We propose coherent (Schr?dinger cat-like) states adapted to the parity symmetry providing a remarkable variational description of the ground and first excited states of vibron models for finite (N)-size molecules. Vibron models undergo a quantum shape phase transition (from linear to bent) at a critical value ξ _c of a control parameter. These trial cat states reveal a sudden increase in vibration-rotation entanglement linear (L) and von Neumann (S) entropies from zero to L (N) _(cat)(ξ)?1-2/ ∈πN(to be compared with L ~((N)) _(max)(ξ) = 1 1/(N + 1)) and S ~((N)) _(cat)(ξ)?1/2 log _2(N+1), respectively, above the critical point, ξ > ξ _c, in agreement with exact numerical calculations. We also compute inverse participation ratios, for which these cat states capture a sudden delocalization of the ground-state wave packet across the critical point. Analytic expressions for entanglement entropies and inverse participation ratios of variational states, as functions of N and ξ, are given in terms of hypergeometric functions.
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