Choice of Coordinate System

From Equation (A9.4) we see that the electron potential energy will have the same value at any point a distance r from the nucleus, i.e. any point on the surface of a sphere of radius r. Equation (A9.4) also shows that, in Cartesian coordinates, r actually depends on all three components of the axis system, which makes the direct solution of the Schrodinger equation quite difficult. However, if we transform to the spherical polar coordinate system illustrated in Eigure A9.1, then the distance from the origin r becomes a single coordinate. In our problem, the potential energy of the electron is then a much simpler function than in the Cartesian case. 

The Laplacian operator in spherical polar coordinates becomes 

To make the manipulation of equations involving the Laplacian easier, we will define more compact operators which contain differentials with respect to radial and angular degrees of freedom  

Direct comparison of Equations (A9.5) and (A9.6) gives the functional forms of the new operators subscripts have been included to show which variables they included. 

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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-... 

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 ... 

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. 

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. 

Each of the symmetry operations we have defined geometrically can be represented by a matrix. The elements of the matrices depend on the choice of coordinate system. Consider a water molecule and a coordinate system so oriented that the three atoms lie in the x-z plane, with the z—axis passing through the oxygen atom and bisecting the H-O-H angle, as shown in Figure 5.1. 

The free-energy density should not depend on the choice of coordinate system [i.e., /( , VO should not depend on the gradient s direction] and therefore L = 0 and K will be a symmetric tensor.5 Furthermore, if the homogeneous material is isotropic... 

Pople et al. 25> pointed out that while the results obtained for two-center integral evaluation in a full Roothaan S.C.F.M.O. treatment are independent of the choice of axis, the same is not true in simplified versions. Such integrals are sensitive to the choice of coordinate system and the hybridization of the orbitals. 

Fig. 6.6. Schematic of realization of alignment-orientation Stark conversion, (a) Choice of coordinate systems, (b) Possible realization scheme for AB molecules seeded in a free jet of X atoms, (c) Symbolic polar plot of J distribution (dashed line refers to initial cylindrical symmetry over beam axis z ).

Arbitrariness One hears comments that graph descriptors being arbitrary, do not have a deep physicochemical significance A misconception here is in confusing an input with an output of a mathematical treatment of a problem Surely die choice of descriptors selected to characterize grains is at the disposal of an investigator, hence arbitary But the same is true of a choice of coordinate systems used to describe a system of classical physics, or the choice of basis functions to compute a molecular structure in quantum chemistry I In each case we speak of input information, and the outcome of the analysis undertaken (when a complete basis is taken) does not depend on the choice of descriptors (coordinates) However, the amount of work, even its mere feasibility, will differ A poor selection of basis functions may... 

Note that the dimensionality of the flow also depends on the choice of coordinate system and its orientation. The pipe flow discussed, for example, is one-dimensional in cylindrical coordinates, but two-dimensional in Cartesian coordinates illustrating the importance of choosing the most appropriate coordinate system. Also note that even in this simple flow, the velocity cannot be uniform across the cross section of the pipe because of the no-slip condition. However, at a well-rounded entrance to the pipe, the velocity profile may be approximated as being nearly uniform across the pipe, since the velocity is nearly constant at all radii except very close to the pipe wall. 

The appearance of dot products in these expressions means that any ( /, is independent of our choice of coordinate system when evaluating s-vector components. As the most directly obtained components we shall prefer those of the molecular system. 

They are called invariants because they are independent of the choice of coordinate system. 

These integrals, which are called action integrals, can be calculated only for conditionally periodic systems that is, for systems for which coordinates can be found each of which goes through a cycle as a function of the time, independently of the others. The definite integral indicated by the symbol is taken over one cycle of the motion. Sometimes the coordinates can be chosen in several different ways, in which case the shapes of the quantized orbits depend on the choice of coordinate systems, but the energy values do not. 

One use of the symmetry coordinate classification is that it can tell us which types of distortion are expected to be coupled to other types. Each representative point can be associated with a value of the molecular potential energy, a structural invariant B that must be independent of such matters as the choice of coordinate system or the way the labels of the various atoms or bonds have been chosen. Consider, for the spiro-ketal example, the general quadratic energy expression for the S2 and Sg coordinates, which both transform as B2, symmetric with respect to the (yz) plane, antisymmetric with respect to the (xz) plane ... 

By proper choice of coordinate system, McCabe-Thiele diagrams can be applied to exchange of ion G in an ion-exchange resin, with ion A in solution. 

The matrices R and the vectors t constitute a representation of the group of the symmetry operations linked to the choice of coordinate system and its origin. For a given group, there is an infinite number of representations, each corresponding to a particular coordinate system. 

The term trigonal refers to a crystal system (defined by the presence of a unique 3 or 3 axis) the term rhombohedral refers to a choice of coordinate system a, b, c as well as a Bravais lattice (Section 2.6.1). 

If necessary, the symbols may be augmented by the addition of axes of order 1 to indicate a specific orientation of the symmetry elements with respect to the coordinate system. Hence the symbol 121 designates the group 2 with the b axis parallel to the twofold axis, i.e. the traditional choice of coordinate system for the monoclinic class there are no symmetry elements parallel to a or to c. The symbol 112 designates the same group 2 with c parallel to the twofold axis. The symbol mil designates the group m with a normal to the mirror plane. The symbol 3m 1... 

It is instructive to note that the origin of these rules lies in the choice of coordinate system and that they may be derived without using structure factors. Any centered cell may be transformed into a primitive cell which, in general, does not convey the symmetry of the motif and which may thus be unsatisfactory from this point of view (Sections 1.4.1 and 2.6.1). Thus the transformation a = (a — b)/2, b = (a -h b)/2 transforms a C cell into a diamond-shaped primitive cell. The indices hkl transform in a covariant manner (Section 1.2.4), h = h — k)/2, k = h- - k)/2. As W and k must be integers according to the Laue equations, then h- -k must be an even number. The indices with h- -k odd do not correspond to a reciprocal lattice vector (Fig. 3.39). 

INVARIANCE WITH RESPECT TO THE CHOICE OF COORDINATE SYSTEM... 

The idea of a tensor may be characterized by using equation (4.7). A tensor is defined by its transformation properties. In other words, a tensor represents an entity whose description is independent of the choice of coordinate system. 

There are an infinite number of ways to reconstruct the same system from parts. These ways are not equivalent in practical calculations if, for any reason, we are unable to compute all the interactions in the system. However, if we have a theory (in our case the multipole method) that is able to compute the interactions, including the long-range forces, then it turns out the final result is virtually independent of t he choice of unit cell motif. This arbitrariness of choice of subsystem looks analogous to the arbitrariness of the choice of coordinate system. The final results do not depend on the coordinate system used, but still the numerical results (as well as the effort to get the solution) do. 

Taking into account our coordinate system, the vector pointing the coordinate system origin a from is / = (0, 0, i ). Then we can express Edip-dip in a very useful form independent of any choice of coordinate system (cf., e.g., pp. 147, 764) ... 

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