Degrees of freedom for motion

The dimension of Evib (x, y) is 24 (total number of degrees of freedom for motion in the xy-plane) - 2 (translation of the molecule in the x and y directions) - 1 (rotation of the molecule about the z axis, Rz) = 21. Furthermore, the representation of the out-of-plane vibrations of benzene can be easily obtained ... 

Motion of the dispersed naththalene triplet probe in P-Np is evident at 180K. By analogy with previous observations of the PMMA system(]Q) such mobility might be concomitant with the onset of the g-relaxation. It is apparent from dynamic mechanical data at IHz that resolution of the a and g relaxations is difficult(M). it may be that the use of molecular probes and labels such as 1-VN (as described above) offers a more sensitive sensory exploration of the matrix than is afforded by macroscopic techniques. It is also possible that the degrees of freedom for motion imbued in the matrix do not result from the motion of ester groups as generally inferred in PMA and PMMA for the g-relaxation but that conformational changes within the it-butyl substituent itself, constitute sufficient release of free volume within the matrix to allow rotational mobility of both the 1-VN label and the dispersed naphthalene probe. 

This formula arises from the fact that each atom is capable of motion in each of three dimensions (the x, y, and z axes of the Cartesian coordinate system). Thus, n atoms will have 3ra total degrees of freedom, but of these 3 correspond to motion of the molecule as a whole through space (translation) and 3 to rotation of the molecule as a whole about its 3 (2 for a linear molecule) principal axes. This leaves 3w-6 (3w-5 for a linear molecule) degrees of freedom for motions of the atoms with respect to each other (vibration). 

Note that a molecule with N atoms has three degrees of freedom for motion (3N). For all three-dimensional objects, there are three axes of translation (x, y, z) and three axes of rotation (as there are three axes of inertia). If we eliminate these six kinds or types of motion (as they are nonvibrational), we are left with 3N-6 vibrational types of motion. (Note If the molecule is linear like many polymers, there are 3N-5 types of motion.) Each other kind of motion is vibrational in nature and has a specific frequency associated with it. As long as the bonds do not break and the vibrations have motions (amplitudes) of about 10-15% of the average distance between atoms, the vibrations are considered harmonic. Any harmonic is considered to be the superposition of two or more vibrations of the molecule and carries the term normal vibration. 

In Chapter 1 it was noted that the number of vibrational modes of a molecule can be calculated by counting the degrees of freedom of the atoms (three per atom for X,Y and Z movement) and subtracting the degrees of freedom for motion of the molecule as a whole, three for its translation and (for nonlinear molecules) three for rotation. This was used in Section 5.2 to arrive at a reducible representation for the basis of nine atomic degrees of freedom for H2O, the classic C2V molecule. The characters for this representation were given in Table 5.1. We can now apply the reduction formula to identify the irreducible representations for the three vibrations of HjO. 

For reactants having complex intramolecular structure, some coordinates Qk describe the intramolecular degrees of freedom. For solutions in which the motion of the molecules is not described by small vibrations, the coordinates Qk describe the effective oscillators corresponding to collective excitations in the medium. Summation rules have been derived which enable us to relate the characteristics of the effective oscillators with the dielectric properties of the fi edium.5... 

Each atom within a molecule has three degrees of freedom for its motion in three-dimensional space. If there are N atoms within a molecule there are 3N degrees of freedom. However, the molecule as a whole has to move as a unit and the x, y, z transitional motion of the entire molecule reduces the degrees of freedom by three. The molecule... 

E. The degrees of freedom for A include the particle in the box translational motion along the reaction coordinate which replaces the so-called imaginary frequency. The spacing between particles in the box states is small and depends on the length l of the box and reduced mass i of the transition state. The density of the translational states corresponding to an energy x in the reaction coordinate is... 

Fig. 2 Schematic representation of the so-called semiclassical treatment of kinetic isotope effects for hydrogen transfer. All vibrational motions of the reactant state are quantized and all vibrational motions of the transition state except for the reaction coordinate are quantized the reaction coordinate is taken as classical. In the simplest version, only the zero-point levels are considered as occupied and the isotope effect and temperature dependence shown at the bottom are expected. Because the quantization of all stable degrees of freedom is taken into account (thus the zero-point energies and the isotope effects) but the reaction-coordinate degree of freedom for the transition state is considered as classical (thus omitting tunneling), the model is ealled semielassieal.

The number of internal degrees of freedom for any system may be reduced by a transformation to center-of-mass coordinates. For example, the system of n + 1 particles with 3(n + 1) degrees of freedom is reduced n pseudoparticles with 3n degrees of freedom, with the 3 leftover degrees of freedom describing the motion of the center of mass. 

In all the approaches mentioned below, it is assumed that the correlation function can be factorized into a product of correlation functions for the three degrees of freedom rotational motion, translational diffusion and electron spin dynamics. 

The 3N degrees of freedom for nuclear motion are divided into 3 translational, 3 (or 2) rotational, and 3N-6 (or 3N-5) vibrational (degrees of freedom. (The translations and rotations are often called nongenuine vibrations.) The 9 irreducible representations in (9.104) include the 3 translations and the 3 rotations. To find the symmetry species of the 3 vibrations, we must find the symmetry species of the translations and rotations. 

We deal first with electronic-nuclear coupling in systems with few atoms, and therefore few degrees of freedom for nuclear motions, so that we can concentrate on the first mentioned challenge. The structure and properties of a molecule in stationary states are well described within the Born-Oppenheimer picture in which the disparity in masses of nuclei and the electron, with mn me and the similarity of Coulomb forces on nuclei and electrons, Fn Fe, mean that within a short time interval At, changes in velocities satisfy Avn — Fn/(mnAt) Ave = Fnj(mnAt) so that an interaction involving small velocities to begin with, and lasting a short time, would be described by slow nuclei. The well known Born-Oppenheimer prescription is then to construct the electronic Hamiltonian Hq for fixed nuclear positions Q = (Ri,. ..,Rn), to calculate electronic states Q) for electron... 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

We place the dividing surface between reactants and products at a critical separation rc and calculate the rate constant according to Eq. (6.8) (assuming, as in Section 4.1.2, that the atoms react with a probability of one when r = rc). In the relevant partition function for the activated complex, there are both translational degrees of freedom for the center-of-mass motion and rotational degrees of freedom... 

The more recent theories of chemical conversions [59-61] take into account the fact that the process of overcoming the activation barrier involves a cooperative change of more than one degree of freedom for the starting reagents subsystem. For the surface processes this is expected to lead to a need for considering the dynamics of the solid atom motion and, at least, the model should include information on Debye frequencies for its atoms (see, e.g., Ref. [62]). An additional inconvenience of the models for the elementary surface processes is associated with the fact that the frequencies of the surface atom oscillations differ from those inside the solid. Consideration of the multiphonon contributions to the probabilities that the elementary process can take place results in a significant modification of its rate constant up to the complete disappearance of the activation form of the temperature dependence [63,64]. 

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