Nonadiabatic dynamics near metal surfaces: Decoupling quantum equations of motion in the wide-band limit

When an electron in a localized orbital approaches a continuum, resonant charge transfer can occur. For instance, when a beam of ions is scattered from a metal surface, neutralization can take place through the transfer of an electron either from ion to metal—in the case of a negative ion—or from metal to ion—in the case of a positive ion. Similarly, a neutral atom may be ionized by removing an electron from the metal surface. In this paper, we introduce the wide-band diabatic dynamics (WBDD) method for simulating the quantum dynamics of such systems by decoupling the equations of motion for the ionic and neutral diabats. We show numerically that our approximation accurately describes the quantum nuclear dynamics near the metal surface. Although we treat the case of ion neutralization, our method is general, and can be applied to any quantum state interacting with a continuum via a position-dependent interaction, provided that the wide-band approximation holds.

Figure 1

The diabatic curves ${ϵ}_a\left(R\right)$ (solid curves), metal band (cross-hatched area), and coupling function $V\left(R\right)$ (dashed curve) for our first model system. The model system parameters were chosen to resemble the neutralization of a hydrogen anion at a metal surface. Depending on the choice of $E_a$, the adatom diabat can stay within the conduction band of the metal, or can dip above or below the metal conduction band edge.

Figure 2

(Color online) The population and the expectation values for position and momentum on the adatom state $\mid a⟩$ vs time using exact quantum dynamics (solid line), WBDD (dashed line), and Ehrenfest dynamics (dash-dotted line) for the model shown in Fig. 1. Position is given in Å and momentum in units of Å amu/fs. The WBDD method gives an excellent approximation to the exact quantum dynamics.

Figure 3

(Color online) The population and the expectation values for position and momentum on two different metal diabats $\mid k⟩$ vs time using exact quantum dynamics (solid line), WBDD (dashed line), and Ehrenfest dynamics (dash-dotted line). The WBDD method again gives an excellent prediction of the population, position, and momentum in a given diabatic electronic state $\mid k⟩$. The two metal diabats selected corresponded to $E_k=1.09,1.52\mathrm{eV}$. These results also show that the adatom trajectory depends on the electronic state of the system. This dependence cannot be obtained by the Ehrenfest method, which can only give average values for nuclear properties.

Figure 4

(Color online) The population $⟨{\psi }_a\left(t\right)\mid {\psi }_a\left(t\right)⟩$ of the adatom diabat vs time given an offset of $E_a=3.5\mathrm{eV}$ and $E_a=-1.5\mathrm{eV}$ using exact quantum dynamics (solid line), WBDD (dashed line), and Ehrenfest dynamics (dash-dotted line). In these systems, there are regions in which the adatom diabat is outside the metal band (see Fig. 1); thus, the wide-band approximation does not always hold. The WBDD method still performs reasonably well, but clear divergences from the exact result are visible due to the non-Markovian nature of the real dynamics.

Figure 5

The diabatic curves ${ϵ}_a\left(R\right)$ (solid curves), metal band (cross-hatched area), and coupling function $V\left(R\right)$ (dashed curve) for our second model system. The model system parameters were chosen to resemble the ionization of neutral hydrogen atom at a metal surface.

Figure 6

(Color online) The population and the expectation values for position and momentum on the adatom state $\mid a⟩$ vs time using exact quantum dynamics (solid line), WBDD (dashed line), and Ehrenfest dynamics (dash-dotted line) for the model shown in Fig. 5. Position is given in Å and momentum in units of Å amu/fs. The Ehrenfest method gives an inaccurate prediction of the population of state $\mid a⟩$ because the single Ehrenfest trajectory traps on the surface and continues to decay, as can be seen in 6b, 6c.

Figure 7

(Color online) The population and the expectation values for position and momentum on two different metal diabats $\mid k⟩$ vs time using exact quantum dynamics (solid line), WBDD (dashed line), and Ehrenfest dynamics (dash-dotted line). The two metal diabats selected corresponded to $E_k=-0.26,0.24\mathrm{eV}$. The average classical trajectory predicted by the Ehrenfest method traps at the surface, while the exact and WBDD methods predict that trapping will be dependent on the electronic state. The $\mid a⟩$ state is reflected [see Fig. 6b] while the trapping of the $\mid k⟩$ states depends on their position in the band [see 7b].