Very useful review has been presented [9] for the Green’s function expressions of
field equations. Namely, the explicit form of the Green’s functions of the KleinGordon-Fock and Dirac equations is derived, and then the decay rate of the solution
to the linear equations is estimated.
An overview of the two-time Green’s function method is presented in [10]. This
[10] method allows one to calculate level shifts in two-electron highly-charged ions
by including in principle all QED effects, for any set of states (degenerate, quasidegenerate or isolated) and as example, an evaluation of the contribution of the
screened self-energy to a finite-sized effective Hamiltonian, that yields the energy
levels through diagonalization, is presented.
It is obvious that a development of the effective numerical algorithms for calculating the electron Green’s function for the Dirac equation with arbitrary potential
and complex energy is of great importance for modern relativistic quantum
chemistry [1–31]. Our interest to this problem is connected with running computing
the self-energy corrections to atomic levels in the heavy atomic systems and
multicharged ions (c.f. [23, 48–68]). Besides, we believe that the similar problem
will be arisen in the quantum chemistry of the heavy and super heavy (ZX, Z > 100)
molecules in some time [23]. In this task all master formulas are conserved, except
of the symmetry of the task. It is obvious that the latter in the molecular case differ
from the atomic symmetry.
From the viewpoint of the modern quantum calculations of the heavy atoms and
multicharged ions it should be mentioned that calculation of the self-energy corrections to the atomic levels energies (radiation widths) have been carried out for
low states of the hydrogen-like ions and nuclear charge Z < 110 (c.f. [32]). The
nuclear finite size correction is usually implemented to the calculation scheme (for
example, the relativistic Hartree-Fock or Dirac-Fock methods) by means of using
several nuclear models (model of the homogeneous charged sphere, the Gauss
model and the Fermi-model) [24–54]. The screening of a nucleus by atomic
(molecular) electrons is usually taken into account for within the Dirac-Fock
(Dirac-Kohn-Sham) approximations. By the way, studying the excited heavy and
super heavy ðZ ~
[ 173Þ quantum systems with an accurate modelling the nuclear
potential (in a whole, nuclear effects in quantum calculations) and accounting for
the screening of a nucleus (nuclei) by electrons and radiative effects of the electron
shell polarization and probably the Dirac equation non-linear terms remain by very
actual problem of the modern theory of multi-electron systems. One could mention
here the known difficulties of the modern calculation procedures of quantum
chemistry, in particular, in calculating the radiation (self-energy) correction to
levels energies of the heavy atomic and molecular systems.
In order to treat correctly a problem of calculating the self-energy correction to
atomic (molecular) levels energies in the relativistic quantum chemistry it is
important to have an effective algorithm of calculating the electron Green’s function
for the Dirac equation. As it is well known, the radial electron Green’s function is
presented as a combination of the fundamental solutions of the Dirac equation. One
can mention the Whittaker functions as the fundamental solutions of the Dirac
Relativistic Quantum Chemistry …
199
field equations. Namely, the explicit form of the Green’s functions of the KleinGordon-Fock and Dirac equations is derived, and then the decay rate of the solution
to the linear equations is estimated.
An overview of the two-time Green’s function method is presented in [10]. This
[10] method allows one to calculate level shifts in two-electron highly-charged ions
by including in principle all QED effects, for any set of states (degenerate, quasidegenerate or isolated) and as example, an evaluation of the contribution of the
screened self-energy to a finite-sized effective Hamiltonian, that yields the energy
levels through diagonalization, is presented.
It is obvious that a development of the effective numerical algorithms for calculating the electron Green’s function for the Dirac equation with arbitrary potential
and complex energy is of great importance for modern relativistic quantum
chemistry [1–31]. Our interest to this problem is connected with running computing
the self-energy corrections to atomic levels in the heavy atomic systems and
multicharged ions (c.f. [23, 48–68]). Besides, we believe that the similar problem
will be arisen in the quantum chemistry of the heavy and super heavy (ZX, Z > 100)
molecules in some time [23]. In this task all master formulas are conserved, except
of the symmetry of the task. It is obvious that the latter in the molecular case differ
from the atomic symmetry.
From the viewpoint of the modern quantum calculations of the heavy atoms and
multicharged ions it should be mentioned that calculation of the self-energy corrections to the atomic levels energies (radiation widths) have been carried out for
low states of the hydrogen-like ions and nuclear charge Z < 110 (c.f. [32]). The
nuclear finite size correction is usually implemented to the calculation scheme (for
example, the relativistic Hartree-Fock or Dirac-Fock methods) by means of using
several nuclear models (model of the homogeneous charged sphere, the Gauss
model and the Fermi-model) [24–54]. The screening of a nucleus by atomic
(molecular) electrons is usually taken into account for within the Dirac-Fock
(Dirac-Kohn-Sham) approximations. By the way, studying the excited heavy and
super heavy ðZ ~
[ 173Þ quantum systems with an accurate modelling the nuclear
potential (in a whole, nuclear effects in quantum calculations) and accounting for
the screening of a nucleus (nuclei) by electrons and radiative effects of the electron
shell polarization and probably the Dirac equation non-linear terms remain by very
actual problem of the modern theory of multi-electron systems. One could mention
here the known difficulties of the modern calculation procedures of quantum
chemistry, in particular, in calculating the radiation (self-energy) correction to
levels energies of the heavy atomic and molecular systems.
In order to treat correctly a problem of calculating the self-energy correction to
atomic (molecular) levels energies in the relativistic quantum chemistry it is
important to have an effective algorithm of calculating the electron Green’s function
for the Dirac equation. As it is well known, the radial electron Green’s function is
presented as a combination of the fundamental solutions of the Dirac equation. One
can mention the Whittaker functions as the fundamental solutions of the Dirac
Relativistic Quantum Chemistry …
199