 \big<\mathrm{\hat n}_i^{\vphantom{\dagger}} \mathrm{\hat n}_j^{\vphantom{\dagger}}\big>                      =   \big<\mathrm{\hat c}_i^\dagger \mathrm{\hat c}_i^{\vphantom{\dagger}} \mathrm{\hat c}_j^\dagger \mathrm{\hat c}_j^{\vphantom{\dagger}}\big>                      =   \big<\mathrm{\hat c}_j^\dagger \mathrm{\hat c}_i^\dagger \mathrm{\hat c}_i^{\vphantom{\dagger}} \mathrm{\hat c}_j^{\vphantom{\dagger}}\big>                        