Article ID: 2016-0014
水素原子のDirac方程式は厳密解が得られている.そのスピノルの形状を知ることは,重元素を含む分子での相対論効果を議論するうえで重要である.s, p, d, fの各スピノルは特徴のある形状をしているが,本論文の目的は電子密度の3D等値面図を図鑑にすることで各スピノルの特徴を一望できるようにすることにある.Schrödinger方程式の厳密解の図も併記しており,相対論では非相対論と比較して節が少なくなっていることが容易に見てとれる.Z = 100の<r > を計算し,その詳細を分析した.相対論的効果で<r > が減少するが,その効果は一様ではなく,s1/2とp1/2で大きい.描画と計算に用いたMathematicaノートブックは本論文とともに提供しており,改変可能な教材として利用できる.
1s+ density.
1s(l = 0, m = 0 α)/1s+ (j = 1/2, mj = 1/2)
2p– density.
2p(l = 1, m = 1 β)/2p– (j = 1/2, mj = 1/2)
2p+ density.
2p(l = 1, m = 1 α)/2p+ (j = 3/2, mj = 3/2)
2p+ density.
2p(l = 1, m = 0 α)/2p+ (j = 3/2, mj = 1/2)
3d– density.
3d(l = 2, m = 2 β)/3d– (j = 3/2, mj = 3/2)
3d– density.
3d(l = 2, m = 1 β)/3d– (j = 3/2, mj = 1/2)
3d+ density.
3d(l = 2, m = 2 α)/3d+ (j = 5/2, mj = 5/2)
3d+ density.
3d(l = 2, m = 1 α)/3d+ (j = 5/2, mj = 3/2)
3d+ density.
3d(l = 2, m = 0 α)/3d+ (j = 5/2, mj = 1/2)
4f– density .
4f(l = 3, m = 3 β)/4f– (j = 5/2, mj = 5/2)
4f– density
4f(l = 3, m = 2 β)/4f– (j = 5/2, mj = 3/2)
4f– density
4f(l = 3, m = 1 β)/4f– (j = 5/2, mj = 1/2)
4f+ density .
4f(l = 3, m = 3 α)/4f+ (j = 7/2, mj = 7/2)
4f+ density
4f(l = 3, m = 2 α)/4f+ (j = 7/2, mj = 5/2)
4f+ density
4f(l = 3, m = 1 α)/4f+ (j = 7/2, mj = 3/2)
4f+ density
4f(l = 3, m = 0 α)/4f+ (j = 7/2, mj = 1/2)
Contraction ratio (<r > nonrel − <r > rel)/<r > nonrel ×100) versus Z.
l | 0 | 1 | 1 | 2 | 2 | 3 | 3 |
j | 1/2 | 1/2 | 3/2 | 3/2 | 5/2 | 5/2 | 7/2 |
κ | –1 | 1 | –2 | 2 | –3 | 3 | –4 |
Labels | s 1/2 | p 1/2 | p 3/2 | d 3/2 | d 5/2 | f 5/2 | f 7/2 |
Labels | s+ | p– | p+ | d– | d+ | f– | f+ |
Labels | mj = 1/2 | mj = 3/2 | mj = 5/2 | mj = 7/2 |
s1/2 (s+) |
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p1/2 (p–) |
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p3/2 (p+) |
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d3/2 (d–) |
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d5/2 (d+) |
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f5/2 (f–) |
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f7/2 (f+) |
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* The upper and lower parts denote the large and small components, respectively.
Labels | h | cmax | nc | v |
1s | 1 | 2.04E-1 | 6 | 1.E-4 |
2p | 6 | 2.04E-3 | 6 | 3.E-4 |
3d | 13 | 2.04E-4 | 6 | 3.E-5 |
4f | 22 | 2.64E-5 | 6 | 1.E-4 |
5g | 40 | 1.30E-5 | 6 | 4.E-6 |
<r > | <r2> | ||||
Labels | n | rel | nonrel | rel | nonrel |
s + | 1 | 1.4999734 | 1.5 | 2.9999068 | 3 |
2 | 5.9998835 | 6.0 | 41.998496 | 42 | |
3 | 13.499802 | 13.5 | 206.99421 | 207 | |
4 | 23.999721 | 24.0 | 647.98545 | 648 | |
5 | 37.499641 | 37.5 | 1574.9706 | 1575 | |
6 | 53.999561 | 54.0 | 3257.9481 | 3258 | |
p – | 2 | 4.9998835 | 5.0 | 29.998735 | 30 |
3 | 12.499802 | 12.5 | 179.99461 | 180 | |
4 | 22.999721 | 23.0 | 599.98601 | 600 | |
5 | 36.499641 | 36.5 | 1499.9713 | 1500 | |
6 | 52.999561 | 53.0 | 3149.9490 | 3150 | |
p + | 2 | 4.999973 | 5.0 | 29.999707 | 30 |
3 | 12.499926 | 12.5 | 179.99804 | 180 | |
4 | 22.999884 | 23.0 | 599.99430 | 600 | |
5 | 36.499842 | 36.5 | 1499.9877 | 1500 | |
6 | 52.999802 | 53.0 | 3149.9774 | 3150 | |
d – | 3 | 10.499926 | 10.5 | 125.99836 | 126 |
4 | 20.999884 | 21.0 | 503.99478 | 504 | |
5 | 34.499842 | 34.5 | 1349.9883 | 1350 | |
6 | 50.999802 | 51.0 | 2933.9782 | 2934 | |
d + | 3 | 10.499973 | 10.5 | 125.99940 | 126 |
4 | 20.999941 | 21.0 | 503.99739 | 504 | |
5 | 34.499912 | 34.5 | 1349.9936 | 1350 | |
6 | 50.999884 | 51.0 | 2933.9874 | 2934 | |
f – | 4 | 17.999941 | 18.0 | 359.99779 | 360 |
5 | 31.499912 | 31.5 | 1124.9941 | 1125 | |
6 | 47.999884 | 48.0 | 2609.9881 | 2610 | |
f + | 4 | 17.999973 | 18.0 | 359.99899 | 360 |
5 | 31.499949 | 31.5 | 1124.9966 | 1125 | |
6 | 47.999926 | 48.0 | 2609.9925 | 2610 |
Fm99+ (exact) | Fm (Hartree-Fock) | |||||||
Labels | n | <r > rel | <r > nonrel | Cont. (%) a | <r > relb | <r > nonrelc | Cont. (%) a | <r > rel-NHFd |
s + | 1 | 0.0118 | 0.015 | 21.1 | 0.0120 | 0.0151 | 20.6 | 0.0120 |
2 | 0.0468 | 0.060 | 22.1 | 0.0495 | 0.0630 | 21.5 | 0.0493 | |
3 | 0.1121 | 0.135 | 16.9 | 0.1293 | 0.1558 | 17.0 | 0.1291 | |
4 | 0.2076 | 0.240 | 13.5 | 0.2790 | 0.3265 | 14.5 | 0.2786 | |
5 | 0.3331 | 0.375 | 11.2 | 0.5716 | 0.6630 | 13.8 | 0.5708 | |
6 | 0.4886 | 0.540 | 9.5 | 1.2748 | 1.5426 | 17.4 | 1.2724 | |
7 | 0.6741 | 0.735 | 8.3 | 4.1333 | 4.9329 | 16.2 | 4.1226 | |
p – | 2 | 0.0368 | 0.050 | 26.5 | 0.0396 | 0.0531 | 25.5 | 0.0395 |
3 | 0.1021 | 0.125 | 18.3 | 0.1209 | 0.1476 | 18.1 | 0.1209 | |
4 | 0.1976 | 0.230 | 14.1 | 0.2761 | 0.3247 | 15.0 | 0.2761 | |
5 | 0.3231 | 0.365 | 11.5 | 0.5895 | 0.6871 | 14.2 | 0.5894 | |
6 | 0.4786 | 0.530 | 9.7 | 1.4085 | 1.7422 | 19.2 | 1.4102 | |
p + | 2 | 0.0472 | 0.050 | 5.5 | 0.0509 | 0.0531 | 4.2 | 0.0509 |
3 | 0.1174 | 0.125 | 6.1 | 0.1411 | 0.1476 | 4.4 | 0.1411 | |
4 | 0.2180 | 0.230 | 5.2 | 0.3136 | 0.3247 | 3.4 | 0.3136 | |
5 | 0.3488 | 0.365 | 4.4 | 0.6677 | 0.6871 | 2.8 | 0.6678 | |
6 | 0.5096 | 0.530 | 3.8 | 1.6989 | 1.7422 | 2.5 | 1.6994 | |
d – | 3 | 0.0974 | 0.105 | 7.2 | 0.1213 | 0.1276 | 4.9 | 0.1213 |
4 | 0.1980 | 0.210 | 5.7 | 0.3076 | 0.3182 | 3.3 | 0.3076 | |
5 | 0.3288 | 0.345 | 4.7 | 0.7329 | 0.7513 | 2.5 | 0.7330 | |
d + | 3 | 0.1023 | 0.105 | 2.6 | 0.1271 | 0.1276 | 0.4 | 0.1271 |
4 | 0.2040 | 0.210 | 2.8 | 0.3185 | 0.3182 | –0.1 | 0.3185 | |
5 | 0.3361 | 0.345 | 2.6 | 0.7606 | 0.7513 | –1.2 | 0.7607 | |
f – | 4 | 0.1740 | 0.180 | 3.3 | 0.3017 | 0.2998 | –0.6 | 0.3017 |
5 | 0.3061 | 0.315 | 2.8 | 1.0751 | 1.0043 | –7.0 | 1.0757 | |
f + | 4 | 0.1773 | 0.180 | 1.5 | 0.3069 | 0.2998 | –2.3 | 0.3069 |
5 | 0.3098 | 0.315 | 1.6 | 1.1184 | 1.0043 | –11.4 | 1.1192 |
a Contraction ratio, i.e., (<r > nonrel − <r > rel)/<r > nonrel ×100.
b Ref [23].
c Ref [24].
d Numerical Hartree-Fock. Ref [25].