The Schroedinger equation is
$$ \hat{H}\vert \Psi_0\rangle = E\vert \Psi_0\rangle, $$and multiplying the latter from the left with \( \langle \Phi_0\vert \) gives
$$ \langle \Phi_0\vert \hat{H}\vert \Psi_0\rangle = E\langle \Phi_0\vert \Psi_0\rangle=E, $$and subtracting from this equation
$$ \langle \Psi_0\vert \hat{H}_0\vert \Phi_0\rangle= W_0\langle \Psi_0\vert \Phi_0\rangle=W_0, $$and using the fact that the both operators \( \hat{H} \) and \( \hat{H}_0 \) are hermitian results in
$$ \Delta E=E-W_0=\langle \Phi_0\vert \hat{H}_I\vert \Psi_0\rangle, $$which is an exact result. We call this quantity the correlation energy.