Higher-order CSEs

An important consequence of the equivalence of the 2-CSE and higher-order CSEs with the Schrodinger equation is that the CSEs may be applied to the study not only of the ground-state but also of excited states. 

Another important hierarchy of equations is obtained by applying the (MCM) to the matrix representation of the Liouville-von Neumann Equation LVNE) [12,13]. In this way the p-order Contracted Liouville-von Neumann Elquation (p-CLVNE) is obtained [4]. It will be shown here that The structure of a particular p-CSE, that involves the higher order CSE s for a given state, can be replaced by an equivalent set of equations, 1-CSE and 1-CLVNE, but for the whole spectrum, i. e. involving all the states. 

Equation (2) describes the effect of defocus f and spherical aberration Cs on the object function and X is the electron wavelength. Higher order aberrations can be considered. An attenuation of the electron wave is taken into account by a spatial damping envelope... 

A molecule may possess higher order rotational axes. Consider the eclipsed form of the molecule ferrocene (Fig. 3.8a). which has a Cs axis through the iron atom and perpendicular to the cyclopentadienyl rings. Now consider the staggered form of... 

Figure 4.2 exhibits the complete failure of the LDA and PW91-GGA for atomic uc. The CS potential also has little in common with the exact vc. The closure approximated potential corresponding to (4.44) is the first DFT potential which at least qualitatively follows the exact uc. The need for the inclusion of higher-order contributions is apparent from the overestimation of the shell structure of vc. In addition, the asymptotic 1 /r4-behaviour of the exact vc is not reproduced. Clearly, perturbative correlation functionals like E cannot be the final answer to the question of which Ec should be combined with the exact Ex. 

The construction of the higher order nonstandard unsplit-field PMLs in curvilinear meshes initiates from the division of a 3-D space, Q, into two areas (separated with a nonplanar interface f) such that Q = f cs U 2pml> where f2cs refers to the computational domain and 2pml is the area occupied by the PML under research. Considerable in the procedure is the extension of the stretched-coordinate theory to nonstandard models [30]. This is performed through the following steps ... 

Measurements of k, ti, tz, and D have been used to study the concentration-dependence of Lij. It is usually found that at low concentrations L11 and Lzz are linear functions of cs whereas Liz is a higher order function of cs which rapidly goes to zero as cs -> 0 (see Harned and Owen, 1950). 

The eoneentration profiles are similar to those in Figure 4.7, and the effeetiveness faetors are plotted in Figure 4.8. For a given (p, the effeetive-ness faetor for second-order is lower than for a first-order reaction, since a decrease in average concentration has more effect for the higher-order reactions. The solution for second-order reactions applies only where the rate is proportional to the square of the concentration of a single reactant, which is rarely the case. For a bimolecular reaction with a rate proportional to both reactant concentrations, r = k2C Cs, the solution for second-order kinetics can be used if the initial concentrations are equal and if the diffusivities of A and B are nearly the same. If reactant B is in considerable excess, the reaction could be considered pseudo-first-order with k = k2Cg and Eq. (4.31) used for a reasonable approximation of rj. 

The hardness concepts have recently been used as indices of aromaticity [97] and of the orientation of electrophilic aromatic substitution [98]. The principle of maximum hardness [20] requires higher-order derivatives of the electronic energy with respect to the electron population variables, and especially the hardness derivative Sq/0N [99]. Applications of the EE procedure and the CS concepts to the structural and reactivity problems of solids and clusters are becoming routine [40, 41, 47, 100, 101] and new sensitivity indicators of reactivity, addity/basisity in crystal chemistry are being developed [102]. A novel CS-type approach to the chemical reactivity has recently been proposed by Tachibana and Parr [103]. 

In comparison to the 1 1 complexes, the enniatin sandwiches display a higher ion selectivity, their actual stability constants decreasing in the order > Cs > Na. Besides adducts of 2 1 stoichiometry, a 3 2 club sandwich has been proposed for the Cs complex... 

To construct a CS-QDT formulation, one shall treat the effects of system-bath coupling H [Eq. (2.1c)] to the second order exactly for not only the reduced density operator p t) evolution, but also the initial canonical state of the total composite system, Pt( o —oo) = p (T), before the external field excitation. Various CS-QDT formulations differ at their partial resummation schemes for the higher order contributions. We have recently arrived at three forms of CS-QDT in terms of differential equations of motion [38]. Two of them are in principle equivalent to the conventional second-order COP [Eq. (1.2)] and POP [Eq. (1.3)] formulations (cf. AppendixB). For the sake of clarity, we shall present here only the unconventional one that may be particularly suitable for the numerical study of non-Markovian dissipation in the presence of external time-dependent fields. 

Obviously, 0 )2 contains not only the complete second-order contributions, but also a partial sum from all higher orders. Moreover, we may obtain the second-order expression for the coordinate response function, [x( )]2 = ([9(0) (0)])25 either via the direct application of linear response theory to the CS-QDT formulation in Sec. 4.2, or, equivalently, from the comparison between Eq. (4.11b) and Eq. (4.6d). 

Shown in Fig. 5 are the values of qt = Tr[qp t)] evaluated with the exact QDT (solid), the POP-CS-QDT (dash), and the CODDE (dot) formulations. Note that the POP-CS-QDT and the CODDE differ only at their correlated driving-dissipation contributions. The correlated driving-dissipation dynamics in the POP-CS-QDT are characterized by Xe t) [Eq. (4.11b) or Eq. (4.13)] that accounts only for the second-order system-bath coupling effects. However, the correlated dynamics in the CODDE involves also the higher order contributions. This may account for the observed fact in Fig. 5 that the CODDE s result appears better than that of POP-CS-QDT. On... 

Note that in Figure 2.11, the initial FTA produces only one CS, which is a 3 order CS, indicating that the probability of system failure should be small. Note also that the updated FTA produces two CSs the original 3 order CS and an SPF CS. Depending upon the probability of the SPF CCF event, the overall FTA probability could be much higher than the initial FTA probability, as demonstrated in this example. 

CS order refers to the quantity of elements within a CS. Usually, the higher the CS order, the lower the probability of the CS, since CSs of order 2 or greater represent AND gate conditions, and the input probabilities are multiplied together. 

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