6 General Scheme of Calculation for a Three-Electron
System
Taking into account for the further application of the algorithm in calculation of the
radiative corrections to the levels energies of the heavy Li-like multicharged ions let
us describe briefly the calculation procedure for three-electron systems. More
detailed description of all method is given in Refs. [59, 60].
One may consider the Dirac-Fock type equations for a three-electron system
1s
2 nlj. Formally they fall into one-electron Dirac equations for the orbitals 1s and
nlj with the potential:
VðrÞ ¼ 2V rj1s
ð
ÞþV rjnlj
ð
ÞþV ex ðrÞ þ V rjR
ð Þ;
ð46Þ
where VðrjRÞ includes the electrical and the polarization potentials of a nucleus; the
components of the Hartree potential:
V rji
ð Þ ¼
1
Z
Z
d~ r
0
q rji
ð Þ= ~ r À~ r
0
j
j
ð47Þ
Here q rji
ð Þ is a distribution of the electron density in the state |i>, V ex is the
exchange inter-electron interaction.
The main exchange effect will be taken into account if in the equation for the
1s orbital we assume
V r
ð Þ ¼ V rj1s
ð
ÞþV rjnlj
ð
Þ
ð48Þ
and in the equation for the nlj orbital
V r
ð Þ ¼ 2 V rj1s
ð
Þ
ð49Þ
The rest of the exchange and correlation effects are taken into account for in the
first two orders of the PT by the total inter-electron interaction [45–50, 59, 60]. The
used expression for q rj1s
ð
Þ coincides with the precise one for a one-electron relativistic atom with a point nucleus. The finiteness of a nucleus and a presence of the
second 1s electron are included effectively into the energy E 1s .
Actually, for determination of the properties of the outer nlj electron one iteration is sufficient. Refinement resulting from second iteration (by evaluations) does
not exceed correlation corrections of the higher orders omitted in the present calculation. The relativistic potential of core (the “screening” potential)
2V
1
ð Þ rj1s
ð
Þ ¼ V scr
ð50Þ
has correct asymptotic at zero and in the infinity; at a ! 0 it changes to an
appropriate potential constructed on the basis of the hydrogen-like functions. The
other details can be found in the papers [59, 60].
Relativistic Quantum Chemistry …
211
System
Taking into account for the further application of the algorithm in calculation of the
radiative corrections to the levels energies of the heavy Li-like multicharged ions let
us describe briefly the calculation procedure for three-electron systems. More
detailed description of all method is given in Refs. [59, 60].
One may consider the Dirac-Fock type equations for a three-electron system
1s
2 nlj. Formally they fall into one-electron Dirac equations for the orbitals 1s and
nlj with the potential:
VðrÞ ¼ 2V rj1s
ð
ÞþV rjnlj
ð
ÞþV ex ðrÞ þ V rjR
ð Þ;
ð46Þ
where VðrjRÞ includes the electrical and the polarization potentials of a nucleus; the
components of the Hartree potential:
V rji
ð Þ ¼
1
Z
Z
d~ r
0
q rji
ð Þ= ~ r À~ r
0
j
j
ð47Þ
Here q rji
ð Þ is a distribution of the electron density in the state |i>, V ex is the
exchange inter-electron interaction.
The main exchange effect will be taken into account if in the equation for the
1s orbital we assume
V r
ð Þ ¼ V rj1s
ð
ÞþV rjnlj
ð
Þ
ð48Þ
and in the equation for the nlj orbital
V r
ð Þ ¼ 2 V rj1s
ð
Þ
ð49Þ
The rest of the exchange and correlation effects are taken into account for in the
first two orders of the PT by the total inter-electron interaction [45–50, 59, 60]. The
used expression for q rj1s
ð
Þ coincides with the precise one for a one-electron relativistic atom with a point nucleus. The finiteness of a nucleus and a presence of the
second 1s electron are included effectively into the energy E 1s .
Actually, for determination of the properties of the outer nlj electron one iteration is sufficient. Refinement resulting from second iteration (by evaluations) does
not exceed correlation corrections of the higher orders omitted in the present calculation. The relativistic potential of core (the “screening” potential)
2V
1
ð Þ rj1s
ð
Þ ¼ V scr
ð50Þ
has correct asymptotic at zero and in the infinity; at a ! 0 it changes to an
appropriate potential constructed on the basis of the hydrogen-like functions. The
other details can be found in the papers [59, 60].
Relativistic Quantum Chemistry …
211