Cartesian reference frame

Figure 7.2 Transforming the Cartesian reference frame a) cylindrical cross section of the screw and barrel with flow out of the surface of the page, and b) the unwound rectangular channel with a stationary barrel and the Cartesian coordinate frame positioned on the screw, is the velocity of the screw core in the z direction and it is negative...

In practice, MC simulations are primarily applied to collections of molecules (e.g., molecular liquids and solutions). The perturbing step involves the choice of a single molecule, which is randomly translated and rotated in a Cartesian reference frame. If the molecule is flexible, its internal geometry is also randomly perturbed, typically in internal coordinates. The ranges on these various perturbations are adjusted such that 20-50% of attempted moves are accepted. Several million individual points are accumulated, as described in more detail in Section 3.6.4. 

Fig. 9. Symmetries of four-, six-, eight-, and twelve-coordinated ligands relative to Cartesian reference frame for d wave functions of Figure 2.

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

If the magnetic-field gradient is applied in direction r in polar coordinates, which forms an angle

spin density Mo(r, 5) along direction s, which is perpendicular to r (cf. Fig. 5.4.1),... 

The Cartesian reference frames p, a, b, and r are defined as follows p denotes a laboratory-fixed frame in which the laser polarization vectors are fixed a signifies a frame of reference that has been chosen to have its origin at center A similarly b has its origin at center B r represents a frame in whieh the R vectors are rotationally invariant. 

We consider an infinitely long cylinder of radius a oriented along the z-axis of the Cartesian reference frame. It has a uniform charge density cr(z, p>) = —cTc > 0. 

Consider a straight, long, and slender beam, as shown in Fig. 1. A Cartesian reference frame is placed such that the x-axis is along neutral axis of the beam. Thus, for a rectangular cross section, if h is the beam thickness, the position y will vary in the range —hjl [Pg.3369]

The values given correspond to Cartesian reference frame shown in Fig. 3.6. 

Reprinted from Ref. [148] with kind permission from Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington 0X5 IGB, UK, Copyright (1995). The values given correspond to Cartesian reference frame shown in Fig. 3.2. 

FIG. 5.2. Definition of Cartesian reference frames for AB2 (Dooh) AB2 (C2v) molecules. 

Fig. 9.2. Cartesian reference frame, numbering of atoms, and definition of internal coordinates and of bond unit vectors for SO2.

An invariant of the bond polarizability tensor with respect to reorientations of the Cartesian reference frame is now defined as the trace of the product... 

In Cartesian reference frames the momentum vector can be expressed as ... 

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