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