Central well potential

Figure 6. The deterministic potential (80) of the driven Henon-Heiles system. A central well is separated from three asymptotic channels by three saddle points SA, SB, and Sr. Also indicated is the surface, Eq. (82), on which initial conditions are sampled. (From Ref. 40.)...

The next debate in the literature was whether these molecules have C2v or Cs symmetry. The nuclear motion of a C2v symmetric structure would be described by a single-well potential (see Figure 10). The alternative is a rapid interconversion of two valence tautomers, each of Cs symmetry. This would occur via the C2v structure as transition state (see Figure 11). In this case the motion of the central sulfur would be described by a double-well potential, and dioxathiapentalene and trithiapentalene would be misnomers for (3//-l,2-oxathiol-3-ylidene)acetaldehyde 180 and (3/7-1,2-dithiol-3-ylidene)thioacetaldehyde 181. One advantage of C2v symmetry is aromatic stabilization from the 1071 electrons <2001CRV1247>. The alternative Cs symmetry has the advantage of avoiding a hypervalent sulfur. 

Biphenyl, terphenyl, and quaterphenyl all have room-temperature structures that behave similarly, and also all have low-temperature phases in which only one conformer is present. The torsion angle about the central C-C bond in biphenyl is estimated to be about 10° in the low-temperature form, which is appreciably less than that in the gas phase. In the case of terphenyl, it has been established (41) that the room-temperature structure is disordered each molecule librates in a double-well potential, with the barrier height being about 0.6 kcal/ mol. In the low-temperature form the molecule is stabilized in one of the two minima of the well, and has its terminal rings rotated in the same sense, so that the molecule conserves its center of symmetry. This alternation of rotations between adjacent rings is found (42) also in the low-temperature form of quaterphenyl and results in the molecule being noncentric (even though the crystal structure has a center of symmetry). 

In order to interpret the observation of reactions which have low efficiencies, we have suggested a double well potential surface model (8) illustrated in Figure 2. This is the simplest model which is consistent with available data. At the low pressures typically used in ICR, long collision times ensure that the system contains its initial total energy throughout the reaction. The efficiency for an exothermic reaction is given by /(k j + k ). Passage over the central barrier (k )... 

Let be the vibronic ground state localized in the left well of the double-well potential, and IP2 be the right-side ground state. The reference symmetry group of the undistorted molecule AB2 is D. An off-center displacement of the central atom A along the axis B - B reduces this symmetry to This means that the local symmetry... 

The design of the structure of the cluster composed of two SiOj tetrahedrons linked by the O atom was done elsewhere [192, 193]. A double-well potential (DWP) of the central O atom is shown in Figure 19. When R is close to equilibrium for crystal, the barrier would not emerge ( = 0), and the O atom has one equilibrium position. Tunneling states are formed at 4.2 A. According to ref. 193, the DWP energy levels are described by an approximate relation ... 

Figure 19. Double-well potential of central oxygen atom in OsSi-O-SiOj cluster imitating structure of fused quartz.

Figure 5.58 shows the water-cut curve of several central wells in the tested zone. This figure shows that water cut dropped after each alternate injection, but that the magnitude of the drop decreased, indicating lower potential for improvement. 

It was from these data that the potential function for the proton was modelled as the superposition of a double well (+ 0.8 A) with a barrier of 150-200 cm and a very narrow central well with a dissociation threshold of 1200-1400 cm, see Fig. 9.16. The energy gain upon hydrogen bond formation is a maximum when the proton is localised in the narrow central well in the totally symmetric structure. 

Fig. 24.4 Well potential additional to the potential of an anion produced by the Coulomb potential from nuclei and electron clouds surrounding the anion excluding the central anion s self-contribution CaCu02. The well depth is relative to the maximum averaged over the sub-subdomain described in the text. In the molecular case, the reference potential is zero at the region apart infinitely from the molecule...

The findings for MX3 species with 23 VEs are to be contrasted to those reported for XO3 molecules with 23 VEs, which were found to have a D3h stmcture for central atoms from the first and second row, such as NO3 and PO3, but a C2V double-well potential for central atoms from the third row, such as AsOs, mainly due to lower excitation energies [3]. 

Efforts to use relativistic dynamics to describe nuclear phenomena began in the 1950s with application to infinite nuclear matter. Johnson and Teller [Jo 55] developed a nonrelativistic field theory for interacting nucleons and neutral, scalar mesons which served as a catalyst for Duerr, who, in a landmark paper [Du 56], developed a relativistic invariant version of the Johnson and Teller model which included both scalar and vector meson fields. He showed that nuclear saturation and the strong spin-orbit potential of the shell model could be readily understood. He also predicted a single particle potential which qualitatively reproduced the real part of the central optical potential well depth and its energy dependence for incident kinetic energies up to 200 MeV. 

Note that making the upper limit of integration from R -negligible error because of the low-temperature accessible experiment." The intermolecular potential U r) can be obtained by inverting the experimental Langmuir constant data numerically. In principle, Eq. [14] has an infinite number of solutions however, there exists a unique central-well solution that has physical meaning. The central-well solution has the functional form... 

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