Recent experiments showed that the conductance of Pt nanocontacts and nanowires is measurably reduced by adsorption of CO. We present DFT calculations of the electronic structure and ballistic conductance of a Pt monoatomic chain and a CO molecule adsorbed in an on-top position. We find that the main electronic molecule-chain interaction occurs via the $5sigma$ and $2pi^{star}$ orbitals of the molecule, involved in a donation/back-donation process similar to that of CO on transition-metal surfaces. The ideal ballistic conductance of the monoatomic chain undergoes a moderate reduction by about 1.0 G_0 (from 4 G_0 to 3.1 G_0) upon adsorption of CO. By repeating all calculations with and without spin-orbit coupling, no substantial spin-orbit induced change emerges either in the chain-molecule interaction mechanism or in the conductance.
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