Cartesian dimensions transformations

Appendix Angle-Action Transformations in Two Cartesian Dimensions... 

APPENDIX ANGLE-ACTION TRANSFORMATIONS IN TWO CARTESIAN DIMENSIONS... 

As this point it is important to note that the B matrix is usually not square and therefore cannot be inverted. In fact, it transforms from Cartesian (dimension 3n) to internal coordinates (dimension 3n - 6). In simple cases, six dummy coordinates (Tx, Ty, Tz, Rx, Ry, Rz) may be added to the 3 - 6 internal ones in order to obtain in invertible square matrix. However, in some cases the symmetry of the problem makes it necessary to introduce redundant non-linearly independent coordinates (6 CCC angles for benzene or 6 HCH angles for CHq). Gussoni et al. (1975) has shown that it is possible to use the transposed matrix instead of the inverse one and that this choice is the only one which ensures invariance of the potential energy upon coordinate transformation. We can therefore write... 

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... 

We have seen that the overall translational motion of a system of two particles along the x-axis is separable from the vibrational motion. In a three-dimensional picture of the system, translational motion is also separable, but the coordinate transformation is different. In three Cartesian dimensions, the positions of the two particles can be specified as xyyyZ and xyyyZ. The separation distance between the two particles, r, is then... 

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... 

The most important new feature of the Lorentz transformation, absent from the Galilean scheme, is this interdependence of space and time dimensions. At velocities approaching c it is no longer possible to consider the cartesian coordinates of three-dimensional space as being independent of time and the three-dimensional line element da = Jx2 + y2 + z2 is no longer invariant within the new relativity. Suppose a point source located at the origin emits a light wave at time t = 0. The equation of the wave front is that of a sphere, radius r, such that... 

In our description of spin reorientational relaxation processes, tensorial quantities are used for which it is necessary to know the transformation properties concerning rotation. A clear and compact formulation is obtained by replacing the cartesian components with a representation in terms of irreducible spherical components. It is known that any representation of the group of rotations can be developed into a sum of irreducible rqpre-sentations D of dimension 2/ +1. If for the description of general rotation R(U) we use the Euler angles Q = (a, p, y), this rotation will be defined by... 

By removing two C3 rotations (2C3, implying both the 2S3 improper rotations, 2C2 rotations and 2ctv reflections removal) of D h group (dimension h = 12) we obtain the point group 2 of lower dimension Qi = 4) which is a subgroup of D h (be careful - C3 principal axis of D h and C2 principal axis of C2v must be coincident with cartesian z axis, ct/, mirror plane of D h is transformed into plane... 

Because of the restrictions imposed on the values of the rotation angles (see Table 1.4), sincp and cos(p in Cartesian basis are 0, 1 or -1 for one, two and four-fold rotations, and they are 1/2 or Vs/2 for three and six-fold rotations. However, when the same rotational transformations are considered in the appropriate crystallographic coordinate system, all matrix elements become equal to 0, -1 or 1. This simplicity (and undeniably, beauty) of the matrix representation of symmetry operations is the result of restrictions imposed by the three-dimensional periodicity of crystal lattice. The presence of rotational symmetry of any other order (e.g. five-fold rotation) will result in the non-integer values of the elements of corresponding matrices in three dimensions. 

The quantities relevant to the rotationally averaged situation of randomly oriented species in solution or the gas phase must necessarily be invariants of the rotational symmetry. Accordingly, they must transform under the irreducible representations of the rotation group in three dimensions (without inversion), R3, just like the angular momentum functions of an atom. The polarisability, po, is a second-rank cartesian tensor and gives rise to three irreducible tensors (5J), (o), a(i),o(2), corresponding in rotational behaviour to the spherical harmonics, with / = 0,1,2 respectively. The components W, - / < m < /, of the irreducible tensors are given below. 

The dimensions for a descriptor are dehned in two groups — Cartesian distance and 2D property — where the minimum, maximum, and resolution of the vector in the hrst dimension of the descriptor can be dehned. The track bars are adapted automatically to changes for example, resolution is calculated and minimum-maximum dependencies are corrected. When the binary checkbox is clicked, only selections are possible that result in dyadic vector length (i.e., the dimension is a factor of 2"). This feature prevents the complicated adjustment of all settings to gain a binary vector that is necessary for transformations. 

Although we have introduced irreducible spherical tensors, we do not yet have a formalism which admits ready generalization to D dimensions. This can be accomplished by transforming the spherical tensor to a Cartesian tensor. The spherical components of a tensor are related by a unitary transformation to Cartesian components [11,12,13]. For example consider a spherical harmonic of / = 1 (a spherical tensor of rank 1) written as a three component vector... 

Finally, before leaving this section, we note another important aspect of the Liouville equation regarding transformation of phase space variables. We noted in Chap. 1 that Hamilton s equations of motion retain their form only for so-called canonical transformations. Consequently, the form of the Liouville equation given above is also invariant to only canonical transformations. Furthermore, it can be shown that the Jacobian for canonical transformations is unity, i.e., there is no expansion or contraction of a phase space volume element in going from one set of phase space coordinates to another. A simple example of a single particle in three dimensions can be used to effectively illustrate this point.l Considering, for example, two representations, viz., cartesian and spherical coordinates and their associated conjugate momenta, we have... 

The cylindrical polar coordinate system uses the variables p and Cartesian coordinates. The equations needed to transform from Cartesian coordinates to cylindrical polar coordinates areEqs. (3.3) and (3.4). Equations (3.1) and (3.2) are used for the reverse transformation. 

The underlying idea behind the complex coordinate rotation (CCR) method " that is suggested by the Balslev-Combes theorem is a complex scaling of the Cartesian coordinates in the Hamiltonian operator, each by the same complex phase factor x xe. This transformation defines a new, complex-scaled Hamiltonian, H H 0). In one dimension (for simplicity), the complex-scaled Hamiltonian is... 

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