Dielectric fluctuations

We are interested in fluctuations of the nuclear polarization. As we have just seen, the probability for such a fluctuation is determined by the difference between the free energy of our equilibrium system and the free energy of a fictitious equilibrium state in which Pn was restricted by some means to a given value. In addition, we assume that the fluctuations relevant to our process are those for which the instantaneous value of Pn corresponds to some charge distribution p of the solute that will produce this value of Pn at equilibrium. We are particularly interested in such fluctuations of F because, as will be seen, these are the ones that lead to the charge rearrangement. 

Specifically, let the initial state of our system ( state 0 ) be characterized by an electronic charge distribution po and by the nuclear polarization FnO that is in equilibrium with it. In the final state ( state 1 ) the electronic charge distribution is Pl and the nuclear polarization is P . Starting in state 0 we are interested in fluctuations in the nuclear polarization about P q in the direction from po to pi. 

We will assign to such fluctuations a parameter 0 that defines a fictitious charge distribution pe according to 

In turn defines a nuclear polarization as that polarization obtained in an equilibrium system (state P) in which the charge distribution is pe. Now, in state 0 (where p = po) this Pne is a fluctuation from equilibrium that is characterized by the parameters pi and P. 

To obtain the probability for such a fluctuation we introduce another state, state t, which is a fictitious restricted equilibrium system in which (1) the charge distribution is Po and (2) the nuclear polarization is P e, that is same as in the equilibrium state P in which the charge density is pe. We want to calculate the free energy difference, AGo z, between the restricted equilibrium state t and the fully equilibrated state 0. This is the reversible work needed to go, at constant temperature and pressure, from state 0 to state t. 

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The deviation of nitrobenzene from the solid line (the slope = 1) in Figure 2 is probably attributed to the frequency dependent dielectric friction for the reaction dynamics around the barrier top, i.e., the much slower dielectric fluctuation of nitrobenzene (tl - 6 ps at 298K) compared with the ET rate hardly works as friction for the barrier crossing. In such case, the friction is shows tl (a[Pg.400]

Continuum dielectric-linear response theory yields a free energy surface for dielectric fluctuations which is a parabola in the reaction coordinate 3. This harmonic oscillator property is quite nonobvious and very significant. 

At this stage, the theoiy did not yield the mass (a parameter of dimensionality [mass] X [length] ) associated with the dielectric fluctuations, which is needed to determine )s that appears in the adiabatic rate expression. On the other hand, short of the nonadiabatic coupling itself, all parameters needed to calculate the nonadiabatic rate from Eq. (16.51) have been identified within this dielectric theory. 

Equation (3.2.3) can be expressed in terms of the spatial Fourier transform of the dielectric fluctuation... 

In order to calculate a light-scattering spectrum we must have a model for the mechanism by which dielectric fluctuations decay. The remainder of this book is devoted primarily to the study of these fluctuations. 

Four different polarization directions are defined in Fig. 3.4.3. The specific components of the dielectric fluctuations or polarizability fluctuations that are responsible for each of these spectral components are ... 

The long-time decay of the dielectric fluctuations is characterized by an angle-dependent diffusion coefficient that decreases linearly with q2 for values of q small compared with qi and q%. 

The preceding formulas are based on an approximate molecular theory of light scattering. It is more precise to use Eq. (3.2.13), which involves the fluctuations of the spatial Fourier components of the dielectric fluctuations. Although the dielectric con-... 

The dielectric constant eo of a pure fluid in total equilibrium is in general a function of the density p0 and temperature T0 that is, eo = e(p0, To). This is called the dielectric equation of state. Clearly on the local level, there are small fluctuations in the local density and temperature so that we can write p(r, t) = p0 + 5p(r, t) and T(t, t)=T0 + ST(t, t). Thus if we assume local equilibrium, the dielectric equation of state can be used to determine the local value of the dielectric constant. Accordingly we write e(r, t) = e(p0 + p r, f) To + ST(t, t)). Since these fluctuations are expected to be quite small, this can be expanded in a (rapidly convergent) power series in these fluctuations. The local dielectric fluctuation <5c(r, t) = e(r, t) — e(p0, T0) is then to first order in the fluctuations dp, ST,... 

The primary variables are by definition those fluctuations which contribute directly to the dielectric fluctuation. The secondary variables are those variables that are dynamically coupled to the primary variables. 

In the case of condensed phases, fluctuations in the dielectric constant may contain a slow time-dependence caused by slowly varying density fluctuations on length-scales of the same order of magnitude as the wavelength of light. The dielectric fluctuations may then be replaced by fluctuations in the number density Ap(r t) according to... 

As we showed, cf. (2.23) and (2.20), the dynamic observable in an OKE experiment, and also in the forward depolarized light scattering (see also Sect. 2.4.2) is the correlation of the susceptibility or dielectric fluctuations ... 

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