Translational motion invariant coordinates

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

The next developments of the FC approach were in papers by (R. A.) Marcus,41 49 and a later series from the Soviet Union. About the same time Hush50 introduced other concepts, to be discussed below. The early work of Marcus41 considered the Inner Sphere to be invariant with frozen bonds and vibrational coordinates up to the time of electron transfer. The classical subsystem for ion activation has its ground state floating on a continuum of classical levels, i.e., vibrational-librational-hindered translational motions of solvent molecules in thermal equilibrium with the ground state of the frozen solvated ion. 

There is a close connection between symmetry and the constants of the motion (these are properties whose operators commute with the Hamiltonian H). For a system whose Hamiltonian is invariant (that is, doesn t change) under any translation of spatial coordinates, the linear-momentum operator p will commute with H, and p can be assigned a definite value in a stationary state. An example is the free particle. For a system with H invariant under any rotation of coordinates, the operators for the angular-momentum components commute with H, and the toted angular momentum and one of its components are specifiable. An example is an atom. A linear molecule has axial symmetry, rather than the spherical synunetry of an atom here only the axial component of angular momentum can be specified (Chapter 13). 

It is expected that constitutive equations should be invariant to relative rigid rotation and translation between the material and the coordinate frame with respect to which the motion is measured, a property termed objectivity. In order to investigate this invariance, the coordinate transformation... 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

To see why this is the case, we first consider the portion of the response that arises from llsm. According to Equation (10), we can express (nsm(t) nsm(0)> in terms of derivatives of llsm with respect to the molecular coordinates. Since in the absence of intermolecular interactions the polarizability tensor of an individual molecule is translationally invariant, FIsm is sensitive only to orientational motions. Since the trace is a linear function of the elements of n, the trace of the derivative of a tensor is equal to the derivative of the trace of a tensor. Note, however, that the trace of a tensor is rotationally invariant. Thus, the trace of any derivative of with respect to an orientational coordinate must be zero. As a result, nsm cannot contribute to isotropic scattering, either on its own or in combination with flDID. On the other hand, although the anisotropy is also rotationally invariant, it is not a linear function of the elements of 11. The anisotropy of the derivative of a tensor therefore need not be zero, and nsm can contribute to anisotropic scattering. 

A third requirement is less absolute but still provide a useful consistency check for models that reduce to simple Brownian motion in the absence of external potentials The dissipation should be invariant to translation (e.g. the resulting friction coefficient should not depend on position). Although it can be validated only in representations that depend explicitly on the position coordinate, it can be shown that Redfield-type time evolution described in such (position or phase space) representations indeed satisfies this requirement under the required conditions. 

It is always possible to make a transformation of the translation-free coordinates such that the rotational motion can be expressed in terms of three orientation variables, with the remaining motions expressed in terms of variables (usually called internal coordinates) which axe invariant un-... 

Whatever way it is intended to go in specifying the electronic coordinates, they must be specifiable as translationally invariant so that the centre of mass motion can be separated from Schrodinger s equation for the system. It is only the translationally invariant part of the Hamiltonian that can have a bound state spectrum and thus be relevant to both the scattering and the bound molecule problem. 

Clamped nudei calculations are usually imdertaken so as to yield a potential that involves no redimdant coordinates. Thus a translationally invariant electronic Hamiltonian like (O Eq. 2.24) would actually generate a more general potential than this. A clamped nuclei potential is, therefore, more properly associated with the electronic Hamiltonian after the separation of rotational motion like (O Eq. 2.34) than with the merely translationally invariant one. With this choice again, the minimum consistent product approximation is... 

This definition allows us to draw the curve at any later time moment if its initial form is known. To analyze the general properties of the curve s motion, we need however to derive some dynamical equations which govern the motion of the curve in this model. Our description is constructed in terms of the so-called natural equation k = k l,t), which gives the curvature k of the curve element as a function of its arc length I measured from the end point. The advantage of such description is its invariance with respect to any translations and rotations of the curve in the plane all curves that differ only by a shift or a rotation are described by the same natural equation. Note that the arc length I provides thus an internal coordinate of the points on a curve. 

Since the internal coordinates r represent changes of distances and angles, they are unaltered by pure translations or rotations of the molecule. Therefore, the potential energy is automatically invariant to those motions. 

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