CYLINDRICAL SYMMETRY

In the case of cylindrical symmetry, the third component $J_z$ of the total angular-momentum is conserved and provides a good quantum number $\Omega_k$. The HF s.p. wave functions in the axial limit can be written as [23]

$$V_k(\bbox{r}st) = V_k^+ (rzt) e^{i \Lambda^- \phi} \chi_{+1/2}(s)$$

$$+ V_k^- (rzt) e^{i \Lambda^+ \phi} \chi_{-1/2}(s),$$ (20)

where $\Lambda^{\pm} = \Omega_k \pm 1/2$ and $(r,\phi,z)$ are the cylindrical coordinates defining the three-dimensional position vector, $\bbox{r} =( r \cos\phi, r \sin\phi, z )$ and $z$ is the chosen symmetry axis. In the following section, the expressions in the cylindrical coordinate basis under the axial symmetry are provided for the local particle-hole densities. In the second section, expressions of the derivatives of the local densities, which are used in the evaluation of the HF potentials, are given.