Calculation of matrix elements

To calculate the coupled matrix elements in Eqs. (23)-(25), we use an intermediate LS scheme:

$${ \langle [\mu\mu^\prime]J_k\vert\bar{V}\vert[\nu\nu^\prime]J_k\rangle }$$

$$= \sum_{LL^\prime S} \hat{j}_\mu \hat{j}_{\mu^\prime} \hat{j}_\nu \... ...L^\prime \\ 1/2 & 1/2 & S \\ j_\nu & j_{\nu^\prime} & J_k \end{array}\right\}$$

$$\times \langle (l_\mu l_{\mu^\prime})LS;J_k\vert\bar{V}\vert(l_\nu l_{\nu^\prime})L^\prime S;J_k\rangle,$$ (46)

$$\hat{j}_\mu \equiv \sqrt{2j_\mu + 1}.$$ (47)

Eq. (38) gives

i) proton-proton or neutron-neutron matrix elements:

$${ \langle (l_\mu l_{\mu^\prime})LS;J_k\vert \bar{V}^{\mbox{\footn... ...ox{\footnotesize {Skyrme}}} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle }$$

$$= \left\{ a_0+c_0 +(2S(S+1)-3)(b_0+d_0) \right\} \langle (l_\mu l_{... ...dmath$r$\unboldmath }}^\prime)\vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+\left\{ a_1+c_1 +(2S(S+1)-3)(b_1+d_1)\right\} \langle (l_\mu l_{... ...{\boldmath$k$\unboldmath }}^2 \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+\left\{ a_2+c_2+(2S(S+1)-3)(b_2+d_2) \right\} \langle (l_\mu l_{... ...ox{\boldmath$k$\unboldmath }} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+\left\{ a_3+c_3+(2S(S+1)-3)(b_3+d_3) \right\} \langle (l_\mu l_{... ...math$r$\unboldmath }}^\prime) \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+2e_3 \langle (l_\mu l_{\mu^\prime})LS;J_k\vert\rho_{10}({\mbox{\... ...egin{array}{cc} (-1) , & {\rm proton} \\ 1 , & {\rm neutron}\end{array}\right.$$

$$+f_3 \langle (l_\mu l_{\mu^\prime})LS;J_k\vert\rho^2_{10}({\mbox{... ...math$r$\unboldmath }}^\prime) \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+(a_4+c_4) \langle (l_\mu l_{\mu^\prime})LS;J_k\vert({\mbox{\bold... ...{\boldmath$k$\unboldmath }} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle ,$$ (48)

ii) proton-neutron matrix elements:

$${ \langle (l_\mu l_{\mu^\prime})LS;J_k\vert \bar{V}^{\mbox{\footn... ...ox{\footnotesize {Skyrme}}} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle }$$

$$= \{ a_0-c_0+(2S(S+1)-3)(b_0-d_0) \} \langle (l_\mu l_{\mu^\prime})... ...math$r$\unboldmath }}^\prime) \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+\left\{ a_1-c_1 +(2S(S+1)-3)(b_1-d_1) \right\}\langle (l_\mu l_{... ...{\boldmath$k$\unboldmath }}^2 \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+\left\{ a_2-c_2 + (2S(S+1)-3)(b_2-d_2) \right\} \langle (l_\mu l... ...ox{\boldmath$k$\unboldmath }} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+\left\{ a_3-c_3 +(2S(S+1)-3)(b_3-d_3) \right\} \langle (l_\mu l_... ...math$r$\unboldmath }}^\prime) \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+f_3 \langle (l_\mu l_{\mu^\prime})LS;J_k\vert\rho^2_{10}({\mbox{... ...math$r$\unboldmath }}^\prime) \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle$$

$$+(a_4-c_4) \langle (l_\mu l_{\mu^\prime})LS;J_k\vert({\mbox{\bold... ...{\boldmath$k$\unboldmath }} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle .$$ (49)

We use the canonical (and real) radial wave functions $R_{\mu}(r)$, the angular wave functions $Y_{l_\mu l_\mu^z}(\Omega)$, and the spin wave functions to write the nontrivial matrix elements included in Eqs. (49) and (50) as

$${ \langle (l_\mu l_{\mu^\prime})LS;J_k\vert\delta({\mbox{\boldmat... ...th$r$\unboldmath }}^\prime)\vert (l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle }$$

$$= \int dr\:r^2 R_\mu(r) R_{\mu^\prime}(r) R_\nu(r) R_{\nu^\prime}(r... ...ht]^\ast_{L0} \left[ Y_{l_\nu}(\Omega) Y_{l_{\nu^\prime}}(\Omega)\right]_{L0} ,$$

$$\int d\Omega \left[ Y_{l_\mu}(\Omega) Y_{l_{\mu^\prime}}(\O... ...cc}l_\nu & l_{\nu^\prime} & L \\ 0 & 0 & 0\end{array}\right) ,$$ (50)

$${ \langle (l_\mu l_{\mu^\prime})LS;J_k\vert \delta({\mbox{\boldma... ...boldmath$k$\unboldmath }}^2 \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle }$$

$$= -\frac{1}{4}\delta_{LL^\prime} \int d\Omega \left[ Y_{l_\mu}(\Ome... ...ight]^\ast_{L0} \left[ Y_{l_\nu}(\Omega) Y_{l_{\nu^\prime}}(\Omega)\right]_{L0}$$

$$\times \int dr\:r^2 R_\mu(r) R_{\mu^\prime}(r) \left\{ \left[ \le... ...}{dr}-\frac{l_\nu(l_\nu+1)}{r^2} \right)R_\nu(r)\right]R_{\nu^\prime}(r)\right.$$

$$\left.+R_\nu(r)\left[ \left( \frac{d^2}{dr^2}+\frac{2}{r}\frac{d}... ...{l_{\nu^\prime}(l_{\nu^\prime}+1)}{r^2}\right)R_{\nu^\prime}(r) \right]\right\}$$

$$-\sum_{\Delta l_\nu=0,1} \sum_{\Delta l_{\nu^\prime}=0,1} \frac{1... ..._{\nu}-1+2\Delta l_{\nu} & L \\ l_\nu & l_{\nu^\prime} & 1 \end{array}\right\}$$

$$\times\int dr\:r^2 R_\mu(r) R_{\mu^\prime}(r) \left\{ (l_\nu+1-\Delta l_\nu)\frac{R_\nu(r)}{r} +(-)^{\Delta l_\nu}\frac{dR_\nu(r)}{dr}\right\}$$

$$\times \left\{ (l_{\nu^\prime}+1-\Delta l_{\nu^\prime})\frac{R_{\... ...prime}(r)}{r} +(-)^{\Delta l_{\nu^\prime}}\frac{dR_{\nu^\prime}(r)}{dr}\right\}$$

$$\times \delta_{LL^\prime} \int d\Omega \left[ Y_{l_\mu}(\Omega) Y... ...Delta l_{\nu^\prime} }(\Omega) Y_{l_\nu-1+2\Delta l_\nu} (\Omega)\right]_{L0} ,$$ (51)

$$\langle (l_\mu l_{\mu^\prime})LS;J_k\vert \mbox{\boldmath$k$... ...k$\unboldmath }})_0\vert\vert (l_\nu l_{\nu^\prime})L\rangle ,$$ (52)

$${ \langle (l_\mu l_{\mu^\prime})LS;J_k\vert i({\mbox{\boldmath$\s... ...{\boldmath$k$\unboldmath }} \vert(l_\nu l_{\nu^\prime})L^\prime S;J_k \rangle }$$

$$= (-)^{1+L^\prime+J_k}4 \sqrt{3} \left\{ \begin{array}{ccc} 1 & L^\... ...\mbox{\boldmath$k$\unboldmath }})_1\vert\vert (l_\nu l_{\nu^\prime})L'\rangle .$$ (53)

The square brackets around products of spherical harmonics and the parentheses surrounding products of operators indicate angular-momentum coupling.

To evaluate Eqs. (54) and (55), one can use

$${ \langle (l_\mu l_{\mu^\prime})L\vert\vert( \mbox{\boldmath$k$\u... ...\mbox{\boldmath$k$\unboldmath }})_I\vert\vert (l_\nu l_{\nu^\prime})L'\rangle }$$

$$= \left[\frac{1}{4}\sum_{\Delta l_\mu=0,1} \sum_{\Delta l_\nu=0,1} ... ...}{r} +(-)^{\Delta l_\mu}\frac{dR_\mu(r)}{dr} \right\} R_{\mu^\prime}(r) \right.$$

$$\times \left\{ (l_\nu+1-\Delta l_\nu)\frac{R_\nu(r)}{r} +(-)^{\Delta l_\nu}\frac{dR_\nu(r)}{dr} \right\} R_{\nu^\prime}(r)$$

$$\times\sqrt{l_\mu+\Delta l_\mu} \sqrt{l_\nu+\Delta l_\nu} \sqrt{2... ...-1} \hat{ l_{\mu^\prime} } \sqrt{2l_\nu+4\Delta l_\nu-1} \hat{ l_{\nu^\prime} }$$

$$\times \frac{1}{4\pi}\hat{l}^2_{\mu\mu^\prime} \hat{L} \hat{L}^\p... ...& L & 1 \\ l_\mu & l_\mu+2\Delta l_\mu-1 & l_{\mu^\prime} \end{array} \right\}$$

$$\times \left\{ \begin{array}{ccc}l_{\mu\mu^\prime} & L^\prime & 1... ... l_\mu-1 & l_{\mu^\prime} & l_{\mu\mu^\prime} \\ 0 & 0 & 0 \end{array} \right)$$

$$\times \left. \left( \begin{array}{ccc}l_\nu +2\Delta l_\nu-1 & l... ...arrow \nu^\prime] -(-)^{l_\mu+l_{\mu^\prime}+L}[\mu \leftrightarrow \mu^\prime]$$

$$+[\mu \leftrightarrow \mu^\prime {\rm\ and\ } \nu \leftrightarrow \nu^\prime ] ,$$ (54)

where for reduced matrix elements we have used the convention

$$\langle LL_z \vert\hat{O}_{lm}\vert L^\prime L^\prime_z \ran... ...ngle \langle L\vert\vert\hat{O}_l\vert\vert L^\prime \rangle ,$$ (55)

and made the abbreviation

$$[A_{\mu\mu^\prime\nu\nu^\prime}] -(-)^{l_\nu+l_{\nu^\prime}+L^\pr... ... + [\mu \leftrightarrow \mu^\prime {\rm\ and\ } \nu \leftrightarrow \nu^\prime]$$

$$\equiv A_{\mu\mu^\prime\nu\nu^\prime} - (-)^{l_\nu+l_{\nu^\prime}+L^\pri... ...\mu^\prime}+L}A_{\mu^\prime\mu\nu\nu^\prime} + A_{\mu^\prime\mu\nu^\prime\nu} .$$ (56)

Eq. (51), modified to include additional factors in the radial integral, can also be used (together with the subsequent equations) to evaluate the matrix elements of the terms involving $\rho_{00}^\alpha({\mbox{\boldmath$r$\unboldmath }})$ in $\bar{V}^{\rm eff}_{\rm Skyrme}$, the Coulomb-exchange interaction, and the contributions of the pairing functional to the effective ph, pp, and 3p1h interactions. The Coulomb-direct term can be evaluated in a similar but slightly more complicated way, via a multipole expansion.

In the main part of this paper we used the Skyrme functional SkM$^\ast$, which is usually parameterized as in interaction in terms of coefficients $t_0,t_1,t_2,t_3,x_0,x_1,x_2, x_3,$ and $W_0$. The relations between these coefficients and those used here, if no terms are neglected, are [79,63]

$$\begin{array}{ll} C_0^{\rho} = \frac{3}{8} t_0 +\frac{3}{48... ...ac{3}{4} W_0, & C_1^{\nabla J} = -\frac{1}{4} W_0 . \end{array}$$ (57)

In the HF fits that originally determined the SkM$^\ast$ parameters, the effects of $C_t^T$ (the ``$J^2$ terms'') were neglected because of technical difficulties. These terms have often been included in subsequent RPA calculations. To maintain self consistency here, we have set them to zero both in the HFB calculation and in the QRPA.