Nonrelativistic ionization energy levels of a helium atom
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Keywords:
Key words, variational method, expansion, variational principle, Schrӧdinger equation, inverse iteration method. PACS numbers, 31.15.-p, 34.50.Fa.Abstract
The nonrelativistic ionization energy levels of a helium atom are calculated for S, P, and D states. The calculations are based on the variational method of “exponential” expansion. Variational wave functions of bound states were obtained by solving the Schrodinger equation for the quantum three body problem with Coulomb interaction using a variational approach based on exponential expansion with the parameters of exponents being chosen in a pseudorandom way. The convergence of the calculated energy levels is studied as a function of the number of basis functions (N). This allows us to claim that the obtained energy values (including the values for the states with a nonzero angular momentum) are accurate to 20 significant digits.References
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[2] M. Hori, A. Soter, D. Barna, A. Dax, R. Hayano, S. Friedreich, B. Juhasz, Th. Pask, Widmann, D. Horvath, L. Venturelli, and N. Zurlo. Two_photon laser spectroscopy of antiprotonic helium and the antiproton_to_electron mass ratio // Nature. – 2011. – Vol. 475. – P. 484.
[3] J.L. Friar. The structure of light nuclei and its effect on precise atomic measurements // Can. J. Phys. – 2002. – Vol. 80. – P. 1337.
[4] M.C. Reed, B. Simon. Methods of Modern Mathematical Physics. – New York: Academic Press 1978. – Vol. 4. – 325 p. (In Russian: – M.: Mir, 1982)
[5] T. Kato. Fundamental properties of Hamiltonian operators of Schrodinger type // Trans. Am. Math. Soc. – 1951. – Vol. 70. – P. 195.
[6] D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonskii. Quantum Theory of Angular Momentum.– Singapore: World Sci., 1988. – P. 526. (In Russian:– Leningrad: Nauka, 1975).
[7] R.A. Sack, C.C.J. Roothaan, W. Kolos. Recursive evaluation of some atomic integrals // J. Math. Phys. A. – 1967. – Vol. 8. – P. 1093.
[8] G.W.F. Drake. Angular integrals and radial recurrence relations for two-electron matrix elements in Hylleraas coordinates // Phys. Rev. A.– 1978. – Vol. 18. – P. 820.
[9] V.D. Efros. The three-body problem: generalized exponential expansion; arbitrary states in a correlated basis, and the binding energy of muonic molecules // Sov. Phys. JETP. – 1986. – Vol. 63. – P. 5.
[10] C.M. Rosenthal. The reduction of the multidimensional Schrӧdinger equation to a one-dimensional integral equation // Chem. Phys. Lett. – 1971. – Vol. 10. – P. 381-386.
[11] J.J. Griffin, J.A. Wheeler. Collective motions in nuclei by the method of generator coordinates // Phys. Rev. – 1957. – Vol. 108. – P. 311.
[12] A.J. Thakkar, V.H. Smith. Compact and accurate integral-transform wave functions. The 11S state of the helium-like ions from H– through Mg10+ // Phys. Rev. A – 1977. – Vol. 15. – P. 1.
[13] J.H. Bartlett. The helium wave equation // Phys. Rev. – 1937. – Vol. 51. – P. 661.
[14] V.A. Fok. On Schrӧdinger equation for helium atom // Izv. Akad. Nauk SSSR, Fiz. – 1954. – Vol. 18. –P. 161.
[15] V.I. Korobov. Coulomb three-body bound-state problem: variational calculations of nonrelatistic energies // Phys. Rev. A. – 2000. – Vol. 61. – P. 064503.
[16] V.I. Korobov. Nonrelativistic ionization energy for the helium ground state // Phys. Rev. A. – 2002. – Vol. 66. – P. 024501.
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Aznabayev, D., Bekbaev, A., Ishmukhamedov, I., Korobov, V., & Turmaganbet, A. (2016). Nonrelativistic ionization energy levels of a helium atom. Physical Sciences and Technology, 2(1). Retrieved from https://phst.kaznu.kz/index.php/journal/article/view/63
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Theoretical Physics and Astrophysics
