Induced Dipole Propagation

The situation is illustrated in Fig. VI-8. Continued step-by-step propagation leads to the general term 

The locus of successive U values can be represented by an exponential relationship 

It is thus seen that the dipole-induced dipole propagation gives an exponential rather than an inverse x cube dependence of U x) with x. As with the dispersion potential, the interaction depends on the polarizability, but unlike the dispersion case, it is only the polarizability of the adsorbed species that is involved. The application of Eq. VI-43 to physical adsoiption is considered in Section XVII-7D. For the moment, the treatment illustrates how a long-range interaction can arise as a propagation of short-range interactions. 

Derive the expression for the electric field around a point dipole, Eq. VI-5, by treating the dipole as two charges separated by a distance d, then moving to distances X d. 

The atomic unit (AU) of dipole moment is that of a proton and electron separated by a distance equal to the first Bohr orbit, oq. Similarly, the au of polarizability is Oq [125]. Express and o for NH3 using both the cgs/esu and SI approach. 

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An important alternative to SCF is to extend the Lagrangian of the system to consider dipoles as additional dynamical degrees of freedom as discussed above for the induced dipole model. In the Drude model the additional degrees of freedom are the positions of the moving Drude particles. All Drude particles are assigned a small mass mo,i, taken from the atomic masses, m, of their parent atoms and both the motions of atoms and Drude particles (at positions r, and rdj = r, + d, ) are propagated... 

Macroscopic solvent effects can be described by the dielectric constant of a medium, whereas the effects of polarization, induced dipoles, and specific solvation are examples of microscopic solvent effects. Carbenium ions are very strong electrophiles that interact reversibly with several components of the reaction mixture in addition to undergoing initiation, propagation, transfer, and termination. These interactions may be relatively weak as in dispersive interactions, which last less than it takes for a bond vibration (<10 14 sec), and are thus considered to involve "sticky collisions. Stronger interactions lead to long-lived intermediates and/or complex formation, often with a change of hybridization. For example, onium ions are formed with -donors. Even stable trityl ions react very rapidly with amines to form ammonium ions [41], and with water, alcohol, ethers, and esters to form oxonium ions. Onium ion formation is reversible, with the equilibrium constant depending on the nucleophile, cation, solvent, and temperature (cf., Section IV.C.3). 

Figure 3. Scheme of the LIDDI interaction a traveling laser field induces dipole moments in the atoms, thereby causing the interatomic interaction. The laser field is propagating in the direction x along which the atoms are weakly confined. The electric field vector is in the xy plane. 

FIGURE 5.71 The Rayleigh-Debye-Gans theory is based on the assumptions that (1) the incident beam propagates without being affected by the particles, and (2) the scattered hght, received by the detector, is a superposition of the beams emitted from the induced dipoles in the different parts of the particle. 

If we consider the picture of the London-Van Der Waals forces as given above, viz., as an attraction between the temporary dipole of one atom and the dipole induced by it of the second atom, the finite velocity of propagation of electromagnetic actions causes the induced dipole to be retarded against the inducing one by a time equal to rn/c (if r is the distance between the two atoms and n the refractive index of the medium for the frequency coupled with the temporary dipole). The reaction of the induced dipole on the first one again is retarded by the same time, and if in this total time-lag of 2rn/c the direction of the first dipole is altered by 90 ", the force exerted is exactly nullified, and by a change of 180 " even reverted from an attraction into a repulsion. 

Figure 8.15 I An external electric field distorts the shape of the electron cloud around a molecule or atom, creating a transient or induced dipole. The distorted molecule can then act as an external field on a nearby molecule, propagating the effect. Dispersion forces result from the mumal attractions among such induced dipoles in a collection of molecules.

Refraction. As previously mentioned, the interaction between matter and an electric field results in the formation of induced dipole moments Irom elements having a polarizability a. These dipole moments slow down the propagation of the incident beam from its velocity in vacuum (c) to a lower velocity in the material traversed (5 ). A consequence of the medium polarizability is the refraction of the incident beam. The refractive index of the material (n) is defined as the ratio of the velocity of the electromagnetic radiation in vacuum to that in the material ... 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

The effect of electromagnetic radiation on matter is to induce a dipole. In a transparent dielectric medium, only the velocity of electromagnetic radiation is reduced, depending on the refractive index of the medium, which is determined by its density. The propagation constant of electromagnetic waves is given by... 

The expectation value of the property A at the space-time point (r, t) depends in general on the perturbing force F at all earlier times t — t and at all other points r in the system. This dependence springs from the fact that it takes the system a certain time to respond to the perturbation that is, there can be a time lag between the imposition of the perturbation and the response of the system. The spatial dependence arises from the fact that if a force is applied at one point of the system it will induce certain properties at this point which will perturb other parts of the system. For example, when a molecule is excited by a weak field its dipole moment may change, thereby changing the electrical polarization at other points in the system. Another simple example of these nonlocal changes is that of a neutron which when introduced into a system produces a density fluctuation. This density fluctuation propagates to other points in the medium in the form of sound waves. 

The energy-dependent — r r polarization propagator is thus the dynamic polarizability tensor, i.e. the induced electric dipole moment in units of an external electric field. 

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