Conical Coordinates

As a partial check on the derivations in the conical coordinates, it should be possible to recover two, easily identified, special cases—the radial flow between parallel disks and the axial Poiseuille flow in an cylindrical annular gap. The parallel-disk flow (Section 5.5) is the case where 0 = 0, with x taking the role of r and y taking the role of z. In this case, h = De/2 + x = r. The momentum equations become... 

In this new conical coordinate system the conditions for frictional contact can be written as... 

Fig 3. Separation, stick and slip in conical coordinate system. 

Use now this equation to describe liquid film flow in conical capillary. Let us pass to spherical coordinate system with the origin coinciding with conical channel s top (fig. 3). It means that instead of longitudinal coordinate z we shall use radial one r. Using (6) we can derive the total flow rate Q, multiplying specific flow rate by the length of cross section ... 

Molecular aspects of geometric phase are associated with conical intersections between electronic energy surfaces, W(Q), where Q denotes the set of say k vibrational coordinates. In the simplest two-state case, the W Q) are eigen-surfaces of the nuclear coordinate dependent Hermitian electronic Hamiltonian... 

Hence, the expression of Eq. (5) indicates that, in a polar coordinate system, Eq. (4) will remain unchanged even if the position of the conical intersection is shifted from the origin of the coordinate system. 

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. 

Single surface calculations with proper phase treatment in the adiabatic representation with shifted conical intersection has been performed in polai coordinates. For this calculation, the initial adiabatic wave function tad(9, 4 > o) is obtained by mapping t, to) ittlo polai space using the relations,... 

At this point, it is important to note that as the potential energy surfaces are even in the vibrational coordinate (r), the same parity, that is, even even and odd odd transitions should be allowed both for nonreactive and reactive cases but due to the conical intersection, the diabatic calculations indicate that the allowed transition for the reactive case ate odd even and even odd whereas in the case of nomeactive transitions even even and odd odd remain allowed. 

Reactive State-to-State Transition Probabilities when Calcnladons are Performed Keeping the Position of the Conical Intersection at the Origin of the Coordinates... 

Reactive State-to-State Transition ftobabilides when Calculations are Performed by Shifting the Position of Conical Intersection from the Origin of the Coordinate System... 

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. 

It would be convenient for obtaining the expressions of the gradient of the hyperangle Jacobi coordinates to inboduce the physical region of the conical intersection in the following manner ... 

The gradient of v l with respect to Jacobi coordinates (the vector potential) considering the physical region of the conical intersection, is obtained by using Eqs. (C.6-C.8) and after some simplification ( ) we get,... 

Figure 4. The H3 and H4 loops. Ac the center, the conical intersections are shown schematically an equilateral triangle for H3 and a perfect tetrahedron for Kt, <2p> Jid Q, are the phase-preserving and phase-inverting coordinates, respectively.

The two coordinates defined for H4 apply also for the H3 system, and the conical intersection in both is the most symmetric structure possible by the combination of the three equivalent structures An equilateral triangle for H3 and a perfect tetrahedron for H4. These sbnctures lie on the ground-state potential surface, at the point connecting it with the excited state. This result is generalized in the Section. IV. 

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