Coordination dependence

Haynes G R and Voth G A 1995 Reaction coordinate dependent friction in classical activated barrier crossing dynamics when it matters and when it doesn t J. Chem. Phys. 103 10 176... 

The time- and coordinate-dependent wavefiinction for any given state is... 

The fitting parameters in the transfomi method are properties related to the two potential energy surfaces that define die electronic resonance. These curves are obtained when the two hypersurfaces are cut along theyth nomial mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

Laser Raman diagnostic teclmiques offer remote, nonintnisive, nonperturbing measurements with high spatial and temporal resolution [158], This is particularly advantageous in the area of combustion chemistry. Physical probes for temperature and concentration measurements can be debatable in many combustion systems, such as furnaces, internal combustors etc., since they may disturb the medium or, even worse, not withstand the hostile enviromnents [159]. Laser Raman techniques are employed since two of the dominant molecules associated with air-fed combustion are O2 and N2. Flomonuclear diatomic molecules unable to have a nuclear coordinate-dependent dipole moment caimot be diagnosed by infrared spectroscopy. Other combustion species include CFl, CO2, FI2O and FI2 [160]. These molecules are probed by Raman spectroscopy to detenuine the temperature profile and species concentration m various combustion processes. 

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... 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

Polymerization Reactions. The polymerization of butadiene with itself and with other monomers represents its largest commercial use. The commercially most important polymers are styrene—butadiene mbber (SBR), polybutadiene (BR), styrene—butadiene latex (SBL), acrylonittile—butadiene—styrene polymer (ABS), and nittile mbber (NR). The reaction mechanisms are free-radical, anionic, cationic, or coordinate, depending on the nature of the initiators or catalysts (194—196). 

Our results are included in Tables I and II along with available experimental data for comparison. For Ni, calculations were performed by including the coordination dependent energy term, U(,on[Pg.263]

A and A = 0.1 eV. The adiabatic ground potential energy surface is shown in Fig. 11. The present results (solid line) are in good agreement with the quantum mechanical ones (solid circles). The minimum energy crossing point (MECP) is conventionally used as the transition state and the transition probability is represented by the value at this point. This is called the MECP approximation and does not work well, as seen in Fig. 10. This means that the coordinate dependence of the nonadiabatic transmission probability on the seam surface is important and should be taken into account as is done explicitly in Eq. (18). 

The laser parameters should be chosen so that a and p can make the nonadiabatic transition probability V as close to unity as possible. Figure 34 depicts the probability P 2 as a function of a and p. There are some areas in which the probabilty is larger than 0.9, such as those around (ot= 1.20, p = 0.85), (ot = 0.53, p = 2.40), (a = 0.38, p = 3.31), and so on. Due to the coordinate dependence of the potential difference A(x) and the transition dipole moment p(x), it is generally impossible to achieve perfect excitation of the wave packet by a single quadratically chirped laser pulse. However, a very high efficiency of the population transfer is possible without significant deformation of the shape of the wave packet, if we locate the wave packet parameters inside one of these islands. The biggest, thus the most useful island, is around ot = 1.20, p = 0.85. The transition probability P 2 is > 0.9, if... 

The laser intensities are taken to be the possible lowest. The intensity in case (b) is almost three times larger than the others. This is simply due to the fact that the transition dipole moment exponentially decays from the equilibrium position and also the potential energy difference increases. Note again that the coordinate-dependent level approximation works well. In order to demonstrate the selectivity the time evolution of the wave packets on the excited state are shown in Fig 41. As a measure of the selectivity, we have calculated the target yield by... 

The potential for coordination depends on the oxy substituents.82 Alkoxy substituents are usually chelated, whereas highly hindered silyloxy groups usually do not chelate. Trimethylsiloxy groups are intermediate in chelating ability. The extent of chelation also depends on the Lewis acid. Studies with a-alkoxy and (3-alkoxy aldehydes with lithium enolates found only modest diastereoselectivity.83... 

By integrating out the coordinate dependence, they also obtain an approximate quantum-corrected formula for the momentum distribution which leads to the definition of an effective temperature. That is, the approximate distribution is still Gaussian in the momenta, but with an increased temperature for each particle... 

Given the density matrix, the average of a coordinate-dependent function F(x) is then... 

The activation parameters of Table 22 suggest that the energy, the charge distribution, and the location of the relevant transition structures (TS) along the reaction coordinate depend significantly on the nature of the moving AOH moiety. When A = H, the racemization and isomerization TS are located early on the reaction coordinate, whereas they are late with A = CH3 and are characterized by comparatively stronger H-bond interactions. 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

---