\hat{a}^+_i\hat{a}_i\hat{b}^+_j\hat{b}_j=\langle \hat{a}^+_i\hat{a}_i\rangle \hat{b}^+_j\hat{b}_j+
 \langle\hat{b}^+_j\hat{b}_j\rangle
 \hat{a}^+_i\hat{a}_i+\langle \hat{a}^+_i\hat{b}^+_j \rangle\hat{a}_i\hat{b}_j+\langle\hat{a}_i\hat{b}_j\rangle
 \hat{a}^+_i\hat{b}^+_j+ \langle \hat{a}^+_i\hat{b}_j\rangle \hat{a}_i\hat{b}^+_j+\langle \hat{a}_i\hat{b}^+_j \rangle
   \hat{a}^+_i\hat{b}_j