Journal of Computer Chemistry, Japan
Online ISSN : 1347-3824
Print ISSN : 1347-1767
ISSN-L : 1347-1767

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Visualization of the Exact Solution of Dirac Equation
Yasuyo HATANOShigeyoshi YAMAMOTOHiroshi TATEWAKI
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Article ID: 2016-0014

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Abstract

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,s1/2p1/2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.

Figures
Figure 1.

 1s+ density.

1s(l = 0, m = 0 α)/1s+ (j = 1/2, mj = 1/2)

Figure 2.

 2p density.

2p(l = 1, m = 1 β)/2p (j = 1/2, mj = 1/2)

Figure 3.

 2p+ density.

2p(l = 1, m = 1 α)/2p+ (j = 3/2, mj = 3/2)

Figure 4.

 2p+ density.

2p(l = 1, m = 0 α)/2p+ (j = 3/2, mj = 1/2)

Figure 5.

 3d density.

3d(l = 2, m = 2 β)/3d (j = 3/2, mj = 3/2)

Figure 6.

 3d density.

3d(l = 2, m = 1 β)/3d (j = 3/2, mj = 1/2)

Figure 7.

 3d+ density.

3d(l = 2, m = 2 α)/3d+ (j = 5/2, mj = 5/2)

Figure 8.

 3d+ density.

3d(l = 2, m = 1 α)/3d+ (j = 5/2, mj = 3/2)

Figure 9.

 3d+ density.

3d(l = 2, m = 0 α)/3d+ (j = 5/2, mj = 1/2)

Figure 10.

 4f density .

4f(l = 3, m = 3 β)/4f (j = 5/2, mj = 5/2)

Figure 11.

 4f density

4f(l = 3, m = 2 β)/4f (j = 5/2, mj = 3/2)

Figure 12.

 4f density

4f(l = 3, m = 1 β)/4f (j = 5/2, mj = 1/2)

Figure 13.

 4f+ density .

4f(l = 3, m = 3 α)/4f+ (j = 7/2, mj = 7/2)

Figure 14.

 4f+ density

4f(l = 3, m = 2 α)/4f+ (j = 7/2, mj = 5/2)

Figure 15.

 4f+ density

4f(l = 3, m = 1 α)/4f+ (j = 7/2, mj = 3/2)

Figure 16.

 4f+ density

4f(l = 3, m = 0 α)/4f+ (j = 7/2, mj = 1/2)

Figure 17.

 Contraction ratio (<r > nonrel − <r > rel)/<r > nonrel ×100) versus Z.

Tables
Table 1.  Quantum numbers and labels.
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+
Table 2.  The angular part of the exact solution of the Dirac equation.*
Labels mj = 1/2 mj = 3/2 mj = 5/2 mj = 7/2
s1/2 (s+) ( Y 0 , 0 0 ) 1 3 ( Y 1 , 0 2 Y 1 , 1 )
p1/2 (p) 1 3 ( Y 1 , 0 2 Y 1 , 1 ) ( Y 0 , 0 0 )
p3/2 (p+) 1 3 ( 2 Y 1 , 0 Y 1 , 1 ) 1 5 ( 2 Y 2 , 0 3 Y 2 , 1 ) 1 3 ( 3 Y 1 , 1 0 ) 1 5 ( Y 2 , 1 4 Y 2 , 2 )
d3/2 (d) 1 5 ( 2 Y 2 , 0 3 Y 2 , 1 ) 1 3 ( 2 Y 1 , 0 Y 1 , 1 ) 1 5 ( Y 2 , 1 4 Y 2 , 2 ) 1 3 ( 3 Y 1 , 1 0 )
d5/2 (d+) 1 5 ( 3 Y 2 , 0 2 Y 2 , 1 ) 1 7 ( 3 Y 3 , 0 4 Y 3 , 1 ) 1 5 ( 4 Y 2 , 1 Y 2 , 2 ) 1 7 ( 2 Y 3 , 1 5 Y 3 , 2 ) 1 5 ( 5 Y 2 , 2 0 ) 1 7 ( Y 3 , 2 6 Y 3 , 3 )
f5/2 (f) 1 7 ( 3 Y 3 , 0 4 Y 3 , 1 ) 1 5 ( 3 Y 2 , 0 2 Y 2 , 1 ) 1 7 ( 2 Y 3 , 1 5 Y 3 , 2 ) 1 5 ( 4 Y 2 , 1 Y 2 , 2 ) 1 7 ( Y 3 , 2 6 Y 3 , 3 ) 1 5 ( 5 Y 2 , 2 0 )
f7/2 (f+) 1 7 ( 4 Y 3 , 0 3 Y 3 , 1 ) 1 9 ( 4 Y 4 , 0 5 Y 4 , 1 ) 1 7 ( 5 Y 3 , 1 2 Y 3 , 2 ) 1 9 ( 3 Y 4 , 1 6 Y 4 , 2 ) 1 7 ( 6 Y 3 , 2 Y 3 , 3 ) 1 9 ( 2 Y 4 , 2 7 Y 4 , 3 ) 1 7 ( 7 Y 3 , 3 0 ) 1 9 ( Y 4 , 3 8 Y 4 , 4 )

* The upper and lower parts denote the large and small components, respectively.

Table 3.   Mathematica code for density plot.
Table 4.  Parameters for density plot.
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
Table 5.  Expectation values of r for Z = 1.
<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
Table 6.  Expectation values of r for Z = 100.
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].

References
 
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