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
A.V. Glushkov, A.A. Svinarenko, O.Yu. Khetselius,
V.V. Buyadzhi, T.A. Florko and A.N. Shakhman
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
R R
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
R
dn 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-2p 1/2
transition for the Li-like ions of argon, iron, krypton and uranium are presented and
compared with available theoretical and experimental results.
A.V. Glushkov (&) Á A.A. Svinarenko Á O.Yu. Khetselius Á V.V. Buyadzhi Á T.A. Florko Á
A.N. Shakhman
Odessa State Environmental University (OSENU), L’vovskaya Str, 15,
Odessa 65016, Ukraine
e-mail: glushkovav@gmail.com; dirac13@mail.ru
© Springer International Publishing Switzerland 2015
M.A.C. Nascimento et al. (eds.), Frontiers in Quantum Methods and Applications
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 29,
DOI 10.1007/978-3-319-14397-2_12
197
Advanced Approach to the Construction
of the Green Function of the Dirac
Equation with Complex Energy
and Mean-Field Nuclear Potential
A.V. Glushkov, A.A. Svinarenko, O.Yu. Khetselius,
V.V. Buyadzhi, T.A. Florko and A.N. Shakhman
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
R R
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
R
dn 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-2p 1/2
transition for the Li-like ions of argon, iron, krypton and uranium are presented and
compared with available theoretical and experimental results.
A.V. Glushkov (&) Á A.A. Svinarenko Á O.Yu. Khetselius Á V.V. Buyadzhi Á T.A. Florko Á
A.N. Shakhman
Odessa State Environmental University (OSENU), L’vovskaya Str, 15,
Odessa 65016, Ukraine
e-mail: glushkovav@gmail.com; dirac13@mail.ru
© Springer International Publishing Switzerland 2015
M.A.C. Nascimento et al. (eds.), Frontiers in Quantum Methods and Applications
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 29,
DOI 10.1007/978-3-319-14397-2_12
197