Advanced Relativistic Energy Approach in Spectroscopy …
7
[118–120], however, the M matrix elements in the relativistic representation are
complex; the corresponding imaginary parts determine values of radiation widths.
The total energy shift of the state is usually presented in the form:
δ E = Re δ E + i Imδ E
Im δ E = −
2
( 1 )
where is interpreted as the level width including the radiation and autoionization
widths simultaneously. In this approach, the whole calculation of the energies and
decay probabilities of a non-degenerate excited state is reduced to the calculation
and diagonalization of the complex matrix M. In the papers of different authors, the
Reδ E calculation procedure has been generalized for the case of nearly degenerate
states, whose levels form a more or less compact group. One of these variants has
been previously [114, 115] introduced: for a system with a dense energy spectrum,
a group of nearly degenerate states is extracted and their matrix M is calculated and
diagonalized. If the states are well separated in energy, the matrix M reduces to one
term, equal to δ E. The non-relativistic secular matrix elements are expanded in a PT
series for the interelectron interaction. The complex secular matrix M is represented
in the form [114, 115]:
M = M
(0)
+ M
(1)
+ M
(2)
+ M
(3)
.
(2)
Here M
(0) is the energy contribution of the vacuum diagrams of all order of
PT, and M
(1) , M
(2) ,M
(3) those of the one-, two- and three- quasiparticle diagrams
respectively. M
(0) is a real matrix, proportional to the unit matrix. It determines only
the general level shift. It is usually assumed M
(0)
= 0. The diagonal matrix M
(1)
can be presented as a sum of the independent one-quasiparticle contributions. For
simple systems (such as alkali atoms and ions) the one-quasiparticle energies can
be taken from the experiment. Substituting these quantities into (5) one could have
summarized all the contributions of the one-quasiparticle diagrams of all orders of
the formally exact relativistic PT. However, the necessary experimental quantities
are not often available.
The first two order corrections to ReM
(2) have been analyzed previously [113–
116] using the Feynman diagrams technique. The contributions of the first-order diagrams have been completely calculated. In the second order, there are two kinds of
diagrams: polarization and ladder ones. The polarization diagrams take into account
the quasiparticle interaction through the polarizable core, and the ladder diagrams
account for the immediate quasiparticle interaction. An effective forms for the twoparticle polarizable operator have been proposed in Refs. [115, 180–182]. The technique of determination of the matrix elements of these polarization potentials has
been presented in [114–135].
As usual, a multielectron atom is described by the Dirac relativistic Hamiltonian
(the atomic units are used):
7
[118–120], however, the M matrix elements in the relativistic representation are
complex; the corresponding imaginary parts determine values of radiation widths.
The total energy shift of the state is usually presented in the form:
δ E = Re δ E + i Imδ E
Im δ E = −
2
( 1 )
where is interpreted as the level width including the radiation and autoionization
widths simultaneously. In this approach, the whole calculation of the energies and
decay probabilities of a non-degenerate excited state is reduced to the calculation
and diagonalization of the complex matrix M. In the papers of different authors, the
Reδ E calculation procedure has been generalized for the case of nearly degenerate
states, whose levels form a more or less compact group. One of these variants has
been previously [114, 115] introduced: for a system with a dense energy spectrum,
a group of nearly degenerate states is extracted and their matrix M is calculated and
diagonalized. If the states are well separated in energy, the matrix M reduces to one
term, equal to δ E. The non-relativistic secular matrix elements are expanded in a PT
series for the interelectron interaction. The complex secular matrix M is represented
in the form [114, 115]:
M = M
(0)
+ M
(1)
+ M
(2)
+ M
(3)
.
(2)
Here M
(0) is the energy contribution of the vacuum diagrams of all order of
PT, and M
(1) , M
(2) ,M
(3) those of the one-, two- and three- quasiparticle diagrams
respectively. M
(0) is a real matrix, proportional to the unit matrix. It determines only
the general level shift. It is usually assumed M
(0)
= 0. The diagonal matrix M
(1)
can be presented as a sum of the independent one-quasiparticle contributions. For
simple systems (such as alkali atoms and ions) the one-quasiparticle energies can
be taken from the experiment. Substituting these quantities into (5) one could have
summarized all the contributions of the one-quasiparticle diagrams of all orders of
the formally exact relativistic PT. However, the necessary experimental quantities
are not often available.
The first two order corrections to ReM
(2) have been analyzed previously [113–
116] using the Feynman diagrams technique. The contributions of the first-order diagrams have been completely calculated. In the second order, there are two kinds of
diagrams: polarization and ladder ones. The polarization diagrams take into account
the quasiparticle interaction through the polarizable core, and the ladder diagrams
account for the immediate quasiparticle interaction. An effective forms for the twoparticle polarizable operator have been proposed in Refs. [115, 180–182]. The technique of determination of the matrix elements of these polarization potentials has
been presented in [114–135].
As usual, a multielectron atom is described by the Dirac relativistic Hamiltonian
(the atomic units are used):