Curve-crossings

Sawada S and Metiu H 1986 A multiple trajectory theory for curve crossing problems obtained by using a Gaussian wave packet representation of the nuclear motion J. Chem. Phys. 84 227-38... 

This situation arises when the electronic wave function of the transition state is described by the out-of-phase combination of the two base functions. If the electronic wave function of the transition state is described by the in-phase coinbination. no curve crossing occurs. 

In the study of (electronic) curve crossing problems, one distinguishes between a situation where two electronic curves, Ej R), j — 1,2, approach each other at a point R = Rq so that the difference AE[R = Rq) = E iR = Rq) — Fi is relatively small and a situation where the two electronic curves interact so that AE R) Const is relatively large. The first case is usually treated by the Landau-Zener fonnula [87-92] and the second is based on the Demkov approach [93]. It is well known that whereas the Landau-Zener type interactions are... 

Schulten, K. Curve crossing in a protein coupling of the elementary quantum process to motions of the protein. In Quantum mechanical simulation methods for studying biological systems, D. Bicout and M. Field, eds. Springer, Berlin (1996) 85-118. 

The technique most often used (i.e., for an atom transfer) is to hrst plot the energy curve due to stretching a bond that is to be broken (without the new bond present) and then plot the energy curve due to stretching a bond that is to be formed (without the old bond present). The transition structure is next dehned as the point at which these two curves cross. Since most molecular mechanics methods were not designed to describe bond breaking and other reaction mechanisms, these methods are most reliable when a class of reactions has been tested against experimental data to determine its applicability and perhaps a suitable correction factor. 

If the system can only be modeled feasibly by molecular mechanics, use the potential energy curve-crossing technique or a force held with transition-structure atom types. 

Drawing pseudo-binaryjy—x phase diagrams for the mixture to be separated is the easiest way to identify the distillate product component. A pseudo-binary phase diagram is one in which the VLE data for the azeotropic constituents (components 1 and 2) are plotted on a solvent-free basis. When no solvent is present, the pseudo-binaryjy—x diagram is the tme binaryjy—x diagram (Eig. 8a). At the azeotrope, where the VLE curve crosses the 45° line,... 

An important system in distillation is an azeotropic mixture. An azeotrope is a liquid mixture which when vaporized, produces the same composition as the liquid. The VLE plots illustrated in Figure 11 show two different azeotropic systems one with a minimum boiling point and one with a maximum boiling point. In both plots, the equilibrium curves cross the diagonal lines. 

These are azeotropic points where the azeotropes occur. In other words, azeotropic systems give rise to VLE plots where the equilibrium curves crosses the diagonals. Both plots are however, obtained from homogenous azeotropic systems. An azeotrope that contains one liquid phase in contact with vapor is called a homogenous azeotrope. A homogenous azeotrope carmot be separated by conventional distillation. However, vacuum distillation may be used as the lower pressures can shift the azeotropic point. Alternatively, an additional substance may added to shift the azeotropic point to a more favorable position. When this additional component appears in appreciable amounts at the top of the column, the operation is referred to as an azeotropic distillation. When the additional component appears mostly at the bottom of the column, the operation is called extractive distillation. 

These curves have some interesting properties. At any given pH, evidently Fhs + Fs = 1. At the point where the two curves cross, Fhs = Fs = 0.5, and from Eqs. (6-61) and (6-62), at this point [H ] = K, or pH = pK. That this point corresponds to the inflection point can be shown by taking the second derivative d F/dpH and setting this equal to zero one finds pHj n = pf a- In the limit as [H ] becomes much greater than K. Fhs approaches unity and Fs approaches zero... 

If the inverse in Eq. (2.8) does not exist then the metric is singular, in which case the parameterization of the manifold of states is redundant. That is, the parameters are not independent, or splitting of the manifold occurs, as in potential curve crossing in quantum molecular dynamics. In both cases, the causes of the singularity must be studied and revisions made to the coordinate charts on the manifold (i.e. the way the operators are parameterized) in order to proceed with calculations. 

The potential energy curves (Fig. 1), the non-adiabatic coupling, transition dipole moments and other system parameters are same as those used in our previous work (18,19,23,27). The excited states 1 B(0 ) and 2 B( rio) are non-adiabatically coupled and their potential energy curves cross at R = 6.08 a.u. The ground 0 X( Eo) state is optically coupled to both the and the 2 R( nJ) states with the transition dipole moment /ioi = 0.25/xo2-The results to be presented are for the cw field e(t) = A Yll=o cos (w - u pfi)t described earlier. However, for IBr, we have shown (18) that similar selectivity and yield may be obtained using Gaussian pulses too. 

Figure 1.4 Two-dimensional curve-crossing potential energy diagram of reacting system with similar potential energies before and after reaction (schematic).

A series of measurements in which the pump wavelength is varied reveal that at some energies the oscillations predominate for times beyond lOps, whilst at others the decay of population by curve-crossing wins out within 400 fs or so. The time resolution of the experiment is in this example is determined by the convolution of the two laser pulse widths, here about 125fs. 

The advantages of this generalized TSH method can be summarized as follows (1) both types of transitions in the potential curve crossing problems. 

In the above numerical examples the held parameter F is taken to be the laser frequency and the nonadiabatic transition used is the Landau-Zener type of curve-crossing. The periodic chirping method, however, can actually be more... 

Zhu and Nakamura proved that the intriguing phenomenon of complete reflection occurs in the ID NT type potential curve crossing [1, 14]. At certain discrete energies higher than the bottom of the upper adiabatic potential, the particle cannot transmit through the potential from right to left or vice versa. The overall transmission probability P (see Fig. 45) is given by... 

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