A Simpler Notation

The description of the two-state system,/= 2, was introduced earlier in Sections 7.1 and 7.2. Here, we present some quite obvious results for systems with nn direct interactions only. Since we discuss only a restricted group of events, we use a simpler notation for the correlations. Thus, instead of g(sj = a,S2 = P), we simply use g 2) or gi2(2) to denote pair correlations (between the event site i occupied and site i + 1 occupied ). Also, we shall always refer to the X. 0 limit as the correlation and omit specific notation for this limit. 

Although any information can be expressed by elements, sometimes a simpler notation would be sufficient. In XML, information without further internal structure can be provided inside start-tags as an attribute. The attribute syntax starts with its name and is restricted by the same rules as element names followed by an equal sign (=). The value of an attribute is given in text delimited by apostrophes (. . ) or quotes ( ... ). Names of elements are separated from attributes by white-space characters (spaces, tabulators, new lines). The same element is prohibited from having several attributes of the same name (<molecule name= benzene id= i4 />). 

This expansion in powers of A can be viewed from a more general perspective. The counting and arranging of powers of A is a formal operation, so we can carry out that analysis on the basis of a simpler notation, e.g. 

We rewrite Equation 6.6ab using a simpler notation, which we adopt for the remainder of the text ... 

We use the N2/W(110) system as an example of the construction of the modified four-body LEPS PES. Since the molecule is homonuclear, the parameters for the A = B = N atoms are identical. We use the explicit subscripts for the relevant bodies, but note that the original work (Kara and DePristo 1988a, 1988b, 1988c) used a simpler notation which is also followed... 

In Appendix B the "" (12) referred to in this chapter are written in a more explicit and familiar manner. We remark that in Section II only dipolar interactions were considered and a simpler notation was used. However, when higher multipole moments are included, a more systematic approach is necessary, and rotational invariants are particularly convenient and theoretically useful. In the present notation the D 2) and A(12) functions of Section II are " (12) and "°(12), respectively. 

If the angle between A and B is the same as that between a and b, a simpler notation introduced by... 

Nickel atoms form a face-centered cubic array (fee). In fact, using a simpler notation, we can describe the crystalline array of nickel only with the coordinates... 

To state clearly the problem at hand it is necessary to introduce initially a detailed notation for the composition of a crystal. For much of the later manipulations it is possible to use a very much simpler, abbreviated version of the notation. From the point of view of thermodynamics, the composition of an imperfect crystal is specified when the number of atoms of each of the different chemical species present is given. Let atoms which appear in a perfect crystal be denoted by a subscript 0, and let N0 denote the 2V0 atoms of a different species (2V01, N02,. . ., N0a), all of which species appear in the perfect crystal, i.e. 

Having clearly stated in detail the microscopic composition, we now introduce a simpler, abbreviated notation which is convenient for the subsequent manipulations. The set of numbers Njf may be relabelled to give in their place a set of numbers Nx, a typical member of the set being N which is the number of atoms of kind s on sublattice number r. The other five sets of numbers, Ng", No, Nf, Nf, Nf", which specify completely the defect composition of the crystal, will be similarly relabelled to give a set of numbers Na, a typical member of the set being N, . The number is the number of defects of type s, and they are situated on the sublattice number r. (By the definitions employed, one kind of defect can only appear on one sublattice but one sublattice may contain more than one kind of defect. Although r is specified by s,... 

In the previous section, the diffusion equation for motion of a large number of particles was developed [see eqn. (211)]. When the solvent is at rest and both hydrodynamic forces and inter-reactants potential energy terms can be ignored, the equation becomes much simpler. This equation provides the basis for the analysis by Wilemski and Fixman. They chose to consider just one excited A molecule in a volume, V, together with m quencher molecules. The co-ordinates of all these molecules are rA and rQj, rQ2... rQ,n at a time f. Initially, thefluorophorand quencher molecules were positioned at rA° and rqj... r m. As a shorthand notation, these co-ordinates are called (r) and r0, respectively. The fluorophor was excited at time f°. 

For consistency with notation found in many other sources, we express (17.3) in a simpler manner ... 

A comparison of sets A and A R,c) provides a measure of the symmetry aspect R for set A with respect to center c. In principle, any similarity measure is applicable for this comparison, for example, the Hausdorff metric hi A, A R,c)) provides a valid measure of the symmetry aspect R for set A with respect to center c. If the center of mass is chosen for c, then the simpler notations A R) and hi A, yd(R)) are used. 

If the center of mass of the fuzzy set A is chosen as reference point c, then the notation A(R,c) is replaced by the simpler notation A(R). 

For the purpose of describing interfacial transport phenomena it is convenient to describe the 2D surface geometry using a 3D reference system with origo located outside the surface in the CV in which the surface is embedded (e.g., [63], chap. 3 [217], sect. 11.4 [62], pp. 486-492). This framework is named extrinsic and connects the 2D surface and the ambient 3D bulk phases. The intrinsic perspective of an observer located internally within the surface is then used as a component part providing a simpler description of the interface in terms of a 2D representation which is independent of the 3D space in which the interface is embedded. Due to the mathematical complexity involved, the notation of the 2D differential geometry is examined briefly. A more comprehensive introduction can be found in, for instance, references [63, 217, 25, 129, 1, 62, 94, 199, 241, 232, 29]. 

The necessity to incorporate goals makes notations like QOC or DRL a good starting point for a Decision Ontology. We have opted for the more expressive, but also more complex DRL. However, QOC can be regarded as a subset of DRL there is no necessity for a decision modeler to apply the entirety of DRL concepts when a simpler QOC-like representation is sufficient. 

Two-state quantum beats (discussed in Section 6.5.3 in a simpler but less powerful notation, and in Sections 9.2.1, 9.2.2 and 9.3.2), provide the simplest illustration of the density matrix formalism. The key ideas are the creation by E of a coherent superposition state, which produces off-diagonal elements of p(t) (known as coherences ), and the simultaneous selection of the desired coherence and destruction of unwanted coherences by D. It is instructive to consider the essential features of E and D by which specific coherences axe selectively created and detected. 

Our research began with a simpler 2,5-cyclohexadiene as shown in Scheme 1.2. Using the n-x configuration as outlined in Scheme 1.1, but utilizing the shorthand notation here for simplicity, we can understand the rearrangement of 4,4-diphe-nyl-2,5-cyclohexadiene to afford the bicyclic photoproduct actually obtained. The final step involves a type A zwitterion whose rearrangement is known to proceed as shown. 

The expressions for T, Eq. (2.28), and V, Eq. (2.31), can be written in a simpler form using matrix notation. Using the column vector q, whose components are the 3N mass-weighted Cartesian displacement coordinates. 

The notation involving the generic symbol <) is a little redundant when discussing electron-repulsion integrals only over the basis functions and there is no danger of confusion with integrals over other functions, so we drop the symbol 0 and refer to the integrals over basis functions by the simpler notation rs, tu). In this notation the above equalities become simply... 

The RPA, together with a simpler variant, the so-called Tamm-Dancoff Approximation, is now used in all forms of many-body theories (nuclear, solid state and molecular) in a wide variety of notations and formulations. It is an interesting exercise to consult these publications and contrast the notation and techniques used in this area with the ones used in this work. In this respect the book The nuclear many-body problem by P. Ring and P. Schuck (Springer-Verlag, 1980) might prove a useful starting point. 

Hirschfelder, in discussing a simpler and more schematic model of stepwise activation in unimolecular reactions assumed the absorption coefficient a kplk = 1 in Hirschfelder notation) to be 0.5 and obtained the equation... 

Therefore, the real part of the complex solution to the complex form of Equation [IJ or [7] is the same as the real solution of the real form of Equation [1] or [7]. The two main advantages of complex notation are (a) simpler algebra in solving Equations [1] or [7], and (b) automatic separation of the in-phase and 90 -out-of-phase components (as the real and imaginary parts of a complex amplitude). 

These equations can be evaluated at all AM-2 collocation points. Equations (58) and (59) can be expressed in a simpler matrix notation, where the — superscript represents a one-dimensional matrix and the = superscript represents a two-dimensional matrix. 

This coefficient is known as a Clebsch-Gordan (CG) coupling coefficient and denoted by the 3F bracket (T ay a/lpfclT K). It indicates how the orbital irreps TL and Fb have to be combined to yield a product ket that transforms as Fy). The CG-coefficients can be determined by using projection operators. The results are listed in Appendix F. It is often possible to obtain these results by a simpler procedure. We illustrate this for the components of the T g two-electron state, obtained in Eq. (6.9). The z-component of this state is the only component that is totally symmetric imder the C4 splitting field. It is clear that this symmetry can be obtained only by multiplying the egc) and ) components, since these are both antisymmetric and thus will form a symmetric product. From here on we will adopt for the product functions the usual notation of small letters for the orbitals and capital letters for the coupled states. Hence ... 

In Chapter 1, we used subscripts A and B to indicate antibonding and bonding orbitals ( a> b)> but from now on we can use the simpler notation of an asterisk ( ) to indicate an antibonding orbital. Thus, in Figure 2.5,0 is the bonding molecular orbital sp + Is and o is the antibonding molecular orbital sp — Is. 

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