Surfaces overlap matrix elements

Pt surfaces tend to restructure into overlayers with an even higher density of Pt atoms than the close-packed (111) surface [21]. The Pt atoms are closer to each other on the reconstructed surfaces than in the (111) surface. The overlap matrix elements and hence the bandwidth are therefore larger, the d bands are lower and consequently these reconstructed surfaces bind CO even weaker than the (111) surface. The reconstructed Pt surfaces are examples of strained overlayers. The effect of strain can be studied theoretically by simply straining a slab. Examples of continuous changes in the d band center and in the stability of adsorbed CO due to strain are included in Figure 4.10. The effect due to variations in the number of layers of a thin film of one metal on another can also be described in the d band model [22,23]. 

The overlap matrix elements are 5n = 522 = 4 and Sn = —2. Thus if we know the various exchange d Coulomb integrals we can evaluate the lower adiabatic surface. From equations (4.65), and (4.66) above, we see that all we need to do is to relate these integrals to the diatomic fragment singlet and triplet curves. Thus for the two gas-phase atoms a and b we have... 

The delta function, 5, limits the analysis to elastic processes. The tunneling matrix element, M, is determined by the overlap of the surface wave functions of the two metal subsystems at a particular separation surface, which also reflects the energy-lowering resonance associated with the interplay of the two states. The tunneling current may be found by summing over... 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

Rates for nonradiative spin-forbidden transitions depend on the electronic spin-orbit interaction matrix element as well as on the overlap between the vibrational wave functions of the molecule. Close to intersections between potential energy surfaces of different space or spin symmetries, the overlap requirement is mostly fulfilled, and the intersystem crossing is effective. Interaction with vibrationally unbound states may lead to predissociation. 

Simple metals Transition metals Interatomic interaction. See Overlap interaction Interatomic matrix elements. See Matrix elements Interbond matrix elements, 1441f Interfaces. See also Surfaces... 

These expressions are the usual perturbation formulae. Here f and Ej denote the energies of the two interacting MOs, Sy their overlap integral, Hy their interaction matrix element and Eq the mean of the orbital energies. Since in a comparative analysis of a given surface the overlap is the main variable quantity, we have expressed the various formulae in terms of Sy. To this purpose we have made use of the assumption that the interaction matrix element Hy is proportional to the overlap integral Sy, i.e. Hy = KSy (K < 0). 

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