The sum rule for Gamow-Teller is obtained as follows $$ \sum_{f,\mu} \vert \langle f\vert \sum_{k} \sigma_{k,\mu} t_{k-} \vert i\rangle\vert^{2} - \sum_{f,\mu} \vert \langle f\vert \sum_{k} \sigma_{k,\mu } t_{k+} \vert i\rangle\vert^{2} $$ $$ = \sum_{f,\mu}\langle i\vert \sum_{k} \sigma_{k,\mu} t_{k+} \vert f\rangle\langle f\vert \sum_{k'} \sigma_{k',\mu} t_{k'-} \vert i\rangle $$ $$ - \sum_{f,\mu} \langle i\vert \sum_{k} \sigma_{k,\mu } t_{k-} \vert f\rangle\langle f\vert\sum_{k'} \sigma_{k',\mu } t_{k'+} \vert i\rangle $$ $$ = \sum_{\mu} \left[\langle i\vert \left(\sum _{k} \sigma_{k,\mu} t_{k+} \right) \left( \sum_{k'} \sigma_{k',\mu} t_{k'-}\right) - \left( \sum_{k} \sigma_{k,\mu} t_{k-} \right) \left( \sum_{k'} \sigma_{k',\mu} t_{k'+} \right) \vert i\rangle \right] $$ $$ = \sum_{\mu } \langle i\vert \sum_{k} \sigma ^{2}_{k,\mu } \left[ t_{k+} t_{k-} - t_{k-} t_{k+} \right] \vert i\rangle = 3 \langle i\vert \sum_{k} \left[ t_{k+} t_{k-} - t_{k-} t_{k+} \right] \vert i\rangle $$ $$ = 3\langle i\vert T_{+} T_{-} - T_{-} T_{+}\vert i\rangle= 3\langle i\vert 2T_{z}\vert i\rangle = 3(N_{i}-Z_{i}). $$