Relation of a†a terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion

We investigate the relation of $a^\dagger a$ terms in the collective operator to the higher-order terms in the adiabatic self-consistent collective coordinate (ASCC) method. In the ASCC method, a state vector is written as $e^{i\hat G(q,p,n)}|\phi(q)\rangle$ with $\hat G(q,p,n)$ which is a function of collective coordinate $q$, its conjugate momentum $p$ and the particle number $n$. According to the generalized Thouless theorem, $\hat G$ can be written as a linear combination of two-quasiparticle creation and annihilation operators $a^\dagger_\mu a^\dagger_\nu$ and $a_\nu a_\mu$. We show that, if $a^\dagger a$ terms are included in $\hat G(q,p,n)$, it corresponds to the higher-order terms in the adiabatic expansion of $\hat G$. This relation serves as a prescription to determine the higher-order collective operators from the $a^\dagger a$ part of the collective operator, once it is given without solving the higher-order equations of motion.