THO and local densities

 

In calculations using the Skyrme force, or in any other calculation that relies on the local density approximation, we can simplify the THO methodology of Sec. 2.5 even further. Indeed, suppose the mean-field calculation in question relies on knowing the density matrix $\rho _{\alpha \alpha ^{\prime }}$in the THO basis. Then the spatial nonlocal density can be expressed as

$$\rho (\mbox{{\boldmath {$r$ }}}_{1},\mbox{{\boldmath {$r$ }}}... ...a ^{\prime }}^{*}\left( \mbox{{\boldmath {$r$ }}}_{2}\right) ,$$ (30)

and the corresponding standard local densities [19] as  

   

$$\rho(\mbox{{\boldmath {$r$ }}}) \rho (\mbox{{\boldmath {$r$ }}},\mbox{{\boldmath {$r$ }}})$$ = (31)

$$\tau (\mbox{{\boldmath {$r$ }}}) \sum_{k=x,y,z}\left[ \nabla _{k}^{(1)}\nabla _{k}^{(2)}\rho (\mbo... ...}}}_{2})\right] _{\mbox{{\boldmath {$r$ }}}_{1}=\mbox{{\boldmath {$r$ }}} _{2}}$$ = (32)

$$j_{k}(\mbox{{\boldmath {$r$ }}}) \frac{1}{2i}\left[ \left( \nabla _{k}^{(1)}-\nabla _{k}^{(2)}\rig... ...}}}_{2})\right] _{\mbox{{\boldmath {$r$ }}}_{1}= \mbox{{\boldmath {$r$ }}}_{2}}$$ = (33)

where

$$\nabla _{k}^{(i)}=\frac{\partial }{\partial (\mbox{{\boldmath {$r$ }}}_{i})_{k}},$$ (34)

for i=1 or 2, and k=x, y, or z. To simplify the notation in Eq. (30), we have neglected the spin and isospin degrees of freedom and, consequently, have shown only the spin-independent densities (2.6). Analogous formulae for the spin-dependent densities $\mbox{{\boldmath {$s$ }}}$, $\mbox{{\boldmath {$T$ }}}$, and J_kl [19] are straightforward.

A direct calculation of the derivatives in Eqs. (2.6) [after inserting the THO wave functions (9) or (24) into the nonlocal density matrix (30)] is prohibitively difficult. Fortunately, nothing of the sort is necessary. It is enough to note that the densities (2.6) serve almost uniquely to define the central, spin-orbit, and effective-mass terms of the mean-field Hamiltonian (see, e.g., Refs. [19,15]), and that these terms are in turn used to calculate matrix elements through integrals of the type (29). Therefore, the densities (2.6) have to be effectively known only at selected points $x^{\prime }$, $y^{\prime }$, $z^{\prime}$ (the Gauss-quadrature nodes) of the inverse LST.

Towards this end, we insert the THO wave functions into the nonlocal density (30), which gives

$$\rho(\mbox{{\boldmath {$r$ }}}_{1},\mbox{{\boldmath {$r$ }}}_... ... {$r$ }}} _{2}^{\prime }(\mbox{{\boldmath {$r$ }}}_{2})\Big) ,$$ (35)

with

$$\rho^{\prime }(\mbox{{\boldmath {$r$ }}}_{1}^{\prime },\mbox{... ... }}^{*}\left( \mbox{{\boldmath {$r$ }}}_{2}^{\prime }\right) .$$ (36)

The density matrix $\rho ^{\prime }(\mbox{{\boldmath {$r$ }}}_{1}^{\prime },\mbox{{\boldmath {$r$ }}} _{2}^{\prime })$ is a standard object expressed in terms of ordinary HO wave functions, and it can be calculated using methods that are employed in any code that works in the HO basis. Likewise, the corresponding local densities  

   

$$\rho ^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime }) \rho ^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime }, \mbox{{\boldmath {$r$ }}}^{\prime })$$ = (37)

$$\tau_{km}^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime }) \left[ \nabla _{k}^{(1)^{\prime }}\nabla _{m}^{(2)^{\prime }}\rho... ...mbox{{\boldmath {$r$ }}}_{1}^{\prime }=\mbox{{\boldmath {$r$ }}}_{2}^{\prime }}$$ = (38)

$$j_{k}^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime }) \frac{1}{2i}\left[ \left( \nabla _{k}^{(1)^{\prime }}-\nabla _{k}... ...mbox{{\boldmath {$r$ }}}_{1}^{\prime }=\mbox{{\boldmath {$r$ }}}_{2}^{\prime }}$$ = (39)

can be calculated without any reference to the THO basis. The only complication is that now we have to calculate the complete kinetic energy tensor density $\tau _{km}^{\prime }$(38), while finally only its trace (32) is needed. Inserting expression (35) into (2.6), and expressing the differential operators (34) as

$$\nabla_{k}^{(i)}=\sum_{m=x,y,z}D_{k}^{m}\nabla _{m}^{(i)^{\prime }},$$ (40)

for

$$D_{k}^{m}\equiv \frac{\partial \mbox{{\boldmath {$r$ }}}_{m}^... ...}}\mbox{{\boldmath {$r$ }}}_{m} \mbox{{\boldmath {$r$ }}}_{k},$$ (41)

we obtain that  

   

$$\rho (\mbox{{\boldmath {$r$ }}}(\mbox{{\boldmath {$r$ }}}')) D\rho ^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime }),$$ = (42)

$$\tau (\mbox{{\boldmath {$r$ }}}(\mbox{{\boldmath {$r$ }}}')) D{\displaystyle\sum_{kmn}}D_{n}^{k}D_{n}^{m} \tau _{km}^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime })$$ =

$$+\frac{1}{2}{\displaystyle\sum_{km}}\left[ \nabla _{k}D\right] D_... ...nabla _{m}^{\prime }\rho ^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime })\right]$$

$$+\frac{1}{4}D^{-1}\left[ \mbox{{\boldmath {$\nabla$ }}}D\right] ^{2}\rho (\mbox{{\boldmath {$r$ }}}^{\prime }),$$ (43)

$$j_{k}(\mbox{{\boldmath {$r$ }}}(\mbox{{\boldmath {$r$ }}}')) D{\displaystyle\sum_{m}}D_{k}^{m}j_{m}^{\prime }(\mbox{{\boldmath {$r$ }}}^{\prime }).$$ = (44)

To use formulae (2.6), we must calculate the Jacobi matrix D_k^m and its determinant D at points $\mbox{{\boldmath {$r$ }}}(\mbox{{\boldmath {$r$ }}}')$; however, this need be done only once for all iterations. On the other hand, no inverse LST needs to be performed for the densities, because expressions (2.6) give directly the values of the local densities at the inverse LST points, as required in matrix-element integrals of the type (29).