As described in IP.I (Hummer et al. 1993), the basic methods are derived from atomic collision theory and the coupled channel approximation or the close coupling (CC) method. The computational method is based on the powerful R-matrix formalism that enables efficient, accurate, and large-scale calculations of compound (bound and continuum), state wavefunctions of the (e + ion) system at all positive or negative energies (in accordance with the terminology of collision theory the "ion'' core is often referred to as the "target ion''). At positive energies the "channels'', characterized by the spin and angular quantum numbers of the (e + ion) system, describe the scattering process with the free electron interacting with the target ion. However, at negative total energies of the (e + ion) system, the solutions of the close coupling equations occur at discrete eigenvalues of the (e + ion) Hamiltonian that correspond to pure bound states (all scattering channels are then "closed''). The positive and negative energy solutions yield many atomic parameters of practical interest: electron impact excitation cross sections, photoionization and recombination cross sections, and radiative transition probabilities.
The non-relativistic (N + 1)-electron Hamiltonian for the N-electron target
ion and a free electron is
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(1) |
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(2) |
In the coupled channel or close coupling (CC) approximation
the wavefunction expansion,
, for a total spin and angular symmetry
or
,
of the (N+1) electron system
is represented in terms of the target ion states as:
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(3) |
where is the target ion wave function in a specific state
or level
, and
is the wave function for
the (N+1)th electron in a channel labeled as
; ki2 is the
incident kinetic energy. In the second sum the
's are correlation
wavefunctions of the (N+1) electron system that (a) compensate for the
orthogonality conditions between the continuum and the bound orbitals,
and (b) represent additional short-range correlation that is often of
crucial importance in scattering and radiative CC calculations for each
. In the relativistic BPRM calculations the set of
are recoupled to obtain (e + ion) states with total
, followed by
diagonalisation of the (N + 1)-electron Hamiltonian, i.e.
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(4) |
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(5) |
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(6) |
Using the energy difference, Eji, between the initial and final states, the oscillator strength, fij, for the transition can be obtained from S as
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(7) |
and the Einstein's A-coefficient, Aji, as
![]() |
(8) |
where is the fine structure constant, and gi, gj are the
statistical weight factors of the initial and final states,
respectively. In terms of c.g.s. unit of time,
![]() |
(9) |
where s is the atomic unit of time.
The BP Hamiltonian in the present work (Eq. 2) does not include the full Breit-interaction in that the two-body spin-spin and spin-other-orbit terms are not included. A discussion of these terms is given by Mendoza et al. in a recent IP paper (IP.XXXIII, 1998). Their study on the intercombination transitions in C-like ions shows that the effect of the two-body Breit terms, relative to the one-body operators, decreases with Z such that for Z = 26 the computed A-values with and without the two-body Breit terms differ by less than 0.5%. However, the differences towards the neutral end of the C-sequence is up to about 20%.
For the few-electron systems it is possible to calculate transition energies and probabilities including electron-correlation, relativistic, and QED effects. Several investigators have done very elaborate ab intio calculations. Highly accurate calculations for transition probabilities have been carried out by Drake (1979, 1988), Lin et al. (1977a,b) for helium-like ions and Yan et al. (1998), and Johnson et al. (1996) for lithium-like ions. In addition to these latest works, relativistic calculations have been done using perturbation theory and the 1/Z expansion method by Vainshtein & Safronova (1985) for He-like and Li-like ions. The calculations by Drake (1988) on the He-sequence include quantum electrodynamic (QED) corrections (screened nuclear charge for the Bethe logarithm) in a more accurate manner than the earlier works. In an elaborate relativistic calculation, using many-body perturbation theory up to third order, Johnson et al. (1996) obtain transition probabilties for a few transitions in Fe XXIV. In the work on Li-like ions, Yan et al. (1998) employ a fully correlated Hylleraas-type variational method and relativistic corrections derived from the relativistic many-body perturbation theory by Johnson et al. (1996). Yan et al. (1998) investigate both the relativistic and the finite nuclear mass corrections and show those to be important for high accuracy. The slight differences between the calculations by Yan et al. (1998) and earlier work by Johnson et al. (1996) are attributable essentially to the nuclear term not included in the latter calculations. In addition, multi-configuration Dirac-Fock (MCDF) calculations, that include the relativistic one-body operators exactly but not the Breit terms, have been carried out by Cheng et al. (1979). Other previous works are also discussed later.
We compare with available data from previous calculations. The agreement between the present values and these previous calculations shows that the relativistic and QED terms omitted from the BP Hamiltonian (Eq. 2) do not affect the transition probabilities of the highly charged ions considered herein by more than a few percent - the accuracy sought in the present large-scale calculations intended for applications in laboratory and astrophysical plasmas.
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