Freedom, degrees of, for molecule

For gas molecules, the heat capacity is a constant equal to C = (n/2)pkB where n is the number of degrees of freedom for molecule motion, p is the number density, and kB is the Boltzmann constant. The rms speed of molecules is given as v = V3kBTlm, whereas the mean free path depends on collision cross section and number density as = (pa)-1. When they are put together, one finds that the thermal conductivity of a gas is independent of p and therefore independent of the gas pressure. This is a classic result of kinetic theory. Note that this is valid only under the assumption that the mean free path is limited by inter-molecular collision. 

Moreover, the very limited applicability of the COM separation is not the only one problem another difficulty arises from the introduction of the degrees of freedom for molecules, which are absent in atoms. Therefore Kutzelnigg [5] writes The adiabatic approximation with COM separation is good for atoms, because it takes care that the electrons participate in the COM motion. However, it is unbalanced for molecules, because it favours the COM motion with respect to other motions dominated by the nuclei, such as rotation and vibration, where the participation of the electrons is less trivial anyway than in the COM motion. The partial participation of the electtons in these motions is ignored in the adiabatic approximation both with and without COM separation. ... 

As stated earlier, the main motivation for using either PCA or PCA is to construct a low-dimensional representation of the original high-dimensional data. The notion behind this approach is that the effective (or essential, as some call it [33]) dimensionality of a molecular conformational space is significantly smaller than its full dimensionality (3N-6 degrees of freedom for an A-atom molecule). Following the PCA procedure, each new... 

At higher frequencies (above 200 cm ) the vibrational spectra for fullerenes and their cry.stalline solids are dominated by the intramolecular modes. Because of the high symmetry of the Cgo molecule (icosahedral point group Ih), there are only 46 distinct molecular mode frequencies corresponding to the 180 6 = 174 degrees of freedom for the isolated Cgo molecule, and of these only 4 are infrared-active (all with Ti symmetry) and 10 are Raman-active (2 with Ag symmetry and 8 with Hg symmetry). The remaining 32 eigcnfrequencies correspond to silent modes, i.e., they are not optically active in first order. 

Collisions at low ion energies (where Equation 1 can be applied) lead to a short-lived complex between the ion and the molecule—i.e., both collision partners move with the same linear velocity in the direction of the incident ion. The decay of the complex may be described by the theory of unimolecular rate processes if its excess energy can fluctuate between the various internal degrees of freedom. For example, the isotope effect in the reaction of Ar+ with HD may be explained by the properties of... 

There are 78 vibrational degrees of freedom for TgHg and it has been shown that the molecule has 33 different fundamental modes under Oh symmetry, 6 are IR active, 13 are Raman active, and 14 vibrations are inactive. The experimental fundamental IR active vibrational frequencies have been assigned as follows 2277 (v Si-H), 1141 (vas Si-O-Si), 881 5 O-Si-H), 566 ( s O-Si-O), 465 (v O-Si-O), and 399 cm ( s O-Si-O). These generally agree well with calculated values The IR spectrum recorded in the solid state shows bands at 2300 and 2293 cm ... 

Consider a particle (a molecule or atom) for which the different degrees of freedom are independent and the energy is simply the sum of the energies contained in the different degrees of freedom. We can then write the partition function as the product of the partition functions for the various degrees of freedom. For an atom this is rather trivial ... 

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... 

Figure 10.2 Translational, rotational, and vibrational degrees of freedom for a simple diatomic molecule.

The transition state theory of reaction rates [21] provides the link between macroscopic reaction rates and molecular properties of the reactants, such as translational, vibrational, and rotational degrees of freedom. For an extensive discussion of transition state theory applied to surface reactions we refer to books by Zhdanov [25] and by Van Santen and Niemantsverdriet [27]. The desorption of a molecule M proceeds as follows ... 

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... 

For simplicity, we assume that the internal PF of the ligand does not change upon binding, and that all the molecules involved in the process (C.14) are devoid of translational degrees of freedom (localized molecules). 

For a molecule with N atoms, its 3iV degrees of freedom would be split into three translational degrees of freedom (corresponding to x-, y-, and z-directions), and three rotational degrees of freedom for nonlinear molecules and two for linear ones. Therefore, 3N—6 and 3N— 5 vibrational degrees of freedom exist for nonlinear and linear molecules, respectively. Vibrational frequencies can be obtained from convenient tabulations (see, for example, Shimanouchi, 1972 Chase et al., 1985). 

Since both S and Cy depend on the logarithm of Q, they can be evaluated as a sum of contributions for each degree of freedom. For example, for a nonlinear molecule, the following expressions represent its S and c ... 

The utility of Eq. (9.49) depends on the ease with which the Hessian matrix may be constructed. Methods that allow for the analytic calculation of second derivatives are obviously the most efficient, but if analytic first derivatives are available, it may still be worth the time required to determine the second derivatives from finite differences in the first derivatives (where such a calculation requires that the first derivatives be evaluated at a number of perturbed geometries at least equal to the number of independent degrees of freedom for tlie molecule). If analytic first derivatives are not available, it is rarely practical to attempt to construct the Hessian matrix. 

Suppose that we consider a gas like nitrogen that can form a diatomic molecule. There are still three degrees of freedom for each atom, so each nitrogen molecule would have six degrees of freedom, and it would take six numbers to specify the positions of the atoms composing the molecule. There are 3 at degrees of freedom per molecule, where nat is the number of atoms per molecule. 

The number of angles required to specify a molecule s orientation depends on whether it is linear or nonlinear. It takes only two angles, 0 and , to specify the orientation for a linear molecule, as illustrated in Fig. 8.2. Thus there are two rotational degrees of freedom for a linear molecule. It takes three angles, 6, 0, and nonlinear molecule in space, so a nonlinear molecule has three rotational degrees of freedom. 

Secondly, the orientation of the molecules at the moment of impact may be of importance, in addition to which, in assuming one degree of freedom for each molecule in calculating the energy of collision, we made only an approximate allowance for the influence of the direction of approach. This would all tend to reduce the effectiveness of the collisions, perhaps by 2 or 3 times. 

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