I've not come across any expression involving $\langle\Omega|f|\Omega\rangle$ in Srednicki's QFT book (please correct me if these exist there). On the other hand, they are abound in Chapter 7 of Peskin&Schroeder, in relation to the LSZ reduction formula. Here $|\Omega\rangle$ is the ground state of the interacting theory and $f$ is anything you can imagine that makes sense.
Srednicki only uses $\langle0|f|0\rangle$ where $|0\rangle$ is the ground state of the free theory.
How come this is so? Can one always go from one expression to the other by rewriting in terms of creation and annihilation operators?
Srednicki also derives the LSZ but he only uses the above-mentioned $|0\rangle$.