\hat{a}^+_i\hat{a}^+_i\hat{a}_i\hat{b}^+_j=\langle \hat{a}^+_i\hat{a}^+_i \rangle\hat{a}_i\hat{b}^+_j+\langle
  \hat{a}_i\hat{b}^+_j\rangle \hat{a}^+_i\hat{a}^+_i+2\langle \hat{a}^+_i\hat{a}_i\rangle
  \hat{a}^+_i\hat{b}^+_j+2
  \langle \hat{a}^+_i\hat{b}^+_j \rangle \hat{a}^+_i\hat{a}_i