F. Variational Derivation of Self-Energies

We consider the time evolution of the GREEN's function under the action of the time-independent HAMILTONian $\hat{H}=\hat{H}_0+\hat{H}^\mathrm{int}$ and the time-dependent external perturbation $\hat{H}^\mathrm{ext}$. The latter is included through the evolution operator $\hat{S}^\mathrm{ext}_\mathrm{C}$

$$\begin{array}{ll}\displaystyle G(12) \ &\displaystyle = \ -\f... ... \hat{\psi}^\dagger_{\mathrm{H}}(2)\} \rangle_0 \ , \end{array}$$ (F.1)

where the abbreviation $1\equiv ({\bf {r}_1},t_1)$ is used. To obtain the equation of motion, one can take the derivative of the GREEN's function with respect to time

$$\begin{array}{ll}\displaystyle i\hbar \partial_{t} G(12) \ = ... ... \hat{\psi}^\dagger_{\mathrm{H}}(2)\} \rangle_0 \ . \end{array}$$ (F.2)

The first contribution results from $\partial_t \theta(t_1,t_2)$ (see Section 3.7.1). Because of the anti-commutation relation of the field operators it can be reduced to $\delta_{1,2}=\delta_{{\bf r_1},{\bf r_2}}\delta_{t_1,t_2}$. The equation of motion for the field operator, $i\hbar\partial_{t_1}\hat{\psi}_\mathrm{H}(1)=[\hat{\psi}_{\mathrm{H}}(1),\hat{H}]_{-}$, has been used in the second term, and the third contribution results from $\partial_t \hat{S}^\mathrm{ext}_\mathrm{C}$. Inserting the commutator with the HAMILTONian, one obtains

$$\begin{array}{ll}\displaystyle \left[i\hbar \partial_{t_1} \ ... ... \ - i\hbar\int_\mathrm{C} d3 \ V(1-3)\ G(1323) \ , \end{array}$$ (F.3)

where the two-particle GREEN's function $G(1234)$ is defined by

$$\begin{array}{ll}\displaystyle G(1234) \ = \ {\left(-\frac{i}... ... \hat{\psi}^\dagger_{\mathrm{H}}(3) \}\rangle_0 \ . \end{array}$$ (F.4)

To evaluate the two-particle GREEN's functions, one can express it as products of single-particle GREEN's functions $G(12)$, yielding an infinite perturbation expansion [203,282,205]. This can be accomplished by utilizing the GREEN's functions as generating functional. The two-particle GREEN's function can be expressed by means of functional derivatives of the single-particle GREEN's functions with respect to the external potential. Based on the variational method, the electron-electron and electron-phonon self-energies are derived next.
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