\Sigma_{\alpha\beta}({\bf t}')}
        {\partial {{\bf t}'}}
   = 0  \: . 
\label{eq:euler}$$ Note that the equation involves, apart from ${\bf G}_0$, quantities of the reference system $H'$ only. The linear response of the self-energy of $H'$ due to a change of the hopping ${\bf t}'$ can be calculated along the lines of Ref. . It turns out that $\partial {\bf \Sigma}({\bf t}') / \partial {\bf t}'$ is given by a two-particle Green’s function of $H'$. Since ${\bf G}' = {\bf G}[{\bf \Sigma}({\bf t}')]$, the exact self-energy of the system $H$ is determined by the condition that the bracket in (\[eq:euler\]) be zero. Hence, one can consider Eq. (\[eq:euler\]) to be obtained from the [*exact*]{} equation that determines the “vector” ${\bf \Sigma}$ in the self-energy space through [*projection*]{} onto the hypersurface of ${\bf t}'$ representable trial self-energies ${\bf \Sigma}({\bf t}')$ by taking the scalar product with vectors $\partial {\bf \Sigma}({\bf t}') / \partial {\bf t}'$ tangential to the hypersurface.

An analysis of the second derivative $\partial^2 \Omega_{\bf t} [{\bf \Sigma}({\bf t}')] / 
\partial t'_{\alpha\beta} \partial t'_{\gamma\delta}$ shows that a stationary point is not an extremum point in general. This feature is shared with the time-dependent DF approach, [@RG84] the Green’s-function approach [@BK61] and 