Variation after K-projection in antisymmetrized molecular dynamics for low-energy dipole excitations in 10Be and 16O

A novel method of variation after $K$-projection is proposed to study dipole excitations in the framework of antisymmetrized molecular dynamics with the $\beta$-constraint. The method is applied to $^{10}$Be and $^{16}$O to describe low-energy dipole excitations. In the application to ${}^{10}\textrm{Be}$, two dipole states in the low-energy region are obtained. For ${}^{16}\textrm{O}$, the $1_1^-$ and $1_2^-$ states are obtained with remarkable dipole strength. The $1_1^-$ state is characterized by significant toroidal dipole (TD) strength and compressional dipole strength, whereas the $1_2^-$ state has significant TD strength and shows a developed $\alpha + {}^{12}\textrm{C}$ cluster structure. Dipole properties in $^{16}$O are discussed by analyzing the current densities of the dipole transitions.