Surface, adiabatic

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

Figure 5, Sketch of a conical intersection. The vectors x and X2 are the GD and DC respectively, that lift the degeneracy of the two adiabatic surfaces, The plane containing these vectors is known as the branching space.

Conical intersections can be broadly classified in two topological types peaked and sloped [189]. These are sketched in Figure 6. The peaked case is the classical theoretical model from Jahn-Teller and other systems where the minima in the lower surface are either side of the intersection point. As indicated, the dynamics of a system through such an intersection would be expected to move fast from the upper to lower adiabatic surfaces, and not return. In contrast, the sloped form occurs when both states have minima that lie on the same side of the intersection. Here, after crossing from the upper to lower surfaces, recrossing is very likely before relaxation to the ground-state minimum can occur. 

The Hamiltonian provides a suitable analytic form that can be fitted to the adiabatic surfaces obtained from quantum chemical calculations. As a simple example we take the butatriene molecule. In its neutral ground state it is a planar molecule with D2/1 symmetry. The lowest two states of the radical cation, responsible for the first two bands in the photoelectron spectrum, are and... 

Figure 4. Four iriceractirig adiabatic surfaces presented in terms of four adiabatic curves. The points Cj j = 1,2,3, stand for the three conical intersections.

The mathematical definition of the Born-Oppenheimer approximation implies following adiabatic surfaces. However, software algorithms using this approximation do not necessarily do so. The approximation does not reflect physical reality when the molecule undergoes nonradiative transitions or two... 

Non-Black-Surface Enclosures In the following discussion we are concerned with enclosures containing gray sources and sinks, radiatively adiabatic surfaces, and no absorbing gas. The calculation of interchange between a source and a sink under conditions involving successive multiple reflections from other source-sink surfaces in the... 

Electron-tunneling Model. Several models based on quantum mechanics have been introduced. One describes how an electron of the conducting band tunnels to the leaving atom, or vice versa. The probability of tunneling depends on the ionization potential of the sputtered element, the velocity of the atom (time available for the tunneling process) and on the work function of the metal (adiabatic surface ionization, Schroeer model [3.46]). 

First, the frozen Gaussian wavepackets just before the transition on the initial adiabatic surface i are expanded as ... 

Comparing J with the thermal energy, one can differentiate between the two processes. For adiabatic surfaces the coupling J kbT, while in the non-adiabatic case J kbT. A quantitative description of the two types of process can be made using the Landau-Zehner parameter g (Zulicke, 1985 Onuchic et al., 1986) defined in (5). Here, o)c is the frequency of the vibrational... 

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