Dipole time-correlation function

If the rotational motion of the molecules is assumed to be entirely unhindered (e.g., by any environment or by collisions with other molecules), it is appropriate to express the time dependence of each of the dipole time correlation functions listed above in terms of a "free rotation" model. For example, when dealing with diatomic molecules, the electronic-vibrational-rotational C(t) appropriate to a specific electronic-vibrational transition becomes ... 

Whenever the absorbing species undergoes one or more processes that depletes its numbers, we say that it has a finite lifetime. For example, a species that undergoes unimolecular dissociation has a finite lifetime, as does an excited state of a molecule that decays by spontaneous emission of a photon. Any process that depletes the absorbing species contributes another source of time dependence for the dipole time correlation functions C(t) discussed above. This time dependence is usually modeled by appending, in a multiplicative manner, a factor exp(-ltl/x). This, in turn modifies the line shape function I(co) in a manner much like that discussed when treating the rotational diffusion case ... 

An important signature of the dynamics of water molecules is the reorientation of its dipole vector that can be probed by dielectric and NMR measurements. We have calculated the single molecule dipole-dipole time correlation function (TCF), defined as,... 

Figure 18. (a) <(5V(5V(t)>. The solid curve at the bottom is the total time correlation function and the curves labeled 1-3 are the contributions of individual shells 1-3. The curve marked /i is the single-particle dipole time correlation function (Cld(t) shown for comparison. From Ref. 57 with permission, from J. Chem. Phys. 89, 5044 (1988). Copyright 1988, American Physical Society. 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Computer simulation studies have explored translational and rotational dynamics in micellar solutions and shown that both translational and rotational dynamics in the hydration layer of micelles (Stem layer) are significantly slower than that in the bulk. The dipole-dipole time correlation function (which measures the rotational dynamics) shows the appearance of a long-time tail of the time constant in the 100 ps range or above. The dependence of the rotational dynamics on the probe location has also been investigated and it was found that the dynamics becomes faster as the probe moves away from the surface [7]. 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

In this form, one says that the time dependenee has been reduee to that of an equilibrium averaged (n.b., the Si pi I)i ) time correlation function involving the eomponent of the dipole operator along the external eleetrie field att = 0(Eo p) and this eomponent at a different time t (Eo p (t)). 

In effect, i is replaced by the vibrationally averaged electronic dipole moment iave,iv for each initial vibrational state that can be involved, and the time correlation function thus becomes ... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

To complete the description and get the connection with the solute emission and absorption spectra, there is need of the correlation functions of the dipole operator pj= (a(t)+af(t))j and, consequently, the differential equation for the one solute mode has to be solved. The reader is referred to [133] for detailed analysis of this point as well as the equations controlling the relaxation to equilibrium population. The energy absorption and emission properties of the above model are determined by the two-time correlation functions ... 

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

Figure 4. Water dipole (a) and the water H-H vector (b) orientational time correlation functions. In both panels, the dotted line is for the first layer, the solid thick line is for bulk water, the dashed line is for the second layer, and the thin solid line is for the third layer from the Pt( 100) surface (T = 300 K).

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

M = (Mx,My,M ) is the dipole moment of the system. Moreover, the indices i, j designate the Cartesian components x, y, z of these vectors, ()q realizes an averaging over all possible realizations of the optical field E, and () realizes an averaging over the states of the nonperturbed liquid sample. Two three-time correlation functions are present in Eq. (4) the correlation function of E(t) and the correlation function of the variables/(q, t), M(t). Such objects are typical for statistical mechanisms of systems out of equilibrium, and they are well known in time-resolved optical spectroscopy [4]. The above expression for A5 (q, t) is an exact second-order perturbation theory result. 

Fig. 8 Reorientational time correlation function of the water dipole, C (<), for water molecules in the three segments of the protein.

Fig. 3.51. Logarithmic plot of the normalized induced dipole moment correlation function, C(t), for hydrogen-argon mixtures at 165 K. Measurements at 90 amagat ( ) 450 amagat ( ) and 650 amagat (o). The broken lines at small times represents the portion of C(t) affected by the smoothing of the wings of the spectral profiles. Reproduced with permission by the National Research Council of Canada from [109].

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

S. Mukamel In general, multipulse experiments depend on a multitime correlation function of the dipole operator [1], The term x(n) depends on a combination of n + 1 time correlation functions. Their behavior for large n will depend on the model. In some cases (e.g., the accumulated photon echo used by Wiersma) the multiple-pulse sequence is simply used to accumulate a large signal and the higher... 

Dipole-Dipole Relaxation and Time-Correlation Function. 65... 

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