Conic equations

As mentioned before it is conjectured that in projective relativity theory the coefficients gij of the conic equation are gravitational potentials and the coefficients of the hyperplane equation are electromagnetic potentials. We shall see, in fact, that the closest field equations for the 7, 3 are a combination of the classical Einstein gravitation equations and the Maxwell field equations. 

For reactors that are not cylindrical but conical, equations can still be established for the case with piecewise-constant characteristics. The diffusion equation is a mass balance that must now take into account the shape and the fact that the surface area of a transversal slice is proportional to for a conical reactor. Between x and x+ dx, the gas content is e dxc(x),... 

It has been verified that the packing of binary mixtures of spheres can be accurately described by the Westman conic equation" Therefore, the specific volume of any binary mixture of particles can be written as... 

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 ... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

While the presence of sign changes in the adiabatic eigenstates at a conical intersection was well known in the early Jahn-Teller literature, much of the discussion centered on solutions of the coupled equations arising from non-adiabatic coupling between the two or mom nuclear components of the wave function in a spectroscopic context. Mead and Truhlar [10] were the first to... 

In this chapter, we discussed the significance of the GP effect in chemical reactions, that is, the influence of the upper electronic state(s) on the reactive and nonreactive transition probabilities of the ground adiabatic state. In order to include this effect, the ordinary BO equations are extended either by using a HLH phase or by deriving them from first principles. Considering the HLH phase due to the presence of a conical intersection between the ground and the first excited state, the general fomi of the vector potential, hence the effective... 

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. 

In this equation, the gradient term U(qx)Wtta (Rx)U(qx) Vr,z (Rx) = W > (R x) Vr x (Rx) still appears and, as mentioned before, introduces numerical inefficiencies in its solution. Even though a truncated Bom-Huang expansion was used to obtain Eq. (53), wJja (Rx), although no longer zero, has no poles at conical intersection geometries [as opposed to the full W (Rx) matrix]. 

The ADT matrix for the lowest two electronic states of H3 has recently been obtained [55]. These states display a conical intersection at equilateral triangle geometi ies, but the GP effect can be easily built into the treatment of the reactive scattering equations. Since, for two electronic states, there is only one nonzero first-derivative coupling vector, w5 2 (Rl), we will refer to it in the rest of this... 

A chemical reaction takes place on a potential surface that is determined by the solution of the electronic Schrddinger equation. In Section, we defined an anchor by the spin-pairing scheme of the electrons in the system. In the discussion of conical intersections, the only important reactions are those that are accompanied by a change in the spin pairing, that is, interanchor reactions. We limit the following discussion to these class of reactions. 

The Solution for a Single Conical Intersection The curl equation for a two-state system is given in Eq. (26) ... 

Equation (165) yields the two components of t(<7, 0), the vectorial non-adiabatic coupling temi, for a distribution of two-state conical intersections expressed in terms of the values of the angular component of each individual non-adiabatic coupling term at the closest vicinity of each conical intersection. These values have to be obtained from ab initio treatments (or from perturbation expansions) however, all that is needed is a set of these values along a single closed circle, each surrounding one conical intersection. 

To study the two isolated conical intersections, we have to treat two-state curl equations that are given in Eq. (26). Here, the first 2 x 2 x mahix contains the (vectorial) element, that is, X012 and the second 2 x 2 x mahix contains X023- As before each of the non-adiabatic coupling terms, X012 and X023 has the following components ... 

Equation (171) is the an explicit curl equation for a coupling that does not has a source of its own but is formed due to the interaction between two real conical intersection. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

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