Visualization of the Exact Solution of Dirac Equation

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，s_1/2とp_1/2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, m_j = 1/2)

Figure 2.

 2p_– density.

2p(l = 1, m = 1 β)/2p_– (j = 1/2, m_j = 1/2)

Figure 3.

 2p₊ density.

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

Figure 4.

 2p₊ density.

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

Figure 5.

 3d_– density.

3d(l = 2, m = 2 β)/3d_– (j = 3/2, m_j = 3/2)

Figure 6.

 3d_– density.

3d(l = 2, m = 1 β)/3d_– (j = 3/2, m_j = 1/2)

Figure 7.

 3d₊ density.

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

Figure 8.

 3d₊ density.

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

Figure 9.

 3d₊ density.

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

Figure 10.

 4f_– density .

4f(l = 3, m = 3 β)/4f_– (j = 5/2, m_j = 5/2)

Figure 11.

 4f_– density

4f(l = 3, m = 2 β)/4f_– (j = 5/2, m_j = 3/2)

Figure 12.

 4f_– density

4f(l = 3, m = 1 β)/4f_– (j = 5/2, m_j = 1/2)

Figure 13.

 4f₊ density .

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

Figure 14.

 4f₊ density

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

Figure 15.

 4f₊ density

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

Figure 16.

 4f₊ density

4f(l = 3, m = 0 α)/4f₊ (j = 7/2, m_j = 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+)	$\begin{array}{c}\left(\begin{array}{c}{Y}_{0,0}\\ 0\end{array}\right)\\ \frac{1}{\sqrt{3}}\left(\begin{array}{c}-Y_{1,0}\\ \sqrt{2}{Y}_{1,1}\end{array}\right)\end{array}$
p1/2 (p–)	$\begin{array}{c}\frac{1}{\sqrt{3}}\left(\begin{array}{c}-Y_{1,0}\\ \sqrt{2}{Y}_{1,1}\end{array}\right)\\ \left(\begin{array}{c}{Y}_{0,0}\\ 0\end{array}\right)\end{array}$
p3/2 (p+)	$\begin{array}{c}\frac{1}{\sqrt{3}}\left(\begin{array}{c}\sqrt{2}{Y}_{1,0}\\ Y_{1,1}\end{array}\right)\\ \frac{1}{\sqrt{5}}\left(\begin{array}{c}-\sqrt{2}{Y}_{2,0}\\ \sqrt{3}{Y}_{2,1}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{3}}\left(\begin{array}{c}\sqrt{3}{Y}_{1,1}\\ 0\end{array}\right)\\ \frac{1}{\sqrt{5}}\left(\begin{array}{c}-Y_{2,1}\\ \sqrt{4}{Y}_{2,2}\end{array}\right)\end{array}$
d3/2 (d–)	$\begin{array}{c}\frac{1}{\sqrt{5}}\left(\begin{array}{c}-\sqrt{2}{Y}_{2,0}\\ \sqrt{3}{Y}_{2,1}\end{array}\right)\\ \frac{1}{\sqrt{3}}\left(\begin{array}{c}\sqrt{2}{Y}_{1,0}\\ Y_{1,1}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{5}}\left(\begin{array}{c}-Y_{2,1}\\ \sqrt{4}{Y}_{2,2}\end{array}\right)\\ \frac{1}{\sqrt{3}}\left(\begin{array}{c}\sqrt{3}{Y}_{1,1}\\ 0\end{array}\right)\end{array}$
d5/2 (d+)	$\begin{array}{c}\frac{1}{\sqrt{5}}\left(\begin{array}{c}\sqrt{3}{Y}_{2,0}\\ \sqrt{2}{Y}_{2,1}\end{array}\right)\\ \frac{1}{\sqrt{7}}\left(\begin{array}{c}-\sqrt{3}{Y}_{3,0}\\ \sqrt{4}{Y}_{3,1}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{5}}\left(\begin{array}{c}\sqrt{4}{Y}_{2,1}\\ Y_{2,2}\end{array}\right)\\ \frac{1}{\sqrt{7}}\left(\begin{array}{c}-\sqrt{2}{Y}_{3,1}\\ \sqrt{5}{Y}_{3,2}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{5}}\left(\begin{array}{c}\sqrt{5}{Y}_{2,2}\\ 0\end{array}\right)\\ \frac{1}{\sqrt{7}}\left(\begin{array}{c}-Y_{3,2}\\ \sqrt{6}{Y}_{3,3}\end{array}\right)\end{array}$
f5/2 (f–)	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}-\sqrt{3}{Y}_{3,0}\\ \sqrt{4}{Y}_{3,1}\end{array}\right)\\ \frac{1}{\sqrt{5}}\left(\begin{array}{c}\sqrt{3}{Y}_{2,0}\\ \sqrt{2}{Y}_{2,1}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}-\sqrt{2}{Y}_{3,1}\\ \sqrt{5}{Y}_{3,2}\end{array}\right)\\ \frac{1}{\sqrt{5}}\left(\begin{array}{c}\sqrt{4}{Y}_{2,1}\\ Y_{2,2}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}-Y_{3,2}\\ \sqrt{6}{Y}_{3,3}\end{array}\right)\\ \frac{1}{\sqrt{5}}\left(\begin{array}{c}\sqrt{5}{Y}_{2,2}\\ 0\end{array}\right)\end{array}$
f7/2 (f+)	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}\sqrt{4}{Y}_{3,0}\\ \sqrt{3}{Y}_{3,1}\end{array}\right)\\ \frac{1}{\sqrt{9}}\left(\begin{array}{c}-\sqrt{4}{Y}_{4,0}\\ \sqrt{5}{Y}_{4,1}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}\sqrt{5}{Y}_{3,1}\\ \sqrt{2}{Y}_{3,2}\end{array}\right)\\ \frac{1}{\sqrt{9}}\left(\begin{array}{c}-\sqrt{3}{Y}_{4,1}\\ \sqrt{6}{Y}_{4,2}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}\sqrt{6}{Y}_{3,2}\\ Y_{3,3}\end{array}\right)\\ \frac{1}{\sqrt{9}}\left(\begin{array}{c}-\sqrt{2}{Y}_{4,2}\\ \sqrt{7}{Y}_{4,3}\end{array}\right)\end{array}$	$\begin{array}{c}\frac{1}{\sqrt{7}}\left(\begin{array}{c}\sqrt{7}{Y}_{3,3}\\ 0\end{array}\right)\\ \frac{1}{\sqrt{9}}\left(\begin{array}{c}-Y_{4,3}\\ \sqrt{8}{Y}_{4,4}\end{array}\right)\end{array}$

* 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

• [1]   W. R. Johnson, "Atomic Structure Theory: Lectures on Atomic Physics", Springer, Berlin (2006).
• [2]   H. A. Bethe, E. E. Salpeter, "Quantum Mechanics of One- and Two-Electron Atoms", Dover Publ. Inc., Mineola, NY (2008).
• [3]   K. Suzuki, O. Kikuchi, "Denshi no Kido", Kyoritsu Shuppan, Tokyo (1984), in Japanese.
• [4]   S. Tokita, J. Comput. Chem. Jpn., 15, A7 (2016).
• [5]   P. A. M. Dirac, Proc. R. Soc. Lon. Ser.-A 117, 610–624 (1928).
• [6]   K. G. Dyall, K. Faegri, Jr., "Introduction to Relativistic Quantum Chemistry", Oxford Univ. Press, New York (2007).
• [7]   W. Gordon, Z. Phys., 48, 11 (1928).
• [8]   C. G. Darwin, Proc. R. Soc. Lon. Ser.-A 118, 654–680 (1928).
• [9]   F. D. Pidduck, J. Lond. Math. Soc., s1-4, 163 (1929).
• [10]   J. Mawhin, A. Ronveaux, Arch. Hist. Exact Sci., 64, 429 (2010).
• [11]   R. P. Martínez-y-Romero, Am. J. Phys., 68, 1050 (2000).
• [12]   H. E. White, Phys. Rev., 38, 513 (1931).
• [13]   R. E. Powell, J. Chem. Educ., 45, 558 (1968).
• [14]   A. Szabo, J. Chem. Educ., 46, 678 (1969).
• [15]   I. P. Grant, "Relativistic Quantum Theory of Atoms and Molecules", Springer, New York (2007).
• [16]   Wolfram Research, Inc., Mathematica, Version 10.3, Champaign, IL (2015).
• [17]   http://www.wolfram.com/cdf-player/
• [18]   L. Pauling, E. B. Wilson, Jr., "Introduction to Quantum Mechanics", Dover Publ. Inc., Mineola, NY (1985).
• [19]   S. Yamamoto, H. Tatewaki, Chem. Phys., 295, 47 (2003).
• [20]   V. M. Burke, I. P. Grant, Proc. Phys. Soc., 90, 297 (1967).
• [21]   N. Takeda, Y. Hatano, S. Yamamoto, H. Tatewaki, Comput. Theor. Chem., 1045, 99 (2014).
• [22]   R. H. Garstang, D. F. Mayers, Proc. Camb. Philos. Soc., 62, 777 (1966).
• [23]   T. Koga, H. Tatewaki, O. Matsuoka, J. Chem. Phys., 119, 1279 (2003), See also http://www.nsc.nagoya-cu.ac.jp/~htatewak/uncntrct_rel_gtf.html.
• [24]   T. Koga, H. Tatewaki, T. Shimazaki, Chem. Phys. Lett., 328, 473 (2000), We have calculated <r> using the HF orbitals given at http://www.nsc.nagoya-cu.ac.jp/~htatewak/uncontracted_gtf.html.
• [25]   L. Visscher, K. G. Dyall, At. Data Nucl. Data Tables, 67, 207 (1997), See also http://www.few.vu.nl/~visscher/FiniteNuclei/FiniteNuclei.htm.
• [26]   H. Moriyama, H. Tatewaki, S. Yamamoto, J. Chem. Phys., 138, 224310 (2013).
• [27]   H. Tatewaki, S. Yamamoto, H. Moriyama, "Low-lying excited states of lanthanide diatomics studied by four-component relativistic configuration interaction methods", in: "Computational methods in lanthanide and actinide chemistry", M. Dolg (Ed.), 89–119, Wiley, UK (2015).
• [28]   Supplement 1. 5g_density.and.Figures.zip.
• [29]   Supplement 2. radial_density.pdf.
• [30]   Supplement 3. Dirac.Hdens.plot.v0d.nb.