A non-practical way of solving the eigenvalue problem
We see that on the right-hand side we have the energy \( E \). This leads to a non-linear equation in the unknown coefficients.
These equations are normally solved iteratively ( that is we can start with a guess for the coefficients \( C_i^a \)). A common choice is to use perturbation theory for the first guess, setting thereby
$$
C_{i}^{a}=\frac{\langle i | \hat{f}| a\rangle}{\epsilon_i-\epsilon_a}.
$$
The observant reader will however see that we need an equation for \( C_{jk}^{bc} \) and \( C_{jkl}^{bcd} \) as well.
To find equations for these coefficients we need then to continue our multiplications from the left with the various
\( \Phi_{H}^P \) terms.