基于改进变分法对原子激发态精确波函数的研究

熊庄; 汪振新; Naoum C. Bacalis

doi:10.7498/aps.63.053104

摘要

传统的利用变分原理求解Schrdinger方程获得原子激发态波函数的方法是基于HUM理论（Hylleraas-Undheim and MacDonald theorem），在有限的N维Hilbert空间中，通过求解久期方程的高阶根获得激发态的近似波函数. 在我们前期的工作中已指出，由于HUM方法的几个内禀缺陷限制，它将导致在相同的函数空间中，由传统变分法得到的激发态波函数的质量远差于足够好的基态波函数. 进一步地，为了避免基于HUM方法的变分缺陷，本文提出了新的变分函数，并证明其试探激发态波函数在其本征态处具有局域极小值，因而可以通过变分极小无限制的逼近该本征态. 在此基础上，利用广义的Laguerre类型轨道（GLTO）在组态相互作用的框架下，分别编写了基于传统HUM理论和新变分函数的关于求解原子近似波函数的计算程序，并且利用该程序计算了氦原子（He）在1S（e），1P（o）态下相应的基态及激发态近似波函数及对应的能量值和径向平均值，并与已有文献中结果进行比较，计算结果显示了HUM理论的缺陷及新变分函数优越性，并就进一步提高激发态的精度指明了方向.

关键词:

Abstract

For the computation of excited states, the traditional solutions of the Schredinger equation, using higher roots of a secular equation in a finite N-dimensional function space, by the Hylleraas-Undheim and MacDonald (HUM) theorem, we found that it has several restrictions which render it of lower quality, relative to the lowest root if the latter is good enough. In order to avoid the variational restrictions, based on HUM, we propose a new variational function and prove that the trial wave function has a local minimum in the eigenstates, which allows to approach eigenstates unlimitedly by variation. In this paper, under the configuration interaction (CI), we write a set of calculation programs by using generalized laguerre type orbitals (GLTO) to get the approximate wave function of different states, which is base on the HUM or the new variational function. By using the above program we get the approximate wave function for 1S (e), 1P (o) state of helium atoms (He) through the different theorems, the energy value and radial expectation value of related states. By comparing with the best results in the literature, the theoretical calculations show the HUM's defects and the new variational function's superiority, and we further give the direction of improving the accuracy of excited states.

Keywords:

作者及机构信息

1.
东南大学空间科学与技术研究院, 南京 210096;

2.
东南大学能源热转换及其过程测控教育部重点实验室, 南京 210096;

3.
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Vasileos Constantinou 48, GR-116 35 Athens, Greece

基金项目: 国家自然科学基金重大研究计划（批准号：91334205）和国家自然科学基金联合基金（批准号：11178008）资助的课题.

Authors and contacts

1.
Space Science and Technology Research Institute, Southeast University, Nanjing 210096, China;

2.
Key Laboratory of Energy Thermal Conversion and Control, Ministry of Education, Southeast University, Nanjing 210096, China;

3.
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Vasileos Constantinou 48, GR-116 35 Athens, Greece

Funds: Project supported by the Major Research Plan of the National Natural Science Foundation of China (Grant No. 91334205), and the Joint Funds of the National Natural Science Foundation of China (Grant No. 11178008).

参考文献

[1]	Lee T D 2005 J. Stat. Phys. 121 1015
[2]	Friedberg R, Lee T D, Zhao W Q 2006 Chin. Phys. 15 1909
[3]	Bunge C F 2006 J. Chem. phys. 125 014107
[4]	Cioslowski J 1987 J. Chem. phys. 86 2105
[5]	Gou B C, Chen Z, Lin C D 1991 Phys. Rev. A 43 3260
[6]	Kallman T R, Palmer P 2007 Rev. Mod. Phys. 79 79
[7]	Eidelsberg M, Crifo-Magnant F, Zeippen C J 1981 Astron. Astrophys. Suppl. Ser. 43 455
[8]	Dalgarno A 1979 Adv. At. Mol. Phys. 15 37
[9]	Qing Bo, Cheng Cheng, Gao Xiang, Zhang Xiao Le, Li Jia Ming 2010 Acta Phys. Sin. 59 7 (in Chinese) [青波, 程诚, 高翔, 张小乐, 李家明 2010 物理学报 59 7]
[10]	Hylleraas E, Undheim B 1930 Z. Phys. 65 759
[11]	McDonald J K L 1933 Phys. Rev. 43 830
[12]	Pilar F L 1968 Elementary Quantum Chemistry (Dover: McGraw-Hill Companies) p240
[13]	Harald Friedrich 1990 Theoretical Atomic Physics (Berlin: Springer-Verlag) p45
[14]	Newton R G 1982 Scattering Theory of Waves and Particles (2nd Ed.) (New York, Berlin, Heidelberg: Spring-Verlag) p326
[15]	Bacalis N C, Xiong Z, Karaoulanisc D 2008 Journal of Computational Methods in Sciences and Engineering 8 277
[16]	Chen M K 1994 J. Phys. B: At. Mol. Opt. Phys. 27 865
[17]	Bacalis N C 2007 Computation in Modern Science and Engineering CP963 Vol.2 Part A pp6-9
[18]	Li Z M, Xiong Z, Dai L L 2010 Acta Phys. Sin. 59 11 (in Chinese) [李尊懋, 熊庄, 代丽丽 2010 物理学报 59 11]
[19]	Ma Y, Xiong Z, Wang Z X 2013 Chinese Journal of Computational Physics 30 2 (in Chinese) [马迎, 熊庄, 汪振新 2013 计算物理 30 2]

施引文献

[1]	Lee T D 2005 J. Stat. Phys. 121 1015
[2]	Friedberg R, Lee T D, Zhao W Q 2006 Chin. Phys. 15 1909
[3]	Bunge C F 2006 J. Chem. phys. 125 014107
[4]	Cioslowski J 1987 J. Chem. phys. 86 2105
[5]	Gou B C, Chen Z, Lin C D 1991 Phys. Rev. A 43 3260
[6]	Kallman T R, Palmer P 2007 Rev. Mod. Phys. 79 79
[7]	Eidelsberg M, Crifo-Magnant F, Zeippen C J 1981 Astron. Astrophys. Suppl. Ser. 43 455
[8]	Dalgarno A 1979 Adv. At. Mol. Phys. 15 37
[9]	Qing Bo, Cheng Cheng, Gao Xiang, Zhang Xiao Le, Li Jia Ming 2010 Acta Phys. Sin. 59 7 (in Chinese) [青波, 程诚, 高翔, 张小乐, 李家明 2010 物理学报 59 7]
[10]	Hylleraas E, Undheim B 1930 Z. Phys. 65 759
[11]	McDonald J K L 1933 Phys. Rev. 43 830
[12]	Pilar F L 1968 Elementary Quantum Chemistry (Dover: McGraw-Hill Companies) p240
[13]	Harald Friedrich 1990 Theoretical Atomic Physics (Berlin: Springer-Verlag) p45
[14]	Newton R G 1982 Scattering Theory of Waves and Particles (2nd Ed.) (New York, Berlin, Heidelberg: Spring-Verlag) p326
[15]	Bacalis N C, Xiong Z, Karaoulanisc D 2008 Journal of Computational Methods in Sciences and Engineering 8 277
[16]	Chen M K 1994 J. Phys. B: At. Mol. Opt. Phys. 27 865
[17]	Bacalis N C 2007 Computation in Modern Science and Engineering CP963 Vol.2 Part A pp6-9
[18]	Li Z M, Xiong Z, Dai L L 2010 Acta Phys. Sin. 59 11 (in Chinese) [李尊懋, 熊庄, 代丽丽 2010 物理学报 59 11]
[19]	Ma Y, Xiong Z, Wang Z X 2013 Chinese Journal of Computational Physics 30 2 (in Chinese) [马迎, 熊庄, 汪振新 2013 计算物理 30 2]

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