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Consider the helium dimer cation.  Its configuration is Sigma_g^2 Sigma_u^1, which corresponds to a bond order of 1/2.  The equivalent localized description has two non-bonding orbitals in the alpha space (from Sigma_g^1 Sigma_u^1) and a bond in the beta (Sigma_g^1).  So the half bond order comes from the beta space, with no contribution from the alpha.

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