Calculation of the Final State Wavefunctions

The wavefunctions for the initial bound state the resonant states j , and the neutral thresholds are generated in separate valence RCI calculations. The valence RCI calculation for the resonant state / yields energy positions of unperturbed resonant states, i.e., the E s in equation 1.3. Since virtual orbitals are used for the correlation configurations that capture not only the bound orbitals but also a portion of the continuum orbitals, it s important to avoid in cj) the correlation configurations that are equivalent to the continuum state. For example, in Ce [5], 4/ 5d 6s vp and 4/ 5d 6s vf were excluded from the basis set for resonant state 4/ 5d 6p. Otherwise, the variational optimization for 4/ 5d 6s 6p may collapse into the continuum 4/ 5d 6s ep (f) in which it lies. 

Since the wavefunctions for the resonant states and the neutral thresholds are prepared by separate RCI calculations, non-orthonormality (NON) must be taken into account when evaluating the Cl matrix V (equation 1.2). This is treated partially by using common DF radials and common first set of virtual orbitals for the two states. The second set of virtual orbitals, however, is optimized separately. Experience has shown the radial overlaps between the second virtual orbital and its corresponding orbitals of the same symmetry are very close to 1 (e.g., larger than 0.97). Orbitals of different symmetry are orthogonal. Usually two sets of virtual orbitals suffice. 

In addition to extensive shell scripts and codes that prepare data and manage runs, which have greatly improved our efficiency in carrying out computations, following are two important modifications made to continuum codes that have facilitated the accuracy of our calculations. 

The continuum wave function solver of Perger et al. [58,59] allows only one basis function from the DF configuration for the core when generating the radial function for the 

Some of the above approximations can be removed from future calculations. The full treatment of NON can be realized numerically. Actually, in our code [63] for differential oscillator strength, NON is fully treated following the formalism of King et al. [64,65]. Also, the orthogonality between (j and V oE can be checked and evaluated by modifying the current code [61]. 

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