Calculation of derivatives

The collective mass involves either derivatives of the density matrices or the mean-field potentials. It should be stressed that these derivatives must be calculated in the original single-particle basis $\vert n\rangle$ as the canonical basis (5) varies with $\{q_i\}$. In the following, we show how to evaluate the collective derivatives in the one-dimensional case of single collective coordinate, the quadrupole deformation $q$. To this end, we approximate the derivative of the density operator $\rho$ or $\kappa$ at a deformation point $q=q_0$ by means of the Lagrange three-point formula for unequally spaced points $q_0-\delta q$, $q_0$, and $q_0+\delta q'$ [12,9,16]:

$$\biggr( \frac {\partial \rho} {\partial q} \biggr)_{q=q_0} \approx \frac { -\delta q'} { \delta q(\delta q+\delta q') } \rho(q_0-\delta q) + \frac {\delta q-\delta q'} {\delta q\,\delta q'} \rho(q_0)$$

$$+ \frac { \delta q} { \delta q' (\delta q+\delta q') } \rho(q_0+\delta q').$$ (52)

The reason for the use of unequally spaced grid in Eq. (52) is purely numerical: the constrained HFB equations cannot be precisely solved at a requested deformation point $q$.

The corresponding matrix element in the canonical basis can be expressed through the matrices $D_{n\nu}$ of the canonical transformation (5):

$$\bigg( \frac {\partial \rho} {\partial q} \bigg)_{\mu \nu} \!\!\! \approx \frac { -\delta q'} { \delta q(\delta q+\delta q') } \sum_{n_1 n_2} D^\ast_{n_1 \mu} (\rho(q_0-\delta q))_{n_1 n_2} D_{n_2 \nu}$$

$$+ \frac {\delta q-\delta q'} {\delta q\,\delta q'} v^2_\mu \delta_{\mu\nu}$$ (53)

$$+ \frac { \delta q} { \delta q' (\delta q+\delta q') } \sum_{n_1 n_2} D^\ast_{n_1 \mu}(\rho(q_0+\delta q'))_{n_1 n_2} D_{n_2 \nu}.$$

It should be noted that the canonical matrix $D_{n\nu}$ in the above expression corresponds to the deformation point, $q_0$, at which the mass is evaluated. Furthermore, as mentioned above, the density matrices at the three deformation points in (52) need to be calculated using the single-particle basis $\vert n\rangle$ with the same basis deformation.