Fixed Relative Molecular Orientations

The limit of nearly free relative orientations is not of much physical interest in the case of electrostatic forces. The intermolecular interaction is a Boltzmann-weighted average of the form 

The trace is zero, leaving four independent components, a number which can only be further reduced by special choice of origin. The dipole-quadrupole has no centre of symmetry. The dipole is symmetric to Tr(f) for all however the quadrupole (II.ll) has no symmetry plane in common, as may be seen by applying the third of Eq. (II.7) to and [viz. to the real forms of Q(2,l) and Q 2,—1)]. 

All cfv reflections transform the dipole-quadrupole to its enantiomeric form the simplest choices of reflection plane are those transforming either or Qy°z into its negative. 

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For all but spherically symmetrical molecules, van der Waals forces are anisotropic. The polarizabihties of most molecules are different in different molecular directions because the response of electrons in a bond to an external field will usually be anisotropic. A consequence of this effect is that the dispersion force between two molecules will depend on their relative molecular orientation. In nonpolar liquids, the effect is of minor importance because the molecules are essentially free to tumble and attain whatever orientation is energetically favorable. However, in sohds, hquid crystals, and polar media, the effect can be important in determining the relative fixed orientation between molecules, thereby affecting or controlling specific conformations of polymers or proteins in solution, critical transition temperatures in liquid crystals and membranes, and so on. Repulsive forces in polar molecules are also orientation dependent, and are often of greater importance in controlling conformations and orientations. 

Naturally occurring molecular ensembles such as proteins from photosyntlietic systems (plants, algae, photosyntlietic bacteria, etc) are usually relatively rigid systems tliat contain various cliromophores and hold tliem at fixed positions and orientations relative to each otlier. That is why, despite tire numerous energy jumps between tire cliromophores, tlie resulting emitted fluorescence is polarized. The extent of tliis polarization tlius affords invaluable infonnation about tlie internal stmcture of molecular complexes. 

The explicit definition of water molecules seems to be the best way to represent the bulk properties of the solvent correctly. If only a thin layer of explicitly defined solvent molecules is used (due to hmited computational resources), difficulties may rise to reproduce the bulk behavior of water, especially near the border with the vacuum. Even with the definition of a full solvent environment the results depend on the model used for this purpose. In the relative simple case of TIP3P and SPC, which are widely and successfully used, the atoms of the water molecule have fixed charges and fixed relative orientation. Even without internal motions and the charge polarization ability, TIP3P reproduces the bulk properties of water quite well. For a further discussion of other available solvent models, readers are referred to Chapter VII, Section 1.3.2 of the Handbook. Unfortunately, the more sophisticated the water models are (to reproduce the physical properties and thermodynamics of this outstanding solvent correctly), the more impractical they are for being used within molecular dynamics simulations. 

To date, only a few solution calculations for carbohydrates have been attempted (one such study of mannitol and sorbitol is described in the chapter by Grigera in this volume), but the results of these early studies bear out the expectation that solvation effects in carbohydrate systems can be both significant and difficult to predict. In the case of pyranoid rings, molecular solvation is further complicated by the close juxtaposition of these groups in essentially fixed relative orientations (assuming no conformational changes in the ring). Under such circumstances, molecular stereochemistry could play important physical roles, as is... 

In Section 2 it was established theoretically that, relative to some molecule-fixed reference axes, a molecular order (alignment) tensor with five independent parameters was sufficient to describe the molecular orientation upon which the observed dipolar couplings depend. Based on knowledge of the order tensor and the molecular structure, it is possible to predict the corresponding RDCs. From this relationship, one might anticipate that... 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

Examination of the structure of the packing of 3-centre carbon dioxide at different pore sizes revealed a clear pattern of change with respect to pore size, associated with layer formation. Both the density profiles and the profiles of molecular orientation showed a monolayer of carbon dioxide oriented parallel to the pore walls at small pore size below about H = 0.71 nm. However, with increase in pore width above 0.68 nm there is a tendency for the flat molecules to rotate, in order to permit additional molecules to adsorb. Above about 0.71 nm an additional layer is formed, with molecules near the wall tending to tie flat and those at the centre tending to rotate relative to the axis. This pattern of behaviour is repeated as the pore size is increased. Fig. 3. depicts snapshots of the structure at different pore sizes. They reveal the formation of a central relatively flat layer, followed by rotation and subsequent separation of two distinct rotated layers. This pattern is consistently followed as the pore size increases, and in this way additional layers are created. This was confirmed fix)m the simulation results, by examination of profiles of density and the molecular orientation. Simulations with the 3-centre fluid without charges at the sites were also conducted, and yielded similar trends as given by the fluid with charges, with only a small reduction in capacity. Thus, the difference in liehavior compared to the U fluid is clearly related to the different molecular shape represented by the 3-center fluid, and not to electrostatic effects. 

The values of and (aj have the property of being invariant to a change in molecular orientation relative to the space-fixed coordinate system. In liquids, the molecular orientation is not fixed. By averaging over all orientations of the polarizability ellipsoid it can be shown that for natural unpolarized incident light the depolarization ratio p is... 

In principle, simulations which use full explicit solvent to account for the natural environment of proteins should give more realistic behavior than those employing any of the above approximations. However, the realization of this in practice depends on the quality of the water model used in the simulation. A simple model that has proven relatively successful in many recent simulations is the TIP3P model for water in which the atoms have fixed charge and fixed relative orientation. More complex models use more points than three to distribute charge or attempt to account for polarization. The most sophisticated water model yet proposed is that by Millot and Stone, which attempts a full description of the relevant physics for intermolecular potentials. Unfortunately, as yet this is too complex to be used in molecular dynamics simulations. 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

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