An intuitive interpretation

In this section, we present an intuitive interpretation for the derivative rale, which we have derived with mathematical rigor in the previous sections. Consider first the effect of a p, tip state centered at x = j . The state has two lobes with same weight but opposite phase. If at a point of the positive lobe, xo + Ax, the tunneling matrix element picks up a value proportional to i ((xo + Ax), then at the corresponding point Xo — Ax the tunneling matrix element picks up a value proportional to - i[i(x() - Ax). The total contribution is then 

A similar interpretation can be applied to other cases. For example, the state dxy has four lobes, as shown in Fig. 1.8. For a tip state centered at xo, yo, any tunneling matrix element must have the following form  

As we have discussed in Chapter 2, a direct consequence of the Bardeen tunneling theory (or the extension of it) is the reciprocity principle If the electronic state of the tip and the sample state under observation are interchanged, the image should be the same. An alternative wording of the same 

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Wolfe and collaborators attempted to understand the origin of the gauche effect by using an intuitive interpretation of good quality ab initio calculations. 

The variety of methods of naming azo compounds which has been in use for many years may lead to considerable confusion, especially when attempts are made to name structural formulas of highly substituted dye molecules with several azo linkages. Furthermore, in regard to the older dye literature, an intuitive interpretation of an author s intention frequently seems more productive than a detailed analysis of the system of nomenclature which he may be using. An effort is made in this chapter to conform to either the IUPAC or the Chemical Abstracts system [la]. 

This result also lends itself to an intuitive interpretation temperature equilibration is a fast phenomenon and T = T2 - a line in the (T), T2) coordinate system -is the equilibrium manifold of the fast dynamics. 

The small parameter A,-1 in this perturbation scheme represents the ratio between the time scales of two subsystems h n, ,) and h(p,x), so an intuitive interpretation for the estimate given in Eq. (4) is rather easy. If the time scale of vibrational degrees of freedom is much faster than that of translational degrees of freedom, then the energy transfer between two subsystems hardly occurs. As X oo, the freezing of energies of subsystems is achieved, which is a sort of the adiabatic limit. 

For large N hence mean field theory is very good, apart from a very narrow region close to the critical point. An intuitive interpretation to this finding can be given as follows [2]. Each polymer coil is a random walk of radius a /N and hence takes a volume adv/Nd. But the density of this polymer chain in this region taken by the polymer is rather small, namely of the order of N/(adv/Nd) = a d/v/Nd 2. Since the segment density is just a d, in the melt... 

Linear Free-Energy and Related Mathematical Models. The most direct connection between an intuitive interpretation of a given set of SARs and its... 

It is helpful to develop an intuitive interpretation of these bounds on the optimal echelon base-stock levels. Note that 0y can be expressed as... 

If thus the problem is not well-posed in the admissible region it can happen that, even if we have found some z where B(z) is of full column rank and y uniquely determined, at least one of the (psudo)standard deviations assessed by linear approximation is great and the estimated yj is uncertain the measurement is inappropriate for estimating this yj. which is an intuitive interpretation of the not-well-posedness . It can then also happen that in the course of successive approximations, some approximation yf escapes from the amissible region and the solution gets lost. 

An intuitive interpretation of Darcy s law is that of a fluidodynamic equivalent of Ohm s law for electric conduction, with the volumetric flow of the fluid, the Kpir] ratio, and pressure taking the place of electric current, conductivity and applied voltage bias, respectively. The pressure gradient, dpldx, across the porous material can often be approximated as AplAx, where Ap and Ax indicate the overall pressure drop and the thickness of the sample, respectively. An apparent flow resistance, Rp [m ], can be defined as ... 

Now we introduce curvy steps. An intuitive interpretation of what is done here is to expand the Taylor series of the exponential transformation to higher orders, such that the step directions are no longer straight lines, but instead they are curved. Invoking the Baker-(Campbell-)Hausdorff lemma (see, e.g.. Ref. 123), the unitary transformation of the density matrix can be written as... 

Let us examine a few sample flowcharts to illustrate further the sort of questions we shall be studying. Later we shall give formal definitions of schemes and of other concepts, but right now let us rely on an intuitive understanding of "flowchart", "interpretation", "equivalence". 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

An intuitive physical interpretation of the correlation terms in the 2-RDM is that the two electrons undergo virtual excitations in such a way that when one goes from C into , the other one undergoes the opposite transition. 

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

The following summary provides a recommended approach to the interpretation of an unknown spectrum which may be adopted until experience has developed an intuitive appreciation of the characteristics of infrared spectra. It should be used in association with the more detailed notes which follow, describing the way in which characteristic group frequencies arise and the variations in frequency position which accompany environmental changes. 

There is probably no other concept that contributed to the development of chemistry so remarkably as the ill-defined, qualitative concept of similarity. Not despite but rather because of a certain fuzziness, the applicability of this concept is extremely broad and touches practically all areas of chemistry [1, 2]. An example would be the Mendeleev periodic law, the disclosure of which was closely connected with the effort to systematise the similarities in the properties of elements. From the intuitively interpreted meaning of similarity arises also one of the most powerful chemical principles - the principle of analogy, on the basis of which a wealth of fundamental chemical notions were introduced. 

Two proofs for the HSAB principle were provided under the restriction of a common chemical potential of the reaction partners [83, 84]. Later on, a local HSAB principle was provided by Gazqu z and Mendez [85], They showed that the interaction between two chemical species will not necessarily occur through their softest atoms, but through those whose softnesses are approximately equal. In Section 4.2, an intuitive application of the HSAB concept is provided, followed by an application of the local HSAB principle in the interpretation of regioselectivity in Diels Alder reactions. 

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