                                 + 0 \big<\mathrm{\hat n}_{i\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat n}_{j\sigma}^{\vphantom{\dagger}}\big>                                 - 0 \big<\mathrm{\hat c}_{i\sigma}^\dagger \mathrm{\hat c}_{j\sigma}^{\vphantom{\dagger}}\big> \big<\mathrm{\hat c}_{j\sigma}^\dagger \mathrm{\hat c}_{i\sigma}^{\vphantom{\dagger}}\big>   \nonumber \\ & &                                 + 0 \big<\mathrm{\hat n}_{i\sigma}^{\vphantom{\dagger}}\big> \