2 Theory

2.1 Target wavefunctions

The nineteen lowest LS states of the target ion, i.e. 1s^{2 1}S, 1s2s ^1,3S, 1s2p ^1,3P, 1s3s ^1,3S, 1s3p ^1,3P, 1s3d ^1,3D, 1s4s ^1,3S, 1s4p ^1,3P, 1s4d ^1,3D and 1s4f ^1,3F, are included in the expansion of the wavefunction in the R-matrix calculation. Each state is represented by a configuration interaction wavefunction. Ten basis orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f are used. The radial part of the orbital is expressed in the Slater form

$$P_{nl}(r) = \sum_{j=1}^{K} C_{j}r^{p_{j}}\exp(-\xi_{j}r).$$ (1)

The coefficients C_j and the parameters p_j and $\xi_{j}$for all the orbitals are the same as those of Nakazaki et al. ([1993]). The excitation energies were obtained from Hibbert's ([1975]) CIV3 program and have been already presented by Nakazaki et al. ([1993]). The resulting energies are in fair agreement with the experimental values determined by Martin et al. ([1990]) and the oscillator strengths obtained agree with those of other calculations (Lin et al. [1977]; Drake [1979]). Table 1 gives a list of the configurations included in our calculation.

  

Table 1: Configurations mixed for each target symmetry

2.2 Collision calculations

The total wavefunction representing the electron-ion collision system is expanded in a sphere with radius $r_{\rm a}$ as follows:

$$\begin{eqnarray} \Psi_{k} = {\cal A} \sum_{ij}c_{ijk}\Phi_{i}(\rm 1,\cdot\cdot\c... ...onumber \\ +\sum_{j}d_{jk}\phi_{j}(\rm 1,\cdot\cdot\cdot,\it N+1),\end{eqnarray}$$ (2)

where $\cal A$ is the anti-symmetrization operator, $\Phi_i$ the channel function representing the target state coupled with the spin and angular functions for the scattered electron, u_ij the continuum basis orbitals for the scattered electron, and $\phi_{j}$ three-electron bound configurations formed from the ten bound orbitals. The coefficients c_ijk and d_jk are determined by diagonalizing the total Hamiltonian of the whole system with the basis set expansion defined by Eq. (2).

We use the computer code of Berrington et al. ([1995]) to calculate the R-matrix on the boundary of the sphere ($r_{\rm a}$= 4.9 a.u.). We include 39 continuum orbitals for each angular momentum l to ensure convergence in the energy range considered. The calculations are made for all the partial waves with total angular momentum L = 0 up to 14. In the outer region of the sphere, a set of close-coupling equation is solved using the asymptotic code STGFJJ of Seaton. STGFJJ computes reactance matrices K by matching the solutions in the inner and the outer regions at the boundary $r=r_{\rm a}$. The collision calculation is carried out over the energy range from the 1s2s ³S threshold to 480 Ryd. The LS coupled K-matrices are transformed to those for intermediate coupling using the code JAJOM (Saraph [1978]), which yields collision strengths between fine-structure levels. The term-coupling coefficients are constructed from the unitary matrices that diagonalize the non-relativistic Hamiltonian and the Breit-Pauli Hamiltonian including the one-body mass-correction, Darwin and spin-orbit terms. Contributions from the partial waves higher than $L\!=\!14$ are taken into account with an extrapolation.

2.3 Calculation of rate coefficients

Rate coefficients for a transition from level i to level f are defined as

$$C(i\rightarrow f) = { 8.629\ 10^{-6}\over{g_{i}\ \sqrt{T_{\r... ...\rm e}){\mit\Upsilon}(i \rightarrow f) ~{\rm cm}^3{\rm s}^{-1},$$ (3)

where g_i is the statistical weight of the initial level $i, \ \Delta E_{if}$ the respective transition energy, $T_{\rm e}$the electron temperature, and k Boltzmann's constant. Assuming a Maxwellian electron velocity distribution, the effective collision strength ${\mit\Upsilon}(i \rightarrow f)$ is defined as

$${\mit\Upsilon}(i \rightarrow f) = \int_0^{\infty}\,{\mit\Ome... ..._{f}/kT_{\rm e}) \rm d\it({\large \varepsilon}_{f}/kT_{\rm e}),$$ (4)

where ${\large \varepsilon}$_f is the energy of the electron after the collision, and ${\mit\Omega}(i,f)$ is the collision strength. It should be noted that $\mit\Upsilon$ is symmetric over the transition $i \rightarrow f$ and $f \rightarrow i$. To obtain the rate coefficient at high temperature, we fit the present collision strengths in the non-resonance region to either of the following two analytic functions  

$${\mit\Omega}(i,f) = A + B/X + C/X^2 + D/X^3 + E\ln X,$$ (5)

and  

$${\mit\Omega}(i,f) = F_0/X^2 + \sum_{j=1}^{4} F_j \exp(-j{\alpha} X),$$ (6)

where X = $k^2_i/(\Delta E_{if})$, where k²_i is the incident energy in Ryd and $\Delta E_{if}$ is given in Ryd. For a dipole-allowed transition, the first one (5) is used with D=0 and E=4g_if_if/$\Delta E_{if}$, where f_if is the oscillator strength for $i \rightarrow f$.The form (5) with E=0 is used for optically forbidden cases, unless the spin changes. For the spin-forbidden cases, the second one (6) is employed. The parameters A, B, C, D, F_j (j=0-4) and $\alpha$are determined by the least square method. Thus the effective collision strength is evaluated with the collision strengths calculated in the resonance region and those fitted above that.