Test run description

The simplest way to run the code HOSPHE (v1.00) consists in providing the namelist input data in the following form:

./hosphe << end > pb208.n50.out
&input AN=126,AZ=82,noscmax=50,
icm=1,icoudir=-1,icouex=-1,keta_J=1,intera="SLy4",ordermax=2,
epsilon=1.e-7,itermax=1000,
Flag_read_ini_dm = .false.,restart = 0,
alpha=0.65,ngrid=-80,boscil=-2,hbarom=-1/
end

The script above is provided in the distribution file of the code HOSPHE (v1.00). By executing the script, one obtains the output file ``pb208.n50.out'', which is also provided in the distribution file. The main section of this file, which gives the total energies in $^{208}$Pb calculated for the maximum HO shell included in the basis of $N_0=50$ and SLy4 Skyrme functional [14], reads

      Kin.prot    Kin. neut.     Tot. kin.
   1337.059947   2529.116266   3866.176214

    T=0 Skyrme    T=1 Skyrme   Tot. Skyrme
  -6405.081099    106.598348  -6298.482751

        Energy     HF Energy    Rearr. ene
  -1635.692396  -1635.692396  -1221.821085

     Cou. tot.     Cou. dir.     Cou. exc.
    796.614142    827.882912    -31.268770

Figure 1: Total energy of $^{208}$Pb as a function of the maximum HO shell included in the basis $N_0$. Upper and lower panels show in the logarithmic scale differences $E(N_0)-E_0$ for two different values of $E_0$. Large full dots and small empty circles give results calculated by using the codes HOSPHE (v1.00) and HFODD (v2.40h) [15], respectively. Up to $N_0=36$, where the HFODD calculations could have been performed, one sees a perfect agreement between the results given by the two codes. Solid lines give results of the exponential-decay fits.

In Figs. 1 and 2, we present results of similar calculations performed for $^{208}$Pb and the HO bases of $N_0=10$-70. Fig. 1 shows the convergence of the total energy in function of $N_0$. It turns out that the energy converges exponentially to the limiting value of $E_0$, namely,

$$E(N_0)=E_0+E_1 \exp(-a N_0) .$$ (118)

However, as shown in the two panels of Fig. 1, two different values of $E_0$ and $a$ are obtained for the regions of $N_0$ below and above $N_0=38$. A rather rapid convergence ($a=0.172(3)$) to $E_0=-1635.719$, which is seen below $N_0=38$, is followed by a slower convergence ($a=0.1068(8)$) to $E_0=-1635.69405$. Since the least-square fit of the limiting values $E_0$ is ill-conditioned, no error estimates can be obtained for them.

By considering convergence patterns in a few more cases for different options and nuclei, we found that the trend with two different slopes is not a general feature. In the few cases we looked at, we found that it is only above 40-50 shells that the rate seems to stabilize to an exponential convergence. These results show that, in general, its not possible to find the extrapolated limit of energy just by calculating only a few points of the curve for some small numbers of shells.

Figure 2: The HOSPHE (v1.00) CPU times required for calculations performed for $^{208}$Pb and the standard Skyrme functional SLy4, shown as functions of the maximum HO shell included in the basis $N_0$. Circles and squares show results obtained by using the codes HOSPHE (v1.00) and HFODD (v2.40h) [15], respectively. The doubly logarithmic scale in the Figure, shows that these times scale as different powers of $N_0$, as indicated in the figure.

Fig. 2 shows the dependence of CPU times on $N_0$, obtained on the AMD Opteron Processor 2352 running at 2100MHz clock speed. First, one can see that the spherical-basis code HOSPHE (v1.00) is, of course, orders of magnitude faster that the 3D code HFODD (v2.40h) [15]. For $N_0=36$, the former needs only 20sec of CPU time while the latter needs 250 000sec. Second, for both codes, the dependencies on $N_0$ are clearly given by power lows indicated in the figure. Strangely enough, these power lows are different for calculations performed below and above $N_0=20$. At the moment, no explanation for such a timing pattern could be found.