Soft potential

Similar structural results are obtained if one donor is neutral (e.g. DMAP) and the other donor is anionic (Cl ) (structure 21). The adducts possess extremely soft potentials and in solution the P NMR shifts are strongly temperature dependent [59]. It confirms the assertion that the adducts are very labile. 

This normalized hardness scale runs from zero (for a totally soft potential when = 1) to unity (for a totally hard potential when = oo). 

Fig. 4.2 (a) The full curve shows the normalized pair potential, / , versus the normalized interatomic distance, R/Rh, for the degree of normalized hardness, = 2 corresponding to X = 2. The two dashed curves show the repulsive and attractive contributions respectively. The shaded region delineates the hard-core potential with = 1 corresponding to X = oo. The two vertical arrows mark the equilibrium nearest-neighbour distances for = 1 and respectively, (b) The normalized pair potential, / , versus the normalized interatomic distance for different values of the degree of normalized hardness, . Note that = 0 corresponds to a totally soft potential, ah = 1 to a totally hard potential. 

Despite the quantitative difference between aromatics and antiaromatics, the qualitative picture is the same The combination of a, T-distortive ground state and a -attractive excited state along the Kekule modes is manifested in soft potentials in the ground state and steep ones in the twin excited state. This in turn is expressed as exalted frequencies and symmetrized geometries in the twin excited state. 

We shall consider what is arguably the archetypal example of the NIRM strategy the Einstein Solid Method (ESM) [28]. The ESM provides a simple way of computing the free energies of crystalline phases and therefore addresses questions of the relative stability of competing crystalline structures. We describe its implementation for the simplest case where the interparticle interaction is of hard-sphere form it is readily extended to deal with particles interacting through soft potentials [29]. 

If a soft potential surface describes the approach of the axial ligand to the transition metal center, then it is not too surprising to see some short axial metal-ligand distances in a couple of instances (as in the example of K2CuF4 in Table VI) where possibly, because of crystal... 

In spite of the fact that the decay after excitation of the hard-potential itinerant oscillator is similar to the experimental computer simulation result of Figs. 7 and 8, we do not believe that it is the reduced model equivalent to the one-dimensional many-particle model under study. As remarked above, indeed, the e(r) function is not correctly reproduced by this reduced model. The choice of a virtual potential softer than the linear one seems also to be in line with the point of view of Balucani et al. They used an itinerant oscillator with a sinusoidal potential, which is the simplest one (to be studied via the use of CFP) to deal with the soft-potential itinerant oscillator. Note that the choice... 

Another systematic study of transport for simple fluids, this time the study of hard and soft potential fluids, has been undertaken Heyes and colleagues. 

Coulomb matrix element and that of the soft potential enables us to... 

An important feature of these clusters is that their structures are independent of the electron count of the surface polyhedron. Since radial metal-metal bonding predominates, there is a soft potential energy surface for cluster rearrangement. This has been experimentally observed for gold clusters129. Theoretical studies of bulk metals have indicated that alternative close packed (and related) structures are also separated by small energy differences130. ... 

DFT has been much less successful for the soft repulsive sphere models. The definitive study of DFT for such potentials is that of Laird and Kroll [186] who considered both the inverse power potentials and the Yukawa potential. They showed that none of the theories existing at that time could describe the fluid to bcc transitions correctly. As yet, there is no satisfactory explanation for the failure of the DFTs considered by Laird and Kroll for soft potentials. However, it appears that some progress with such systems can be made within the context of Rosenfeld s fundamental measures functionals [130]. 

The growth of orientational correlations and the slow down of collective orientational dynamics were subsequently investigated using a soft potential [106]. Allen and Warren (AW) studied a system consisting of N = 8000 particles of ellipsoids of revolution, interacting with a version of the Gay-Beme potential, GB (3, 5, 1, 3), originally proposed by Berardi et al. [107]. AW computed the direct correlation function, c( 1, 2), in the isotropic phase near the I-N transition. The direct correlation function is defined through the Omstein-Zemike equation [108]... 

It has been common to display the results of computer simulation studies of systems approaching the glassy state as simple linear plots of D versus T, usually in dimensionless reduced units natural to the simulation For example, for soft potentials. 

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