Potential minimum

Figure Al.1.5. Ground state wavefimetion of the double-well oseillator, as obtained in a variational ealeulation usmg eight basis funetions eentred at the origin. Note the spurious oseillatory behaviour near the origin and the loeation of the peak maxima, both of whieh are well inside the potential minima.

Since the prewetting transition may occur only for weakly attractive surfaces [146], we must choose an appropriate value for the parameter This value has been set as follows e = 3 /3/2 j = 6. This corresponds to a relative strength of the fluid-fluid and fluid-surface potential minima close to that for Ar in contact with sohd carbon dioxide [147]. The second parameter in Eq. (144), zq, was set to 0.8[Pg.219]

This particular potential energy surface seems very clean-cut, because there is a single minimum in the range of variables scanned. The chances are that this minimum is a local one, and a more careful scan of the potential surface with a wider range of variables would reveal many other potential minima. 

Another and more accurate microwave method makes use of frequency measurements.1 11 16 28 31 Acetaldehyde is an example of the class of molecules to which this method has been applied. Here there are three equivalent potential minima because of the... 

Fig. 3-2. Molecular electrostatic potential with 6-31G //3-21G basis set in the molecular plane of (ii)-nitrous acid. Black dots refer to four different protonation sites in potential minima. For values of isopotential contours see Nguyen and Hegarty, 1984.

In order to solve Eq. (34), we use the method of characteristics and consider a family of classical trajectories on the inverted potential q(p, x), p(P, x), where P is an (A — 1)-dimensional parameter to characterize the trajectory and x is the time running for the infinite interval along the trajectory, where x = — oo corresponds to the minimum of the potential q(p, —oo) = q ,p(p, —oo) = 0. The solution we want is the trajectory that connects the two potential minima and along which the action becomes minimum. This is called the instanton trajectory and belongs to the above mentioned family qo(x) = q(Po At q close to the potential minimum q , the momentum p(q) is linear with respect to the deviation (q — q ) and Wo(q) is quadratic. 

Let us first start with the simple ID problem of the symmetric potential V —x) = V x) with 2 equivalent potential minima ix. In our treatment, the principal exponential factor VTo(x) does not depend on the energy E and thus the Hamilton-Jacobi equation does not change. Putting... 

Wiener, J. J. M., J. S. Murray, M. E. Grice, and P. Politzer. 1997. Relationships Between Bond Dissociation Energies, Electronic Density Minima and Electrostatic Potential Minima. Mol. Phys. In press. 

Here an additional distinction is to be made between thermodynamic averages of a conformational observable such as the internal energy, which converges well if potential minima are correctly sampled, and statistical properties such as free energies, which depend on the entire partition function. 

Since the potential of mean force is a statistical property, it is insufficient to calculate it directly by importance sampling which, by design, emphasizes potential minima... 

In such cases, the MEHMC method could be employed in combination with an enhanced sampling method that deforms the effective energy surface (but preserves the location of the potential minima), such as that in [29, 97]. Likewise, it may be worthwhile to explore the use of a reversible multiple-time-scale molecular dynamics propagator [103] with MEHMC to accelerate the dynamical propagation. 

Schnitker et al. s (1986) finding, based on classical molecular dynamics simulation, of a large density (4.4 ml-1 at 10°C) of local potential minima qualifying as trapping sites. 

A static electrical field in free space cannot have potential minima and hence cannot trap a charged particle. By using oscillating fields, a pseudopotential can be formed as illustrated in Fig. 17.8, where amass with a suitable charge to mass ratio can be trapped. 

Some Remarks on Reaction Pathways. So far, hypersurface calculations with close correlation to (spectroscopic) molecular data have been considered. Outside the potential minima, the uncer-... 

An important addition compared to previous models was the parameterization of the internucleosomal interaction potential in the form of an anisotropic attractive potential of the Lennard-Jones form, the so-called Gay-Berne potential [90]. Here, the depth and location of the potential minimum can be set independently for radial and axial interactions, effectively allowing the use of an ellipsoid as a good first-order approximation of the shape of the nucleosome. The potential had to be calibrated from independent experimental data, which exists, e.g., from the studies of mononucleosome liquid crystals by the Livolant group [44,46] (see above). The position of the potential minima in axial and radial direction were obtained from the periodicity of the liquid crystal in these directions, and the depth of the potential minimum was estimated from a simulation of liquid crystals using the same potential. 

In some thermodynamic models there are also potential minima associated with different site occupations, even though the composition may not vary, e.g., a phase with an order/disorder transformation. This must be handled in a somewhat different fashion and the variation in Gibbs energy as a function of site fraction occupation must be examined. Although this is not, perhaps, traditionally recognised as a miscibility gap, there are a number of similarities in dealing with the problem. In this case, however, it is the occupation of sites which govern the local minima and not the overall composition, per se. 

Hereafter, we will assume uniaxial anisotropy, of easy-axis type, given by Eq. (3.4) (if not otherwise indicated), since it is the simplest symmetry that contains the basic elements (potential minima, barriers) responsible for the important role of magnetic anisotropy in superparamagnets. Experimental evidence for uniaxial anisotropy is given in Refs. 15 and 16. 

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