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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.
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Keywords:
- variational method /
- wavefunction of excited states /
- variational function /
- configuration interaction
[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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[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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