_{4}^{2+}}$ (tetrahedron) $ \mathrm{2 Li + 2 Li^{+}}$   -38.9 -18.8  



Table 6.9: Comparison of bond lengths (angstroms) of s-like geometries.
  no correlation   with correlation
  eFF HF   eFF B3LYP exact
$ \mathrm {H_{2}}$ 0.780 0.735   0.778 0.744 0.741
LiH 1.594 1.607   1.556 1.593 1.596
BeH 1.377 1.341   1.357 1.341 1.343
$ \mathrm{BeH_{2}}$ 1.333 1.332   1.315 1.327 1.334
$ \mathrm{H_{3}}$ (linear) 0.989 0.913   0.975 0.927 0.930
$ \mathrm{H_{4}}$ (square) 1.119 1.131   1.085 1.168 1.220
$ \mathrm{H_{3}^{+}}$ (triangle) 0.875 0.870   0.873 0.882 0.889
$ \mathrm{H_{4}^{2+}}$ (tetrahedron) 1.183 1.225   1.180 1.254  
$ \mathrm{Li_{2}}$ 2.675 2.785   2.569 2.707 2.673
$ \mathrm{Li_{2}^{+}}$ 2.971 3.141   2.869 3.092  
$ \mathrm{Li_{3}^{+}}$ (triangle) 2.892 3.044   2.754 2.953  
$ \mathrm{Li_{4}^{2+}}$ (tetrahedron) 3.339 3.547   3.129 3.426  



Table 6.10: Comparison of bond length differences (angstroms) upon adding correlation for s-like geometries.
  eFF corr B3LYP-HF exact-HF
$ \mathrm {H_{2}}$ -0.002 0.009 0.005
LiH -0.038 -0.014 -0.011
BeH -0.020 0.001 0.002
$ \mathrm{BeH_{2}}$ -0.018 -0.005 0.002
$ \mathrm{H_{3}}$ (linear) -0.014 0.014 0.017
$ \mathrm{H_{4}}$ (square) -0.034 0.037  
$ \mathrm{H_{3}^{+}}$ (triangle) -0.002 0.012 0.019
$ \mathrm{H_{4}^{2+}}$ (tetrahedron) -0.003 0.029  
$ \mathrm{Li_{2}}$ -0.105 -0.077 -0.112
$ \mathrm{Li_{2}^{+}}$ -0.102 -0.049  
$ \mathrm{Li_{3}^{+}}$ (triangle) -0.138 -0.090  
$ \mathrm{Li_{4}^{2+}}$ (tetrahedron) -0.210 -0.121  



Table 6.11: Comparison of atomic correlation energies.
    correlation energy (kcal/mol)
  spin eFF B3LYP-HF exact [21]
He s -0.0297 -0.0531 -0.0420
Li d -0.0390 -0.0593 -0.0454
Be s -0.0745 -0.0993 -0.0940
B d -0.1130 -0.1338 -0.1240
C t -0.1522 -0.1695 -0.1551
N q -0.1928 -0.2023 -0.1861
O t -0.3050 -0.2830 -0.2539
F d -0.4144 -0.3589 -0.3160
Ne s -0.5278 -0.4283 -0.3810


next up previous contents
Next: Bibliography Up: Development of an electron Previous: Conclusion   Contents
Julius 2008-04-29