Dipoles, average over orientations

The calculation of the molar polarizabilities, often involves statistical mechanical averaging over orientational distributions of the molecules. An important example is the distribution function w caused by dipole orientation in an externally applied static electric field E° because it describes the process of electric poling of NLO-phores. To second order in the field, the dipolar contributions to this (normalized) function are given by (100),... 

In this case we expect the dipole-induced dipole contribution to the attraction to be greater than the dipole-dipole contribution. How can the induced dipole effect be greater than the permanent dipole effect This counter-intuitive result stems from the fact that while the dipole-dipole interaction is averaged over orientations that are repulsive as well as attractive, the dipole-induced dipole interaction is attractive at all orientations. 

Figure 1.20. (a) Angles 0 0y, and y, describing the relative orientation of the electronic transition dipole moments s between two dye molecules, (b) Relative orientations of the electronic transition dipole moments between two equal dye molecules in the channels of zeolite L. (c) Angular dependence of the orientation factor k2 under the anisotropic conditions (b) and averaged over y. 

With the exception of the orientation factor, all the parameters in this equation may be obtained within reasonable error by direct experimental measurement or by estimation. The problem of setting reasonable values for k2, which may vary from 0 to 4 for orientations in which the dipole moments are orthogonal or parallel, respectively, is nontrivial. A value of , which is an unweighted average over all orientations, is often used. Dale et al.(53) have examined this problem in great detail and have shown that a k2 value of is never justified for energy transfer in macromolecules because it is impossible for the donors and acceptors to achieve a truly isotropic distribution. They do provide an experimental approach, using polarized emission spectroscopy, to estimate the relative freedom of motion for the donor and acceptor that allows reasonable limits to be set for k2. 

This is an attractive interaction, again changing as R. Now, suppose we average over all possible orientations of the two dipoles, giving equal weight to each orientation. We find... 

The dipole-dipole interactions, frequently referred to as Keesom interactions, are historically included in the van der Waals interactions, even though they are purely electrostatic. For molecules that are free to orient themselves, the dipole-dipole interactions must be averaged over the molecular orientations, as the angular dependence of the interaction energy is comparable to the Boltzmann energy kBT (Israelachvili 1992, p. 62). With the averaging of the Keesom... 

Polypeptides are electrically polar, carrying permanent dipoles at the planar CO-NH groups of the backbone chain and generally at some atomic groups of the side-chains. Because of the vector nature of dipoles, we must speak of the mean-square dipole moment, averaged over all possible conformations of the backbone chain and all accessible orientations of the side-chains when the dipolar nature of a polypeptide in solution is considered. The of a polypeptide thus may depend on what conformation the molecule assumes in a given solvent. 

Equilibrium electrostatic interactions between a solute and a solvent are always nonpositive - tliey are zero if the solute is characterized by no electrical moments (e.g., a noble gas atom) and negative otherwise, i.e., attractive. It is easiest to visualize the electrostatic interactions as developing in a stepwise fashion. Consider a solute A characterized by electrical moments for simplicity, consider only die dipole moment. When A passes from the gas phase into a solvent, the solvent molecules, if diey have permanent moments of their own, reorient so that, averaged over thermal fluctuations, their own dipole moments oppose that of the solute. In an isotropic liquid with solvent molecules undergoing random thermal motion, the average electric field at any point will be zero however, the net orientation induced by the solute changes this, and the lield induced by introduction of the solute is sometimes called the reaction field . 

Triplet—triplet energy transfer from benzophenone to phenanthrene in polymethylmethacrylate at 77 and 298 K was studied by steady-state phosphorescence depolarisation techniques [182], They were unable to see any clear evidence for the orientational dependence of the transfer probability [eqn. (92)]. This may be due to the relative magnitude of the phosphorescence lifetime of benzophenone ( 5 ms) and the much shorter rotational relaxation time of benzophenone implied by the observation by Rice and Kenney-Wallace [250] that coumarin-2 and pyrene have rotational times of < 1 ns, and rhodamine 6G of 5.7 ns in polymethyl methacrylate at room temperature. Indeed, the latter system of rhodamine 6G in polymethyl methacrylate could provide an interesting donor (to rose bengal or some such acceptor) where the rotational time is comparable with the fluorescence time and hence to the dipole—dipole energy transfer time. In this case, the definition of R0 in eqn. (77) is incorrect, since k cannot now be averaged over all orientations. 

Generally, dielectrics differ from conductors because they have no charges to move freely. However, a small displacement (compared to atomic dimensions) of electrons in their parent molecules may occur when this dielectric is embedded in an electric field. This effect is called the polarization, and a polarized dielectric will create its own field that can be compared with an external field imposed. We have also to note that if the dielectric molecules possess a permanent dipole moment, the orientational polarization takes place, revealing an average (over the molecular ensemble), nonzero dipole moment. As a consequence, the polarization field may alter the net, outer field significantly. 

Polarizability (of a molecule) — There are numerous different mechanisms that contribute to the total polarizability of a molecule. The three most important of these are termed electron polarizability, molecular-distortion polarizability, and orientation polarizability. All these parameters are measured as statistical averages over large numbers of molecules present in the bulk phase. (1) -> Electron polarizability a is a measure of the ease with which electrons tend to be displaced from their zero-field positions by the applied -> electric field. Thus, the electron polarizability of a molecule is defined as the ratio of induced dipole moment pincj (coulomb meters) to the inducing electric field E (volts per meter) ... 

Because the orientations and spatial locations of the two chromophores may vary over the ensemble of molecules, and because they can also change during the time the donor is in the excited state, the measured effect of is usually an average over the appropriate spatial/temporal distributions. Whenever Eo and the acceptor dipole moment pA have parallel orientations, the rate of FRET is maximum for that placement in space for the acceptor relative to the donor. For pA oriented parallel to Ed, the possible maximum values of are between 1 and 4 that is, the actual maximum value depends on the value of 0 ) (see below). The minimum value of is zero for every position of the acceptor relative to the donor whenever Ed and the acceptor dipole moment pA are oriented perpendicular to each other. 

The dynamics of the dipole-dipole interaction tensor was averaged over each proton of the complex, and had a correlation time around 50 ps. Due to the rigidity of the complex, the decay in the TCP was caused by reorientation of the whole complex and the wagging motion of the water molecules. The fluctuations of symmetry was studied both from the individual water molecules distortion from their ideal symmetry positions and from the symmetry modes of the complex. The symmetry modes were well defined for the oxygen atoms in the water molecule, which show small distortions. The orientations of the water molecules, on the other hand, were too widely distributed from such an analysis to be meaningful. The time scale of the symmetry modes was in the sub-picosecond regime, much too fast to be correlated to the dipole-dipole interaction tensor. Hence, the decomposition of the total TCP into a spin part and a space part is well motivated. 

An axially symmetric molecule is characterized by its linear polarizability in the principal axes a x and a y = a" and a" = af/. It is a good approximation to assume that its second- and third-order polarizability tensors each have only one component and respectively, which is parallel to the z principal axis of the molecule. For linear and nonlinear optical processes, the macroscopic polarization is defined as the dipole moment per unit volume, and it is obtained by the linear sum of the molecular poiarizabilities averaged over the statistical orientational distribution function G(Q). This is done by projecting the optical fields on the molecular axis the obtained dipole is projected on the laboratory axes and orientational averaging is performed. The components of the linear and nonlinear macroscopic polarizabilies are then given by ... 

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