Electronic subsystems stationary states

Equation (7) represents a crucial result for theoretical chemistry if the electronic wave function q) is stationary while the time evolution of the polyatomic system is in progress, the nuclei move in the field of force, the potential of which is equal to the energy of one of the eigenstates of the electronic subsystem. In this connection, the potential function Wm = Wm(Q) is referred to as the potential energy surface (PES) corresponding to the mth electronic state of the polyatomic system (10-12). 

In this section we will examine conditions implying stationary states of the electronic subsystem during the time the evolution of the entire polyatomic system is in progress. The nuclear motion will be treated classically (20,21). 

Our problem of seeking electronic stationary states corresponding to the evolution of the nuclear subsystem turned out to be identical with the problem of finding stationary solutions to Eq. (11), or of establishing conditions implying time independence of the expansion coefficients Am(t) in Eq. (13). 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

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. 

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