Reduced effective representation

The SPC model does not include an induction energy term. However, a revised version of this model, SPC/E, is based on a mean-field form (see earlier section on Electrostatics) for the induction term in Eq. [1]. Parameter changes resulted in a 3% increase of the charges. The revised potential generates a more stable hydrogen bond network than does SPC, as well as a self-diffusion coefficient that is in good agreement with experiment. The inclusion of the mean-field correction is also used by two other effective potentials, the and the reduced effective representation (RER) potentials. 

Structural constraints imposed suppress the effect of noise and allow the results to be easily interpreted because the parameters, which can be viewed as reduced dimensionality representation of the data, reduce complexity of the analysis and have clearly associated meaning. 

Fig 3 16 The effect of introducing a weak potential into the ID lattice is to lift ttie degeneracy of the energy levels near to the edge of the Brillouin zone (shown in both extended-zone and reduced-zone representation)... 

If experimental measurements of neither the dif sion coefiBcient nor the viscosity exist, then the binary diffusion coefficient can be estimated with the aid of a representation of the reduced effective cross section such as that provided by the correlations... 

The functional form of the HDMR expansion and its use for global sensitivity analysis was aheady discussed in Sect. 5.5.5, but a similar approach can also be taken to develop reduced model representations. The purpose is to create a fast equivalent operational model (FEOM) based on the HDMR, giving sufficient accuracy with respect to the full chemical model, but with much lower computational expense. HDMR builds approximations recursively, based on the assumption that high-order-correlated effects of the inputs are expected to have negligible impact on the output. Applications have shown that the order of the correlations between the independent variables dies off rapidly, and therefore, only a few terms are usually required to represent even highly nonlinear input-output relationships. 

Equation 6 shows that the adsorption of component 1 at a partial pressureis reduced in the presence of component 2 as a result of competition for the available surface sites. There ate only a few systems for which this expression (with 5 1 = q 2 = 5 ) provides an accurate quantitative representation, but it provides useful quaUtative or semiquantitative guidance for many systems. In particular, it has the correct asymptotic behavior and provides expHcit recognition of the effect of competitive adsorption. For example, if component 2 is either strongly adsorbed or present at much higher concentration than component 1, the isotherm for component 1 is reduced to a simple linear form in which the apparent Henry s law constant depends onp. ... 

Fig. 8.3 Schematic representation of the stress corrosion cracking mechanism of the pit (after Pickering and Swann ). (a) Tubular pits initiated at solute-rich slip step. The pits may, but need not necessarily, follow the slip plane once they are initiated, (b) Ductile tearing along a plane containing the tubular pits. The stress is increased across the plane because of the reduced cross section and the stress raising effect...

These so-called Pareto-based techniques do not force consolidation over multiple criteria in advance and aim to return a representation of the set of optimal compounds. They support discussion between team members who may have different views on the downstream impacts of different risk factors perhaps, for example, one team member may know that there is a reliable biomarker for one potential side-effect. This would then mean that assessing this risk need not consume much development time and cost, and the risk factor can have a reduced weighting within the target product profile being evolved by the team. 

Figure 20.6 Schematic representation of the effects of 5-HT reuptake inhibitors on serotonergic neurons, (a) 5-HT is released at the somatodendritic level and by proximal segments of serotonergic axons within the Raphe nuclei and taken up by the 5-HT transporter. In these conditions there is little tonic activation of somatodendritic 5-HTia autoreceptors. At nerve terminals 5-HTib receptors control the 5-HT synthesis and release in a local manner, (b) The blockade of the 5-HT transporter at the level of the Raphe nuclei elevates the concentration of extraneuronal 5-HT to an extent that activates somatodendritic autoreceptors (5-HTia). This leads to neuronal hyperpolarisation, reduction of the discharge rate and reduction of 5-HT release by forebrain terminals, (c) The exposure to an enhanced extracellular 5-HT concentration produced by continuous treatment with SSRIs desensitises Raphe 5-HTia autoreceptors. The reduced 5-HTia function enables serotonergic neurons to recover cell firing and terminal release. Under these conditions, the SSRI-induced blockade of the 5-HT transporter in forebrain nerve terminals results in extracellular 5-HT increases larger than those observed after a single treatment with SSRIs. (Figure and legend taken from Hervas et al. 1999 with permission)...

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

Applying the adiabatic approximation, we restrict the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes an undamped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

We shall give here a brief summary of our previous work [71,72] that was concerned with the introduction of the relaxation phenomenon within the adiabatic treatment of the Hamiltonian (77), as was done in the undamped case by Witkowski and Wojcik [74]. Following these authors, we applied the adiabatic approximation and then we restricted the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes a damped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

Figure 3.1. Schematic representation of dimensional reduction for a framework of corner-sharing MX6 octahedra. The M and X atoms are represented by black and white spheres, respectively. In a) though d), reaction with AbX incorporates additional X atoms into the M—X framework, progressively reducing the connectedness and effective dimensionality of the M—X framework. In d), after incorporating n units of AbX (n > 2), the structure is reduced to isolated oligomeric or monomeric components. For clarity, the A atoms are not shown in the figure. [Adapted with permission from [Ref. 16]. Copyright 2001 American Chemical Society.]...

FIGURE 32. Schematic representation of the geometry changes of a hypothetical model of two facing n-systems with HOMOs Ta and n t,. The neutral molecule is represented in the centre. Upon ionization (removal of an electron from the HOMO it ), the antibonding interactions which prevail in n are reduced, and the distance R decreases. As a consequence, the IT. /t overlap and cr increase. Conversely, upon electron ejection from n+ (or on 7T+ - %- excitation), the bonding interaction in 7T+ is diminished, which has the opposite effect on R and cr as described above... 

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