When we look at the expectation value of \( \langle \mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle \), we can rewrite this expression in terms of the spin \( \mathbf{S}=\mathbf{s}_1+\mathbf{s}_2 \), resulting in $$ \langle\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle=2(S^2-s_1^2-s_2^2)=2S(S+1)-3, $$ where we \( s_1=s_2=1/2 \) leading to $$ \left\{ \begin{array}{cc} \langle\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle=1 & \mathrm{if} \hspace{0.2cm} S=1\\ \langle\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2\rangle=-3 & \mathrm{if} \hspace{0.2cm} S=0\\\end{array}\right. $$