Coupled electronic/nuclear motion, local

Fig. 9.23. Square-well model for electronic mixing between two discrete states. The displacement toward resonance is derived from modulation of the energy levels by the coupling of the electronic levels to the nuclear motion of the surrounding medium. In configuration A, the electron is localized at the donor site B corresponds to the condition of quantum resonance between the two states C corresponds to the nuclear configuration in which the electron becomes localized on the acceptor site (Reprinted from R. J. D. Miller, G. McLendon, A. J. Nozik, W. Schmickler, and F. Willig, Surface Electron Transfer Processes, p. 4, copyright 1995 VCH-Wiley. Reprinted by permission of John Wiley Sons, Inc.)...

In situ STM of metalloproteins with localized low-lying redox levels can be expected to follow ET patterns similar to metalloprotein ET in homogeneous solution and at electrochemical surfaces. The redox level is thus strongly coupled to the protein and solvent environment. A key notion is that the vacant local level (oxidized form) at equilibrium with the environmental nuclear motion is located well above the Fermi levels of both the substrate and tip, whereas, the occupied level (reduced form) at equilibrium is located well below the Fermi levels. Another central notion is that the local redox level at the transition metal centre is still much lower than environmental protein or solvent electronic levels. The redox level therefore constitutes a pronounced indentation in the tunnel barrier. This alone would strongly enhance tunnelling. Configurational fluctuations in the environment can, secondly take the redox level to such low values that temporary physical population occurs. This requires nuclear activation but can still be favourable due to the much shorter electron tunnel distances... 

If the regions of nonadiabatic behavior are well localized in the configuration space M, an (F — l)-dimensional hypersurface can be defined at which the nonadiabatic transitions may take place this hypersurface is referred to as the crossing seam. The coupled relations, Eq. (14), describing the corresponding nonseparable electronic and nuclear motion, are to be solved at the seam. Elsewhere, the evolution of the polyatomic system can be then treated adiabatically (49,50). 

Oscillations in time of quantal states are usually much faster than those of the quasiclassical variables. Since both degrees of freedom are coupled, it is not efficient to solve their coupled differential equations by straightforward time step methods. Instead it is necessary to introduce propagation procedures suitable for coupled equations with very different time scales short for quantal states and long for quasiclassical motions. This situation is very similar to the one that arises when electronic and nuclear motions are coupled, in which case the nuclear positions and momenta are the quasiclassical variables, and quantal transitions lead to electronic rearrangement. The following treatment parallels the formulation introduced in our previous review on this subject [13]. Our procedure introduces a unitary transformation at every interval of a time sequence, to create a local interaction picture for propagation over time. 

The most common interactions are dipolar coupling between two spins, chemical shielding and quadrupolar interaction for spins I> 1. The chemical-shielding interaction arises from the motion of electrons around a nucleus induced by the external applied static magnetic field. This motion generates local magnetic fields that modify the total field experienced by the nucleus, and are characteristic of the local chemical environment of the nuclei. All nuclei with a spin /> 1 possess a quadrupolar moment Q that interacts with the EFG Vrs — d2 V/dXfdx at the nuclear site. The EFG tensor Vrs is a symmetric tensor with zero trace Tr( E) = A E = 0 (from Laplace equation). 

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