Comparison of coordinate-space HFB calculations and configuration-space HFB+HO calculations

The main differences between the results of coordinate-space HFB calculations and those from configuration-space HFB+HO calculations can be seen in plots of the corresponding local density distributions. A typical example is shown in Fig. 1, where the densities and their logarithmic derivatives from coordinate-space HFB calculations (solid lines) are compared with those from a configurational HFB+HO calculation. Although the calculations were done for a specific spherical nucleus and Skyrme interaction, the features exhibited are generic. Note that the coordinate-space HFB calculations were carried out in a box of 30fm, so that the logarithmic derivative of the density obtained in that calculation shows a sudden drop near the box edge.

Invariably, the logarithmic derivative $\rho ^{\prime }/\rho$ associated with the coordinate-space HFB solution shows a well-defined minimum near some point $R_{{\rm min} }$ in the asymptotic region, after which it smoothly approaches a constant value $-k$, where

$$k=2\kappa =2\sqrt{2m(E_{{\rm min} }-\lambda' )/{ \hbar^2 }}$$ (19)

is associated with the HFB asymptotic behavior for the lowest quasiparticle state that has the corresponding quasiparticle energy $E_{{\rm min}}$ (see Eq. (17) and Ref. [6]). This property is clearly seen in the upper panel of Fig. 1. One can also see that the HFB+HO densities and logarithmic derivatives are in almost perfect agreement with the coordinate-space results up to (or around) the distance $R_{{\rm min} }$. We conclude, therefore, that the HFB+HO densities are numerically reliable up to that point.

Moreover, the HFB value of the density decay constant $k$=2$\kappa$, when calculated from Eq. (19), is also correctly reproduced by the HFB+HO results. It is not possible to distinguish between the values of $k$ that emerge from the coordinate-space and harmonic-oscillator HFB calculations, both values being shown by the same line in the upper panel of Fig. 1.

Figure 1: Logarithmic derivative of the density (upper panel), and the density in logarithmic scale (lower panel), as functions of the radial distance. The coordinate-space HFB results (solid line) are compared with those for the HFB+HO method (denoted $\bar{\rho}$) with $N_{\rm sh}$= 8, 12 and 20 HO shells, as well with the approximation (denoted $\tilde{\rho}$) given by Eq. (25) (small circles).

Soon beyond the point $R_{{\rm min} }$, the HFB+HO density begins to deviate dramatically from that obtained in the coordinate-space calculation. For relatively small numbers of harmonic oscillator shells $N_{{\rm sh}}$, the logarithmic derivative of the HFB+HO density goes asymptotically to zero following the gaussian behavior of the harmonic oscillator basis. The resulting HFB+HO density does not develop a minimum around the point $R_{{\rm min} }$, as seen from the $N_{{\rm sh}}=8$ curve shown in the upper panel of Fig. 1. When the number of harmonic oscillator shells $N_{{\rm sh}}$ increases, the HFB+HO solution tries to capture the correct density asymptotics. Due to the gaussian asymptotic of the basis, however, the logarithmic derivative of the HFB+HO density only develops oscillations around the exact solution (see the $N_{{\rm sh}}=12$ and $20$ curves in the upper panel of Fig. 1). As a result, the logarithmic derivative of the HFB+HO density is very close to the coordinate-space result around the mid point $R_m=(R_{{\rm max}}-R_{{\rm min}})/2$, where $R_{{\rm max}}$ is the position of the first maximum of the logarithmic derivative after $R_{{\rm min} }$.

In summary, the following HFB+HO quantities agree with the coordinate-space HFB results: (i) the value of the density decay constant $k$; (ii) the local density up to the point $R_{{\rm min} }$ where the logarithmic derivative $\rho ^{\prime }/\rho$ shows a clearly-defined minimum; (iii) the actual value of this point $R_{{\rm min} }$; (iv) the value of the logarithmic derivative of the density at the point $R_m$ defined above. In fact, the last of the above is not established nearly as firmly as the first three; nevertheless, we shall make use of it in developing our new formulation of the HFB+THO method.

Beyond the point $R_m$, the HFB+HO solution fails to capture the physics of the coordinate-space results, especially in the far asymptotic region. It is this incorrect large-$r$ behavior that we now try to cure by introducing the THO basis.