Physical intuition

Another connnon approximation is to construct a specific fonn for the many-body waveftmction. If one can obtain an accurate estimate for the wavefiinction, then, via the variational principle, a more accurate estimate for the energy will emerge. The most difficult part of this exercise is to use physical intuition to define a trial wavefiinction. 

When the stress that can be bom at the interface between two glassy polymers increases to the point that a craze can form then the toughness increases considerably as energy is now dissipated in forming and extending the craze structure. The most used model that describes the micro-mechanics of crazing failure was proposed by Brown [8] in a fairly simple and approximate form. This model has since been improved and extended by a number of authors. As the original form of the model is simple and physically intuitive it will be described first and then the improvements will be discussed. 

A little bit of physical intuition as to how the vortices form in the first place may help in explaining the behavior as TZ is increased still further. We know that u = 0 at the cylinder s surface. We also know that the velocity increases rapidly as we get further from that surface. Therefore vortices are due to this rapid local velocity variation. If the variation is small enough, there is enough time for the vorticity to diffuse out of the region just outside the cylinder s surface and create a large von Karman vortex street of vorticity down stream [feyn64]. 

These assumptions concerning steel were often based on physical intuition and in... 

By a brilliant physical intuition B. van der Pol succeeded finally (1920) in establishing his equation (which is given in Section 6.11) but, not having any mathematical theory at his disposal, he determined the nature of Ike solution by the graphical method of isoclines. It became obvious that the problem, which was a real stumbling block for many years, had been finally solved, at least in principle. 

The mass of the catalyst does not appear. However, physical intuition or the Sq terms in Equations (10.11) and (10.16) suggest that doubling the amount... 

The compaction should be based on physically intuitive criteria and should require minimum a priori assumptions. 

Equation (51) has a clear physical interpretation. Recalling the lineshape for a single excitation route, where fragmentation takes place both directly and via an isolated resonance [68], p oc (e + q)2/( 1 + e2), we have that 8j3 is maximized at the energy where interference of the direct and resonance-mediated routes is most constructive, e = (q I c(S )j2. In the limit of a symmetric resonance, where q —> oo, Eq. (51) vanishes, in accord with Eq. (53) and indeed with physical intuition. The numerator of Eq. (51) ensures that 8]3 has the correct antisymmetry with respect to interchange of 1 and 3 and that it vanishes in the case that both direct and resonance-mediated amplitudes are equal for the one-and three-photon processes. At large detunings, e —> oo, and 8j3 of Eq. (51) approaches zero. 

In simulating transient flows in pipeline networks, the importance of accuracy cannot be over-emphasized. Because the transient behaviors are less well-understood, they are often rich in surprises. Physical intuition affords less guidance in these situations than in steady-state phenomena. Rachford and Dupont (R2) provided two instructive and deceptively simple examples to illustrate the interaction between regulators and compressors and the oscillatory response which can produce pressures higher than the supply pressure through reinforcement. 

These considerations raise a question how can we determine the optimal value of n and the coefficients i < n in (2.54) and (2.56) Clearly, if the expansion is truncated too early, some terms that contribute importantly to Po(AU) will be lost. On the other hand, terms above some threshold carry no information, and, instead, only add statistical noise to the probability distribution. One solution to this problem is to use physical intuition [40]. Perhaps a better approach is that based on the maximum likelihood (ML) method, in which we determine the maximum number of terms supported by the provided information. For the expansion in (2.54), calculating the number of Gaussian functions, their mean values and variances using ML is a standard problem solved in many textbooks on Bayesian inference [43]. For the expansion in (2.56), the ML solution for n and o, also exists, lust like in the case of the multistate Gaussian model, this equation appears to improve the free energy estimates considerably when P0(AU) is a broad function. 

In the method of constraints, a force of the form AV is applied at each step such that remains constant throughout the simulation. To see how A can be related to dA/d , we recall that the free energy A is the effective or average potential acting on . From physical intuition, it should be true that -dA/d is the average of the force acting on . In a constraint simulation, this force is equal to -A. Therefore we can expect to have d/l/d - - (A). We now make this statement more rigorous. 

Equation (3.40) tells us that the symmetry adaptation (3.39) from LMOs to CMOs is purely decorative, with no energetic consequence. As physical intuition suggests, we are perfectly justified in describing the Hearth—Hmooh system in terms of localized atom-like functions, each weakly perturbed by its remote twin, rather than as a completely delocalized MO of symmetry-adapted form. 

The four equations [(7.9)-(7.12)] are simplified using chemical and physical intuition. Two examples are given. In the first (Sections 7.63-7.6.6) the case where ionic point defects are more important than electrons and holes is considered, and in the following sections (Sections 7.7.1-7.7.5) the reverse case, where electronic defects are more important than vacancies, is described. 

The four Eqs. [(8.3)—(8.6)] are simplified using chemical and physical intuition and appropriate approximations to the electroneutrality Eqs. (8.7) and (8.10). Brouwer diagrams similar to those given in the previous chapter can then be constructed. However, by far the simplest way to describe these equilibria is by way of polynomials. This is because the polynomial appropriate for the doped system is simply the polynomial equation for the undoped system, together with one extra term, to account for the donors or acceptors present. For example, following the procedure described in Section 7.9, and using the electroneutrality equation for donors, Eq. (8.9), the polynomial appropriate to donor doping is ... 

The quantities that best represent a particular property can often be rationalized on the basis of physical intuition. For example, those that reflect interactions between like molecules, such as heats of sublimation and vaporization, can be expressed well in terms of molecular surface area and the product vofot. A large value for this product means that each molecule has both significantly positive and significantly negative surface potentials, which is needed to ensure strongly attractive inter-molecular interactions, with consequently higher energy requirements for the solid —> gas and liquid —> gas transitions. 

Having recognized the close quantitative connection between optical properties and collision dynamics at this stage of development of atomic theory is a remarkable example of Bohr s physical intuition. 

The ability to efficiently minimize the total energy of a collection of atoms is central to perhaps the majority of all DFT calculations. Before moving ahead, you should reread this section with the aim of summarizing the pitfalls we have identified in the two simple examples we examined. Developing a strong physical intuition about why these pitfalls exist and how they can be detected or avoided will save you enormous amounts of effort in your future calculations. 

If we were only interested in bulk copper and its oxides, we would not need to resort to DFT calculations. The relative stabilities of bulk metals and their oxides are extremely important in many applications of metallurgy, so it is not surprising that this information has been extensively characterized and tabulated. This information (and similar information for metal sulfides) is tabulated in so-called Ellingham diagrams, which are available from many sources. We have chosen these materials as an initial example because it is likely that you already have some physical intuition about the situation. The main point of this chapter is that DFT calculations can be used to describe the kinds of phase stability that are relevant to the physical questions posed above. In Section 7.1 we will discuss how to do this for bulk oxides. In Section 7.2 we will examine some examples where DFT can give phase stability information that is also technologically relevant but that is much more difficult to establish experimentally. 

The forward reaction is favored by the alkaline slurry solutions which result in breakage of the Si—O bonds. In metal CMP, oxidizing slurries are often used, resulting in faster removal rates. Since the contributions of the chemical and mechanical components are not well known, modeling efforts have focused on empirical approaches guided by physical intuition of process mechanisms. 

The physical significance of these boundary conditions is as follows. Equation (26a) represents the fact that just before the tracer is injected into the system the concentration is everywhere zero. Equations (26b) and (26g) are obvious since a finite amount of tracer is injected. Equations (26d) and (26e) follow from conservation of mass at the boundaries between the sections (W4) the total mass flux entering the boundary must equal that leaving. Equations (26c) and (26f) are based on the physically intuitive argument that concentration should be continuous in the neighborhood of any point. These boundary conditions will be used extensively in the subsequent derivations. 

To make further progress, it is standard practice to take this definition of the spectral density and replace it by a continuous form based on physical intuition. A form that is often used for the spectral density is a product of ohmic dissipation qco (which corresponds to Markovian dynamics) times an exponential cutoff (which reflects the fact that frequencies of the normal modes of a finite system have an upper cutoff) ... 

As shown in (7.53), the P-xsap boundary always curves below the linear P-x boundary, in such a manner that the vapor phase is always enriched in the more volatile component, as physical intuition would suggest ... 

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