Extended Eckart Theorem And Accurate Study For Excited Helium Atom ³S^?e? And³P^?o? State:New Variation Method Approach

We extend the Eckart theorem from the ground state to excited states, which introduced an energy augmentation to the variation criterion for excited states. It is shown that the energy of a good excited state trial function can be lower than the eigenvalue. Further, the energy calculated by the trial excited states wave function, which is the closest to its eigenstate through Gram-Schmidt orthonormalization to its ground state, is lower than its 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 and proved that it has a local minimum at the eigenstates, which allows approaching the eigenstate unlimitedly by variation of the trial wave function. Under the Configuration Interaction(CI), the atomic physical computing package using generalized Laguerre Type Orbitals realizes the algorithm. By using the above program,3S(e) and 3P(o) state of Helium atoms (He) are studied. Further, by calculating the energy value and the radial expectation value, The numerical results of the traditional variational method and the new variational function method are compared with the results of the existing literature, and reveals the superiority of the new variational method.The first chapter:Introduction, This chapter brief introduces the research background of the atomic data in the field of physics, and the importance of obtaining the exact approximate wave function to study the physics and atomic structure of the object, So as to explain the significance of this research.The second chapter:Basic theoretical method. This chapter introduces some basic concepts and methods about obtaining the atomic wave function in quantum mechanics and atomic physics, and the related theories of the excited state approximate wave function by using the variation method. Because the numerical method to calculate the minimum value is one of the physical difficulties, this chapter also introduces solving method about the extremum of Function, which is applied to the below mentioned atomic physical computing package. In addition, introducing the Generalized Eckart Theorem and HUM theory, explains that the traditional variation method in finite base condition, based on the HUM theory or Eckart theorem, have intrinsic defects.The third chapter is about Omega function theory. In order to overcome the defects of the traditional variation method, we have proposed a new variation function and show theoretical derivation of the new function.The fourth chapter is a brief introduction to some basic knowledge and the generalized Laguerre type atomic orbital used in the program design; then, The principles involved in the calculation program are introduced in detail.The fifth chapter is the numerical results and analysis. Using the above program studies 3S(e) and 3P(o) state of Helium atoms (He).By calculating the energy value and the radial mean value, the numerical results of the traditional variational method and the new variational variational function method are compared with the results of the existing literature, and reveals the superiority of the new variational method.The sixth chapter:Summary and Outlook, this chapter summarizes the main work and further indicate the need improved research direction, that the new variation function is used to find the approximate wave function of excited state.