Extended Eckart Theorem and New Variation Method for Excited States of Atoms
📝 Abstract
We extend the Eckart theorem, from the ground state to excited statew, 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 Omega-n 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 Triplet-S(even) 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 3rd Triplet-S(even) and 4th Triplet-S(even) 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.
💡 Analysis
We extend the Eckart theorem, from the ground state to excited statew, 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 Omega-n 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 Triplet-S(even) 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 3rd Triplet-S(even) and 4th Triplet-S(even) 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.
📄 Content
Extended Eckart Theorem and New Variation Method for Excited States of Atoms
Zhuang Xiong 1,3 *, Jie Zang1, N.C. Bacalis2，Qin Zhou3 1Space Science and Technology Research Institute, Southeast University, Nanjing 210096, People’s Republic of China 2Theoretical 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 n Ωand 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.
Keywords: Variation Method, Wave function of Excited States, Variation Function,
Configuration Interaction
- Introduction
The electrons around the nucleus in atoms are described by the Schrödinger wave equation. However, for many-particle systems Schrodinger equation cannot be solved exactly, from the time of establishment of quantum mechanics up to now; seeking a precise approximate solution of many-particle Schrodinger equation, for the ground state, and especially for the excited states, has been one of the most challenging research directions in physics [1-7]. Precise atomic structure calculations can not only explain the mechanism of electron correlation effects, but also provide the indispensable parameters of the relevant material science, such as plasma physics, astrophysics [8-10], which controlled nuclear fusion, etc. On the other hand, through the advances in vacuum technology, cryogenic technology, detection technology, etc, which provides more accurate experimental measurements, analyzing these high precision experimental results put higher requirements on theoretical calculations. Therefore, obtaining highly accurate wave functions, especially the wave functions of excited states, becomes an urgent need. Through ab-initio theoretical methods, calculating the atomic structure is based on the variation principle [11]. The traditional standard method, utilizing variation theory to solve the Schrödinger equation for excited state wave functions, is based on the Hylleraas, Undheim, and McDonald [HUM] upper bound theorem [12-14]. In solving the secular equation in finite N-dimensional Hilbert space, the lowest root is the upper-limit of the ground state exact energy, while its higher roots are upper-limits of the corresponding upper excited states exact energies [15, 16] . All states obtained by this method are orthogonal to each other. However, as our previous work[17-21] showed, in N-dimensional Hilbert space this method has inherent restrictions, so that the ‘quality’ of the excited state wave function obtained by optimization will be lower than that of the ground state wave function. In practical applications, it was found：On the one hand, if the ground state is optimized, the accuracy of the excited states, which are orthogonal to the ground state, will be reduced; if an excited state is optimized, then all orthogonal states lower than this excited state will loose accuracy. Thus, by either optimization in the ground state or in the excited state, it is impossible to get acceptable accuracy for the ground and the excited wave functions simultaneously, thereby, it will be difficult to achieve the high precision requirements of the experiment. On the other hand, if the calculation of the ground state and excited states are optimized by variation respectively [15], such as the State-Specific Theory (SST) developed by Nicolaides’ group, based on approximated orthogonality, the energy of the ground state and the excited states can be obtained with good accuracy, but th
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