  We derive variational expressions for the grand potential or action in terms
of the many-body Green function $G$ which describes the propagation of
particles and the renormalized four-point vertex $\Gamma$ which describes the
scattering of two particles in many-body systems. The main ingredient of the
variational functionals is a term we denote as the $\Xi$-functional which plays
a role analogously to the usual $\Phi$-functional studied by Baym (G.Baym,
Phys.Rev. 127, 1391 (1962)) in connection with the conservation laws in
many-body systems. We show that any $\Xi$-derivable theory is also
$\Phi$-derivable and therefore respects the conservation laws. We further set
up a computational scheme to obtain accurate total energies from our
variational functionals without having to solve computationally expensive sets
of self-consistent equations. The input of the functional is an approximate
Green function $\tilde{G}$ and an approximate four-point vertex
$\tilde{\Gamma}$ obtained at a relatively low computational cost. The
variational property of the functional guarantees that the error in the total
energy is only of second order in deviations of the input Green function and
vertex from the self-consistent ones that make the functional stationary. The
functionals that we will consider for practical applications correspond to
infinite order summations of ladder and exchange diagrams and are therefore
particularly suited for applications to highly correlated systems. Their
practical evaluation is discussed in detail.
