Mathematical modeling of changes in density of near-surface ionic layers in semi-infinite metals. Equations for displacements of ionic layers

In this work, we propose a mathematical model for describing the change in the ion density of the near-surface ionic layers of a semi-infinite metal.  Through averaging over the subsystem of conduction electrons, we obtain in the adiabatic approximation an effective Hamiltonian of the ionic subsystem of a semi-infinite metal, which models the effect of the "metal–vacuum" separation surface on the structure of the near-surface ionic layers.  We calculate the free energy of such a model and, by its minimization, obtain an equation for finding the displacements $\mathbf{\xi}_{m}$ of the ionic layer $m$.  We show that in the absence of an inhomogeneous distribution of the electronic subsystem $\mathbf{\xi}_{m}\equiv 0$.