the spreading width of 1p1h states, which we denote $\gamma_{12}$. We define the GQR TD state as the Tamm-Dancoff (TD) state with the largest quadrupole strength among all TD states, where the TD states mean the states obtained in the TDA, i.e., by the diagonalization within the model space limited to the 1p1h configurations. The GQR TD state also couples to 2p2h states, and hence it should have the spreading width due to the coupling. This is similar to $\gamma_{12}$, but we introduce a separate symbol $\Gamma_2$ since the GQR TD state is a special state consisting of a coherent superposition of many unperturbed 1p1h excitations. $d_{\rm 2p2h}$ is the average spacing of background 2p2h states. The residual interaction also acts among the 2p2h states, and the mixing among the 2p2h states causes a spreading width of the 2p2h states, which we denote $\gamma_{22}$.

In the following we shall evaluate all these quantities in order to clarify the damping mechanism of the GQR in $^{208}$Pb (Sec. \[sec:mechanism\] and Sec. \[sec:spreading\_width\] ). We also study whether there are specific states among 2p2h states which strongly couple with the GQR mode (Sec. \[sec:surfacce\_vib\]) and then discuss the difference of the nature associated 