Intersection points

Figure B3.4.16. A generic example of crossing 2D potential surfaces. Note that, upon rotating around the conic intersection point, the phase of the wavefunction need not return to its original value.

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

Conical intersections can be broadly classified in two topological types peaked and sloped [189]. These are sketched in Figure 6. The peaked case is the classical theoretical model from Jahn-Teller and other systems where the minima in the lower surface are either side of the intersection point. As indicated, the dynamics of a system through such an intersection would be expected to move fast from the upper to lower adiabatic surfaces, and not return. In contrast, the sloped form occurs when both states have minima that lie on the same side of the intersection. Here, after crossing from the upper to lower surfaces, recrossing is very likely before relaxation to the ground-state minimum can occur. 

In the nonrelativistic case, at a given the quantity x was shown to be invariant under the hansformation in Eq. (16), for a = y = 0. This invariant, whose value depends on was used to systematically locate confluences, [18-21], intersection points at which two distinct branches of the conical intersection seam intersect. Here, we show that the scalar triple product, gij X is the invariant for q = 3. Since the g t, and h cannot be... 

Before we continue with the construction of the sub-Hilbert spaces, we make the following comment Usually, when two given states fomr conical intersections, one thinks of isolated points in configuration space. In fact, conical intersections are not points but form (finite or infinite) seams that cut through the molecular configuration space. However, since our studies are carried out for planes, these planes usually contain isolated conical intersection points only. 

As we have seen, the sub-Hilbert spaces are defined for the whole configuration space and this requirement could lead, in certain cases, to situations where it will be necessary to include the complete Hilbert space. However, it frequently happens that the dynamics we intend to study takes place in a given, isolated, region that contains only part of the conical intersection points and the question is whether the effects of the other conical intersections can be ignored ... 

Recently, Xu et al. [11] studied in detail the H3 molecule as well as its two isotopic analogues, namely, H2D and D2H, mainly with the aim of testing the ability of the line integral approach to distinguish between the situations when the contour surrounds or does not surround the conical intersection point. Some time later Mebel and co-workers [12,72-74,116] employed ab initio non-adiabatic coupling teiins and the line-integral approach to study some features related to the C2H molecule. 

The flexibility factor k applies to bending in any plane. The flexibility factors k and stress intensification factors shall not be less than unity factors for torsion equal unity. Both factors apply over the effective arc length (shown by heavy centerlines in the sketches) for curved and miter bends and to the intersection point for tees. 

Another entire column with a partially vaporized feed, a hqnid-sidestream rate equal to D and withdrawn from the second stage from the top, and a total condenser is shown in Fig. 13-36. The specified concentrations are Xp = 0.40, Xb = 0.05, and Xo = 0.95. The specified L/V ratio in the top sec tion is 0.818. These specifications permit the top operating hne to be located and the two top stages stepped off to determine the liqnid-sidestream composition Xs = 0.746. The operating line below the sidestream must intersect the diagonal at the blend of the sidestream and the overhead stream. Since S was specified to be equal to D in rate, the intersection point is... 

Turn now to Nomograph 2 and locate in their respective scales the air volume and the calculated system capacity. A straight line between these two points intersects the scale in between them, thus providing at the intersection point the value of the solids ratio, if the sohds ratio exceeds 15, assume a larger hue size. 

The critical speed map shown in Figure 5-15 can be extended to include the second, third, and higher critical speeds. Such an extended critical speed map can be very useful in determining the dynamic region in which a given system is operating. One can obtain the locations of a system s critical speeds by superimposing the actual support versus the speed curve on the critical speed map. The intersection points of the two sets of curves define the locations of the system s critical speeds. 

When the intersection points lie below the 0.5 slope line, the system is said to have a bending critical speed. It is important to identify these points, since they indicate the increasing importance of bending stiffness over support stiffness. 

Find the intersection point of the curves for known values of L and V on the grid. Interpolate, if necessary. Mark this point A. 

Connect point A with a ruler to the known value of Is on the appropriate scale and read the final result F on the intersection point of a ruler with the II scale. 

The use of the nomograph is as follows Find the intersecting point of the curves of continuous phase and dispersed phase viscosities on the binary field (left side of nomograph). A line is drawn from this point to the common scale volume fraction of dispersed phase and continuous phase liquids. The intersection of this line with the Viscosity of Emulsion scale gives the result. 

The fan volume flow q, and its corresponding Ap,g, can be found when a Ap,o( - chart is drawn the duct parabola and experimental Ap, t both equal f(q ) (Eig. 9.47). The experimental curve Apj j = f q ) is called the fan characteristic curve, and the duct static pressure drop dependency on the duct volume flow is the characteristic curve. The characteristic curve intersection point is called fan operating point. 

Beyond the CMC, surfactants which are added to the solution thus form micelles which are in equilibrium with the free surfactants. This explains why Xi and level off at that concentration. Note that even though it is called critical, the CMC is not related to a phase transition. Therefore, it is not defined unambiguously. In the simulations, some authors identify it with the concentration where more than half of the surfactants are assembled into aggregates [114] others determine the intersection point of linear fits to the low concentration and the high concentration regime, either plotting the free surfactant concentration vs the total surfactant concentration [115], or plotting the surfactant chemical potential vs ln( ) [119]. 

The slope of a line from the intersection point of the feed composition, xp with the 45° line on Figure 8-2 is given by q/(q - 1) = - q/(l - q). Physically this gives a good approximation of the mols of saturated liquid that will form on the feed plate by the introduction of the feed, keeping in mind that tmder some thermal conditions the feed may vaporize liquid on the feed plate rather than condense any. 

---