Correlated-Gaussian approach to linear-chain states -Case of four $\alpha$-particles-

We show that correlated Gaussians with good angular momentum and parity provide flexible basis functions for specific elongated shape. As its application we study linear-chain states of four-alpha particles in variation-after-projection calculations in which all the matrix elements are evaluated analytically. We find possible chain states for $J=0^+$, $2^+$, $4^+$ and perhaps $6^+$ with the bandhead energy being about 33 MeV from the ground state of $^{16}$O. No chain states with $J\geq 8$ are found. The nature of the rotational sequence of the chain states is clarified in contrast to a rigid-body rotation. The quadrupole deformation parameters estimated from the chain states increase from 0.59 to 1.07 for $2^+$ to $6^+$. This work suggests undeveloped fields for the correlated Gaussians beyond those problems which have hitherto been solved successfully.