  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.
