RVB Ansatze for the Linear Chain

For the linear chain there is only one Kekule structure in each LR-SPO subspace. Therefore, two main different ansatze including long SP may be considered. See, for instance, Refs. 24, 31 and 35. 

Five-bond-range RVB ansatz The M-bond RVB variational local-ized-site cluster expansion ansatze can be defined [31] as 

Numerical results for such ansatze with connected-cluster patterns involving up to M = 5 bonds have been treated so far. The hierarchy of patterns have been defined as 

The xn, yn and zn are variational parameters submitted to translational-symmetry constraints, vn+m = vn, v = x, y, z, where m = 1 for a regular chain, and m = 2 when alternating translational symmetry conditions are desired. 

Infinite-bond-range RVB ansatze Infinite range RVB ansatze are defined [35] by 

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