Dihedral potentials

In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. 

In this representative dihedral potential, V is the dihedral force constant, n is the periodicity of the Fourier term, (jtg is the phase angle, and (]) is the dihedral angle. 

We can therefore conclude that differences in the structural relaxation between bead-spring and chemically realistic models can be attributed to either the differences in packing that we discussed above or the presence of barriers in the dihedral potential in atomistic models. To quantify the role of dihedral barriers in polymer melt dynamics, we now examine high-temperature relaxation in polymer melts. 

In order to achieve a force field that will provide sufficiently accurate results, the addition of a dihedral potential that adequately describes the complex torsional profile is required. This should allow vdW parameters, especially well-depths, to remain at reasonable magnitudes and at values that reproduce the intermolecular distances found in the solid state. To achieve this, a six-term cosine potential of the form... 

Very often the terms dihedral and torsional are used interchangeably. In a later section, discussing explicit dihedral potentials, we shall specifically reserve the term torsional for an internal rigid rotation, and the term dihedral for the rotation of two vicinal bonds about a middle bond. 

Figure 11 Differential dihedral potential in ethane. The ab initio second derivatives of the energy with respect to the dihedral angle t h are plotted vs. Thh> showing the periodic asymmetric behavior of this coordinate. Although a computational observable, this curve is not experimentally accessible. Recalculated from ref. 118.

These results demonstrate that as defined by Eq. [32] is a formal dihedral potential that couples the 1-4 atoms, because the sum of these interactions over all nonbonded rotating pairs is exactly the total torsional potential (for a rigid rotation). Because of the relationships given in Eqs. [31] and [32], V"dAB) and V.,(AB) are called the differential and integrated dihedral potentials, respectively. 

V and are periodic functions of t and can be represented in terms of a Fourier series. Because of the intrinsic symmetry of the dihedral potential, as well as the symmetry of ethane, only the cosine series exists in this case (see the discussion for propane below ), i.e.,... 

Table 2 Fourier Expansion Coefficients (kcal/mol) of the Differential and Integrated Dihedral Potentials for Ethane...

In summary, this three-body effect, i.e., the dependence of the pairwise interaction on the surroundings, as reflected in a sine series in the Fourier expansion of the dihedral potentials, is something that can be expected in propane in distinction to ethane. The transferability of the interactions from ethane to propane thus also depends on this effect being small. 

Figure 16 Integrated dihedral potentials for V(H"-H) in propane and ethane. The relatively small difference suggests that the interaction is similar for both molecules and that the parameters are therefore transferable.

Figure 17 1-4 H--C integrated dihedral potential in propane. The abrupt change in slope at 80° suggests a stronger steric interaction between C and H than observed in the case of H-"H (see Figure 16). Recalculated from ref. 118.

Figure 18 Torsional potentials in propane as a function of the torsional angle. The torsional angle was arbitrarily chosen as the H —C3 —Cl—C2 dihedral angle (see the inset for notation). Empty squares CH3" H4,H5 interaction. Triangles Me" C3 interaction. Solid line sum of all 1-4 integrated dihedral potentials. Filled squares ab initio points from Ref. (118). The experimental barrier for internal rotation is 3.3 kcal/mol from Ref. (126).

The number of trial sites at any given step, km, can be chosen freely (k = 1 corresponds to an unbiased insertion) and should thus be optimized to improve efficiency [38]. For chain molecules with intramolecular potentials (e.g., bond bending and dihedral potentials), it is more convenient to divide the energy into a bonded part and a nonbonded part. The bonded part can then be used to efficiently bias the selection of trial positions and only the nonbonded part is used for the selection of trial sites and the calculation of the Rosenbluth weights [34,35,37]. Using an isolated chain with intramolecular interactions as the reference state, the excess chemical potential can now be obtained from... 

Fig. 1.10 Left structural diagram of butane. Center coarse-grained model of butane using CH3 and CH2 groups. Right torsion (or dihedral) potential energy showing locations of local minimum corresponding to trans (global minimum) and gauche (secondary local minimum) states...

Typical atomistic models also include so-called dihedral potential, that is, a potential that depends on the dihedral angle

detailed description of this potential, we refer readers to the book of Allen and Tildesley. The typical potential with three minima has a form ... 

We see that effective bonded stress accounts for all stress relaxation after a short time Tnb 30. We interpret this characteristic time as the time at which the stress due to collisions completely relaxes to its equilibrium value. Thus, the only role of nonbonded interactions at long times is the renormalization of the bonded potential. Note however that this is rigorously demonstrated only for the simple KG model without bending or dihedral potentials. The relaxation time of the nonbonded stress inaeases with density and finally diverges at the glass transition. Calculations like these were repeated for longer chains, N = 200, with similar results however, a more detailed analysis for these longer chains is needed. 

FIG. 1 Dihedral potential for polyethylene (PE). Eg is the trans-gauche energy difference and and are the barriers for torsional transitions. 

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