Coordinate System Choice

There is not generally accepted rule for the choice of the coordirrate system for right-handed and left-handed quartz crystals. Coordinate jc-axis is chosen generally in the direction of electrical a-axis and z-axis is in the direction of the optical c-axis. Cady and several other authors use right-handed coordinate system for right-handed quartz and left-handed coordinate system for left-handed quartz. On the contrary. 

According to the crystal variant and coordinate system choice, different signs for various material coefficients could be obtained. For the signs of the involved material coefficients in right-handed and left-handed quartz see Table 7.1. 

Quartz IRE Standard 1949 Cady Voigt IEEE Std. 176-1978 ANSI/IEEE Std. 176-1987  

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SCF is by no means the only framework in which good coordinates for coupled polyatomic vibrations can be defined, or pursued. Stefanski and Taylor31 stressed the importance of coordinate system choice for simple inter-... 

The coordinate system choice is also important in TS optimization. As with minimization, redundant internal coordinates have been shown to be the best choice for TS optimization [52]. Table 10.5 compares the number of optimization steps required for convergence using the three-structure STQN method with Z-matrix and redundant internal coordinates. Clearly, redundant internals work best. In Section 10.3.5.2, we advised that users check the redundant internal coordinate definitions to ensure all of... 

As noted above, the coordinate system is now recognized as being of fimdamental importance for efficient geometry optimization indeed, most of the major advances in this area in the last ten years or so have been due to a better choice of coordinates. This topic is seldom discussed in the mathematical literature, as it is in general not possible to choose simple and efficient new coordinates for an abstract optimization problem. A nonlmear molecule with N atoms and no... 

AVcorr can be evaluated readily from the classical MD simulation for any choice of coordinate system, and it may be possible to determine the modes that give the smallest AVcorr- These should be optimal CSP modes. Work along these lines is in ])rogress in our group. So far, however, the coordi-... 

In many applications, x and q will not necessarily be coordinates of particles but other degrees of freedom of the system under consideration. Typically however, a proper choice of the coordinate system allows the initial quantum state to be approximated by a product state (cf. [11], IIb) ... 

The following three scalars remain independent of the choice of coordinate system in which the components of T are defined and hence are caUed the invariants of tensor T ... 

It turns out that the htppropriate X matrix" of the eigenvectors of A rotates the axes 7t/4 so that they coincide with the principle axes of the ellipse. The ellipse itself is unchanged, but in the new coordinate system the equation no longer has a mixed term. The matrix A has been diagonalized. Choice of the coordinate system has no influence on the physics of the siLuatiun. so wc choose the simple coordinate system in preference to the complicated one. 

If, instead of making an arbiPary choice of the coordinate system, we choose more wisely, the ellipse can be expressed more simply, without cross temis [Eq. 2-43)]... 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

Theoreticians did little to improve their case by proposing yet more complicated and obviously unreUable parameter schemes. For example, it is usual to call the C2 axis of the water molecule the z-axis. The molecule doesn t care, it must have the same energy, electric dipole moment and enthalpy of formation no matter how we label the axes. I have to tell you that some of the more esoteric versions of extended Hiickel theory did not satisfy this simple criterion. It proved possible to calculate different physical properties depending on the arbitrary choice of coordinate system. 

To look ahead a little, there are properties that depend on the choice of coordinate system the electric dipole moment of a charged species is origin-dependent in a well-understood way. But not the charge density or the electronic energy Quantities that have the same value in any coordinate system are sometimes referred to as invariants, a term borrowed from the theory of relativity. 

Since the EFG component E and the asymmetry parameter rj = (Fee Vyy)/V,. in the PAS are invariants of the EFG, the two similar arrangements of the charge q shown in Fig. 4.7a, b must produce the same quadrupole splitting (because energies do not depend on the choice of the coordinate system). 

The values of the rhombicity parameters are conventionally limited to the range 0 < EjD < 1/3 without loss of generality. This corresponds to the choice of a proper coordinate system, for which /)zz (in absolute values) is the largest component of the D tensor, and /) is smaller than Dyy. Any value of rhombicity outside the proper interval, obtained from a simulation for instance, can be projected back to 0 < EID < 1/3 by appropriate 90°-rotations of the reference frame, that is, by permutations of the diagonal elements of D. To this end, the set of nonconventional parameters D and EID has to be converted to the components of a traceless 3x3 tensor D using the relationships... 

It should be noted that the positive sign of this result depends on the choice of a right-handed coordinate system in which the angle is acute. The relation developed here for the volume of a parallelepiped is often employed in crystallography to calculate the volume of a unit cell, as shown in the following section. 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

Since chemical reactions usually show significant nonadiabaticity, there are naturally quantitative errors in the predictions of the vibrationally adiabatic model. Furthermore, there are ambiguities about how to apply the theory such as the optimal choice of coordinate system. Nevertheless, this simple picture seems to capture the essence of the resonance trapping mechanism for many systems. We also point out that the recent work of Truhlar and co-workers24,34 has demonstrated that the reaction dynamics is largely controlled by the quantized bottleneck states at the barrier maxima in a much more quantitative manner than expected. 

We now repeat the derivation of the steady-state heat transport limited moisture uptake model for the system described by VanCampen et al. [17], The experimental geometry is shown in Figure 9, and the coordinate system of choice is spherical. It will be assumed that only conduction and radiation contribute significantly to heat transport (convective heat transport is negligible), and since radiative flux is assumed to be independent of position, the steady-state solution for the temperature profile is derived as if it were a pure conductive heat transport problem. We have already solved this problem in Section m.B, and the derivation is summarized below. At steady state we have already shown (in spherical coordinates) that... 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

The set of components used in a geochemical model is the calculation s basis. The basis is the coordinate system chosen to describe composition of the overall system of interest, as well as the individual species and phases that make up the system (e.g., Greenwood, 1975). There is no single basis that describes a given system. Rather, the basis is chosen for convenience from among an infinite number of possibilities (e.g., Morel, 1983). Any useful basis can be selected, and the basis may be changed at any point in a calculation to a more convenient one. We discuss the choice of basis species in the next section. 

It is obvious that gab z) is independent both of the choice of inertial frame at z, with its corresponding natural coordinate system (v), and the choice of curve x(X). The elements of g are known as the components of the metric tensor in this coordinate system. Expression (39) is the required generalization that allows evaluation of 4> at all points in terms of gab %) and the curve x(A). 

A numerical solution of the Schrodinger equation in Eq. [1] often starts with the discretization of the wave function. Discretization is necessary because it converts the differential equation to a matrix form, which can then be readily handled by a digital computer. This process is typically done using a set of basis functions in a chosen coordinate system. As discussed extensively in the literature,5,9-11 the proper choice of the coordinate system and the basis functions is vital in minimizing the size of the problem and in providing a physically relevant interpretation of the solution. However, this important topic is out of the scope of this review and we will only discuss some related issues in the context of recursive diagonalization. Interested readers are referred to other excellent reviews on this topic.5,9,10... 

In principle, the sum in Eq. [2] contains infinite terms. However, a judicious choice of the coordinate system and basis functions allows for a truncation with finite (N) terms without sacrificing accuracy. Substituting Eq. [2] back to Eq. [1], we have... 

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