-\frac{1}{N}\sum_{\vec{k}}J(\vec{k})\sum_{\vec{k}_1  ,\vec{k}_2}\hat{a}^+_{\vec{k}_1}\hat{a}_{\vec{k}_2  }\langle \hat{a}^+_{\vec{k}-\vec{k}_1+\vec{k}_2}\hat{a}^+_{-\vec{k}}\rangle=-\frac{1}{N}\sum_{\vec{k}}J(\vec{k})\sum_{\vec{k}_1  ,\vec{k}_2}\hat{a}^+_{\vec{k}_1}\hat{a}_{\vec{k}_2  }\langle\hat{a}^+_{\vec{k}}\hat{a}^+_{-\vec{k}}\rangle\delta_{\vec{k}_1,\vec{k}_2}=-\frac{1}{N}\sum_{\vec{k}}J(\vec{k})\sum_{\vec{k}_1  }\hat{a}^+_{\vec{k}_1}\hat{a}_{\vec{k}_1  }\langle\hat{a}^+_{\vec{k}}\hat{a}^+_{-\vec{k}}\rangle= -\sum_{\vec{k}}\hat{a}^+_{\vec{k}}\hat{a}_{\vec{k}}  \frac{1}{N}\sum_{\vec{q}}J(\vec{q})\langle \hat{a}^+_{\vec{q}}\hat{b}^+_{-\vec{q}}\rangle