\hat{b}^+_j\hat{b}_j=\hat{b}^+_j\langle \hat{b}_j \rangle+\langle\hat{b}^+_j\rangle \hat{b}_j-\langle\hat{b}^+_j\hat{b}_j\rangle+(\hat{b}^+_j-\langle \hat{b}^+_j\rangle)(\hat{b}_j-\langle \hat{b}_j\rangle )