Convergence of strength functions

The iterative Arnoldi method is meaningful for the calculation of strength functions only if the number of iterations needed for accurate results is significantly less than the full RPA dimension. To study how many Arnoldi iterations we need for good accuracy, we calculated electromagnetic isoscalar (IS) and isovector (IV) strength functions [16] for doubly magic nuclei. All calculations were performed by implementing the RPA iterative solutions within the computer program HOSPHE [19], which solves the self-consistent equations in the spherical harmonic-oscillator (HO) basis. We studied both the convergence of smoothed strength functions as a function of number of Arnoldi iterations and as a function of the number of HO shells.

We used the same definitions of the $0^+$, $1^-$, and $2^+$ transition operators as in Ref. [17] and the Skyrme functional SLy4 of Ref. [20]. The function we used to smooth the strength functions was also the same as in [16], with $R_{\rm box}=20$ fm. Because the HF ground state of $^{132}{\rm Sn}$ is spherically symmetric, our approximate RPA phonons have good angular momentum. We tested the use of large basis sets up to $40$ HO shells. The HF ground state energies were well converged for all double magic nuclei when $25$ HO shells were used. Below, we present the results only for $^{132}{\rm Sn}$.