Cartesian coordinates transformation

That is, the transformation is represented as successive rotations of y, P, a about the e3, e2, and axes. A positive rotation is a counterclockwise rotation.1 Since R is unitary, it follows that the Cartesian coordinates transform as... 

The <, Uf, Uy can be thought of as polar vector components (as opposed to axial vector components u, Uy, ) and they transform accordingly. When the lattice dynamical problem is treated in terms of the dynamical variable ujtyU ujigUy, Cochran and Pawley have pointed out that the two-molecule interaction force constants 0, (/A , I k ) can be treated as a two-dimensional tensor of dimension six. If S is the cartesian coordinate transformation matrix corresponding to a symmetry transformation, then the six-dimensional transformation matrix is... 

In principle a similarity transformation can be found which will change the cartesian coordinate transformation matrices [Eq. (3.10)] into the completely reduced normal coordinate transformation matrices which include... 

Since in computations of electronic structure theory derivatives of the total energy of molecular systems with respect to geometrical coordinates are best obtained in Cartesian coordinates, transformation of these derivatives to coordinate systems of more spectroscopic use, e.g., internal or normal coordinates, needs to be discussed. Furthermore, it is noted that, due to the lack of analytic higher-derivative methods at correlated levels of computational quantum chemistry, in practice higher-order force constants are usually determined first in a convenient set of internal coordinates. Then, in order to employ varia-tional or perturbational approaches utilizing anharmonic force fields they may n6ed to be expressed in normal coordinates, never known a priori to the calculation. It is thus clear that these usually nonlinear and somewhat complicated transformation equations occupy a central role in anharmonic force field studies. 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

To shift it to some arbirtrary point ( yo,0jo) we first express Eq. (161b) in terms of Cartesian coordinates, and then shift the solution to the point of interest, namely, to (xjo, o)[= ( T/Oi 0yo)]- Once completed, the solution is transformed back to polar coordinates (for details see Appendix F). Following... 

In this appendix, we discuss the case where two components of Xm, namely, x p and XMg (p and q are Cartesian coordinates) are singular in the sense that at least one element in each of them is singular at the point B p = a,q = b) located on the plane formed by p and q. We shall show that in such a case the adiabatic-to-diabatic transformation matrix may become multivalued. 

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... 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

When one of the cartesian coordinates (i.e. x, y, or z) of a centrosymmetric molecule is inverted through the center of symmetry it is transformed into the negative of itself. On the other hand, a binary product of coordinates (i.e. xx, yy, zz, xz, yz, zx) does not change sign on inversion since each coordinate changes sign separately. Hence for a centrosymmetric molecule every vibration which is infrared active has different symmetry properties with respect to the center of symmetry than does any Raman active mode. Therefore, for a centrosymmetric molecule no single vibration can be active in both the Raman and infrared spectrum. 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

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). 

Fig. 2.5. Possible applications of a coupling parameter, A, in free energy calculations, (a) and (b) correspond, respectively, to simple and coupled modifications of torsional degrees of freedom, involved in the study of conformational equilibria (c) represents an intramolecular, end-to-end reaction coordinate that may be used, for instance, to model the folding of a short peptide (d) symbolizes the alteration of selected nonbonded interactions to estimate relative free energies, in the spirit of site-directed mutagenesis experiments (e) is a simple distance separating chemical species that can be employed in potential of mean force (PMF) calculations and (f) corresponds to the annihilation of selected nonbonded interactions for the estimation of e.g., free energies of solvation. In the examples (a), (b), and (e), the coupling parameter, A, is not independent of the Cartesian coordinates, x. Appropriate metric tensor correction should be considered through a relevant transformation into generalized coordinates...

Path integrals can be expressed directly in Cartesian coordinates [1, 2] or can be transformed to Fourier variables [1,2,20,45]. A Fourier path integral method will be used here [20]. The major reason for doing this is that length scales are directly built... 

It has two components, (x, y) in cartesian coordinates, and these can be transformed into plane polar coordinates to give... 

The transformations of a cartesian coordinate system (x, y) in the plane of the circle can be used to generate a representation of the group. The... 

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