2 Theory

In the coupled channel or close coupling (CC) approximation an ion is described in terms of an (e + ion) complex that comprises of a "target'' or core ion, with N bound electrons, and a "free'' (N+1)-th electron that may be either bound or continuum. For a total spin and angular symmetry $SL\pi$ or $J\pi$, of the (N+1)-electron system, the total wavefunction, $\Psi_{\rm E}$, is represented in terms of the target ion states or levels as:

$$% \Psi_{\rm E}({\rm e+ion}) = A \sum_{i} \chi_{i}({\rm ion})\theta_{i} + \sum_{j} c_{j} \Phi_{j},$$ (1)

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 $S_iL_i(J_i)\pi_i \ k_{i}^{2}\ell_i(SL\pi) \ [J\pi]$; k_i² 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 $SL\pi$.

Details of the theory of Breit-Pauli R-matrix method in the close coupling approximation is described in earlier work (Hummer et al. 1993). The method yields the solutions of the relativistic CC equations using the Breit-Pauli Hamiltonian for the (N+1)-electron system to obtain the total wavefunctions $\Psi_{\rm E}({\rm e+ion})$. The Breit-Pauli Hamiltonian is

$$% H_{N+1}^{\rm BP}=H_{N+1}+H_{N+1}^{\rm mass} + H_{N+1}^{\rm Dar} + H_{N+1}^{\rm so},$$ (2)

where H_N+1 is the nonrelativistic Hamiltonian,

$$% H_{N+1} = \sum_{i=1}\sp{N+1}\left\{-\nabla_i\sp 2 - \frac{2Z}{r_i} + \sum_{j>i}\sp{N+1} \frac{2}{r_{ij}}\right\},$$ (3)

and the relativistic effects are included through the additional terms which are the one-body mass correction term, the Darwin term and the spin-orbit term respectively. The other relatively weaker Breit-interaction,

$$% H^{\rm B} = \sum_{i>j}[g_{ij}({\rm so}+{\rm so}')+g_{ij}({\rm ss}')],$$ (4)

representing the two-body spin-spin and the spin-other-orbit interactions is not included.

The eigenvalues of the (N + 1)-electron Hamiltonian are the energies of the states such that,

$$% \begin{array}{l} E = k^2 > 0 \longrightarrow {\rm continuu... ...2} < 0 \longrightarrow {\rm bound~state (\Psi_B)}, \end{array}$$ (5)

where $\nu = n - \mu(\ell)$ is the effective quantum number relative to the core level. $\mu(\ell)$ is the quantum defect.

The oscillator strength is proportional to the generalized line strength defined, in either length form or velocity form, by the equations

$$% S_{\rm L}= \left\vert\left\langle{\mit\Psi}_k \vert\sum_{j=1}^{N+1} r_j\vert {\mit\Psi}_i\right\rangle\right\vert^2$$ (6)

and

$$% S_{\rm V}=\omega^{-2} \left\vert\left\langle{\mit\Psi}_k ... ...l}{\partial r_j}\vert {\mit\Psi}_i\right\rangle\right\vert^2.$$ (7)

In these equations $\omega$ is the incident photon energy in Rydberg units, and $\mit\Psi_i$ and $\mit\Psi_k$ are the wave functions representing the initial and final states respectively.

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

$$% f_{ik} = {E_{ki}\over {3g_i}}S,$$ (8)

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

$$% A_{ki}({\rm a.u.}) = {1\over 2}\alpha^3{g_i\over g_k}E_{ki}^2f_{ik},$$ (9)

where $\alpha$ is the fine structure constant, and g_i=2J_i+1, g_k=2J_k+1 are the statistical weight factors of the initial and final states, respectively. In terms of c.g.s. unit of time,

$$% A_{ki}({\rm s}^{-1}) = {A_{ki}({\rm a.u.})\over \tau_0},$$ (10)

where $\tau_0 = 2.419^{-17}$ s is the atomic unit of time. The lifetime of a level can be obtained from the A-values of the level as,

$$% \tau_k = {1\over A_k},$$ (11)

where A_k is the total radiative transition probability for the level k, i.e.,

$$% A_k = {\sum_i A_{ki}}.$$ (12)