Internal rotation modes

The vibrational wave-numbers , and characteristic temperatures of ethane are given (Hansen and Dennison, J. Chem. Phys. 1952, 20, 317) in table 2. There are six non-degenerate modes including the internal rotation (mode 4) and six pairs of doubly-degenerate modes. The potential energy u restricting the internal rotation has three equal minima and three equal maxima. It may be represented approximately by the form... 

Electron-diffraction data in the gas phase [43] suggest that ferrocene prefers to adapt an eclipsed conformation, with an internal rotational barrier of 0.9 0.3 kcal/mol. The calculated barrier derived from the vibrational frequency of the internal rotational mode is 0.72 kcal/mol [44]. Our LDA/NL calculation finds the eclipsed conformation to be the most stable, with a calculated rotational barrier of 0.69 kcal/mol, in good agreement with experiment. The structure of ferrocene has been studied by several theoretical methods. Our optimized geometrical parameters are similar to those that had been obtained previously by Fan and Ziegler [25] employing the same 1.DA/NL scheme (Table 8). The LDA/NL geometry represents a better fit to experiment than... 

The three translational degrees of freedom of a H atom are converted to three vibrations in the alkane. For methyl radicals, three translations and three rotations are lost and five vibrational and one internal rotation modes are formed. If one assumes that the vibrational modes contribute less to the total entropy than translations and rotations, the loss of entropy is larger for CH3 than for H and the U-factor will be smaller. 

In good agreement with the experimental results, the transverse peak at 195 cm (C-C internal-rotation mode) is much higher than... 

The far-infrared spectra suggested that the internal rotational mode with increasing amplitude has a possibility of soft mode in the transition, which may cause a large change in the CFj dipole or the q ntaneous polarization P of the crystal. In other words, the temperature change in the polarization P may originate from the thermal behavior of the optical phonon. The strain c is related with the acoustic phonon. Therefore, the anomalous thermal behavior of the ultrasonic velocity may be interpreted in such a way that the optical phonon is coupled with the acoustic phonon through the piezoelectric interaction OPi), electrostrictive interaction (yP cX and so on. 

Concerning the proper treatment of torsional anharmonicity, which still represents a challenging aspect for accurate thermochemical calculations of complex molecules [257-265], a hindered-rotor anharmonic oscilattor (HRAO) model has been shown to provide accurate results[62, 72, 117, 204]. The HRAO model is based on a generalization to anharmonic force fields of the hindered-rotor harmonic oscillator (HRHO) model [257] that automatically identifies internal rotation modes and rotating groups during the normal-mode vibrational analysis. 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

As the most notable contribution of ab initio studies, it was revealed that the different modes of molecular deformation (i.e. bond stretching, valence angle bending and internal rotation) are excited simultaneously and not sequentially at different levels of stress. Intuitive arguments, implied by molecular mechanics and other semi-empirical procedures, lead to the erroneous assumption that the relative extent of deformation under stress of covalent bonds, valence angles and internal rotation angles (Ar A0 AO) should be inversely proportional to the relative stiffness of the deformation modes which, for a typical polyolefin, are 100 10 1 [15]. A completly different picture emerged from the Hartree-Fock calculations where the determined values of Ar A0 AO actually vary in the ratio of 1 2.4 9 [91]. 

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

A nonlinear molecule, such as water, methane, or benzene, can rotate about any of three perpendicular axes, and so it has three rotational modes of motion. The average rotational energy of such a molecule is therefore 3 X jkT = ]kT. The contribution of rotation to the molar internal energy of a gas of nonlinear molecules is therefore... 

In general a nonlinear molecule with N atoms has three translational, three rotational, and 3N-6 vibrational degrees of freedom in the gas phase, which reduce to three frustrated vibrational modes, three frustrated rotational modes, and 3N-6 vibrational modes, minus the mode which is the reaction coordinate. For a linear molecule with N atoms there are three translational, two rotational, and 3N-5 vibrational degrees of freedom in the gas phase, and three frustrated vibrational modes, two frustrated rotational modes, and 3N-5 vibrational modes, minus the reaction coordinate, on the surface. Thus, the transition state for direct adsorption of a CO molecule consists of two frustrated translational modes, two frustrated rotational modes, and one vibrational mode. In this case the third frustrated translational mode vanishes since it is the reaction coordinate. More complex molecules may also have internal rotational levels, which further complicate the picture. It is beyond the scope of this book to treat such systems. 

Show that, for the bimolecular reaction A + B - P, where A and B are hard spheres, kTsr is given by the same result as jfcSCT, equation 6.4-17. A and B contain no internal modes, and the transition state is the configuration in which A and B are touching (at distance dAR between centers). The partition functions for the reactants contain only translational modes (one factor in Qr for each reactant), while the transition state has one translation mode and two rotational modes. The moment of inertia (/ in Table 6.2) of the transition state (the two spheres touching) is where p, is reduced mass (equation 6.4-6). 

The A defined in Equation 5.30 is not to be confused with the Helmholtz free energy. Should the A frequencies be limited to the external hindered translations and rotations, vi g = vi g = 0, and this is an additional simplification. In some molecules, however, there are isotope sensitive low lying internal modes (often internal rotations or skeletal bends). In that cases both terms in Equation 5.30 contribute. 

When the temperature of a molecule is increased, rotational and vibrational modes are excited and the internal energy is increased. The excitation of each degree of freedom as a function of temperature can be calculated by way of statis-hcal mechanics. Though the translational and rotational modes of a molecule are fully excited at low temperatures, the vibrational modes only become excited above room temperature. The excitation of electrons and interaction modes usually only occurs at well above combushon temperatures. Nevertheless, dissocia-hon and ionization of molecules can occur when the combustion temperature is very high. 

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. 

A significant application of microwave spectroscopy is in the determination of barriers to internal rotation of one part of a molecule relative to another. Internal rotation is a vibrational motion, but has effects observable in the pure-rotation spectrum. If the barrier to internal rotation is very high, then the internal torsion is just like any other vibrational mode, and the rotational constants are affected in the usual way Bv = Be —... 

The DTO model ignores the overall translations and rotations of the molecule as a whole and refers only to internal vibrational modes. It is therefore incapable of explaining on its own the viscosity of dilute polymer solutions. The enhanced viscosity of dilute polymer solutions is undoubtedly due to a hydrodynamic damping of the polymer as a whole as it translates and rotates in the shear field. This was very well described by Debye (21). We should point out that the Debye viscosity is alternatively derivable from the RB theory. 

The combining of two molecules to form one leads to the loss of one set of rotational and translational entropies. The rotational and translational entropies of the adduct of the two molecules are only slightly larger than those of one of the original molecules, since these entropies increase only slightly with size (Table 2.4). The entropy loss is up to 190 J/deg/mol (45 cal/deg/mol) or 55 to 59 kJ/mol (13 to 14 kcal/mol) at 25°C for the small molecules. This may be offset somewhat by an increase in internal entropy due to new modes of internal rotation and vibration (Figure 2.6). 

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