Large Separations

This shows that the dielectric constant e of a polar solvent is related to the cavity fimction for two ions at large separations. One could extend this concept to define a local dielectric constant z(r) for the interaction between two ions at small separations. 

A drop of a dilute solution (1%) of an amphiphile in a solvent is typically placed on tlie water surface. The solvent evaporates, leaving behind a monolayer of molecules, which can be described as a two-dimensional gas, due to tlie large separation between tlie molecules (figure C2.4.3). The movable barrier pushes tlie molecules at tlie surface closer together, while pressure and area per molecule are recorded. The pressure-area isotlienn yields infonnation about tlie stability of monolayers at tlie water surface, a possible reorientation of tlie molecules in tlie two-dimensional system, phase transitions and changes in tlie confonnation. Wliile being pushed togetlier, tlie layer at... 

At large separation r, equation (C2.6.3) decays as oc r just as the van der Waals interactions between... 

The last approximation is for finite At. When the equations of motions are solved exactly, the model provides the correct answer (cr = 0). When the time step is sufficiently large we argue below that equation (10) is still reasonable. The essential assumption is for the intermediate range of time steps for which the errors may maintain correlation. We do not consider instabilities of the numerical solution which are easy to detect, and in which the errors are clearly correlated even for large separation in time. Calculation of the correlation of the errors (as defined in equation (9)) can further test the assumption of no correlation of Q t)Q t )). 

All heteronuclear diatomic molecules, in their ground electronic state, dissociate into neutral atoms, however strongly polar they may be. The simple explanation for this is that dissociation into a positive and a negative ion is much less likely because of the attractive force between the ions even at a relatively large separation. The highly polar Nal molecule is no exception. The lowest energy dissociation process is... 

Table 11.1 shows an interesting point about CISD. The energy of the dineon pair at the arbitrarily large separation of 5000 pm is exactly twice the energy of two free atoms at the HF-LCAO level of theory, but this is not the case at the CISD level of theory. We say that HF theory scales correctly, whilst CISD does not. 

On the other hand, when a molecule in a vacuum is broken up into a positive ion and a negative ion, the mutual attraction, instead of disappearing rapidly, falls off slowly. At large separations of the ions the mutual potential energy — e2/r approaches zero as 1/r. In Fig. 8a the... 

Since plane waves are delocalised and of infinite spatial extent, it is natural to perform these calculations in a periodic environment and periodic boundary conditions can be used to enforce this periodicity. Periodic boundary conditions for an isolated molecule are shown schematically in Fig. 8. The molecular problem then becomes formally equivalent to an electronic structure calculation for a periodic solid consisting of one molecule per unit cell. In the limit of large separation between molecules, the molecular electronic structure of the isolated gas phase molecule is obtained accurately. 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

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