Specific perturbations

The generic form of the perturbation part of the total Hamiltonian is expressed by (20). In this section we consider specific forms of the perturbation operators Hx appearing in this expression. Note that these operators are time-independent any time-dependence of the perturbation is expressed by the exponentials appearing in the Fourier transform of V(t). We will consider the perturbation operators arising from nuclear spins as well as external electric and magnetic fields. 

A large class of molecular properties arise from the interaction of molecules with electromagnetic fields. As emphasized previously, the external fields are treated as perturbations and so one considers only the effect of the fields on the molecule and not the effect of the molecule on the field. The electromagnetic fields introduced into the electronic wave equation is accordingly those of free space. From (79) one observes that in the absence of sources the electric field has zero divergence, and so both the electric and magnetic fields are purely transversal. It follows that the scalar potential is a constant and can be set to zero. In Coulomb gauge the vector potential is found from the equation 

The corresponding electric and magnetic fields may be found using (83) and 

When the wavelength X lTcjk is long compared to the extent of the molecule the vector potential (193) may be taken to be constant over the molecular volume, that is 

In terms of multipoles only the electric dipole is coupled to the electromagnetic field and so this approximation is termed the electric dipole approximation. It may appear strange that the electromagnetic field, which is transversal, in this approximation is given solely by the scalar potential. It must, however, be remembered that the scalar and vector potentials of (197) do not describe the electromagnetic wave as such. Rather, it models the interaction of the electromagnetic wave with the molecule [65]. 

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Not only can electronic wavefiinctions tell us about the average values of all the physical properties for any particular state (i.e. above), but they also allow us to tell us how a specific perturbation (e.g. an electric field in the Stark effect, a magnetic field in the Zeeman effect and light s electromagnetic fields in spectroscopy) can alter the specific state of interest. For example, the perturbation arising from the electric field of a photon interacting with the electrons in a molecule is given within die so-called electric dipole approximation [12] by ... 

The above properties of /(to) are quite general and do not depend on either the nature of the system or the perturbation. To obtain more specific information, one must of course make use of the specific Hamiltonian H for a system, and the specific perturbation A in which one is interested. The simplest specific properties which one can calculate are the moments p of the spectral density... 

Amphiphilic compounds are also known as potent modifiers of the bilayer intrinsic radius of curvature and utilize this property to act as a non-specific perturbator of membrane protein function [27]. Catamphiphilic drugs that can interact with the head groups or with the scramblases or flippases can change cell functioning. 

In this theory, the dynamics of the intrinsic-surface-confined excitons account surprisingly well—in a natural way, without introducing ad hoc parameters—for the surface emissive properties, and they allow, a contrario, a very sensitive probing of various types of surface disorders, whether residual, accidental, or induced. The disorder may be thermal, substitutional, chaotic owing to surface chemistry, or mechanical owing to interface compression. It may be analyzed as a specific perturbation of the surface exciton s coherence and of its enhanced emissive properties. 

Many aspects of metabolic control are not readily apparent when isolated enzymes are characterized in vitro. The regulatory function of allosteric properties and putative enzyme complexes are best studied in vivo. For this purpose, metabolic engineering can be used as a basic research tool to investigate the effect of specific perturbations on the overall flux through an alkaloid pathway. The first application of metabolic engineering to plant alkaloid biosynthesis involved the transformation... 

This expression, without the specific perturbations given by — er, is what Stone (1985) called the charge susceptibility of a system. It contains all of the information needed to evaluate any of the molecular polarizabilities. The polarizability density, as defined in eqn (8.258), is obtained by integrating the polarization susceptibility over the coordinate r"... 

Usually, the second term is small and the various specific perturbations related to a single site are similar given the experimental precision. If this regularity is observed, the perturbation related to the introduction of site s is practically independent of its anterior environment. The generality of this observation leads us to adopt the hypothesis of local average perturbation pi(s). This results in estimating p(s, by the information perturbation pi(s) related to... 

Figure 5.8. Specific perturbation p(s) and local average perturbation p,(s)...

Within a population, a site s, here B,. is reached by different generation pathways. The anterior environments E, E of this site are different and the associated specific perturbations are not the same p(B,t, E)[) = 0.25 and p(B , E ) = 0.37. However, considering the experimental error 0.09, these perturbations are similar. Consequently the local average perturbation p,(On) is used for estimating the specific perturbations associated with the environments tested and for predicting the specific perturbations p(B ,Ej ), which cannot be directly calculated from the experimental compounds. 

Figure 5.14. Oinerent specific perturbations and detecting cosites...

The specific perturbations associated with sites A, and in their minimal anterior environment, respectively 1.31 and 0.66, are about 0.70 less when these sites are present simultaneously. This additivity deviation is taken into accoimt by introducing a cosite or interaction site A A" during the optimization of the relationship. 

The choice of two border compounds, the minimal compound 10 and the maximal compound 31 associated with the minimal and maximal anterior environments ET and E of site A", makes possible a completely reliable estimation of specific perturbations p(A] , EJ) in all predicted compounds comprising site A] by the local average perturbation p,(A ) = 1/2(p(A , E ) -i-p(Al, E )). The anterior environment of all the predicted compounds lies between those tested and the predictions are all based on a topological interpolation. 

When the perturbations pis E ) and p(s, ") associated with the minimal and maximal anterior environments are similar for each site, then the variable part of each perturbation is almost zero. There then exists a strong probability that no cosite influences the property, and the estimation of each specific perturbation p(s, ) by the average perturbation j5(s)" is completely reliable for all the compounds of the preference. When the specific perturbations p s,E ) and p(s, ") are too far removed from each other, considering the experimental precision, then the minimal compounds associated with the cosites influencing property are introduced into the key population. They enable us to estimate the perturbations associated with these detected cosites, and the estimation is once again completely reliable for all the compounds of the preference. 

The ab initio calculation of NLO properties has been a topic of research for about three decades. In particular, response theory has been used in combination with a number of electronic structure methods to derive so-called response functions [41 8], The latter describe the response of a molecular system for the specific perturbation operators and associated frequencies that characterize a particular experiment. For example, molecular hyperpolarizabilities can be calculated from the quadratic and cubic response functions using electric dipole operators. From the frequency-dependent response functions one can also determine expressions for various transition properties (e.g. for multi-photon absorption processes) and properties of excited states [42]. 

In general, the physical properties of an electron system are defined by referring to a specific perturbation problem and can be classified according to the order of the perturbation effect. For instance, the electric dipole moment is associated with the first-order response to an applied electric field (i.e. the perturbation), the electric polarizability with the second-order response, hyperpolarizabilities with higher-order terms. In addition to dipole moments, there is a number of properties which can be calculated as a first-order perturbation energy and identified with the expectation value... 

One may note that quasienergy derivatives dt listed above are symmetric with respect to the capital indices Ai... A . It must be emphasized that the indices must be thought of as a double index specifying a specific perturbation strength a, as well as a frequency cOa, and, as discussed previously, they must be permuted at the same time in order to retain symmetry. Further simplifications of the expressions for molecular properties are possible using the response equations, e.g. (48), (49) and (50), but generally at the expense of index symmetry and numerical precision [28,9]. As an example, using the response equation the expression for second-order molecular properties (52) can be rewritten as... 

Saxena et al. produced a permselective membrane from a combination of these oppositely functionalized chitosan chains. Both modifications were found to have good hydrophilic interaction with water and the mixed solution resulted in charge neutralization. The resultant membrane was shown to discriminate between different ionic radii, specifically perturbing the diffusive properties of molecules with a larger ionic radius such as Mg and Ca, while facilitating the diffusion of Na, which has a smaller ionic radius. This implies the suitability of the bimodal functionalized chitosan membrane for separation of Na from Ca + and Mg +, allowing for specific detection of Na+. 

Are these in-group spectral effects responsible for the CWD for fertile and fissile isotopes The simple model results (13S) tend to support this assumption. Results of numerical calculations for realistic problems obtained recently by Kier and Zolotar 139) show, however, that even though spectral fine structure effects are significant in the low energy end of the resolved resonance region, they contribute little to the reactivity worth of a Pu and a sample in the ZPR 6 assembly 7 reactor. These results were obtained from a small number of calculations for two specific perturbations. Further investigation seems necessary before the significance of spectral fine structure effects and their connection with the CWD can be firmly established. 

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