In surface catalysis, the adsorption of carbon monoxide on transition-metal electrodes represents the prototype of strong chemisorption. Notwithstanding significant changes in the molecular orbitals of adsorbed CO, spectroscopic experiments highlight a close correlation between the adsorbate stretching frequency and equilibrium bond length for a wide range of adsorption geometries and substrate compositions. In this work, we study the origins of this correlation, commonly known as Badgers rule, by deconvoluting and examining contributions from the adsorption environment to the intramolecular potential using first-principles calculations. Noting that intramolecular anharmonicity is preserved upon CO chemisorption, we show that Badgers rule for adsorbed CO can be expressed solely in terms of the tabulated Herzberg spectroscopic constants of isolated CO. Moreover, although it had been previously established using finite-cluster models that Badgers rule is not affected by electrical conditions, we find here that Badgers rule breaks down when the electrified surface is represented as a periodic slab. Examining this breakdown in terms of anharmonic contributions from the effective surface charge reveals limitations of conventional finite-cluster models in describing electrical conditions at metal electrodes.
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