Torsion potential energy function

Figure 2-108. Derivation of a syrMbolic potential energy function from the torsion angle distribution of a torsion fragment.

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. 

Figure 4. History of the torsion angle C4-C5-C6-06 calculated from a typical MD simulation of a-D-glucopyranose in vacuum using the Rasmussen potential energy function PEF422 (41). (Reproduced from Ref. 9. Copyright 1986 American Chemical Society.)...

Since the torsion itself is periodic, so too must be the torsional potential energy. As such, it makes sense to model the potential energy function as an expansion of periodic functions, e.g., a Fourier series. In a general form, typical force fields use... 

It is common practice to represent the total strain energy, Utotab of a molecule by a set of potential energy functions, including bonding (Eb), valence (Eg) and torsional angle (E interactions as well as nonbonded (Enb) and electrostatic interactions (Ec) (see Eq. 2.2). 

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)]. 

Potential Energy Functions I Bond Length,Valence Angle, Torsion... 

There is a strong correlation between the parameters of different potential energy functions so that they should not be developed or refined in isolation. For example, the barrier to rotation about a bond can be modified by changing the explicit torsion angle term or by changing the nonbonded interactions. Thus, the effect of any change on a force field parameter needs to be tested extensively, i.e.,... 

Potential Energy Functions I Bond Length, Valence Angle, Torsion Angle, Twist Angle and Out-of-Plane Deformation Functions... 

Rotation about the central bond costs energy and there are two components to this one arises from the repulsion between the first and fourth atoms in the sequence and the other is intrinsic to the bond. The former component is included in the nonbonded interactions of the molecular mechanics model but the other component must be explicitly included by means of a torsion angle potential energy function. 

Rigid Molecule Group theory will be given in the main part of this paper. For example, synunetry adapted potential energy function for internal molecular large amplitude motions will be deduced. Symmetry eigenvectors which factorize the Hamiltonian matrix in boxes will be derived. In the last section, applications to problems of physical interest will be forwarded. For example, conformational dependencies of molecular parameters as a function of temperature will be determined. Selection rules, as wdl as, torsional far infrared spectrum band structure calculations will be predicted. Finally, the torsional band structures of electronic spectra of flexible molecules will be presented. 

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