Fields for terms containing additional density dependence

In the Skyrme functional, a dependence on the isoscalar density is usually added to the four terms that are zero order in derivatives, $\rho_{\tau}^{2}$ and $\vec{s}_{\tau}^{2}$ ($\tau=0,1$), that is

$$T_{00,0000}^{\alpha,0000}\left(\tau\right) = C_{00,0000}^{\alpha,0000}\left(\tau\right)\rho_{0}^{\alpha}\rho_{\tau}^{2} ,$$ (61)

$$T_{00,0011}^{\alpha,0011}\left(\tau\right) = C_{00,0011}^{\alpha,0011}\left(\tau\right)\rho_{0}^{\alpha}\vec{s}_{\tau}^{2} .$$ (62)

We use the extra index $\alpha$ on the coupling constant to distinguish the notation from the one used for the density-independent terms.

The contribution to the fields is obtained from the variation of the energy, which we show here only for the first of the above two terms, namely,

$$\begin{eqnarray*} \frac{\delta {\cal E}_{00,0000}^{\alpha,0000}\left(\tau\right)... ...}\rho_{\tau}\delta_{\tau',\tau}\right]\phi_{i}\left(\tau'\right) \end{eqnarray*}$$

which gives the additional contributions

$$\sum_{\tau=0}^1 C_{00,0000}^{\alpha,0000}\left(\tau\right)\l... ...au',0}+2\rho_{0}^{\alpha}\rho_{\tau}\delta_{\tau',\tau}\right]$$ (63)

to $U_{0000,0}\left(\tau'\right)$.