Torsional potential energy functions

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

The local conformational preferences of a PE chain are described by more complicated torsion potential energy functions than those in a random walk. The simulation must not only establish the coordinates on the 2nnd lattice of every second carbon atom in the initial configurations of the PE chains, but must also describe the intramolecular short range interactions of these carbon atoms, as well as the contributions to the short-range interactions from that... 

In the previous section, the adaptation of the RIS model was based on the distance between next-nearest neighbor beads. This approach is obviously inadequate for CH3-CHX-CH2-CHX-CH3, because it necessarily abandons the ability to attribute different conformational characteristics to the meso and racemo stereoisomers. Therefore a more robust adaption of the RIS model to the 2nnd lattice is necessary if one wants to investigate the influence of stereochemical composition and stereochemical sequence on vinyl polymers [156]. Here we describe a method that has this capability. Of course, this method retains the ability to treat chains such as PE in which the bonds are subject to symmetric torsion potential energy functions. 

For butane (Fig. 3.5), using Eq. 3.4 and experimenting with a curve-fitting program shows that a reasonably accurate torsional potential energy function can be created with five parameters, k0 and ki k4 ... 

The crystal and molecular structure data of the three Se8 forms listed in Table II have been determined by X-ray diffraction (27-29, 31). a-, /3-, and y-cyclooctaselenium crystallize in the same space group but differ in the packing of the molecules (see Fig. 4). The average bond distances, bond angles, and torsional angles of the Se8 molecules are identical within the limits of the standard deviation. The torsional angle of 101° is close to the value of 99° observed in the case of Sg (36) and obviously corresponds to the minimum of the torsional potential energy function. The shortest intermolecular distance has been observed in the case of -y-Se8 the value of 334.6 pm is even smaller than the shortest intermolecular contact in orthorhombic cy-clooctasulfur, S8 [337 pm (33)]. 

Figure 6 Harmonic and Fourier torsional potential energy functions compared for a threefold periodicity.

The torsional angles of gauche conformers of a number of molecules are included in this study. However, the prediction of torsional potential energy functions for these molecules are not included although geometrical structures at conformation minima are indeed related to torsional potentials. Further systematic study of these problems is required. 

T. HalilogluandW. L. Mattice, Mapping ofrotational isomeric state chains with asymmetric torsional potential energy functions on ahigh coordination lattice Application to polypropylene, J. Chem. Phys. 

Fig. 1. Torsion potential energy function for the central C—C bond in n-butane. From Ref 4. Copyright 1994 John Wiley Sons, Inc. Reprinted by permission of John Wiley Sons, Inc.

Table 1. Maxima and Minima in the Torsion Potential Energy Function for the Internal C—C Bond in n-Butane...

The t state is preferred in the melt at this temperature. The symmetry of the torsion potential energy function in equation (3) for the C—C hond in polyethylene, U(< ) = U(—< ), produces pg+ j = Pg -,i- Knowledge of all of the p y, is often usefiil in the interpretation of conformation-dependent spectral properties that have a local origin, such as the chemical shift and coupling constants in nmr spectroscopy (17). 

Summation of the elements in the individual colimms reproduces the results in equations (10) and (11). The result ptg i = Ptg- i = Pg t-,i = Pg-t-,i arises from the symmetry of the torsion potential energy function and the absence of a distinguishable direction along the polyethylene chain. The interdependence of the bonds makes the probability for two successive g states dependent on the relationship of their signs, pg+g+.j =pg-g. >pg+g-. =pg-g+ j. 

Figure 1 Torsional potential energy function for an alkane-like tetramer. The solid line corresponds to the original function. The dotted line depicts the modified function that is actually employed during an umbrella sampling simulation to facilitate transitions over energy barriers...

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