Oscillation function friction coefficient

Since DFT calculations are in principle only applicable for the electronic ground state, they cannot be used in order to describe electronic excitations. Still it is possible to treat electronic exciations from first principles by either using quantum chemistry methods [114] or time-dependent density-functional theory (TDDFT) [115,116], First attempts have been done in order to calculate the chemicurrent created by an atom incident on a metal surface based on time-dependent density functional theory [117, 118]. In this approach, three independent steps are preformed. First, a conventional Kohn-Sham DFT calculation is performed in order to evaluate the ground state potential energy surface. Then, the resulting Kohn-Sham states are used in the framework of time-dependent DFT in order to obtain a position dependent friction coefficient. Finally, this friction coefficient is used in a forced oscillator model in which the probability density of electron-hole pair excitations caused by the classical motion of the incident atom is estimated. 

As required by the energy-conservation law reflected by the function in (2.44), each intradoublet transition is accompanied by emission or absorption of a phonon with energy hAo-Equation (2.41) describes either damped oscillations (at tls < 2do) or exponential relaxation ( TLs > 2do). Since ijjls grows with increasing temperature, there may be a cross-over between these two regimes at P such that 2h QoJ Ao)coth P hAo) = 2d . If the friction coefficient... 

Figure 2. The timebehavi< of ttecorrelationfuiictioii (/),Eq. (2.6), with/t>l,

The application of this method requires knowledge of the explicit form of at least one of the two funetions (F(/)F(0)> or y t), in order to find a solution of the equation. Variants of the approaeh, developed up to the present time are based on different ways of modeling of the dissipation term y(t), conneeted with the secondary zone of atoms [ 18-20]. Adelman and Garrison use Debye s model for phonons of the solid and obtain an equation for the dissipation term whieh, ean be solved numerically. Doll and Dion propose y t) as a linear combination of conveniently chosen functions, where the coefficients are determined by numerieal self-consistency. Another possibility is to model the microscopic interactions in the lattice of the solid in order to derive a dissipation term. Tully presents the friction as a white noise or positionally autocorrelated function of a Brownian oscillator, including both oscillation and dissipation terms. 

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