Independent internal rotation

For actual chains, the assumption of independent internal rotations cannot be valid, because of the steric interactions between substituent groups on second-neighboring chain atoms and beyond. Various degrees of approximation exist, corresponding to the number of correlations explicitly considered. As an example, we may give a formula of Ptitsyn and Sharonov [(216) see also Volkenstein (21258)] for chains of the —CH2CX2— type ... 

Accordingly, the conformational problem in this class of polyamides can be reduced to the general problem of molecules with two independent internal rotation angles. Once the values of these angles are known from model diamides, one should be able to specify the preferred conformation adopted by the corresponding P/P or Z)MP/P polyamides. 

Molal thermodynamic properties of Pb(CH3)4 in the ideal gas state comprising S , Cp , AHf, AG , and log Kf are calculated for temperatures between 0 and 1000 K. The calculations are based on rigid-rotator, harmonic oscillator, and independent-internal-rotator approximations [620]. 

The situation with respect to HCN is rather different, because it is no longer vahd to approximate the internal rotational constant, B, as independent of the angle 0. As discussed by Efstathiou et al. [10], the form of the most recent potential energy surface [48, 49] shows that the separation of the H atom from the CN center of mass decreases by about 30% between the HCN configuration and the T-shaped transition state, giving the optimized bending potential energy... 

Figure 2(b) represents the potential surface of the identical system, mapped onto the double-cover space [28], The latter is obtained simply by unwinding the encirclement angle < ), from 0 2ti to 0 4ti, such that two (internal) rotations around the Cl are represented as one in the page. The potential is therefore symmetric under the operation Rin defined as an internal rotation by 2n in the double space. To map back onto the single space, one cuts out a 271-wide sector from the double space. This is taken to be the 0 2ti sector in Fig. 2(b), but any 27i-wide sector would be acceptable. Which particular sector has been taken is represented by a cut line in the single space, so in Fig. 2(b) the cut line passes between < ) = 0 and 2n. Since the single space is the physical space, any observable obtained from the total (electronic + nuclear) wave function in this space must be independent of the position of the cut line. 

When the two monomers are linked by a short flexible chain, intramolecular excimers can be formed. This process is still diffusion-controlled, but in contrast to the preceding case, it is not translational it requires a close approach between the two molecules via internal rotations during the excited-state lifetime. Equations (4.44), (4.45), (4.47) to (4.49) are still valid after replacing k [M] by k because intramolecular excimer formation is independent of the total concentration. Estimation of the local fluidity of a medium can be achieved by means of probes capable of forming intramolecular excimers (see Chapter 8). 

With more than one monocentric unit the overall configuration of molecules with inhibited internal rotations may also be determined by the dihedral angles of the bond system between monocentric units. The latter may no longer be treated as independent. A representation by polycentric configurations is advisable. 

Since none of the lattice models is now clearly superior, the choice for interpretation of spin relaxation in polymers is arbitrary. Familiarity leads us to select the Jones and Stockmayer model so we will now consider application of this model to several well studied polymer systems in order to compare dynamics from polymer to polymer. Also the equations required to consider anisotropic Internal rotation of substituent groups and overall molecular tumbling as independent motions in addition to backbone rearrangements caused by the three-bond jump are available for the Jones and Stockmayer model (13). 

The relationship between internal rotation of substituents and backbone rearrangements can be considered from the interpretation. The time scales of anisotropic internal rotation and backbone rearrangements are well separated in M2PPO. In addition, the concentration and temperature dependences of these two quantities are quite different leading us to conclude that the motions are independent. In PIB and PS, Internal rotation and backbone... 

If the molecule is nonspherical or in the case of internal rotational motions of the fluoro-phore independent of the overall rotational motion of the peptide, the anisotropy decays can be described as a sum of exponentials as in eq 14... 

Derivatives of partially unsaturated six-membered N-heterocycles have been investigated. In the Af-acetyl- and N-benzoylazacyclohex-2-enes and in the analogs 130 with X = O, S, CO, internal rotation of the N-acyl group can be studied (77CJC949 840MR676) independently of ring inversion, the... 

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

It is a familiar fact that the skeletal factor s defined in Eq. (2) can very often be decomposed into two independent parts, depending respectively on the restriction of bond angles and on the hindrance to internal rotations. In the simplest case of the polymethylene chain with an internal rotational potential that is symmetric about the irons conformation, we have... 

Theoretically the factor a differs from unity only because all the possible sets of values of internal rotational coordinates are not equally probable. As we have suggested in our main discussion, the most common causes of energetic bias in rotational isomerism are probably steric, although electrostatic, forces involving polar groups are sometimes obviously involved. In the simplest theories, it is assumed for convenience that each internal rotational angle is governed, independently... 

A nonlinear molecule of N atoms has 32V — 6 internal vibrational degrees of freedom, and therefore 3A — 6 normal modes of vibration (the three translational and three rotational degrees of freedom are not of vibrational spectroscopic relevance). Thus, there are 32V — 6 independent internal coordinates, each of which can be expressed in terms of Cartesian coordinates. To first order, we can write any internal displacement coordinate ry in the form... 

From the mathematical point of view the complexity is reduced because the system of equations which has to be solved is a function defined on the two-dimensional manifold of the control volumes boundary and leads to a dimension reduction. Practically the discretisation of the boundary usually is more simple than the meshing of complex three dimensional volumes. Especially this pertains to the transient flow channel geometry in co-rotating twin screw extruders. The surface meshes for the screws can independently be rotated inside the screw and barrel mesh analogous to the batchwise working internal mixer (Banbury Mixer) shown in the bottom part of Fig. 5.26. 

Consider a solute s with internal rotational degrees of freedom. We assume that the vibrational, electronic, and nuclear partition functions are separable and independent of the configuration of the molecules in the system. We define the pseudo-chemical potential of a molecule having a fixed conformation Ps as the change in the Helmholtz energy for the process of introducing s into the... 

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