equation in a case of the Coulomb potential. As it was noted in Refs. [23, 50–54],
the known expansions to the Taylor expansion with numerical calculating in the
separated blocks are usually used, but there are two significant disadvantages. First
of all, calculating the Whittaker function in the separated block increases the
dimension of the calculation procedure. Secondly, the corresponding Taylor
expansion has a bad convergence for large values of ξ [that is very important for
problem (1)] and contains the significant compensations of the separated terms
(look more detailed explanation in Refs. [23, 50–54]).
Here we present an advanced approach to construction of the electron Green’s
function for the Dirac equation with a non-singular central nuclear potential and
complex energy, which generalizes the known scheme [23, 50–54] by Ivanov et al.
Here firstly the Fermi-model and relativistic mean-field nuclear potentials are used.
Naturally we will also represent the radial Green’s function as a combination of two
fundamental solutions of the Dirac equation. The approach proposed has two new
blocks: (i) a procedure of generating the relativistic electron functions Ψ with
performance of the gauge invariance principle; (ii) the Fermi model for nuclear
finite size potential.
In order to reach the gauge invariance principle performance we use the QED
approach, which has been earlier developed in Refs. [53, 54]. The detailed consideration of the exchange-correlation (the screening and polarization effects, selfenergy correction to mass operator etc.) diagrams is performed within the QED PT
formalism and. In the fourth order of the QED PT [53, 54] there are diagrams,
whose contribution into an imaginary part of the radiation width ImdE for the multielectron system accounts for multi-body correlation effects. A minimization of the
functional ImdE leads to integral-differential density functional Kohn-Sham-like
equations. Further check for the gauge principle performance is realized by means
of the Ward identities. In the numerical procedure we use the effective algorithm,
which has been earlier approbated by us in many calculations of different characteristics of the atomic and molecular systems [45–68]. Within this procedure a
definition of the Dirac equation fundamental solutions is reduced to solving the
single system of the differential equations. This method is usually called as the
method of differential equations by Ivanov-Ivanova (c.f. [45–47]). The general
system in our version includes also the differential equations for the Fermi-model
and relativistic mean-field nuclear potentials and equations for calculating the
integrals of the
R R
dr 1 dr 2 type in formula for definition of the self-energy shift to
atomic levels energies (look below).
Let us remember that following to the Mohr papers [34–36], within the covariant
regularization of the Feynman S-matrix results the self-energy shift to the level
energy can be written as follows (c.f. [50–56] too):
E ¼ E L þ E H K
ð Þ À
1
~ a p Z
3
2
ln K ~ a
2
þ
3
8
b
h i; K ! 1:
ð2Þ
200
A.V. Glushkov et al.
the known expansions to the Taylor expansion with numerical calculating in the
separated blocks are usually used, but there are two significant disadvantages. First
of all, calculating the Whittaker function in the separated block increases the
dimension of the calculation procedure. Secondly, the corresponding Taylor
expansion has a bad convergence for large values of ξ [that is very important for
problem (1)] and contains the significant compensations of the separated terms
(look more detailed explanation in Refs. [23, 50–54]).
Here we present an advanced approach to construction of the electron Green’s
function for the Dirac equation with a non-singular central nuclear potential and
complex energy, which generalizes the known scheme [23, 50–54] by Ivanov et al.
Here firstly the Fermi-model and relativistic mean-field nuclear potentials are used.
Naturally we will also represent the radial Green’s function as a combination of two
fundamental solutions of the Dirac equation. The approach proposed has two new
blocks: (i) a procedure of generating the relativistic electron functions Ψ with
performance of the gauge invariance principle; (ii) the Fermi model for nuclear
finite size potential.
In order to reach the gauge invariance principle performance we use the QED
approach, which has been earlier developed in Refs. [53, 54]. The detailed consideration of the exchange-correlation (the screening and polarization effects, selfenergy correction to mass operator etc.) diagrams is performed within the QED PT
formalism and. In the fourth order of the QED PT [53, 54] there are diagrams,
whose contribution into an imaginary part of the radiation width ImdE for the multielectron system accounts for multi-body correlation effects. A minimization of the
functional ImdE leads to integral-differential density functional Kohn-Sham-like
equations. Further check for the gauge principle performance is realized by means
of the Ward identities. In the numerical procedure we use the effective algorithm,
which has been earlier approbated by us in many calculations of different characteristics of the atomic and molecular systems [45–68]. Within this procedure a
definition of the Dirac equation fundamental solutions is reduced to solving the
single system of the differential equations. This method is usually called as the
method of differential equations by Ivanov-Ivanova (c.f. [45–47]). The general
system in our version includes also the differential equations for the Fermi-model
and relativistic mean-field nuclear potentials and equations for calculating the
integrals of the
R R
dr 1 dr 2 type in formula for definition of the self-energy shift to
atomic levels energies (look below).
Let us remember that following to the Mohr papers [34–36], within the covariant
regularization of the Feynman S-matrix results the self-energy shift to the level
energy can be written as follows (c.f. [50–56] too):
E ¼ E L þ E H K
ð Þ À
1
~ a p Z
3
2
ln K ~ a
2
þ
3
8
b
h i; K ! 1:
ð2Þ
200
A.V. Glushkov et al.