Abstract
In previous papers we proved that, for stationary systems, the geometric elements of the wave described by the Schrödinger equation, namely the characteristic surfaces and their normals, are periodic solutions of the Hamilton-Jacobi equation. In this paper we prove that the Hamilton-Jacobi equation admits periodic solutions with the same geometrical symmetries as the wave function of the system in the case of the beryllium, boron, carbon and oxygen atoms. The above property is a reflection of the fact that for a multielectron atomic system the energetically most favorable geometric configuration minimizes the electron electron repulsion, and it leads to a general semiclassical calculation method, which is in principle valid for more complex systems. We show that this property can be used to compute the energetic atomic values, with the help of the central field method which we developed in previous publications. The relative error of our method is of the order 3×10-3, compared with experimental data for the atoms mentioned above. The accuracy of our method is revealed by a comparison between our theoretical data and values resulting from Hartree-Fock methods.
Similar content being viewed by others
References
A. Popa, J. Phys. A: Math. Gen. 36, 7569 (2003)
A. Popa, J. Chem. Phys. 122, 244701 (2005)
A. Popa, J. Phys. Soc. Jpn 67, 2645 (1998)
A. Popa, J. Phys. Soc. Jpn 68, 763 (1999)
A. Popa, J. Phys. Soc. Jpn 68, 2923 (1999)
A. Popa, Eur. Phys. J. D 49, 279 (2008)
A. Luis, Phys. Rev. A 67, 024102 (2003)
M.D. Poulsen, L.B. Madsen, Phys. Rev. A 72, 042501 (2005)
A. Messiah, Quantum Mechanics (North-Holland Publishing Company, Amsterdam, 1965), Vol. 1
H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, 1950)
J.C. Slater, Quantum Theory of Atomic Structure (McGraw Hill, New York, 1960), Vol. 1
L. Landau, E. Lifchitz, Mécanique Quantique (Éditions Mir, Moscou, 1967)
G. Arfken, Circular Cylindrical Coordinates, in Mathematical Methods for Physicists, 3rd edn. (Academic Press, Orlando FL, 1985), Sect. 2.5
L. Landau, E. Lifschitz, Mechanics (Pergamon Press, London, 1960)
CRC Handbook of Chemistry and Physics, edited by D.R. Lide (CRC Press, Boca Raton, Florida, 2003)
E. Clementi, C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974)
S. Huzinaga, M. Klobukowski, Chem. Phys. Lett. 120, 509 (1985)
C.F. Bunge et al., Phys. Rev. A 46, 3691 (1992)
T. Koga, H. Tatewaki, A.J. Thakkar, Phys. Rev. A 47, 4510 (1993)
E.V.R. de Castro, F.E. Jorge, An. Acad. Bras. Cienc. 73, 511 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Popa, A. Geometrical symmetry of atoms with applications to semiclassical calculation of energetic values. Eur. Phys. J. D 54, 575–583 (2009). https://doi.org/10.1140/epjd/e2009-00191-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjd/e2009-00191-8