c_{j-m}^{+}
(-1)^{l'+j'-m'} \nonumber \\ c_{j'-m'} c_{j'm'},\end{aligned}$$ where the sum over $m$ and $m'$ is restricted to $m$, ${m'}> 0$, and $l$ donates the orbital angular momentum.

In the present work, in addition to the well known particle-hole GT force [@Halbleib67; @Staudt90; @Muto92] $$\label{GrindEQ__3_}
V_{GT}^{ph} =2\chi \sum _{\mu }(-1)^{\mu } Y_{\mu }  Y_{-\mu }^{+},$$ with $$\label{GrindEQ__4_}
Y_{\mu } =\sum _{j_{p} m_{p} j_{n} m_{n} }<j_{p} m_{p} |t{}_ {-}
\sigma _{\mu }  |j_{n} m_{n} >c_{j_{p} m_{p} }^{+} c_{j_{n} m_{n} },$$ the particle-particle GT force [@Soloviev87; @Kuzmin88] $$\label{GrindEQ__5_}
V_{GT}^{pp} =-2\kappa \sum _{\mu }(-1)^{\mu }  P_{\mu }^{+} P_{-\mu
},$$ with $$\begin{aligned}
 \label{GrindEQ__6_}
P_{\mu }^{+} =\sum _{j_{p} m_{p} j_{n} m_{n} }<j_{n} m_{n} |(t{}_{-}
\sigma _{\mu } )^{+} |j_{p} m_{p} >(-1)^{l_{n} +j_{n} -m_{n} }
\nonumber \\ c_{j_{p} m_{p} }^{+} c_{j_{n} -m_{n} }^{+},\end{aligned}$$

is also taken into account.

The capture/decay rate of a transition from the *i*th state of a parent nucleus (Z, N) to the $jth$ state of the daughter nucleus $(Z - 1, N + 1)$ is given by

$$\label{GrindEQ__7_}
\lambda _{ij} \, =\, \ln 2\frac{f_{ij} (T,\rho ,E_{f} )}{(ft)_{ij}
},$$

$f_{ij} $ is the phase space integral. The $(ft)_{ij} $ of an ordinary $\beta ^{\pm } $ decay from the state ${\left| i
\right\rangle} $ of 