Steeply repulsive potential

Figure 1.3. Real-time femtosecond spectroscopy of molecules can be described in terms of optical transitions excited by ultrafast laser pulses between potential energy curves which indicate how different energy states of a molecule vary with interatomic distances. The example shown here is for the dissociation of iodine bromide (IBr). An initial pump laser excites a vertical transition from the potential curve of the lowest (ground) electronic state Vg to an excited state Vj. The fragmentation of IBr to form I + Br is described by quantum theory in terms of a wavepacket which either oscillates between the extremes of or crosses over onto the steeply repulsive potential V[ leading to dissociation, as indicated by the two arrows. These motions are monitored in the time domain by simultaneous absorption of two probe-pulse photons which, in this case, ionise the dissociating molecule.

The systems considered included a steeply repulsive potential, SRP, between the particles with n arbitrary ... 

One conclusion from this study is that although the hard-sphere fluid has been very successful as a reference fluid, for example, in developing analytical equations of state, it is unrealistic in representing the dynamical relaxation processes in real systems, even with very steeply repulsive potentials. Owing to the discontinuity in the hard-sphere potential, this fluid, in fact, is not a good reference fluid for the short time (fast or j9 ) viscoelastic relaxation aspects of rheology. 

This follows from previous work of these authors that studied the transport of fluids interacting via a steeply repulsive potential of the form... 

There are potential technical problems in simulating steeply repulsive potential fluids by MD, and in response a new molecular dynamics algorithm capable of integrating the equations of motion of impulsive-continuous potential systems (e.g. such as a hard sphere core combined with an r tail) has been derived using operator splitting techniques by Houndonougbo et... 

The constancy of the ligand radii, in contrast to van der Waals radii, suggests that gem-inal ligands on molecules of period 2 elements are squeezed together almost to their limit of compressibility. The repulsive interaction between two atoms is usually represented by a steeply rising potential such as that shown in Figure 5.7. This potential is often approximately represented by a function of the type... 

This potential exhibits a steeply repulsive R 12 term that dominates at small R and a weakly attractive R 6 term that dominates in the long-range limit, with empirical parameters a, b (or equivalently, e, cr) that can be adjusted to best approximate the experimental behavior. 

Typical potential energy curves for the interaction of two atoms are illustrated in Figure 11.3. There is characteristically a very steeply rising repulsive potential at short interatomic distances as the two atoms approach so closely that there is interpenetration of their electron clouds. This potential approximates to an inverse twelfth-power law. Superimposed upon this is an attractive potential due mainly to the London dispersion forces. This follows an inverse sixth-power law. The total potential energy is given by... 

Even within the 6-A limit, variations between the predicted atom-atom distribution functions can be quite small. All of the distribution functions predicted by rigid molecule potentials rise from zero more steeply than the experimental curve. There are two reasons for this. First, the rather slow initial increase in the experimental first-neighbor peak is most likely an artifact. Second, the very sharp increase shown by essentially all of the calculated functions is a consequence of the repulsive potential used in these models. This form of repulsion is much too strong, and the softer exponential repulsion gives a slower increase in goo( )-Models that allow for polarization or internal relaxation give a better description of this increase although the value of the maximum is usually overestimated. 

The foremost thermodynamic property associated with any phase boundary is the location of its surface in the p-V-T phase diagram. Most laboratory experiments of glass formation are carried out in a particular V-T plane, usually for atmospheric pressure, and the temperature dependence of volume through the transition is determined. If the glass transition is indeed dictated by the repulsive part of the potential, we expect, at least for simple steeply repulsive systems, that it will occur at the same molecular-reduced volume for many real and model systems and that this will be largely insensitive to the strength of the attractive component of the pair potential relative to kT. 

The rare gas excimer lasers are based on bound-continuum transitions from an excited diatomic species to its dissociative ground state. The observed continuum emission is a superposition of the Franck-Condon factors from the vibrational levels of the upper state. Thus these molecular dissociation lasers display relatively broad fluorescence as a consequence of the steeply repulsive ground-state potential, and there is always a population inversion on such transitions. However, the net gain is significantly lower than that for a bound-bound transition because of the distribution of oscillator strength over the broad fluorescence band. Figure 1 illustrates schematic potential energy curves for such transitions in the excimer and exciplex lasers. 

We should expect similar results for ionic liquid simulations, and that greater accuracy will come with improved treatment of inter- and intramolecular energetics. Second, the simulations have yielded a tremendous amount of quantitative and qualitative information, which has enabled us to understand molten salt systems much better than would have been possible using only experiment or theory. We know that the Coulombic forces lead to order over a much longer length scale than is present in simple liquids [112] and that these forces can lead to the appearance of small voids having lifetimes on the order ofps[112,113]. This is in contrast to normal molecular liquids, where the steeply repulsive part of the potential dominates fluid structure and leads to a more close-packed structure. 

Frenkel defects in general do not play an important role in molecular crystals. The asymmetric shape of the molecules and the steeply increasing repulsive potential between molecules at short intermolecular distances make the occurrence of interstitial molecules in molecular crystals thermodynamically improbable. The... 

Typically one assumes that (i) U is smooth except on a low-dimensional set of singular points, (ii) U is bounded below (which is normally the case if a potential such as Lennard-Jones, steeply repulsive at short range, is incorporated between all pairs of atoms), and (iii) U grows sufficiently rapidly as oo, in the case of an infinite domain. 

When z is small and xrepulsive potential energy will increase less steeply than should follow from eq. (43). The additional exp. (— 2xnegative deviation from the approximated curves in figure 13. 

Up to this point, the formal relation (5.12.8) was general and it applies to any arbitrary division of the pair potential in two parts in (5.12.2). We now choose a more specific division of the pair potential in such a way that pu Xi, Xy) 1. The most successful procedure is to use a hard-sphere cutoff at some R= [Pg.339]

As discussed before, P- 3p transitions occur by means of the crossing at short-range between the 11 and the more steeply repulsive Z potential curves (Fig. 1) [2-6,9,57]. Since the initially prepared state correlated adiabatically with equal probability with the case (a) II and Z curves, we might expect the magnitude of the Ip- P cross sections to depend critically on the strength of the nonadiabatic coupling between the states which correlate at short range one with a Z-like and the other with a Il-like curve, e.g. < /i i3/9R /4> and < Ti l3/3Rl >. 

As shown in the table, the nonclassical Ej , component is vastly smaller than El. The latter contains all significant classical-type steric and electrostatic effects, as well as the energy of Lewis-type covalent bonding, and is typically >99.9% of Etot- Yet as shown in the more complete potential curves of Fig. 9.5, El exhibits only feeble net attraction at large distances (ca. 2kcal/mol near 3.5 A), then turns steeply repulsive well beyond the distance of equilibrium H-bonding. 

The a NLS (symbolically denoted as iiN=Oll) at 1.40 A is particularly noteworthy. Unlike other double-bonded forms in Tables 11.2, 11.3, both 2-c bonds are here of n type (Ux, Uy). The usual ctno bond is replaced by filled o -axis 1-c orbitals (the former cr-bonding hybrids) that are mutually opposed for strong repulsive interaction. This configuration therefore leads to a steeply repulsive region on the C-state potential (ca. 1.30-1.45 A), which forms the apparent inner wall of the F... 

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