What is Nesbet's theorem?
Section 4.1.1 (p 237) of Szabo and Ostlund's Modern Quantum Chemistry also describes this issue. 

Using intermediate normalisation the correlation energy can be written as

$$\begin{align}
\langle \Psi_0|\hat{H}-E_0| \Phi_0\rangle &amp;= \langle \Psi_0|\hat{H}-E_0 \left(| \Psi_0\rangle + \sum_{ar}c_a^r |\Psi_a^r \rangle + \sum_{abrs}c_{ab}^{rs}|\Psi_{ab}^{rs} \rangle +.... \right) \\
&amp;= \sum_{abrs}c_{ab}^{rs}\langle \Psi_0|\hat{H}| \Psi_{ab}^{rs}\rangle
\end{align}$$

$\langle \Psi_0|\hat{H}| \Psi_{a}^{r}\rangle = 0$  according to Brillouin's theorem. 
$$\langle \Psi_0|\hat{H}| \Psi_{abc}^{rst}\rangle = \langle \Psi_0|\hat{H}| \Psi_{ab}^{rs}\rangle \langle \phi_c | \phi_t \rangle = 0$$

and similarly for higher excitations, because $\hat{H}$ is a 2-electron operator.