Given a hamiltonian in the form: $$\hat{H}=\hat{H}_0+\hat{\mathcal{O}}=\sum_{\alpha, \beta, \sigma} t_{\alpha,\beta} \hat{c}^\dagger_{\alpha \sigma}\hat{c}_{\beta \sigma}+\frac{1}{2}\sum_{\alpha,\beta,\gamma,\lambda,\sigma 1, \sigma 2} \mathcal{O}_{\alpha, \beta}^{\gamma,\lambda}\hat{c}^\dagger_{\alpha \sigma_1} \hat{c}^\dagger_{\beta \sigma_2}\hat{c}_{\lambda \sigma_2} \hat{c}_{\gamma \sigma_1}.$$

How are the Feynman rules in frequency space to compute the interacting Green function and the self-energy? I'm having a hard time trying to do this. Basically, I want to understand diagramatic perturbation theory for hamiltonians given in this representation.


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