Localized eigenstates

Figure 1.8 The probability density of several continuum eigenstates of the Hamiltonian in Eq. (24) plotted on the baseline of their corresponding energy. The potential is also plotted for convenience. Note that most continuum states (dashed lines) are delocalized and have a very small amplitude inside the potential well between the two barriers, whereas there are continuum functions that are localized inside the well. The localized eigenstate (solid line) is the same as shown in Figure 1.1.

This Hamiltonian may hold Nioc discrete, localized eigenstates of complex energy with purely exponential decay. The detector is placed at the barrier... 

Kohn-Sham Hamiltonian of equation (10). The functions x ) form the localized basis set for the subsystem a. Note that in equation (32) the occupation number for each localized eigenstate is approximated by the Fermi function with a very low but finite temperature. 

The next significant development in the history of the geomebic phase is due to Mead and Truhlar [10]. The early workers [1-3] concenbated mainly on the specboscopic consequences of localized non-adiabatic coupling between the upper and lower adiabatic elecbonic eigenstates, while one now speaks... 

The local and partial den.sitie.s of. state have been calculated u.sing projections of the plane-wave components of the eigenstates onto spherical waves centred at the atomic sites[.53]. 

Similar to the diatom-diatom reaction, the initial wavefunction is chosen as the direct product of a localized translational wavepacket for R and a specific (JMe) state for the atom-triatom system with a specific rovibrational eigenstate (z/o, Lo,Bo) f°r the triatom ABC ... 

Neglecting the tunneling splitting we can assume the eigenstates Oct) to be localized in the wells of the deformation potential so that... 

Figure 3 (right) compares the life times of the collinear antisymmetric stretch eigenstates in one, two, and three dimensions, for principal quantum numbers N = 2... 8, and for singlet symmetry. Apart from unsystematic fluctuations - which we attribute to local avoided cross-... 

Next we can consider the instantaneous eigenstates themselves At each time step, ic(f) is diagonalized by the orthogonal transformation BtkB. One can get a rough idea about over how many local-mode chromophores the eigenstates extend by considering the inverse participation ratio ... 

As an isolated adsorbate particle approaches a metal surface, the eigenstate density of localized electrons in the paridde increases due to its interaction with metal ions and electrons in the metal and the electron level is broadened into a level band. With an increasing number of metal ions and electrons that interact with an approaching adsorbate particle, the width of the electron level band in the particle increases in proportion to erq) (— k d) where k is the energy barrier... 

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