    \big< \mathrm{\hat c}^{\dagger}_{i\sigma} \mathrm{\hat c}^{\vphantom{\dagger}}_{j\sigma} \big>               \big< \mathrm{\hat c}^{\dagger}_{j\sigma'} \mathrm{\hat c}^{\vphantom{\dagger}}_{i\sigma'} \big>          