High pressure effects on the excitation spectra and dipole properties of Li, Be<sup>+</sup>, and B<sup>2+</sup> atoms under confinement

C. Martínez-Flores; R. Cabrera-Trujillo

doi:10.1063/1.5139099

Matter and Radiation at Extremes, Volume. 5, Issue 2, 24401(2020)

High pressure effects on the excitation spectra and dipole properties of Li, Be⁺, and B²⁺ atoms under confinement

C. Martínez-Flores¹ and R. Cabrera-Trujillo^2、*

¹Departamento de Química, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, C.P. 09340 México D.F., Mexico

²Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Ap. Postal 43-8, Cuernavaca, Morelos, 62251, Mexico

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Figures & Tables(11)

Fig. 1. Wavefunctions for the 1s and 2s states of Li, Be⁺, and B²⁺ atoms as functions of the radial coordinate r for the unconfined atoms. The curves are our results, while the symbols are from Ref. 28.

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Fig. 2. Orbital energies for the 1s, 2s, 2p, 3s, and 3p states of Li, Be⁺, and B²⁺ atoms confined by a spherical impenetrable cavity as a function of the confinement radius R₀: (a) 3s and 3p states; (b) 2s and 2p states; (c) 1s ground state. The crossing points between the ns–np levels are highlighted by circles for better visualization. The curves without symbols are for the ns states, while the curves with symbols are for the np states. For comparison, the HF results of Weiss³³ for the unconfined atom energy levels are also shown at R₀ = 30 a.u (▿).

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Fig. 3. Total HF energy, Eq. (6), as a function of cavity radius R₀ for Li, Be⁺, and B²⁺ atoms. In the case of Li, the symbols show the theoretical results from Sañu-Ginarte et al.²⁹ (×), Le Sech and Banerjee³⁷ (□), Sarsa and Le Sech³⁸ (○), and Sarsa et al.¹³ (▵). The HF results for unconfined atoms as reported by Weiss³³ are shown at R₀ = 5 a.u (▿).

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Fig. 4. Static pressure induced by the cavity as a function of cavity size R₀ for Li, Be⁺, and B²⁺ atoms confined by an impenetrable spherical cavity. The open square symbols (□) indicate the cavity size and pressure at which the 2s → 2p transition occurs. Some naturally occurring pressures are also shown.

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Fig. 5. Dipole oscillator strength for the is → 2p and is → 3p electronic transitions as a function of the impenetrable spherical cavity size R₀ for Li, Be⁺, and B²⁺ atoms for i = 1 (core) and i = 2 (valence) electrons. The curves without symbols are for the is → 2p transition of the valence i = 2 electron, while the curves with symbols are for the i = 1 core electron.

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Fig. 6. Static dipole polarizabilities αs2s (a) and αs1s (b) as functions of cavity size R₀ for Li, Be⁺, and B²⁺ atoms. The solid triangle (▴) and the solid circle (•) at R₀ = 30 a.u. are the HF results of Schwerdtfeger and Nagle¹¹ and Tang et al.,³² respectively.

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Fig. 7. (a) Mean excitation energies I01s (curves with symbols) and I02s (curves without symbols) as functions of cavity size R₀ for Li, Be⁺, and B²⁺ atoms. For comparison, we also show at R₀ = 30 a.u. the values of the free atoms obtained by Oddershede and Sabin²⁷ (○), Kamakura⁴¹ (▵), and Dehmer et al.⁴² (□). (b) Total mean excitation energy I₀.

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Fig. 8. Slater’s X-α parameter α_X as a function of cavity size R₀ for Li, Be⁺, and B²⁺ atoms.

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Table 1. Unconfined ground state properties for free Li, Be⁺, and B²⁺ atoms. We report values for the core (i = 1) and valence (i = 2) electrons for the ground ( $ϵ_{0}^{i s}$ ) and excited ( $ϵ_{0}^{2 p}$ ) orbital energies, the total HF energy E_HF, the DOS $f_{i s 2 p}^{i}$ , the static dipole polarizability $α_{s}^{i}$ , and the mean excitation energy $I_{0}^{i}$ . Slater’s α_X parameter, Eq. (9), takes the values $α_{X}^{L i} = 0.580 02$ , $α_{X}^{{B e}^{+}} = 0.526 32$ , and $α_{X}^{B^{2 +}} = 0.499 22$ .

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Table 1. Unconfined ground state properties for free Li, Be⁺, and B²⁺ atoms. We report values for the core (i = 1) and valence (i = 2) electrons for the ground ( $ϵ_{0}^{i s}$ ) and excited ( $ϵ_{0}^{2 p}$ ) orbital energies, the total HF energy E_HF, the DOS $f_{i s 2 p}^{i}$ , the static dipole polarizability $α_{s}^{i}$ , and the mean excitation energy $I_{0}^{i}$ . Slater’s α_X parameter, Eq. (9), takes the values $α_{X}^{L i} = 0.580 02$ , $α_{X}^{{B e}^{+}} = 0.526 32$ , and $α_{X}^{B^{2 +}} = 0.499 22$ .

	Valence (2s¹)
	Li	Be⁺	B²⁺
$ϵ_{0}^{2 s}$	−0.201 16	−0.672 82	−1.397 22
	(−0.196 32)^a	(−0.666 15)^a	(−1.389 85)^a
$ϵ_{0}^{2 p}$	−0.138 61	−0.547 31	−1.210 49
E_HF	−7.437 49	−14.283 8	−23.38 26
	(−7.432 72)^a	(−14.277 4)^a	(−23.375 9)^a
	(−7.419 23)^c	…	…
$f_{2 s 2 p}^{2 s}$	0.651 27	0.413 15	0.299 63
	(0.767 1)^d	(0.510 9)^d	…
$α_{s}^{2 s}$	171.188	27.383 6	8.994 40
	(164.05)^e	(24.496 6)^f	…
$I_{0}^{2 s}$	3.567 84	12.052 9	25.765 5
	(3.29)^b	…	…

Table 2. Similar to Table I, but for a Li atom confined by an impenetrable spherical cavity for several selected values of the cavity size R₀. The values in parentheses are the theoretical results from Sañu-Ginarte et al.²⁹

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Table 2. Similar to Table I, but for a Li atom confined by an impenetrable spherical cavity for several selected values of the cavity size R₀. The values in parentheses are the theoretical results from Sañu-Ginarte et al.²⁹

R₀	$ϵ_{0}^{1 s}$	$ϵ_{0}^{2 s}$	$ϵ_{0}^{2 p}$	E_HF	$f_{1 s 2 p}^{1 s}$	$f_{2 s 2 p}^{2 s}$	$f_{2 s 3 p}^{2 s}$	$α_{s}^{1 s}$	$α_{s}^{2 s}$	$I_{0}^{1 s}$	$I_{0}^{2 s}$	α_X [Eq. (9)]
0.7	1.320 77	31.446 0	16.934 4	31.131 0	0.990 43	−0.606 48	1.571 51	0.005 35	0.000 16	378.609	…	0.351 35
0.75	0.536 49	26.840 5	14.525 6	25.115 2	0.989 92	−0.605 38	1.571 75	0.006 74	0.000 48	337.266	…	0.357 48
0.79	0.033 49	23.793 8	12.929 2	21.173 3	0.989 19	−0.604 20	1.571 54	0.008 01	0.000 87	309.699	…	0.362 48
0.8	−0.078 06	23.105 9	12.568 3	20.288 0	0.988 97	−0.603 87	1.571 43	0.008 35	0.000 99	303.449	…	0.363 73
1.0	−1.524 94	13.559 0	7.5403 3	8.240 71	0.980 14	−0.593 31	1.564 10	0.016 80	0.006 45	215.303	…	0.389 80
	…	…	…	(8.513 92)	…	…	…	…	…	…	…	…
2.0	−2.749 71	2.002 96	1.295 42	−5.192 26	0.799 73	−0.412 73	1.370 96	0.077 55	0.726 01	111.358	…	0.517 36
	…	…	…	(−5.084 19)	…	…	…	…	…	…	…	…
3.0	−2.791 16	0.415 51	0.343 33	−6.820 14	0.543 18	−0.112 39	1.046 19	0.100 16	21.263 4	103.957	…	0.577 90
4.0	−2.792 33	0.018 79	0.067 24	−7.217 59	0.385 24	0.136 82	0.779 41	0.102 35	58.957 8	103.625	22.106 9	0.588 02
	…	…	…	(−7.185 99)	…	…	…	…	…	…	…	…
5.0	−2.792 36	−0.112 58	−0.040 06	−7.348 98	0.310 79	0.305 19	0.595 47	0.102 45	59.181 9	103.616	11.353 7	0.585 90
	…	…	…	(−7.323 00)	…	…	…	…	…	…	…	…
6.0	−2.792 36	−0.163 46	−0.088 74	−7.399 87	0.279 08	0.418 02	0.465 07	0.102 45	76.600 6	103.616	7.478 42	0.583 19
	…	…	…	(−7.376 79)	…	…	…	…	…	…	…	…
8.0	−2.792 36	−0.194 16	−0.125 16	−7.430 56	0.262 82	0.550 79	0.287 46	0.102 45	118.705	103.616	4.702 04	0.580 69
	…	…	…	(−7.409 85)	…	…	…	…	…	…	…	…
10	−2.792 36	−0.199 96	−0.135 15	−7.436 36	0.261 29	0.614 55	0.172 21	0.102 45	150.076	103.616	3.879 67	0.580 13
	…	…	…	(−7.416 58)	…	…	…	…	…	…	…	…
∞	−2.792 32	−0.201 16	−0.138 81	−7.437 49	0.261 19	0.651 27	0.020 21	0.102 45	171.188	103.613	3.567 84	0.580 02

Table 3. Similar to Table II, but for the Be⁺ and B²⁺ atoms.

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Table 3. Similar to Table II, but for the Be⁺ and B²⁺ atoms.

R₀	$ϵ_{0}^{1 s}$	$ϵ_{0}^{2 s}$	$ϵ_{0}^{2 p}$	E_HF	$f_{2 s 2 p}^{1 s}$	$f_{2 s 2 p}^{2 s}$	$f_{2 s 3 p}^{2 s}$	$α_{s}^{1 s}$	$α_{s}^{2 s}$	$I_{0}^{1 s}$	$I_{0}^{2 s}$	α_X [Eq. (9)]
Be⁺
0.5	2.137 12	60.173 1	31.471 1	60.325 6	0.990 70	−0.608 33	1.572 96	0.001 41	−0.000 21	737.308	…	0.346 04
0.55	0.123 48	48.159 9	25.269 9	44.587 6	0.990 17	−0.607 04	1.573 59	0.001 94	−0.000 14	628.227	…	0.353 88
0.557	−0.107 08	46.742 9	24.537 9	42.747 0	0.990 03	−0.606 80	1.573 60	0.002 02	−0.000 16	615.258	…	0.354 98
0.57	−0.507 74	44.255 4	23.252 5	39.525 3	0.989 73	−0.606 31	1.573 56	0.002 18	−0.000 21	592.433	…	0.357 06
0.6	−1.312 15	39.150 5	20.612 9	32.954 1	0.988 82	−0.604 97	1.573 17	0.002 58	−0.000 36	545.347	…	0.361 89
0.8	−4.145 59	19.084 9	10.201 9	7.861 46	0.973 72	−0.587 66	1.559 61	0.006 20	−0.003 52	356.727	…	0.395 24
1.0	−5.095 47	10.385 1	5.645 46	−2.410 34	0.940 73	−0.553 61	1.525 12	0.011 02	−0.016 18	274.038	…	0.428 96
2.0	−5.663 14	0.637 48	0.342 34	−12.970 4	0.609 97	−0.194 04	1.134 74	0.026 48	−2.156 21	198.783	…	0.530 27
3.0	−5.667 09	−0.414 12	−0.332 76	−14.025 3	0.395 47	0.118 46	0.787 49	0.027 48	18.111 5	197.523	41.1282	0.534 69
4.0	−5.667 10	−0.614 43	−0.488 48	−14.225 6	0.325 82	0.279 47	0.581 18	0.027 49	18.076 4	197.515	20.3702	0.529 12
5.0	−5.667 09	−0.659 95	−0.531 32	−14.271 2	0.308 63	0.357 81	0.441 37	0.027 49	22.364 1	197.515	14.7393	0.526 99
6.0	−5.667 09	−0.670 23	−0.543 24	−14.281 4	0.305 47	0.393 28	0.336 29	0.027 49	25.356 9	197.515	12.8579	0.526 44
10	−5.667 08	−0.672 89	−0.547 30	−14.284 0	0.304 96	0.413 09	0.147 45	0.027 49	27.374 4	197.514	12.0547	0.526 30
∞	−5.667 11	−0.672 82	−0.547 31	−14.283 8	0.304 97	0.413 15	0.117 10	0.027 49	27.383 6	197.505	12.0529	0.526 32
B²⁺
0.4	2.057 86	91.865 3	47.233 0	90.8137	0.990 79	−0.609 09	1.574 08	0.000 57	0.0	1157.60	…	0.345 26
0.425	0.026 86	79.606 1	40.953 1	74.7372	0.990 50	−0.608 35	1.574 57	0.000 70	0.0	1045.47	…	0.350 04
0.43	−0.328 06	77.419 9	39.832 9	71.8863	0.990 41	−0.608 16	1.574 62	0.000 73	0.0	1025.37	…	0.351 00
0.6	−6.736 88	33.751 6	17.428 1	16.4387	0.978 38	−0.593 42	1.565 24	0.002 07	−0.002 80	614.980	…	0.384 90
0.8	−8.761 85	15.098 4	7.7981 9	−5.712 24	0.938 23	−0.551 23	1.523 06	0.004 37	−0.006 89	435.736	…	0.425 00
1.0	−9.319 56	7.291 26	3.712 27	−14.396 3	0.869 37	−0.480 51	1.447 14	0.006 72	−0.030 34	365.245	…	0.460 22
2.0	−9.541 72	−0.670 23	−0.699 29	−22.656 2	0.477 74	−0.018 72	0.938 65	0.010 21	−22.104 5	321.515	…	0.510 64
3.0	−9.541 95	−1.309 56	−1.135 42	−23.295 7	0.352 20	0.204 47	0.646 29	0.010 24	6.908 57	321.394	40.5256	0.501 88
4.0	−9.541 94	−1.387 56	−1.200 28	−23.373 7	0.331 75	0.277 28	0.469 21	0.010 24	8.205 13	321.394	28.5561	0.499 53
5.0	−9.541 93	−1.396 37	−1.209 33	−23.382 5	0.329 76	0.295 81	0.343 88	0.010 24	8.831 21	321.393	26.1855	0.499 23
6.0	−9.541 92	−1.397 17	−1.210 37	−23.383 3	0.329 64	0.299 16	0.264 51	0.010 24	8.970 87	321.392	25.8135	0.499 20
10	−9.541 89	−1.397 23	−1.210 49	−23.383 3	0.329 63	0.299 66	0.186 63	0.010 24	8.993 81	321.390	25.7654	0.499 20
∞	−9.541 58	−1.397 22	−1.210 49	−23.382 6	0.329 65	0.299 63	0.184 99	0.010 24	8.994 40	321.368	25.7655	0.499 22

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C. Martínez-Flores, R. Cabrera-Trujillo. High pressure effects on the excitation spectra and dipole properties of Li, Be⁺, and B²⁺ atoms under confinement[J]. Matter and Radiation at Extremes, 2020, 5(2): 24401

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Paper Information

Category: Fundamental Physics At Extremes

Received: Nov. 20, 2019

Accepted: Jan. 16, 2020

Published Online: Apr. 1, 2020

The Author Email: Cabrera-Trujillo R. (trujillo@icf.unam.mx)

DOI:10.1063/1.5139099

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Table 2. Similar to Table I, but for a Li atom confined by an impenetrable spherical cavity for several selected values of the cavity size R0. The values in parentheses are the theoretical results from Sañu-Ginarte et al.29

Table 2. Similar to Table I, but for a Li atom confined by an impenetrable spherical cavity for several selected values of the cavity size R0. The values in parentheses are the theoretical results from Sañu-Ginarte et al.29

Table 3. Similar to Table II, but for the Be+ and B2+ atoms.

Table 3. Similar to Table II, but for the Be+ and B2+ atoms.

Table 2. Similar to Table I, but for a Li atom confined by an impenetrable spherical cavity for several selected values of the cavity size R₀. The values in parentheses are the theoretical results from Sañu-Ginarte et al.²⁹

Table 2. Similar to Table I, but for a Li atom confined by an impenetrable spherical cavity for several selected values of the cavity size R₀. The values in parentheses are the theoretical results from Sañu-Ginarte et al.²⁹

Table 3. Similar to Table II, but for the Be⁺ and B²⁺ atoms.

Table 3. Similar to Table II, but for the Be⁺ and B²⁺ atoms.