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We investigate the relation of $a^\dagger a$ terms in the collective operatorto the higher-order terms in the adiabatic self-consistent collectivecoordinate (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 ofcollective coordinate $q$, its conjugate momentum $p$ and the particle number$n$. According to the generalized Thouless theorem, $\hat G$ can be written asa 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 termsin the adiabatic expansion of $\hat G$. This relation serves as a prescriptionto determine the higher-order collective operators from the $a^\dagger a$ partof the collective operator, once it is given without solving the higher-orderequations of motion.

Relation of $a^\dagger a$ terms to higher-order terms in the adiabatic expansion for large-amplitude collective motion