Rotation external

To further illustrate the application of Equation 14.35 (the limiting behavior of the low pressure IE), consider the case when only the external rotations are adiabatic (translations do not contribute to the isotope effect). In this case the ratio of Q s reduces to a ratio of ratios of moments of inertia, which, provided the structure does not change on passing from active molecules to activated complex, is unity. In this simplified example, the isotope effect reduces to a simple ratio of the number of states and state densities in the activated complex and energized (active) molecules for the light (1) and heavy (h) molecules. 

For a molecular crystal, the description can be simplified considerably by differentiating between internal and external modes. If there are M molecules in the cell, each with nM atoms, the number of external translational phonon branches will be 3M, as will the number of external rotational branches. When the molecules are linear, only 2M external rotational modes exist. For each molecule, there are 3nM — 6 (3nM — 5 for a linear molecule) internal modes, the wavelength of which is independent of q. Summing all modes gives a total number of N M(3nM — 6) + 6M = 3nN, as required, because each of the modes that have been constructed is a combination of the displacements of the individual atoms. 

Furthermore, physical processes such as internal and external rotation of intermediates will be indispensable elementary processes in some catalytic reactions. In this review, the discussions are focused on rather simple reactions taking place on isolated single sites, but it may be reasonable to assume that more complex reactions demand more complicated prerequisites for active sites. 

For simplicity we assume that the particles are magnetically hard.1 Then, the already developed formalism [Eqs. (4.90) and (4.293)] applies in full under two conditions e is identified with v and the internal relaxation time Tq is replaced by the external rotational diffusion (Debye) time x. In the modified equations, the orientation order parameter is given by (Pi)- Similarly to (4.294), we set... 

Before we come to further determinations of the unknown quantities, we shall estimate here the effect of the internal angular momentum on the motion of the liquid. Let a be the characteristic size of internal structural elements, then Sik pav, <7ik pv/a, where p is the viscosity coefficient. An estimate of the characteristic relaxation time of the balance of the internal and external rotation follows from equation (8.7)... 

For times which are much bigger than the relaxation time, the internal and external rotation are balanced, so equation (8.7) is followed by... 

So, we shall further assume, that the internal and external rotation are balanced in polymer solutions and the stress tensor is symmetric, when there is no external force torque. 

Let us remark that generally the coupling term between the external rotation and the internal motions is not very significant, so that it may be neglected in equation (18), at least in first approximation ... 

Notice that, since the external rotation is relatively slower than the internal motions, the external rotation symmetry group may be expected to be isomorphic with the symmetry point group of the molecule in its most symmetric configuration [4]. As a result, the local full NRG, defined by operator (19), may be expected to be isomorphic to the direct product of the restricted NRG by the symmetry point group of the molecule ... 

A complete study of distorted methylhalide, should to consider the external rotation, as well as the coupling between the external and internal motions. The full Hamiltonian operator may be easily formulated, when we consider ... 

For a complete study of symmetric orthoformic acid, we have to deduce the full NRG. The full Hamiltonian operator for the internal and external rotations, may be derived if we consider ... 

Notice that the full NRG s of this series of molecular systems may not be written as a semi-direct product of a restricted NRG for the internal motions and a symmetry point subgroup for the external rotation, as Altmann presented [10], because of the coupling between both motions. The full NRG s, however, possess the same group structure as the Longuet-Higgins Molecular Symmetry Group, in all the cases. 

The restricted Hamiltonian operator for a non-rigid molecular system may be regarded as a special case of a local Hamiltonian operator in which the external rotation term has been dropped. This case holds only when the external and internal motions are separable. Let us consider some typical examples. 

Taking into account that there are no interaction between the external and internal rotation, the switch subgroup corresponding to external rotation may be completely factorized, so that ... 

The local full NRG appears then as a direct product of two subgroups the restricted NRG corresponding to the internal motions, and a switch subgroup, Ul, corresponding to the external rotation, in accordance with equation (20) ... 

The local full NRG appears then as a direct product of two subgroups the restricted NRG corresponding to the internal motions, and a single switch subgroup, Uf, which corresponds to the external rotation. The restricted NRG is found to be smaller than the full NRG and the external rotation one isomor-pic to the symmetry point group of the molecule, Cs, in its most symmetric configuration. As a result, it may be written as in the case of phenol ... 

Both clusters of transitions are in very good agreement with the experimental values. In addition, the substructure of the observed peaks may be clearly attributed to the existence of torsional microstates, although the external rotational must also contribute. 

Table 3 Moment analysis of the partial densities of states of external coordinates. / is the fraction of unstable modes and aJ,/u the mean frequency for stable/unstable portions of the spectrum for the coordinates indicated, rot, trs, and ext stand for rotational, translational, and external (rotational + translational) coordinates, respectively.

The classical approximation for the rotational density of states of a molecule is familiar from elementary statistical mechanics, where it is common to assume that the rotational states form a continuum in calculating the rotational partition function. For the external rotations of most molecules this approximation is very good. For example the classical approximation for the rotational partition function of an asymmetric top is... 

Qei and Qvtbrot denote electronic and rovibrational partition fimctions, respectively. In general, the contributions of the internal degrees of freedom of A and B cancel in g and gviiroXA)gv iro((B ), such that only contributions Irom the external rotations of A and B and the relative motion, summarized as "transitional modes", need to be considered. Under low temperature quantum conditions, these can be obtained by statistical adiabatic channel (SACM) calculations [9],[10] while classical trajectory (CT) calculations [11]-[14] are the method of choice for higher temperatures. CT calculations are run in the capture mode, i.e. trajectories are followed Irom large separations of A and B to such small distances that subsequent collisions of AB can stabilize the adduct. 

In the case of isotropic potential energy surfaces, such as appropriate (approximately) for ion A - induced dipole B systems, the situation is even simpler. In this case, the external rotational levels of A and B transfer unchanged into W(E,T) and g becomes a centrifugal partition function... 

The separate optimization of the internal orbital rotations at the beginning of each macro-iteration improves convergence considerably. However, this treatment so far neglects the coupling to the internal-external rotations. This coupling creates additional rotations R,y between the internal orbitals when the non-linear equations (53) are solved. Convergence can be further improved... 

The corrections are significant if the absolute value of reaction energy is very large thus, they mainly affect initiation reaction and radical recombinations. The first consideration regards initiation reactions. Unlike the case of gas phase, the entropy change is related to the fact that when the two radicals are formed, they remain caged and cannot fully develop their translational and external rotational degrees of freedom (internal rotations and vibrational frequencies remain more or less the same in the reactant and in the transition state). 

The standard entropy S° is the sum of several contributions, translation, internal rotation, external rotation and vibration. Each rotational contribution can be broken down into an intrinsic term, Vjnl and a logarithmic term taking into account the symmetry number a of the compound (if the molecule is optically active, the standard entropy must also be corrected by adding an asymmetry term related to the chirality 2").11 We can therefore express above... 

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