Formal definitions

we provide the formal definitions of classical absorption and photodissociation cross sections and subsequently we describe a practical way in order to calculate them. 

We define the total classical absorption cross section, save for an unimportant normalization constant, as 

The sin7o term reflects the fact that the intramolecular vector r is expressed in polar coordinates it is especially important for the dissociation of linear molecules and must not be forgotten The delta-function S(Hf — Ef) selects only those points (i.e., trajectories) in the multidimensional phase-space that have the correct energy Ef. It ensures that the quantum mechanical resonance condition Ef = Ei + Ephoton is fulfilled. 

1) Absorption of the photon takes place only if the energy in the excited state, Hf(r0), exactly equals Ef = Et + Ephoton- 

3) Each phase-space point has a weight Pyfi(to), which reflects the particular quantum mechanical state of the parent molecule in the electronic ground state. 

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Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

Comparing this expression with (3.81), one obtains the formal definition of the quasiclassical tunneling matrix element... 

Present formal definitions of intensities and Raman depolarization ratios,... 

We ve already looked briefly at Gaussian input in the Quick Start. Here we present a more formal definition and discuss the various molecule specification options. 

The LST alleviates this problem by systematically approximating the probabilities of Bm, with M > N, from the set of probabilities of smaller blocks, Bi, B2,. .., Bn- In this way, order correlation information is used to predict the statistical properties of evolving patterns for arbitrarily large times. The outline of the approach begins with a formal definition of block probability functions. 

In order to get at least a formal definition of the problem, we will write the exact solution to the Schrodinger equation (Eq. II. 1) in the form... 

A formal definition of the reaction order with respect to the concentration of substance i, C,, is... 

Because proteins are not composed of identical repeating units, but of different amino acids, they do not fall within the formal definition of polymers given at the start of this chapter. They are nevertheless macromolecular and techniques developed for the study of tme polymers have been applied to them with success. However, for the most part they are outside the scope of this book and accordingly will receive very little attention in the chapters that follow. 

It may be useful to reiterate and expand the more formal definition of the terms. 

Baader F. A formal definition for expressive power of knowledge representation languages. In Proceedings of the 9th European Conference on Artificial Intelligence, ECAI-90, Stockholm, Sweden, 1990, p. 53-58. 

The formal definition of a partition function implies that it is a summation over an infinitely high number of terms. Explain why the partition function... 

A formal definition of salt hydrolysis can follow from the description outlined above. Salt hydrolysis may be defined as a reaction in which the anion or the cation of a salt reacts with the solvent water to produce acidity or alkalinity. Evidently, it is the nature of the anion or the cation constituting the salt which will determine whether the solution produced as a result of hydrolysis will be acidic or alkaline. If the matter is examined from these points of view, the following three different cases can arise. 

The hazardous waste listings, the hazardous waste characteristics, and the mixture and derived-from rules are all essential parts of the definition of hazardous waste, but these key elements are all described in different sections of the RCRA regulations. Only one regulatory section unites all four elements to establish the formal definition of hazardous waste. This section is entitled Definition of Hazardous Waste, which states that all solid wastes exhibiting one of the four hazardous characteristics are hazardous wastes. This section also states that all solid wastes listed on one of the four hazardous waste lists are hazardous wastes. Finally, this section explains in detail the mixture and derived-from rules and the regulatory exemptions from these rules. Thus, although the section is entitled Definition of Hazardous Waste, it serves primarily as a guide to the mixture and derived-from rules. 

The rectangular pulse can be generated by subtracting a step function with dead time T from a step function. We can derive the Laplace transform using the formal definition... 

On the negative real axis (-180°), find the "distance" of GCGP from (-1,0). This is the gain margin, GM. The formal definition is... 

Find the frequency where the magnitude Gc(jto)Gp(jco) is 1. This particular frequency is the phase crossover frequency, to = tocp. We then find the angle between GcGp and -180°. This is the phase margin, PM. The formal definition is... 

In this abstraction each edge corresponds to a pipeline network element and each vertex corresponds to a junction connecting two or more elements. It is often convenient to refer to the formal definition of a graph G as the sets... 

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". 

Now we must explain the semantics of the logical connectives. Informally, they have meanings you are probably acquainted with. Connective " a " is "AND", connective " v " is "OR", " 1 " is "NOT", " = " is "IMPLIES", while " = " is "IF AND ONLY IF". Occasionally we use the English words rather than the formal symbols as a matter of convenience. We now give the formal definitions of these concepts. In this definition, recall that we allow members of the domain of I to play the same role as constants or free (but not bound) variable in wffs. 

The formal definition of the electronic chemical hardness is that it is the derivative of the electronic chemical potential (i.e., the internal energy) with respect to the number of valence electrons (Atkins, 1991). The electronic chemical potential itself is the change in total energy of a molecule with a change of the number of valence electrons. Since the elastic moduli depend on valence electron densities, it might be expected that they would also depend on chemical hardness densities (energy/volume). This is indeed the case. 

The formal definition of this quality factor, Q, is the amount of power stored in the resonator divided by the amount of power dissipated per cycle (at 9.5 GHz a cycle time is l/(9.5 x 109) 100 picoseconds). The dissipation of power is through the resonator walls as heat, in the sample as heat, and as radiation reflected out of the resonator towards the detector. The cycle time is used in the definition because the unit time of one second would be far too long for practical purposes within one second after the microwave source has been shut off, all stored power has long been dissipated away completely. 

Students often get the terms, heat, specific heat and temperature confused. Here are the formal definitions ... 

We might properly refer to this value as the apparent Peclet number, because by many formal definitions the Peclet number accounts for the relative importance of advection and molecular diffusion, without mention of hydrodynamic dispersion. 

According to the Oxford English Dictionary, the word molecule seems to have arisen in the seventeenth century in the discussions initiated by the physical speculations of Descartes. Early formal definitions described a molecule as a group of atoms mechanically indivisible (H. E. Roscoe, Elementary Chemistry [1871], p. 169) or a group of atoms drawn and held together by what chemists call affinity (J. Tyndall, Longm. Mag. 1 [1882], 30). 

Following the above observations, the process model can be formulated by TA. For formal definitions of the syntax and semantics of TA, see [15]. TA are used to model the individual resources by resource automata and to describe timing constraints by place automata. The former are used to start and to finish tasks which are uniquely assigned to resources, the latter establish timing constraints of places. 

Let us consider what happens when V moves and we perform the operations in Equation (3.1). This is depicted in Figure 3.2. Here we have displayed the drawing in two dimensions for clarity, but our results will apply to the three-dimensional volume. In Figure 3.2 the volume is shown for time t, and a small time increment later, At, as a region enclosed by the dashed lines. The shaded regions Vj and Vn are the volumes gained and lost respectively. For these volumes, we can represent Equation (3.1) by its formal definition of the calculus as... 

If an arbitrary standard state is marked with, a formal definition of a Raoultian standard state for component A of a solution is... 

Following from Equation (3.3), we say that internal energy is a state function. A more formal definition of state function is, A thermodynamic property (such as internal energy) that depends only on the present state of the system, and is independent of its previous history . In other words, a state function depends only on those variables that define the current state of the system, such as how much material is present, whether it is a solid, liquid or gas, etc. 

This formal definition of Ka can be extended to multi-protic acids. We consider the dissociation to occur in a step-wise manner, the acid losing one proton at a time. Consider, for example, the two-proton donation reactions of sulphuric acid ... 

This chapter discusses packages, the relationships between them, and how to use them effectively. Section 7.1 introduces packages and explains how a package contains names and formal definitions within an informal narrative structure. 

In our view, architecture is not only about Gothic-scale structures but is also about all structures and relationships used down to the level of code.1 The decision to use a four-tier structure, with a thin client, a Web server, a business application server, and a database, is architectural. But, in the extreme, we consider a consistent use of getXQ and setX(x) methods also to be part of the (detailed) architecture. This view leads to a somewhat less formal definition of architecture. 

The effective molarity (EM) is formally the concentration of the catalytic group (RCOO- in [5]) required to make the intermolecular reaction go at the observed rate of the intramolecular process. In practice many measured EM s represent physically unattainable concentrations, and the formal definition is probably relevant only in reactions (which will generally involve very large cyclic transition states) where the formation of the ring or cyclic transition state per se is enthalpically neutral, or in diffusion-controlled processes. For the formation of small and medium-sized rings and cyclic transition states the EM as defined above contains, and may indeed be dominated by, the enthalpy of formation of the cyclic form. This topic has been discussed briefly by Illuminati et al. (1977) and will be treated at greater length in a future volume in this series. 

A short list of invariant lattice complex symbols is reported in the following. For a complete list, for a more systematic description and formal definition, see the International Tables of Crystallography, Hahn (2002). 

If necessary, refer back to Chapter 1 for a formal definition of an enzyme. 

Given the definition of the geometry of the transition states in TST as the highest energy point in the minimum energy pathway from reactants to products, the formal definition of MEP is as follows. The MEP is, in one direction, the path of steepest descents from the transition state to reactants while, in the other direction, it is the path of steepest descents from transition state to products. For reasons which will not be discussed here, the formal definition of MEP includes the statement that the pathway is expressed in mass scaled Cartesian coordinates of the position of the atoms (introduced in Chapter 3, e.g. x is replaced by x = ). This simplifies... 

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