X, y, z coordinate system

The x,y,z coordinate system is defined by the g-matrix principal axes ... 

Assuming identical principal axes for A and P, the Hamiltonian term would have the form in the (x", y", z") coordinate system ... 

Take an N-atomic molecule with the nuclei each at their equilibrium internuclear position. Establish a Cartesian x, y, z coordinate system for each of the nuclei such that, for Xj with i = 1,.., 3N, xi is the Cartesian x displacement of nucleus 1, x2 is the Cartesian y displacement coordinate for nucleus 1, X3 is the Cartesian z displacement coordinate for nucleus 1,..., x3N is the Cartesian z displacement for nucleus N. Use of one or another quantum chemistry program yields a set of force constants I ij in Cartesian displacement coordinates... 

Figure 11.3 Screw dislocation with 6 = [006], C = [001] moving in the +x direction at a constant velocity v. The origin of the primed (x, y, z ) coordinate system is fixed to the moving dislocation.

We now investigate the relationship between Pr and the original nuclear momenta Pa in more detail. We have introduced a redundant coordinate x in the transformation matrix 911(2.40). We therefore need to define our original (3n + 6) coordinates in terms of the (3 n + 7) final coordinates. To do this, we note that the position of the molecular centre of mass in the (X", Y", Z") coordinate system is... 

Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values , 9, x)by

The figure also illustrates how the rotations can be performed in the floating coordinate system as in (32). X Y Z of position (1), rotated through tp about Z then determines XYZ, and the X Y Z coordinate system with Y J Z of position (3), determines XYZ. 

Let us again consider a rotated tZ-basis / defined relative to an X Y Z coordinate system and having the relation to our space fixed function basis given by (1) and (4). We consider a central ion M situated at the origin of the coordinate systems and a ligand Lk on the Z axis. 

Xk is the row of ligator functions [Xa - - Xgc] where the components refer to the primed basis of the X Y Z coordinate system of the ligator Lk. Mk is analogously the row of the M angular functions, referred to the corresponding primed central ion coordinate system. S k and Ck are diagonal matrices of the same form as Ek. 

In Figure 2.8 are shown four representations of M in the old way, the so-called laboratory frame of reference, the normal x,y,z coordinate system as viewed by a stationary... 

Following the same philosophy of the previous graphical approaches, a representation into the 3D spaces was proposed assigning the four nucleic acid bases, the four directions associated with the regular tetrahedron [Randic, Vracko et al., 2000], To specify directions, the origin of the Cartesian (x, y, z) coordinate system was assigned in the center of a cube so that the four comers of the cube, which define the tetrahedral directions, constitute the main axes. Then, each basis is moved along the directions as arbitrarily defined in Table B8. 

The principal directions of the g-tensor coincide with the molecular x-y-z coordinate system. 

The relative motion is described by the vector R pointing from the center of mass (CM) to the departing C-atom. The vector 6 is described by its polar coordinates R,0 and in the space fixed (x,y,z) coordinate system (not shown). The orientation of the internuclear axis of the diatomic product is described by the vector r, which points from A to B. The motion of the internuclear axis is described by polar coordinates r, Y, ), which are defined relative to the vector R as z-8 is . 

For the specific case of H2O, the rotational motion is described by the quantum numbers J for the total angular momentum, for the projection of J on the A-axis and for the projection of J on the C-axis. The set of quantum numbers jK K > describes the motion of the nuclear plane in a space fixed (x,y,z) coordinate system and this motion depends only on the angles 0,4 and < >, The vibrational motion in H2O may be separated into the stretch modes lviV3> and the bending mode by )v2>. The stretch modes viV3> depend on the coordinates r and R, whereas the bending mode v2> depends only on Y. To describe the electronic motion in the ground state we use the symbol leo>. We assume that vibrational, rotational and electronic motion is decoupled and obtain... 

Transition metals (TMs) possess nine orbitals in the valence shell, which are depicted in Scheme 9.R. 1. Letters that describe the orbitals shapes also coincidentally correspond to their spatial properties s, spherical p, polar (namely, pointed in a given direction) and d, dipolar (pointed in two different directions). The directed orbitals have also labels describing their directions in space for example, p refers to a p orbital oriented along the x direction of the x, y, z coordinate system placed around... 

Figure 6.12 I A point P can be specified using three coordinates. Because atoms are spherically symmetric, the spherical polar coordinates r, 0 and (p, are often used rather than the more familiar x, y, z-coordinate system.

The three vectors of the ligand positions can be expressed in a row notation for the primed x, y, z coordinate system as ... 

In the x y z -coordinate system, the orientation direction (i.e., the optical axis) in the initial state is written as... 

A direction in a crystal is given as a set of three integers in square brackets [uvw] u, v, and w correspond to the above definition of the translation vector r. A direction in a cubic crystal can be described also by Miller indices, as a plane can be defined by its normal. The indices of a direction are expressed as the smallest integers which have the same ratio as the components of a vector (expressed in terms of the axis vectors a, b, c) in that direction. Thus the sets of integers 1, 1, 1 and 3, 3, 3 represent the same direction in a crystal, but the indices of the direction are [111] and not [333]. To give another example, the x axis of an orthogonal x, y, z coordinate system has Miller indices [ 100] the plane perpendicular to this direction has indices (100). 

D echocardiographic image data is in general representative of many other approaches, and will be described in detail later. In brief, a 3-D x, y, z) coordinate system is constructed using three of the short-axis images. The transducer position and orientation data from each individual cross-section are then... 

The integer h, k, and I are called Miller indices. To identify a plane or direction, a series of three numbers are used. Figure 2 shows three planes in a cubic lattice nestled into the origin of the x, y, z coordinate system which illustrate the important planes (100), (110) and (111) in the conventional microfabrication process. Each plane is unique, differing in atom count and binding energy between the atoms. Each plane also has different mechanical, chemical and electrical properties. The etch rate is... 

As it is observed from Figure 4, the highest point on the suction cups (the point with maximum y coordinate in x y z system) has the most and earliest tendency to leave the surface, and the lowest point (the point with the least y coordinate in x y z system) has the least. Therefore, it is reasonable to assume that the reaction force acting on the suction cups has linear distribution along y axis shown in Figure 4. Based on this assumption, the reaction force density (f ) on every point of the suction cups can be expressed as a linear function of its y coordinate in x y z coordinate system, and the amount of the reaction force density at extreme points A, and B or ... 

Figure E.2. Definition of the stress tensor for the cartesian (x, y, z) coordinate system, and the polar (r, 6, z) coordinate system.

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