Intersection points calculation

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. 

The potential surfaces of the ground and excited states in the vicinity of the conical intersection were calculated point by point, along the trajectory leading from the antiaromatic transition state to the benzene and H2 products. In this calculation, the HH distance was varied, and all other coordinates were optimized to obtain the minimum energy of the system in the excited electronic state ( Ai). The energy of the ground state was calculated at the geometry optimized for the excited state. In the calculation of the conical intersection... 

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. 

Sorption curves obtained at activity and temperature conditions which have been experienced to be not able to alter the polymer morphology during the test, i.e. a = 0.60 and T = 75 °C, for as cast (A) and for samples previously equilibrated in more severe conditions, a = 0.99 and T = 75 °C (B), are shown in Fig. 13. According to the previous discussion, the diffusion coefficient, calculated by using the time at the intersection points between the initial linear behaviour and the equilibrium asymptote (a and b), for the damaged sample is lower than that of the undamaged one, since b > a. The morphological modification which increases the apparent solubility lowers, in fact, the effective diffusion coefficient. 

Figure 4.26. Shelf-life calculation for active components A and B in a cream see data file CREAM.dat. The horizontals are at the j = 90 (specification limit at t = shelflife) resp. y = 95% (release limit) levels. The linear regression line is extrapolated until the lower 90%-confidence limit for Kfl = a + h x intersects the SLs the integer value of the real intersection point is used. The intercept is at 104.3%.

The molar mass of the degraded sample can easily be calculated by assuming that the overlap parameter is equal to the intersection point of the horizontal with the fine of slope 1. 

The same relations axe more easily obtained from a very simple one-dimensional model, in which only one degree of freedom is considered in this case the two potential energy surfaces reduce to parabolas, and the energy of activation is simply calculated from their intersection point (see Problem 1). 

Calculate the intersection point of these two parabolas and the energy of reorganization. Prove Eqs. (6.6) and (6.7) for this system. 

Matveev, Y.I. and Ablett, S. 2002. Calculation of the Cg and Tg intersection point in the state diagram of frozen solutions. Food Hydrocolloids 16, 419-422. 

The next step is to determine whether the calculated -points are valid or not. The simplest test is to determine which reactions for formation of the sulphur compounds have the highest values of x- i.e. the lowest (most negative) AG. If a potential intersection point, X(ijk), Y(ijk) is valid then the values of x(l,i), xC t 0) and x(1,k) should fulfil three criteria. They should be higher than any other value of x(l g). greater than or equal to zero and equal to each other except for rounding off errors. 

For simplicity intersections between pairs of lines ij and jk can be taken in numerical order, evaluating x(ijk) and Y(ijk) with i < j < k and k n, where n is the number of compounds. The maximum number of intersection points that need to be examined is 3), i.e. n(n-1)(n-2)/6. For Uo compounds it is 9880. This is not a very large number for a computer to handle because the actual calculations are very trivial. Nevertheless, methods are available for improving the programme efficiency as briefly described under program comparisons. 

The coordinates of the two valid intersection points for each real line can be stored in an array. When all intersections have been calculated, this array constitutes the output file to the plotter, which should be programmed to draw straight lines between the coordinate pairs corresponding to a given line. It is useful... 

The intersection points between coexistence lines and of coexistence lines with the boundaries of the diagram are calculated in the same way as already described for simple diagrams. However, in systems of more than one type 1 component quadrupl intersections can occur at which for example HSOq, SOq2, Cu2 and Cu2S coexist. In such a case four values of x will be found to be equal at the intersection point showing that it is the... 

Figure 6.6 Schematic illustration of a two dimensional energy surface with two local minima separated by a transition state. The dark curves are energy contours with energy equal to the transition state energy. The transition state is the intersection point of the two dark curves. Dashed (solid) curves indicate contours with energies lower (higher) than the transition state energy. The MEP is indicated with a dark line. Filled circles show the location of images used in an elastic band calculation.

Fig. 1.15 Schematic of the energy curves in the Marcus-Hush model with a single, global reaction coordinate q such that the potential energy hypersurface reduces to two parabolas and the activation energy can he calculated from the intersection point between them. The electronic coupling (Sect. 1.7.2.2) and the continuum of electronic levels in the metal electrode (Sect. 1.7.2.1) are not shown...

The kink observed around 367 K corresponds to a change of the thermal expansion coefficient from a glassy to a liquid-like state and, by that, marks the position of the glass transition temperature. Usually, the 7g is calculated as a intersection point between two linear dependencies. Nevertheless, a more convenient method is the calculation of the first and second numerical derivatives of the experimental data (Fig. 15b,c). In this case, the Tg is defined as the minimum position in the second numerical derivative plot (Fig. 15c). Down to a thickness of 20 nm, no shifts of 7g as determined by capacitive scanning dilatometry were found (Fig. 16). 

Another possibility of determining the gel point with the help of rheological methods is dynamical mechanical spectroscopy. Analysis of change of dynamic mechanical properties of reactive systems shows that the gel point time may be reached when tan S or loss modulus G" pass a miximum [3,4,13], Some authors proposed to correlate the gel point with the intersection point of the curves of storage and loss moduli, i.e., with the moment at which tan 5 = 1 [14-16], However, theoretical calculations have shown that the intersection point of storage modulus and loss modulus meets the gelation conditions only for a certain law of relaxation behavior of the material and the coincidence erf the moment of equality G = G" with the gel point is a particular case [17]. The variation of the viscosity... 

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