  The generator coordinate method (GCM) has been a well-known method to
describe nuclear collective motions. In this method, one specifies {\it a
priori} the relevant collective degrees of freedom as input of the method,
based on empirical and/or phenomenological assumptions. We here propose a new
extension of the GCM, in which both the basis Slater determinants and weight
factors are optimized according to the variational principle. Applying this
method to $^{16}$O and $^{28}$Si nuclei with the Skyrme functional, we
demonstrate that the optimized bases correspond to excited states along a
collective path, unlike the conventional GCM which superposes only the local
ground states. This implies that a collective coordinate for large amplitude
collective motions is determined in a much more complex way than what has been
assumed so far.
