Translational orientational correlations calculations

Analogous to equation (8) the vibrational densities of states D " u) are calculated for the cases of CO and CS2 at two different pressures, respectively. They are shown in Figure 4 while Table 4 contains information about their positions and widths. Since under the similarity transformation the trace of V is preserved and because the relative band widths are small, the spectra are centred around the natural oscillator frequency uiq. The widths of the distributions depend on the strength of coupling between two oscillators which, apart from factors of positional and orientational correlation, scale linearly with transition dipole (dfijd(,Y and density p (equation (14)). CO is regarded as a reference system because of its simple translational and minor orientational structure. The last column in Table 4 expresses the influence of liquid structure on band width. All densities of states... 

Translation Search. A translation search was done by using the P2 probe molecule oriented by the rotation function studies and refined by the Patterson correlation method. The translation search employed the standard linear correlation coefficient between the normalized observed structure factors and the normalized calculated structure factors (Funinaga Read, 1987 Brunger, 1990). X-ray diffraction data from 10-3 A resolution were used. Search was made in the range x = 0-0.5, y = 0—0.5, and z = 0-0.5, with the sampling interval 0.0125 of the unit cell length. 

Equation (5.161) is very similar to that used by Bagchi et al. in their recent study [76]. However, since their expression is written in terms of the longitudinal ion-dipole direct correlation function and the orientational intermediate scattering function of the solvent in place of Cux k) and F fi k,t) in our formula, its application is limited to the calculation of the dielectric friction. As we have clarified in Sec. 5.3, both the translational and rotational motions of solvent molecules manifest themselves in Fxfi k, t), and Eq. (5.161) can be applied to the calculation of the friction coefficient which comprises the hydrodynamic as well as dielectric contributions. Thus Eq. (5.161) can be regarded as a more general microscopic expression for the friction coefficient. 

The calculation of collisional cross sections for phenomena involving atoms and molecules is particularly difficult because many quantum states of the colliding partners are coupled by the interaction forces. Even in cases involving electronically adiabatic phenomena, where one can assume that the electronic states of the system remain the same while the nuclei move, one must yet deal with the coupling of translational, rotational and vibrational degrees of freedom of the nuclei. The interaction forces furthermore depend intricately on the molecular orientations and on the atomic displacements within molecules, and change extensively with the atomic composition of molecules. It is therefore usually impossible to invoke physical considerations to make a preliminary selection of the quantum states that are relevant to the collision. We describe here the computational aspects of an alternative approach, based on the time evolution of operators for scattering, and on their time-correlation functions, which eliminates the need for basis set expansions. 

While the steric method described above is very efficient, in many cases, geometric criteria alone are insufficient to correctly dock the two molecules. This is especially true when the stmcmre of the receptor is of poor quality or a ligand molecule is relatively small so that shape complementarity is insufficient to specify the correct conformation. To overcome this problem, we decided to build a statistical potential that could be used for additional evaluation of the quality of the match. In order to build the potential, we defined 20 general atom types and built the contact statistics on the basis of the structures of known complexes available in the PDB [171]. After projection of the two molecules onto the grid, every cube is additionally labeled with the properties defined by the atom types that were projected onto it. Once the approximate representation of the system is ready, the best match of these two cube-clusters is determined by exhaustive scanning over the six-dimensional conformational space of the three relative translations and the three rotations. Calculating the value of the correlation function between these two sets of cubes and the value of the potential function, the quality of the particular ligand-receptor orientation is scored. 

In addition to the dipole orientational order, analogous distribution functions for the H-H vector were defined and calculated. The distribution function Ph(0) shows similar behavior to the dipole orientation functions. However, Ph(structure extending into layer C. Thus the presence of the crystal influences the liquid correlations as far away as layer C. Hence, the orientational order induced by the ice crystal propagates at least one and possibly two layers deeper into the water than translational order. This is a potentially important effect in electrochemical systems and biological membranes. 

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