A model with all radial degrees of freedom frozen is used to simulate the $^{2\mathit{S}+1}$${\mathit{S}}^{\mathit{e}}$ intrashell states of four-valence-electron atoms. By inspecting the two-body densities, the latitudinal distribution functions, and by inspecting directly the eigenfunctions in appropriate subspaces, the features of geometric structure and of internal motion have been obtained. The effect of the spatial permutation symmetry and the parity have been emphasized. All even-parity states under consideration prefer the coplanar axial-symmetric structure. Two modes of motion are found, namely, the triple two-body collision mode and the rectangle-square-rectangle mode (or planar double two-body collision mode). It is found that the basic modes in quantum states are associated with periodic solutions of classical mechanics.