Extended Eckart Theorem and New Variation Method for Excited States of Atoms

Zhuang Xiong 1,3 *, Jie Zang1 , N.C. Bacalis2 ，Qin Zhou3

1 Space Science and Technology Research Institute, Southeast University, Nanjing 210096, People’s Republic of China

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

3 School of Economics and Management, Southeast University, Nanjing 210096, People’s Republic of China

* Email: zhuangx@seu.edu.cn


Abstract

 We extend the Eckart theorem, from the ground state to excited states, which introduces an energy augmentation to the variation criterion for excited states. It is shown that the energy of a very good excited state trial function can be slightly lower than the exact eigenvalue. Further, the energy calculated by the trial excited state wave function, which is the closest to the exact eigenstate through Gram–Schmidt orthonormalization to a ground state approximant, is lower than the exact eigenvalue as well. In order to avoid the variation restrictions inherent in the upper bound variation theory based on Hylleraas, Undheim, and McDonald [HUM] and Eckart Theorem, we have proposed a new variation functional Ωn and proved that it has a local minimum at the eigenstates, which allows approaching the eigenstate unlimitedly by variation of the trial wave function. As an example, we calculated the energy and the radial expectation values of 3 () e S Helium atom by the new variation functional, and by HUM and Eckart theorem, respectively, for comparison. Our preliminary numerical results reveal that the energy of the calculated excited states 3 3 () e S and 4 3 () e S may be slightly lower than the exact eigenvalue (inaccessible by HUM theory) according to the General Eckart Theorem proved here, while the approximate wave function is better than HUM.

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https://arxiv.org/ftp/arxiv/papers/1602/1602.08342.pdf