Potential single minimum

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

As would be expected, the enhancement of potential in cylindrical pores turns out to be considerably greater than in dits, as curve (ii) of Fig. 4.9 clearly demonstrates. At R/r = 2 the enhancement is more than 50 per cent, and it is still appreciable when R/r = 3 (R = radius of cylinder). The calculations show that at radii in excess of R = 1086ro, the single minimum (comparable with Fig. 4.8(c)) develops into a ring minimum (i.e. two minima are present in any axial plane, cf. Fig. 4.8(a)). 

The only types of anharmonic potential function we have encountered so far are the two illustrated in Figure 6.38, both of which show only a single minimum. There are, however, some vibrations whose potential functions do not resemble either of those but show more than one minimum and whose term values are neither harmonic, nor are they given by Equation (6.88) or Equation (6.89). Such vibrations can be separated into various types, which will now be discussed individually. 

Consider a potential V x) having a single minimum separated from the continuous spectrum by a sufficiently large barrier satisfying (1.1), e.g., a cubic parabola (fig. 19)... 

Again we use the ImF method in which the tunneling rate is determined by the nontrivial instanton paths which extremize the Eucledian action in the barrier. Let for deflniteness the potential V Q) have a single minimum at = 0, F(0) = 0, separated from the continuous spectrum... 

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. 

If e is now decreased, with the chain in the extended state, the dumbbell nevertheless stays in the stretched state where the potential is the lowest. The transition back to the coiled state occurs only when there is a single minimum on the potential energy curve, i.e. at et = 0.15. Since the critical strain-rate for the stretch-to-coil transition (esc) is much below the corresponding value for the coil-to-stretch transition (eca), the chain stretching phenomenon shows hysteresis (Fig. 11). 

For a potential with a single minimum, a straightforward interpolation scheme suggests itself. We choose xq to be the true minimum, b to be V(xq). The frequency is determined by requiring that eq. IV.6 be satisfied. This condition is found to be ... 

Samec et al. [15] used the AC polarographic method to study the potential dependence of the differential capacity of the ideally polarized water-nitrobenzene interface at various concentrations of the aqueous (LiCl) and the organic solvent (tetrabutylammonium tetra-phenylborate) electrolytes. The capacity showed a single minimum at an interfacial potential difference, which is close to that for the electrocapillary maximum. The experimental capacity was found to agree well with the capacity calculated from Eq. (28) for 1 /C,- = 0 and for the capacities of the space charge regions calculated using the GC theory,... 

Fig. 5 Potential energy hypersurfaces as a function of the reaction coordinate for adiabatic (A, single-minimum potential B, double-minimum potential) and non-adiabatic (C) electron-transfer reactions.

Barbaralene [85] undergoes a rapid Cope rearrangement with a doublewell potential. The radical cation was studied using CIDNP by Roth (1987) after one-electron oxidation of [85] by y or X-irradiation. On the time-scale of the CIDNP experiment ( 10 8s), a single-minimum potential energy surface was found, i.e. bishomoaromatic structure [156] was suggested. 

The presence or absence of a homoaromatic interaction is often based solely on the distance between the non-bonded atoms. Distances greatly over 2.0 A are thought to lead to a p-p overlap that is too small to make any significant contribution. This simplistic approach is not necessarily reliable as shown by Cremer et al. (1991). Their calculations on the homotropylium cation [12] indicate a double-minimum potential energy surface with respect to variations of the C(l)-C(7) distance at the Hartree-Fock level of theory. At the MP4(SDQ) level of theory, only a single-minimum curve was found with the minimum at 2.03 A. The calculated potential energy curves are quite flat in this region. 

Figure 1. Hypothetical single-minimum potential surface for ion-molecule...

In this complex, the energy gap 2J separating upper and lower potential surfaces is estimated to be 0.19 eV. This is evidently too large for the weak-interaction limit to hold. In other systems, J may, in fact, become so large that we have a single minimum, with a stationary delocalized electronic ground state. 

Figure 4. Potential energies for above Hamiltonian for different values of 6. The condition for appearance of a single minimum is 6 (= /2 J) < 1.

An excellent review of bifluorides up to 1980 is available (Gmelin, 1982). An earlier review of very strong hydrogen bonding (Emsley, 1980) assumed that HFj " represented the upper limit of hydrogen bonding and that the proton was in a single minimum potential well. Both assumptions have since been questioned. 

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