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Relativistic Quantum Chemistry: An Advanced Approach to the Construction of the Green Function of the Dirac Equation with Complex Energy and Mean-Field Nuclear Potential

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Frontiers in Quantum Methods and Applications in Chemistry and Physics

Part of the book series: Progress in Theoretical Chemistry and Physics ((PTCP,volume 29))

Abstract

We present an advanced approach to construction of the electron Green’s function of the Dirac equation with a non-singular central nuclear potential and complex energy. The Fermi-model and relativistic mean-field (RMF) nuclear potentials are used. The radial Green’s function is represented as a combination of two fundamental solutions of the Dirac equation. The approach proposed includes a procedure of generating the relativistic electron functions with performance of the gauge invariance principle. In order to reach the gauge invariance principle performance we use earlier developed QED perturbation theory approach. In the fourth order of the QED perturbation theory (PT) there are diagrams, whose contribution into imaginary part of radiation width Im δE for the multi-electron system accounts for multi-body correlation effects. A minimization of the functional Im δE leads to integral-differential Dirac-Kohn-Sham-like density functional equations. Further check for the gauge principle performance is realized by means of the Ward identities. In the numerical procedure we use the effective Ivanova-Ivanov’s algorithm, within which a determination of the Dirac equation fundamental solutions is reduced to solving the single system of the differential equations. This system includes the differential equations for the nuclear potential and equations for calculating the integrals of \( {\iint {dr_{1} dr_{2} } } \) type in the Mohr’s formula for definition of the self-energy shift to atomic levels energies. Such a approach allows to compensate a main source of the errors, connected with numerical integration \( \int {d\xi } \) and summation on χ in the Mohr’s expressions during calculating the self-energy radiative correction to the atomic levels energies. As illustration, data on the nuclear finite size effect and self-energy Lamb shift contributions to the energy of 2s-2p1/2 transition for the Li-like ions of argon, iron, krypton and uranium are presented and compared with available theoretical and experimental results.

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Acknowledgments

One of authors (A.V.G.) would like to thank Professor Marco Nascimento for the invitation to present the invited lecture at the International workshop on Quantum Systems in Chemistry, Physics and Biology (QSCP-XVIII; Paraty, Rio de Janeiro, Brazil). The help in editing the manuscript by Ms. Radhika Sree V. is very much appreciated.

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Authors and Affiliations

  1. Odessa State Environmental University (OSENU), L’vovskaya Str, 15, Odessa, 65016, Ukraine

    A. V. Glushkov, A. A. Svinarenko, O. Yu. Khetselius, V. V. Buyadzhi, T. A. Florko & A. N. Shakhman

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  1. A. V. Glushkov
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  2. A. A. Svinarenko
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  3. O. Yu. Khetselius
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  4. V. V. Buyadzhi
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  5. T. A. Florko
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  6. A. N. Shakhman
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Correspondence to A. V. Glushkov .

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Editors and Affiliations

  1. Department of Physical Chemistry, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    M.A.C. Nascimento

  2. Laboratoire de Chimie Physique, CNRS & UPMC, Paris, France

    Jean Maruani

  3. Department of Quantum Chemistry, Uppsala University, Uppsala, Sweden

    Erkki J. Brändas

  4. Instituto de Fisica Fundamental, CSIC, Madrid, Spain

    Gerardo Delgado-Barrio

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Glushkov, A.V., Svinarenko, A.A., Khetselius, O.Y., Buyadzhi, V.V., Florko, T.A., Shakhman, A.N. (2015). Relativistic Quantum Chemistry: An Advanced Approach to the Construction of the Green Function of the Dirac Equation with Complex Energy and Mean-Field Nuclear Potential. In: Nascimento, M., Maruani, J., Brändas, E., Delgado-Barrio, G. (eds) Frontiers in Quantum Methods and Applications in Chemistry and Physics. Progress in Theoretical Chemistry and Physics, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-14397-2_12

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