Translation, along cartesian axis

For the calculation of the principal moments, the structural description of a particular isotopomer must be translated into Cartesian coordinates. This can be done by an algorithm which has been given an efficient and widely applicable form by Thompson [58], Each atom is considered to be the origin of a local Cartesian base system whose axis directions are fixed by a prescription which references the two preceding atoms in the chain (and a few conventions if necessary). The positional description by the local base systems is then retransformed, in a stepwise fashion along the chain (by a translation plus a rotation), into that of the local base... 

Let us consider mass transfer for a translational flow past a solid spherical particle, where the flow field remote from the particle is the superposition of a translational flow with velocity U and an axisymmetric straining shear flow, the translational flow being directed along the axis of the straining flow. The dimensional fluid velocity components in the Cartesian coordinates relative to the center of the particle have the form... 

The lattice does not have to be cubic. The appearance of the lattice along one coordinate axis does not have to be the same as along the other axes, and the translational axes of symmetry need not even be perpendicular to each other like the Cartesian axes. The essential property is simply that the pattern along each axis continues to repeat in all three dimensions. 

We can now explore further the properties of the frequencies. The force constants fy are second derivatives of the electronic energy with respect to displacements of the Cartesian coordinates of the nuclei from their minimum energy positions (the so-called equilibrium configuration). Suppose that we displace all the nuclei by the same amount along the x-axis of an appropriate coordinate system. This motion would correspond to a translation of the molecule in the X direction. In this displacement, the nuclei do not move with respect to each other and there would be no energy change. The same will happen if we move the molecule in the Y direction or in the Z direction. There are three independent translations which are possible. 

The anisotropic rigid-body displacements of a molecule can be described in terms of translation (T, vibration along a straight-line path), libration (L, vibration along an arc), and screw (S, a combination of vibration and translation that may be regarded as vibration along a helical path). The mean-square amplitude of translational vibration is usually referred to as a system of Cartesian coordinates and unit vectors. S is the mean correlation between libration about an axis and translation parallel to this axis. Each of these three components can be expressed as... 

The CRU rotates through a helical advance angle, a, and a translational advance, d, in replicating itself (see Fig. 1). The hdix is arbitrarily oriented with respect to a Cartesian coordinate syston such that the helix axis is along the z axis and the first atom of the repeat imit is on the x axis at z = 0. The Cartesian... 

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... 

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