Cosine potential

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... 

In the presence of umklapp processes, the charge bosons are no longer free. They are bound by a cosine potential energy term of amplitude g3 /2ir, so the charge excitations have a gap of this magnitude which can be felt at low temperatures. Under those circumstances, only the SDW susceptibility remains singular with ySDW = 1 (see, however, the refinement when umklapp are present in Section IV.B.3.d). 

The powerful continued fraction procedure (CFP) described by Grosso and Pastori Parravicini in Chapter III may be used to solve Eqs. (3) and (4). An alternative approach has been provided by Ferrario et al., who have computed a variety of numerically derived orientation and velocity acFs for a simple cosine potential and the more comphcated cosinal itinerant oscillator, another RMT-allowed structure. In Chapter VI, Ferrario et al. describe deexcitation effects from the two-dimensional disk-annulus itinerant oscillator also studied by Brot and coworkers. ... 

A great deal of the difficulty affecting the translational case can be bypassed simply by considering the rotational one. The cosine potential in such a case naturally fits the requirements that the system be left unchanged by rotating through 2ir the real body around the virtual one [see Eq. (5.69)]. The rotational counterpart of Eq. (S.l) reads... 

Figure 42. Broad-band dielectric loss spectrum of 10% v/v solution of probe molecule CH2CI2 in glassy decalin at 110 K. Filled circles are the experimental data [98], Curve 1 is the best fit for the anomalous diffusion in the double-well cosine potential (a - 1.5, t,v = 8, and 7 = 0.003) curve 2 is the best fit for the normal diffusion ( J. = 1, v = 7, and 7 — 0.001) in the double-well cosine potential. Dashed line (curve 3) is the Cole-Cole equation [Eq. (9)] with a = 2 — a.

The complex susceptibility x( ) yielded by Eq. (9), combined with Eq. (22) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms combined with the matrix continued fraction solution for the fixed center of oscillation cosine potential model treated in detail in Ref. 25. Thus we shall merely outline that solution as far as it is needed here and refer the reader to Ref. 25 for the various matrix manipulations, and so on. On considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential, we have... 

Another approximate formula may also be derived for the cosine potential as follows. On expansion of and pe in Taylor s series, we have from Eqs. (9) and (22)... 

In the limit of large damping and small F (linear response regime), Risken and Volhner find the following for the mobility of a particle moving in a simple cosine potential ... 

For example. East and Radom devised a procedure they call El, which calculates from the MP2/6-31G geometry (MP2 calculations are (Uscussed in Section 15.18) and Svib from HF/6-31G scaled vibrational frequencies and the harmonic-oscillator approximation, except that internal rotations with barriers less than 1.4R7 are treated as free rotations [A. L. L. East and L. Radom,/. Chem. Phys., 106,6655 (1997)]. For 19 small molecules with no internal rotors, their El procedure gave gas-phase 5S,298 values with a mean absolute deviation from experiment of only 0.2 J/mol-K and a maximum deviation of 0.6 J/mol-K. The El procedure was in error by up to Ij J/mol-K for molecules with one internal rotor and by up to 2 J/mol-K for molecules with two rotors. An improved procedure called E2 replaces the harmonic-osdllator potential for internal rotors by a cosine potential calculated using the MP2 method and a large basis set, and reduces the error to 1 J/mol-K for one-rotor molecules. 

We consider first the nucleation under an applied simple shear stress a without the presence of an accompanying mean normal stress We consider the nucleation as a reversible experiment in which the transformation strain in the volume element Qf can be built up by an imposed external shear strain y. As is clear, and as discussed by Orowan (1954), this results in a rise in the Helmholtz free energy AF(y) that initially is given by a quadratic function and for the whole process can be taken as being given by a cosine potential (Kocks et al. 1975) (see also Johnson and Samwer (2005)),... 

Relating the Helmholtz free energy to a cosine potential, as done in Chapter 7, from which the plastic shear resistance r and its particular form are derivable, gives the peak threshold resistance i at the temperature of interest as... 

The validity of the simple cosine function in equation (21) as a description of the restricting potential of a symmetric top has been questioned by Blade and Kimball. They examined the applicability of a two-parabola potential for the internal rotation in ethane, 1,1,1-trifluoroethane, and methanol, and concluded that the simple cosine function is not satisfactory. This conclusion for ethane has been criticized in detail by Pitzer, who has shown that the two-parabola potential does not provide a significantly better fit to the calorimetric data than does the simple cosine potential. Experimental heat capacities for ethane over the range 90 to 305 K are... 

SH and one —CH3 internal rotation in propane-2-thiol were determined from i.r. data, but agreement between calculated and experimental values of entropy and heat capacity could not be obtained by fitting a 3-fold cosine potential to the second methyl torsion. This was attributed to interaction between methyl groups. Satisfactory agreement between the calculated and experimental data was obtained by arbitrarily treating the second methyl torsion as an anharmonic oscillator having fundamental wavenumber v = 257 cm and anharmonicity coefficient X = — 6.3 cm (see p. 279). ... 

Many applications of Kilpatrick and Pitzer s procedure for calculating thermodynamic properties of molecules with compound rotation have been reported. In all cases possible potential energy cross-terms between rotating tops have been neglected. Contributions from internal rotation of symmetric tops have been calculated using the appropriate tables." These tables have also been used in calculations for the internal rotation of asymmetric tops hindered by a simple -fold cosine potential. 3-Fold potential barriers have been assumed in calculations for the —OH rotations in propanol and 1-methylpropanol, the —SH rotations in propane-1-thiol, butane-2-thiol, 2-methylpropane-l-thiol, and 2-methylbutane-2-thiol, the C—S skeletal rotations in ethyl methyl sulphide, diethyl sulphide, isopropyl methyl sulphide, and t-butyl methyl sulphide, and the C—C skeletal rotations in 2,3-dimethylbutane, and 2-methylpropane-l-thiol. 2-Fold cosine potential barriers have been assumed in calculations in the S—S skeletal rotations in dimethyl disulphide and diethyl disulphide. ... 

The approximation of internal rotation as an equilibrium of rotational conformers (p. 296) described by equations (29)—(32) has been followed in calculations of contributions from the C—C skeletal torsion in n-propanol, s-butanol, propane-l-thiol, and butane-2-thiol. Contributions for each of the conformers were calculated assuming 3-fold cosine potentials and using the tables of thermodynamic functions for symmetric tops, -... 

Analysis by a dynamical model that incorporated a large-angle range on either side of the planar anti form gave good agreement with previous data at 225 °C [1]. A three-term cosine potential for internal rotation about the N-N bond was employed. The nozzle temperature was 225 °C. 

Several methods have been proposed for reducing the intensity of repulsive interactions in this initial period, such as use of a cosine potential or a soft core potential.Unfortunately for both these forms the force is actually zero for r = 0 and there is still a finite chance of sites remaining superimposed. One method that has proved quite robust uses a truncated force potential in which the short-range force for neighbours i and j, where i -y > 5, is constrained to be constant below a critical separation rtr i.e.. 

Equation (52) predicts that chain twisting occurs for all values of n, where n is the number of chain units. Williams, Lauritzen and Hoffman replaced the cusp potential of equation (51) by the more realistic cosine potential equation (53) and by a matched parabola potential, equations (54)... 

FIGURE 19 Schematic representation of the potential function and torsional energy levels for a threefold barrier. A simple cosine potential is depicted with three identical minima and maxima. The corresponding eclipsed and staggered configurations for ethane are also indicated. Each torsional energy level is labeled by the torsional quantum number v. The torsional sublevels are denoted by A or E. 

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