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Suppose we want to determine the potential energy between a hydrogen atom and a surface, such as Cu(001), during hydrogen scattering on Cu(001) surface, assuming that the hydrogen atom does not change the surface structure.

I used DFT to performe single-point calculations to find the potential energy at various heights of the hydrogen atom above the surface. The potential energy was calculated using the equation: $$ E_{potential} = E_{H+surface} - E_{H} - E_{surface} $$ The results looked like

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The question then arises: is the method outlined above reliable for computing the H-surface potential energy? Can DFT accurately calculate the potential energy in dynamical processes?

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The real question is not about DFT but the validity of the hypothesis that the scattering process is slow compared to the copper atoms' dynamics.

DFT assumes the adiabatic approximation for the electronic dynamics. I.e., electrons follow the ionic dynamics instantaneously, always remaining in their instantaneous ground state. Considering the typical excitation energies and atomic scattering processes, I would take the validity of such approximation for granted in most cases.

A different story is related to the ionic time scales. The assumption that the hydrogen atom does not change the surface structure does not guarantee that a sequence of static calculations where $Cu$ ionic positions are allowed to relax to their instantaneous equilibrium values accurately represents what is going on in the actual process. Indeed, even keeping the same structure, copper ions may move according to the normal modes of the surface slab, and depending on the energy of the hydrogen atom, the scattering process may interact with such modes.

The most direct way to assess the validity of a static calculation would be to perform a dynamic calculation, using ab initio molecular dynamics, with the parameters relevant to the experimental situation.

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