Higher order effects

The coupled-channel calculations allow for accurate calculations of higher order effects. At high energies the electronic energy loss may be expanded in terms of the projectile charge Zp according to 

For collisions of antiprotons with atomic hydrogen, a quasidipole is formed during the collisions. The dipolar antiproton-proton system does not support bound states for inter-particle distances below 0.64 a.u. [52]. For finite velocities and larger impact parameters b (in the figure, Z = 1) there is still a significant ionization contribution. As can be observed in the figure at 

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The topography of a conical intersection affects the propensity for a nonadiabatic transition. Here, we focus on the essential linear tenns. Higher order effects are described in [10]. The local topography can be detennined from Eq. (13). For T] = 3, Eq. (13) becomes, in orthgonal intersection adapted coordinates... 

Fleming et al. (1985) define A, as the independent failure rate and higher order effects in order of the Greek alphabet (skipping a). The conditional probability that a CCF is shared by one... 

Erom the qualitative analysis in this section, we tentatively conclude that there are several contributions of comparable magnitude to the thermal expansion at low temperatures. Higher order effects may also be present. In this case, it may be more straightforward to estimate the interaction between ripplons as extended membranes without using a multipole expansion, as indeed is done when computing the regular Casimir force between extended plates. 

Table 3.1 Higher order effects on line positions in [V0(H20)5]2+...

The main feature of the CBS (complete basis set) methods (e.g. CBS-Q [15] and CBS-QB3 [20]) is extrapolation to the complete basis set limit at the UMP2 level. Additional calculations [UMP4 and UQ-CISD(T) or UCCSD(T)] are performed to estimate higher-order effects. A scaled ZPVE, together with a size-consistent empirical correction and a spin-contamination correction, are added to yield the total CBS energy of the molecule. 

The above-mentioned nonlinear optical effects can be described by the perturbation of the electromagnetic held intensity under the electric dipole approximation. Actually, this approximation is broken in optical near-helds. Hence, a perturbation effect of multipole such as electric quadrupole or magnetic dipole should also be considered, although such a higher-order effect is normally negligible. Indeed, electric quadrupole contributions can be comparable with electric dipole contributions... 

As is the case for the two-factor interactions, the three-factor interaction is also symmetric in all its variables the interaction effects ABC, ACB, BAC, CAB, BCA and CBA all give the same result. Higher-order effects are calculated by analogous reasoning. 

Full factorial designs can be fractionated by the exclusion of experiments designed to identify higher order effects and such reduced designs are known as fractional factorial designs. 

Although these saturated designs, assume interaction effects to be negligible and only estimate main effects, they have the feature known as confounding where higher order effects can overwrite the main effects. 

Thus, if a method is not rugged to higher order effects this will be observed in the values of the main effects. This effect will be discussed in more detail later in the section on evaluation of results. 

Like main effects, interaction effects can be positive, negative, or zero. Higher order effects are viewed in the same way The three-way interaction effect ABC is the change in the change of the change of response as A goes... 

After this brief survey of the general theory let us discuss in a little more detail the Boltzmann approximation (Eq. (39)) and the equation (Eq. (40)) retaining higher order effects. 

Finally, the problem was resolved by irradiating standards and mixtures of standards in a factorial experiment. The experiment design was a full factorial experiment with three variables, mercury, selenium, and ytterbium, at two levels with replication and with a center point added to test higher order effects. The pertinent information on treatments and levels of variables are shown in Table VII. 

A spectrum of metal compound reactivities in petroleum could arise for several reasons. Nickel and vanadium exist in a diversity of chemical environments. These can be categorized into porphyrinic and non-porphyrinic species vanadyl and nonvanadyl or associated with large asphaltenic groups and small, isolated metal-containing molecules. Each can be characterized by unique intrinsic reactivity. Reaction inhibition which occurs between the asphaltenes and the nonasphaltenes, as well as between Ni and V species, can also contribute to reactivity distributions. The parallel reaction interpretation of the observed reaction order discrepancy is therefore compatible with the multicomponent nature of petroleum. Data obtained at low conversion could appear as first order and only at higher conversions would higher-order effects become obvious. The... 

For f7 ions, the ligand field splitting is zero under first-order conditions, whereas higher order effects account for the splitting of the 7 = % level and for a small pseudocontact shift. 

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