Asymptotic regions

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

An example of a family of classical trajectories together with the caustics are shown in Fig. 21. The caustics in the asymptotic region appear periodically as turning points of the vibrational motion. In the rearrangement region, the trajectories are no more periodic and the caustics appear rather randomly. As is seen, the present method is demonstrated to work well. 

In the asymptotic region, an electron approximately experiences a Z /f potential, where Z is the charge of the molecule-minus-one-electron ( Z = 1 in the case of a neutral molecule) and r the distance between the electron and the center of the charge repartition of the molecule-minus -one-electron. Thus the ip orbital describing the state of that electron must be close to the asymptotic form of the irregular solution of the Schrodinger equation for the hydrogen-like atom with atomic number Z. ... 

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

The quality of the TD-DFT results is determined by the quality of the KS molecular orbitals and the orbital energies for the occupied and virtual states. These in turn depend on the exchange-correlation potential. In particular, excitations to Rydberg and valence states are sensitive to the behavior of the exchange-correlation potential in the asymptotic region. If the exchange-correla-... 

Now we know that the asymptotic decay of the density far from atom B (but not so far that ps still dominates the molecular density p ) is determined by the ionization energy Ib- The asymptotic decay of the density pa is however also determined by the highest occupied Kohn-Sham orbital Kohn-Sham potential cannot go to zero in the asymptotic region of atom B otherwise the density decay around atom B would be determined by 1a- As the Hartree and the external potential go to zero in the outer asymptotic region it follows from Eq. (145) that the only possibility is that the molecular exchange-correlation potential around atom B is given by... 

Eigure 4.1 shows a schematic of an excited-state potential energy surface and superimposed on it is a wavepacket, which is the initial wavepacket for a photodissociation process. An analysis line is drawn perpendicular to the contour lines in the asymptotic region of the surface, where there is no longer any substantial interaction between the separating fragments and the contour... 

As the kinetic energy operator has no off-diagonal elements in the space-fixed axis representation we may more easily correct for the fact that the analysis line is not tmly in the asymptotic region as far as the centrifugal representation is concerned. We do this first by subtracting the residual centrifugal repulsion, I I + l)/(2p / ), from the radial kinetic energy at the analysis line in the product channel. Thus we use the expression... 

In addition, we are comparing these molecules with a minimal basis. With eight valence orbitals and ten electrons, configurations that produce some angular correlation in the 2s shell cannot occur in the asymptotic region. The upshot is that there is just one principal standard tableaux function at long distance, and this is shown in Table 11.13. Because of the antisymmetry in the columns of standard tableaux functions, we see that this function represents two noninteracting N atoms. 

The wave functions change character, of course, during the dissociation process. The asymptotic region is the simplest and we start with that. Table 16.9 shows the most important Rumer tableaux when CO and H2 are well separated from one another. 

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