Cross point

To see physically the problem of motion of wavepackets in a non-diagonal diabatic potential, we plot in figure B3.4.17 a set of two adiabatic potentials and their diabatic counterparts for a ID problem, for example, vibrations in a diatom (as in metal-metal complexes). As figure B3.4.17 shows, if a wavepacket is started away from the crossing point, it would slide towards this crossing point (where where it would... 

Figure B3.4.17. When a wavepacket comes to a crossing point, it will split into two parts (schematic Gaussians). One will remain on the same adiabat (difFerent diabat) and the other will hop to the other adiabat (same diabat). The adiabatic curves are shown by fidl lines and denoted by ground and excited die diabatic curves are shown by dashed lines and denoted 1, 2.

The problem of branching of the wavepacket at crossing points is very old and has been treated separately by Landau and by Zener [H, 173. 174], The model problem they considered has the following diabatic coupling matrix ... 

Figure C3.2.1. A slice tlirough tlie intersecting potential energy curves associated witli tlie K-l-Br2 electron transfer reaction. At tlie crossing point between tlie curves (Afy, electron transfer occurs, tlius Tiarjiooning tlie species,...

Figure 5. A cut across the ground state (GS) and the excited state (ES) potential surfaces of the H4 system. The parameter Qp is the phase preserving nuclear coordinate connecting the H(lll) with the transition state between H(I) and H(1I) (Fig, 4). Keeping the phase of the electronic wave function constant, this coordinate leads from the ground to the excited state. At a certain point, the two surfaces must touch. At the crossing point, the wave function is degenerate.

Worth and Cederbaum [100], propose to facilitate the search for finding a conical intersection if the two states have different symmetiies If they cross along a totally symmetric nuclear coordinate, then the crossing point is a conical intersection. Even this simplifying criterion leaves open a large number of possibilities in any real system. Therefore, Worth and Cederbaum base their search on large scale nuclear motions that have been identified experimentally to be important in the evolution of the system after photoexcitation. 

To continue, we assume the following situation We concentrate on an x-y plane, which is chosen to be perpendicular to the seam. In this way, the pseudomagnetic field is guaranteed to be perpendicular to the plane and will have a nonzero component in the z direction only. In addition, we locate the origin at the point of the singularity, that is, at the crossing point between the plane and the seam. With these definitions the pseudomagnetic field is assumed to be of the form [113]. 

Next we apply this distribution to the case where F = 0, that is, to the case where no fronts have crossed point x. There are several aspects to note about this situation ... 

The shape of the broad absorption curve in Figure 9.17 is typical of that of any dye suitable for a laser. It shows an absorption maximum to low wavelength of the Og band position, which is close to the absorption-fluorescence crossing point. The shape of the absorption curve results from a change of shape of the molecule, from Sq to 5i, in the... 

If the projected pipeline is situated in an area with dc railway lines, rail/soil potential measurements should be carried out at crossing points and where the lines run parallel a short distance apart, particularly in the neighborhood of substations, in order to ascertain the influence of stray currents. Potential differences at the soil surface can give information on the magnitude of stray current effects in the vicinity of dc railway lines. It is recommended that with existing pipelines the measurements be recorded synchronously (see Section 15.5) and taken into account during design. 

The transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

Now we make the usual assumption in nonadiabatic transition theory that non-adiabaticity is essential only in the vicinity of the crossing point where e(Qc) = 0- Therefore, if the trajectory does not cross the dividing surface Q = Qc, its contribution to the path integral is to a good accuracy described by adiabatic approximation, i.e., e = ad Hence the real part of partition function, Zq is the same as in the adiabatic approximation. Then the rate constant may be written as... 

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