I know the relationship between normal and correlated two-particle Green Functions for fermions:$$G_c(1,2,3,4)=\Gamma(1,2,3,4)=G(1,2,3,4)+G(1,3)G(2,4)-G(1,4)G(2,3)$$Also known as irreducible n-particle Green function or n-particle vertex (there are many definitions so I'm confused).
Definitions for G also vary, I use this one: $G(1..2n) = (-1)^{n}\langle T\ c_1\dots c_n c_{n+1}^\dagger\dots c_{2n}^\dagger \rangle$
I need a similar relationship for three-particle functions.
Now I have this formula: $$\Gamma(1,2,3,4,5,6) = G(1, 2, 3, 4, 5, 6)-2 G(1, 4) G(2, 5) G(3, 6) +2 G(1, 4) G(2, 6) G(3, 5)-2 G(1, 5) G(2, 6) G(3, 4) +2 G(1, 5) G(2, 4) G(3, 6)-2 G(1, 6) G(2, 4) G(3, 5)+2 G(1, 6) G(2, 5) G(3, 4)-G(1, 4) G(2, 3, 5, 6)+G(1, 5) G(2, 3, 4, 6)-G(1, 6) G(2, 3, 4, 5)+G(2, 4) G(1, 3, 5, 6)-G(2, 5) G(1, 3, 4, 6)+G(2, 6) G(1, 3, 4, 5)-G(3, 4) G(1, 2, 5, 6)+G(3, 5) G(1, 2, 4, 6)-G(3, 6) G(1, 2, 4, 5)$$But it doesn't always satisfy Wick's theorem for the Hamiltonian I'm working with. Is this formula correct?
Also it would be great to get an explanation of how these formulas appear in many-body theory, from logarithm of time-ordered exponential.