Exact solutions to Stark effect of rigid symmetric-top molecules

Chen Chang-Yuan; Sun Guo-Hua; Wang Xiao-Hua; Sun Dong-Sheng; You Yuan; Lu Fa-Lin; Dong Shi-Hai

doi:10.7498/aps.70.20210214

Abstract

In this work a new scheme is proposed to accurately calculate the rotational energy level of the rigid symmetric-top molecule subjected to the external electric field, and also to obtain the corresponding analytical wave functions. For this purpose, first we use the different forms of function transformation and variable substitution to transform the differential equation of the polar angle θ into a confluent Heun differential equation, and then we use the characteristics of the confluent Heun differential equation and the confluent Heun function to find two linearly dependent solutions of the same eigenstates, which are used to construct the Wronskian determinant to obtain the exact energy spectrum equation. Finally, with the aid of the Maple software, we calculate the eigenvalues for different quantum states, and then substitute the obtained eigenvalues into the unnormalized eigenfunction to obtain the analytical normalized eigenfunction expressed by the confluent Heun function. These results are conducive to the in-depth study of the Stark effect of symmetric-top molecules.

Keywords:

Authors and contacts

1.
School of Physics and Electronic Engineering, Yancheng Teachers University, Yancheng 224007, China
2.
Catedrática CONACYT, CIC, Instituto Politécnico Nacional, CDMX 07700, Mexico
3.
Research Center for Quantum Physics, Huzhou University, Huzhou 313000, China
4.
Laboratorio de Información Cuántica, CIDETEC, Instituto Politécnico Nacional, UPALM, CDMX 07700, Mexico

Corresponding author: Chen Chang-Yuan, chency@yctu.edu.cn ; Dong Shi-Hai, dongsh2@yahoo.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11975196) and the SIP, Instituto Politécnico Nacional (IPN), Mexico (Grant No. 20210414)

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References

[1]	Townes C H, Schawlow A L 1975 Microwave Spectroscopy (New York: Dover Publications, Inc.) pp60−62, 248−251
[2]	Wollrab J E 1967 Rotational Spectra and Molecular Structure (New York: Academic Press) pp15−18, 247−250
[3]	Shirley J H 1963 J. Chem. Phys. 38 2896 Google Scholar
[4]	Tomutza L, Mizushima M 1972 J. Quant. Spectrosc. Radiat. Transfer. 12 925 Google Scholar
[5]	Roeggen I 1972 Atomic Date 4 289 Google Scholar
[6]	Maergoiz A I, Troe J 1993 J. Chem. Phys. 99 3218 Google Scholar
[7]	Maergoiz A I, Troe J, Weiss Ch 1994 J. Chem. Phys. 101 1885 Google Scholar
[8]	Fernández F M 1995 J. Math. Chem. 18 197 Google Scholar
[9]	Burrowst B L, Cohen M, Feldmann T 1995 J. Phys. B 28 4249 Google Scholar
[10]	Andreev S N, Makarov V P 2008 Radiophys. Quantum Electron. 51 718 Google Scholar
[11]	Sun G H, Chen C Y, Taud H, Yáñez-Márquez C, Dong S H 2020 Phys. Lett. A 384 126480 Google Scholar
[12]	Chen C Y, Wang X H, You Y, Sun G H, Dong S H 2020 Int. J. Quantum Chem. 120 e26336 Google Scholar
[13]	Chen C Y, You Y, Wang X H, Lu F L, Sun D S, Dong S H 2021 Results Phys. 24 104115 Google Scholar
[14]	Chen C Y, Sun D S, Sun G H, Wang X H, You Y, Dong S H 2021 Int. J. Quantum Chem. 121 e26546 Google Scholar
[15]	Fiziev P P 2010 J. Phys. A 43 035203 Google Scholar
[16]	Downing C A 2013 J. Math. Phys. 54 072101 Google Scholar
[17]	Al-Gwaiz M A 2008 Sturm-Liouville Theory and its Applications (London: Springer-Verlag London Limited) pp45−46, 55−59
[18]	程建春 2016 数学物理方程及其近似方法(第2版) (北京: 科学出版社) 第109−111, 133−135页 Cheng J C 2016 The Equations for Mathematical Physics and Their Approximate Methods (2nd Ed.) (Beijing: Science Press) pp109−111, 133−135 (in Chinese)
[19]	Wang H Y 2017 Mathematics for Physicists (Beijing: Science Press) pp130−146

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图 1 对称陀螺分子的$ f(\lambda ) $随λ的变化曲线　(a) $ b = 1, \;K = 0, \;M = 0\, , \;1\, , \;2\, , \;3, \, 4 $; (b) $ b = 5, \;K = 1, \;M = 0\, , \;1\, , \;2\, , \;3, \, 4 $; (c) $b = $$ 10, \;K = 2, \;M = 0\, , - 1\, , - 2\, , - 3$; (d) $ b = 20, \;K = 3, \;M = 0\, , - 1\, , - 2\, , - 3 $

Figure 1. Plot of $ f(\lambda ) $ as the function of λ for the symmetric-top molecules: (a) $ b = 1, \;K = 0, \;M = 0\, , \;1\, , \;2\, , \;3, \, 4 $; (b) $b = 5, $$ \;K = 1, \;M = 0\, , \;1\, , \;2\, , \;3, \, 4$; (c) $ b = 10, \;K = 2, \;M = 0\, , - 1\, , - 2\, , - 3 $; (d) $ b = 20, \;K = 3, \;M = 0\, , - 1\, , - 2\, , - 3 $.

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图 2 $ {N_1}{\varTheta _1}(x) $和$ {( - 1)^n}{N_2}{\varTheta _2}(x) $是线性相关的　(a) $ b = 1\, , \;J = 1\, , \;K = 1\, , \;M = 1\, , \;n = 0 $; (b) $b = 1\, , \;J = 1\, , \;K = 1\, , \;M = $$ - 1\, , \;n = 0$; (c) $ b = 10\, , \;J = 3\, , \;K = 1\, , \;M = 1\, , \;n = 2 $; (d) $ b = 1\, 0, \;J = 3\, , \;K = 1\, , \;M = 2\, , \;n = 1 $

Figure 2. Linear dependence relation between $ {N_1}{\varTheta _1}(x) $ and $ {( - 1)^n}{N_2}{\varTheta _2}(x) $: (a) $ b = 1\, , \;J = 1\, , \;K = 1\, , \;M = 1\, , \;n = 0 $; (b) $b = $$ 1\, , \;J = 1\, , \;K = 1\, , \;M = - 1\, , \;n = 0$; (c) $ b = 10\, , \;J = 3\, , \;K = 1\, , \;M = 1\, , \;n = 2 $; (d) $ b = 1\, 0, \;J = 3\, , \;K = 1\, , \;M = 2\, , \;n = 1 $.

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表 1 对称陀螺分子λ的精确值

Table 1. Precise values of λ for the symmetric-top molecules.

J	K	(K, M)	n	b = 1	b = 5	b = 10	b = 20
0	0	(0, 0)	0	–0.15766348	–2.36561388	–6.04507511	–14.18676623
1	0	(0, 0)	1	2.09076065	2.63874703	1.72333341	–2.64067155
	0	(0, 1) (0, –1)	0	1.95033391	0.90475071	–1.52336944	–7.83391897
	1	(1, 1) (–1, –1)	0	1.46550429	–1.13615048	–4.90088894	–13.09274467
		(1, 0) (–1, 0)	0	1.95033391	0.90475071	–1.52336944	–7.83391897
		(1, –1) (–1, 1)	0	2.45930791	3.25240982	2.28859798	–2.10241580
2	0	(0, 0)	2	6.02403166	6.64546248	7.87932729	7.59702750
		(0, 1) (0, –1)	1	6.01157793	6.14932334	5.86920589	3.21233397
		(0, 2) (0, –2)	0	5.97623080	5.42770799	3.91145991	–0.75028417
	1	(1, 2) (–1, –2)	0	5.64610143	3.87965848	1.11394605	–5.42557601
		(1, 1) (–1, –1)	1	5.83444799	5.02308599	3.42628660	–1.26151418
		(1, 0) (–1, 0)	1	6.01157793	6.14932334	5.86920589	3.21233397
		(1, –1) (–1, 1)	1	6.17303459	7.17694539	8.52241501	8.17688040
		(1, –2) (–1, 2)	0	6.31159733	7.08037678	6.95972394	4.30076661
	2	(2, 2) (–2, –2)	0	5.32080726	2.40921111	–1.50915355	–9.82205218
		(2, 1) (–2, –1)	0	5.64610143	3.87965848	1.11394605	–5.42557601
		(2, 0) (–2, 0)	0	5.97623080	5.42770799	3.91145991	–0.75028417
		(2, –1) (–2, 1)	0	6.31159733	7.08037678	6.95972394	4.30076661
		(2, –2) (–2, 2)	0	6.65266958	8.89010555	10.43690550	9.90220896
3	0	(0, 0)	3	12.01112256	12.28609326	13.23851702	16.01072933
		(0, 1) (0, –1)	2	12.00832614	12.20185604	12.68109763	13.00122422
		(0, 2) (0, –2)	1	11.99996510	11.98051777	11.76516026	10.12548434
		(0, 3) (0, –3)	0	11.98611962	11.65790586	10.68518379	7.32389600
	1	(1, 3) (–1, –3)	0	11.73714675	10.44710964	8.38487920	3.22650134
		(1, 2) (–1, –2)	1	11.83176701	11.08791249	9.87421333	6.36151132
		(1, 1) (–1, –1)	2	11.92226344	11.68117484	11.32510413	9.60625434
		(1, 0) (–1, 0)	2	12.00832614	12.20185604	12.68109763	13.00122422
		(1, –1) (–1, 1)	2	12.08964046	12.60886369	13.77998811	16.66271656
		(1, –2) (–1, 2)	1	12.16589908	12.84075061	13.66336275	14.14238561
		(1, –3) (–1, 3)	0	12.23682184	12.90821524	13.10954794	11.68530530
	2	(2, 3) (–2, –3)	0	11.48984080	9.26997167	6.18154889	–0.66884422
		(2, 2) (–2, –2)	1	11.66163328	10.18131826	8.01132314	2.77663392
		(2, 1) (–2, –1)	1	11.83176701	11.08791249	9.87421333	6.36151132
		(2, 0) (–2, 0)	1	11.99996510	11.98051777	11.76516026	10.12548434
		(2, –1) (–2, 1)	1	12.16589908	12.84075061	13.66336275	14.14238561
		(2, –2) (–2, 2)	1	12.32917589	13.62647723	15.47989534	18.60143861
		(2, –3) (–2, 3)	0	12.48932036	14.20615704	15.70403035	16.41950037
	3	(3, 3) (–3, –3)	0	11.24414341	8.12208069	4.05742691	–4.40185593
		(3, 2) (–3, –2)	0	11.48984080	9.26997167	6.18154889	–0.66884422
		(3, 1) (–3, –1)	0	11.73714675	10.44710964	8.38487920	3.22650134
		(3, 0) (–3, 0)	0	11.98611962	11.65790586	10.68518379	7.32389600
		(3, –1) (–3, 1)	0	12.23682184	12.90821524	13.10954794	11.68530530
		(3, –2) (–3, 2)	0	12.48932036	14.20615704	15.70403035	16.41950037
		(3, –3) (–3, 3)	0	12.74368721	15.56368561	18.56353300	21.76261674

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表 2 对称陀螺分子(4, 0, 0)态的λ值

Table 2. Values of λ of the state (4, 0, 0) for the symmetric-top molecules.

b	Perturbation	Maergoiz^[7]	This work	b	Maergoiz^[7]	This work
0.1	20.00006494		20.00006494	100	15.696	15.69559837
1	20.00649351		20.00649533	200	–31.200	–31.20010397
5	20.16233766	20.164	20.16352072	300	–90.598	–90.59767902
10	20.64935065	20.671	20.67080282	400	–156.41	–156.40991571
20	22.59740260	22.975	22.97522235	500	–226.31	–226.30840320
50	36.23376623	27.681	27.68132358	750	–412.26	–412.26234718

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[1]	Townes C H, Schawlow A L 1975 Microwave Spectroscopy (New York: Dover Publications, Inc.) pp60−62, 248−251
[2]	Wollrab J E 1967 Rotational Spectra and Molecular Structure (New York: Academic Press) pp15−18, 247−250
[3]	Shirley J H 1963 J. Chem. Phys. 38 2896 Google Scholar
[4]	Tomutza L, Mizushima M 1972 J. Quant. Spectrosc. Radiat. Transfer. 12 925 Google Scholar
[5]	Roeggen I 1972 Atomic Date 4 289 Google Scholar
[6]	Maergoiz A I, Troe J 1993 J. Chem. Phys. 99 3218 Google Scholar
[7]	Maergoiz A I, Troe J, Weiss Ch 1994 J. Chem. Phys. 101 1885 Google Scholar
[8]	Fernández F M 1995 J. Math. Chem. 18 197 Google Scholar
[9]	Burrowst B L, Cohen M, Feldmann T 1995 J. Phys. B 28 4249 Google Scholar
[10]	Andreev S N, Makarov V P 2008 Radiophys. Quantum Electron. 51 718 Google Scholar
[11]	Sun G H, Chen C Y, Taud H, Yáñez-Márquez C, Dong S H 2020 Phys. Lett. A 384 126480 Google Scholar
[12]	Chen C Y, Wang X H, You Y, Sun G H, Dong S H 2020 Int. J. Quantum Chem. 120 e26336 Google Scholar
[13]	Chen C Y, You Y, Wang X H, Lu F L, Sun D S, Dong S H 2021 Results Phys. 24 104115 Google Scholar
[14]	Chen C Y, Sun D S, Sun G H, Wang X H, You Y, Dong S H 2021 Int. J. Quantum Chem. 121 e26546 Google Scholar
[15]	Fiziev P P 2010 J. Phys. A 43 035203 Google Scholar
[16]	Downing C A 2013 J. Math. Phys. 54 072101 Google Scholar
[17]	Al-Gwaiz M A 2008 Sturm-Liouville Theory and its Applications (London: Springer-Verlag London Limited) pp45−46, 55−59
[18]	程建春 2016 数学物理方程及其近似方法(第2版) (北京: 科学出版社) 第109−111, 133−135页 Cheng J C 2016 The Equations for Mathematical Physics and Their Approximate Methods (2nd Ed.) (Beijing: Science Press) pp109−111, 133−135 (in Chinese)
[19]	Wang H Y 2017 Mathematics for Physicists (Beijing: Science Press) pp130−146

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