Cartesian coordinates vector transformation

Cartesian coordinates by a canonical point transformation. The Langevin equations, or the equivalent FPEs, for the N coupled atoms are most conveniently written in the 3Af-element Cartesian coordinate vector x = (Xj,..., with conjugate momentum vector p = (p,...,Pjjy),... 

The use of redundant coordinates requires extensive modification of the lattice dynamical procedure. It is, however, often worth the additional complication to use redundancies if this facilitates the formulation of symmetry coordinates. When the Wigner projection operator (Wigner, 1931) is used to build such symmetry coordinates, it is necessary to first understand the results of the application of all symmetry operations of the applicable group to the displacement coordinates chosen. This is indeed relatively straightforward for the direction cosine displacement coordinates and therein lies their principal value. These coordinates transform like axial vectors in contrast to cartesian coordinates, which transform like polar vectors. 

A factor-group operation consists of a rotation (a ) and a noninimitive translation (tj [see Slater (1965)]. For the orthorhombic polyethylene crystal of the space group Pnam, symmetry operations are listed in Table IV.l. The symmetry operation R (at the lattice origin) transforms the Cartesian coordinate vector X(g,) as... 

The procedure Merge transforms the internal displacement coordinates and momenta, the coordinates and velocities of centers of masses, and directional unit vectors of the molecules back to the Cartesian coordinates and momenta. Evolve with Hr = Hr(q) means only a shift of all momenta for a corresponding impulse of force (SISM requires only one force evaluation per integration step). 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

Cartesian coordinates is a product of N — 1 Jacobians for the local transformations from polar to Cartesian components for each bond vector Q for j < — 1, times the Jacobian det[A ]p = 1 for the transformation of Q... 

The relationships between the matrices representing the reflection in different coordinate systems are expressible in terms of the matrix S that defines the relationships between the coordinate systems themselves. Suppose x, y) and x, if ) are two pairs of normalized vectors oriented along the axes of two Cartesian coordinate systems related by a hnear transformation ... 

Any number of desired internal coordinates St (t - 1, Nt) can be calculated from the final Cartesian position vectors rjj1, most conveniently by known vector formulae. For the transformation of the covariance matrix of the Cartesian coordinates into that of the internal coordinates, the derivatives of any particular internal coordinate are required with respect to all Cartesian coordinates that actually participate in the motion of this internal coordinate ... 

As mentioned above, the expanded covariance matrix rni of the results contains a zero-filled row and column when a Cartesian coordinate has been kept fixed. A judiciously chosen variance G2 11) can be entered on the respective diagonal of rm, prior to the transformation (Eq. 61) if the fixed coordinate is afflicted with an (estimated) error and the propagation of this error to any of the derived internal coordinates is to be studied. For this reason it may even be practical to carry along in the expanded vector of variables jj an atom that has never been substituted (and will hence drop out of the fit by the application of E), but whose position can be estimated and is required for the calculation of certain bond lengths and angles involving atoms that were substituted. 

Here Jix, Jiy, and J, z are the Cartesian coordinates of the vector J,. As the volume V and the product J, dA are scalars, the divergence is also a scalar quantity. A positive divergence means a source of component i, while a negative divergence indicates a sink, and at points of div J,- = 0, there is no accumulation and no removal of material. Transformation of the surface integral of a flow into a volume integral of a divergence is... 

The particles are numbered from 1 to N with Mi the mass of particle i, R, = [A, Yt Zi) a column vector of Cartesian coordinates for particle i in the external, laboratory fixed, frame, Vr the Laplacian in the coordinates of R, and Ri — Rj the distance between particles i and j. The total Hamiltonian, eqn.(l), is, of course, separable into an operator describing the translational motion of the center of mass and an operator describing the internal energy. This separation is realized by a transformation to center-of-mass and internal (relative) coordinates. Let R be the vector of particle coordinates in the laboratory fixed reference frame. 

An efficient method to visualize the properties of the normal coordinates is to calculate activity measurements, AM. They show which components of the dipole moment vector and the polarizabilty tensor are modulated by the vibration, and the relative sign of the infrared and Raman optical activity (Schrader et al., 1984 Schrader, 1988). The necessary transformation of the eigenvectors (Eq. 5.2-13) needs only seconds of computer time. The AMs are useful to assign vibrations to symmetry species and to check the input of the frequency calculation the symmetry of the Cartesian coordinates of the atoms as well as of the force constant matrix. This program is part of the SPSIM program package (Fischer et al., 1989). 

The coefficients eja governing the mathematical transformation from normal coordinates Qa to atomic Cartesian coordinates 7j provide a transparent description of mode character. The vector ej parallels the motion of atom j in normal mode a, while the sqnared magnitude describes its relative mean squared amplitude. Normalization, according to = then ensures that the mode... 

Herein the vectors are the unit eigenvectors of the Cartesian coordinate system and 1 the right-hand eigenvectors of the value matrix W as discussed in Appendix 4. Formally the transformed differential equation is of the same general type as (11.12) ... 

The governing equations can be transformed directly from Cartesian coordinates into cylindrical coordinates without considering the vector notation. In this appendix the relationships between the Cartesian coordinates and the cylindrical coordinates are defined solely, but the method of coordinate transformation is generic and can thus be applied to any orthogonal coordinate system. 

Thus the transformation matrix for vector components is the transpose of the inverse of A, the transformation matrix for basis vectors. For transformations among Cartesian coordinate systems, we have the special relationships,... 

By definition the components of the second-rank Cartesian tensor ax transform under rotation just like the product of coordinates xy (e.q., see Jeffreys, 1961) The motivation for what ensues springs from the observation that the spherical harmonics Ym (0, ft) (where 6, ft) are the polar and azimuthal angles of the unit vector (r/1 r )) can be written in terms of the coordinates (x, y, z) of the vector r, for example,... 

Vector d represents the normal mode p in internal coordinate space. It can be transformed to Cartesian coordinate space according to Eq. (21) ... 

NRVS data are commonly interpreted within a harmonic approximation, which describes molecular vibrations in terms of independent oscillations along a set of normal coordinates Qa = ej rjni p related through a linear transformation to the Cartesian coordinates rj of atoms j weighted by their masses nij. Molecular rotation and translation lead to six modes having zero frequency. Projection of the transformation coefficients eja onto the direction k of the X-ray wave vector k = (Eo/hc)k determines the recoil fraction... 

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