Many-body perturbation theory
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.