Relation of $a^dagger a$ 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 $e^{ihat G(q,p,n)}|phi(q)rangle$ 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 $a^dagger_mu a^dagger_ u$ and $a_ u a_mu$. 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.