    \mathrm{\hat c}^{\vphantom{\dagger}}_{j\sigma} \big>            - \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big> \nonumber \\    &  = & 0 \sum_{\sigma\sigma'} \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\