Internal molecular degrees of freedom

The expression for the rate constant in Example 6.3 can also be considered as an application of transition-state theory to a bimolecular reaction where the molecules are considered as structureless atoms with the relevant masses. Thus, when we compare this with the proper expression for a bimolecular reaction, we can gain insight into how the internal molecular degrees of freedom affect the rate constant. In Section 4.1.2,... 

At low temperatures the behavior of k(7) for the three alcohols displays a systematic trend. Data for 1-propanol are much higher than those for ethanol and in turn these are higher and less steep than those for methanol. In what follows we interpret such a finding as an evidence of the role played by internal molecular degrees of freedom as sources of strong resonant phonon scattering. 

The following treatment applies to homogeneous systems with constant mass. The dissipative, time-dependent effects are caused in such homogeneous systems by disturbance of the equilibrium of the internal, molecular degrees of freedom. The... 

Mandelshtam-Brillouin s spectrum of molecular liquids is more complex. The internal molecular degrees of freedom, in general, are related to translational motion of molecules, which leads to additional relaxational mechanisms of density fluctuations. 

A diatomic molecule has two atoms and hence six degrees of freedom. It is convenient to separate three degrees of freedom corresponding to the center of mass. The other degrees of freedom are then relative coordinates. In the present case the number of internal molecular degrees of freedom is... 

The most commonly used stochastic methods are the torsional Monte Carlo method11101 and the cartesian stochastic (or random kick) method11111. The two methods differ in the coordinate system in which they operate. The torsional Monte Carlo method uses internal coordinates, while the random kick method uses cartesian coordinates. The advantage of using internal coordinates is that the molecular degrees of freedom are reduced. The reason for choosing torsional angles as the vari-... 

The most Raman-active intramolecular modes are the two modes (ag) encountered in the absorption spectra of Fig. 2.8 397 and 1404 cm-With a resolution of 1 cm- it is difficult to observe any splitting in these modes in addition, their frequencies show very little sensitivity to temperature. All these observations are in good agreement with the fact that the intramolecular modes are little affected by the crystal field. In contrast, the weakest vibrations (torsions, flexions, and internal distortions, such as butterfly and twisting motions) contribute significantly to the determination of the intermolecular modes, securing the thermalization of all the molecular degrees of freedom 45... 

The vibrational levels of a molecule in a condensed matter system are influenced by the surrounding medium through intermolecular interactions. The time-averaged forces exerted by the solvent on a molecular oscillator cause a static shift in the vibrational absorption frequency. The frequency shifts of the vibrational transitions of a molecule between the gas phase and a condensed matter environment is an indicator of the effect of the solvent on the internal mechanical degrees of freedom of a solute. 

Here Mnuc, is the internal matrix element of the radioactive nucleus which we have supplied with the index n Mmol is the molecular matrix element pv, fp, and Prec are, respectively, the neutrino, the / -electron, and the recoil momenta [ o1 and noi are the wave functions of the daughter and the parent molecules which describe the electronic, vibrational, and rotational molecular degrees of freedom. 

Returning to the spin-lattice relaxation studies, the relaxation behavior of type A and B glasses was systematically compared only recently.3,100,101 It turned out that, at least down to T 0.5 Tg, the spin-lattice relaxation in molecular glasses (without internal relaxing degrees of freedom) is essentially determined by the slow dynamics , namely the NCL contribution and the /1-process.70,73,100,101,115 Next, we shall briefly review these studies. 

Fast variable physics underlies many processes other than typical liquid phase chemical reactions. In fact, this physics is expected to be important for most condensed phase molecular processes governed by the motion of internal solute degrees of freedom, that is, those other than molecular translational and rotational coordinates. 

Conceptually related to the molecular mechanics approach are the so-called empirical or conformational energy methods (Scheraga, 1977 Ra-machandran and Sasisekharan, 1968). These treat molecules as a collection of subunits (e.g., peptides) and usually investigate only internal rotational degrees of freedom. Somewhat akin to the empirical methods is the so-called hard-sphere or contact distance method (Ramachandran and Sasisekharan, 1968). This rapid method assumes all atoms are impenetrable balls and can be used to differentiate possible and impossible conformations. No energies are calculated. Space-filling, hand-held molecular models give the same type of information, but less quantitatively. 

A liquid is said to be non-associated if the intra-molecular degrees of freedom (rotational, vibrational, electronic and nuclear) are not majorly disturbed by the proximity of neighboring molecules. These liquids can be treated, as is the case with gases, with independence between the internal motions and the translation of the molecules. 

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