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2 Theory

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
\begin{displaymath}
H_{N+1} = \sum_{i=1}\sp{N+1}\left\{-\nabla_i\sp 2 - \frac{2Z}{r_i}
 + \sum_{j\gt i}\sp{N+1} \frac{2}{r_{ij}}\right\} . \end{displaymath} (1)
Relativistic effects are incorporated into the R-matrix formalism in the Breit-Pauli approximation with the Hamiltonian
\begin{displaymath}
H_{N+1}^{\rm BP}=H_{N+1}+H_{N+1}^{\rm mass} + H_{N+1}^{\rm Dar}
+ H_{N+1}^{\rm so},\end{displaymath} (2)
where HN+1 is the non-relativistic Hamiltonian defined by Eq. (1), together with the one-body mass correction term, the Darwin term and the spin-orbit term resulting from the reduction of the Dirac equation to Pauli form. The mass-correction and Darwin terms do not break the LS symmetry, and they can therefore be retained with significant effect in the computationally less intensive LS calculations. Spin-orbit interaction does, however, split the LS terms into fine-structure levels labeled by $J\pi$,where J is the total angular momentum.

In the coupled channel or close coupling (CC) approximation the wavefunction expansion, $\Psi(E)$, for a total spin and angular symmetry $SL\pi$ or $J\pi$, of the (N+1) electron system is represented in terms of the target ion states as:


\begin{displaymath}
\Psi(E) = A \sum_{i} \chi_{i}\theta_{i} + \sum_{j} c_{j} \Phi_{j},\end{displaymath} (3)

where $\chi_{i}$ is the target ion wave function in a specific state $S_iL_i\pi_i$ or level $J_i\pi_i$, and $\theta_{i}$ is the wave function for the (N+1)th electron in a channel labeled as $S_iL_i(J_i)\pi_i \ k_{i}^{2}\ell_i(SL\pi~or~ \ J\pi)$; ki2 is the incident kinetic energy. In the second sum the $\Phi_j$'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 $SL\pi$. In the relativistic BPRM calculations the set of $SL\pi$are recoupled to obtain (e + ion) states with total $J\pi$, followed by diagonalisation of the (N + 1)-electron Hamiltonian, i.e.
\begin{displaymath}
H^{\rm BP}_{N+1}\mit\Psi = E\mit\Psi.\end{displaymath} (4)
The oscillator strength (or photoionization cross section) is proportional to the generalised line strength (Seaton 1987) defined, in either length form or velocity form, by the equations  
 \begin{displaymath}
S_{\rm L}=
 \left\vert\left\langle{\mit\Psi}_{\rm f}
 \vert\sum_{j=1}^{N+1} r_j\vert
 {\mit\Psi}_i\right\rangle\right\vert^2\end{displaymath} (5)
and  
 \begin{displaymath}
S_{\rm V}=\omega^{-2}
 \left\vert\left\langle{\mit\Psi}_{\rm...
 ...al}{\partial r_j}\vert
 {\mit\Psi}_i\right\rangle\right\vert^2.\end{displaymath} (6)
In these equations $\omega$ is the incident photon energy in Rydberg units, and $\mit\Psi \rm _i$ and $\mit\Psi_{\rm f}$are the wave functions representing the initial and final states respectively. The boundary conditions satisfied by a bound state with negative energy correspond to exponentially decaying partial waves in all "closed'' channels, whilst those satisfied by a free or continuum state correspond to a plane wave in the direction of the ejected electron momentum $\underline{\hat{k}}$and ingoing waves in all open channels.

Using the energy difference, Eji, between the initial and final states, the oscillator strength, fij, for the transition can be obtained from S as


\begin{displaymath}
f_{ij} = {E_{ji}\over {3g_i}}S,\end{displaymath} (7)

and the Einstein's A-coefficient, Aji, as


\begin{displaymath}
A_{ji}({\rm a.u.}) = {1\over 2}\alpha^3{g_i\over g_j}E_{ji}^2f_{ij},\end{displaymath} (8)

where $\alpha$ 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,
\begin{displaymath}
A_{ji}({\rm s}^{-1}) = {A_{ji}\rm (a.u.)\over \tau_0},\end{displaymath} (9)

where $\tau_0 = 2.4191\ 10^{-17}$ 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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