Spectral functions reorientation

For a single subbarrier level we have A0(f) = 1, B0(4) = C0(4) = 0, and the halfwidth of the spectral function is determined solely by the reorientation rate, which is equal to the transition rate to the first excited state of deformation vibrations 164... 

We demonstrate that the spectral function of valence harmonic vibrations of a diatomic group that effects rotational reorientations is broadened by w. The vector of atom C displacements relative to the atom B (see Fig. A2.1) may be represented as x(t)e(t), where x(t) is the change in the length of the valence bond oriented at the time t along the unit vector e(/). Characteristic periods of valence vibrations are much shorter than periods of changes in unit vector orientations. As a consequence, the GF of the displacements defined by Eq. (4.2.1) can be expressed approximately as ... 

The spectral function thus has a Lorentz shape with a halfwidth at the half distribution height equal to the average reorientation frequency w. If expressed in spectroscopic units (cm 1), the halfwidth Avv2 amounts to cd2nca (c0 designates the velocity of light in vacuo). 

B. Spectral Function of a Dipole Reorienting in a Local Axi symmetric Potential... 

The spectral function L(z) involved in Eq. (142) is determined by the profile of the model potential well (in this section it is the rectangular well). It follows from Eq. (148) that if we fix the dimensional quantities, such as frequency v and temperature, then the spectral function L(z) depends also on the lifetime x and the moment of inertia of a molecule I. We consider a gas-like reorientation of a polar molecule determined by a dipole moment p of a molecule in a liquid. Calculation of the moment of inertia I deserves special discussion. 

The main purpose of this section is consideration of the FIR spectra due to the second dipole-moment component, p(f). However, for comparison with the experimental spectra [17, 42, 51] we should also calculate the effect of a total dipole moment ptot. In Refs. 6 and 8 the modified hybrid model44 was used, where reorientation of the dipoles in the rectangular potential well was considered. In this section the effect of the p(f) electric moment will be found for the hat-curved, potential, which is more adequate than the rectangular potential pertinent to the hybrid model. In Section VI.B we present the formula for the spectral function of the hat-curved model modified by taking into account the p(f) term (derivation of the relevant formula is given in Section VI.E). The results of the calculations and discussion are presented, respectively, in Sections VI.C and VI.D. 

The first and second terms in the right-hand part are, respectively, the transverse and longitudinal components of the spectral function. In other words, these terms are stipulated by reorientation of the projections of a dipole moment, which are, respectively, normal and collinear to the potential symmetry axis. The potential under consideration comprises two wells with oppositely directed symmetry axes. Such is the cosine-squared potential... 

Spectral function (33) for longitudinal vibration Spectral function (33) for reorientation of H-bonded molecules... 

In Figure 9 we depict the frequency dependences of the partial absorption coefficients aq(v) and a (v) pertinent to two harmonic-vibration modes. These frequency dependences are calculated from formulas (A6), (21) [24], (25), (28), and (29). When the above-mentioned coupling is accounted for (solid lines in Fig. 9), the spectral functions are taken from Eq. (Al). On the other hand, when the coupling is neglected (open circles in Fig. 9), then Lq and L are found from Eq. (19). We see from Fig. 9a that for both cases the calculated partial absorption a (v) practically coincide. The same assertion is valid also for the partial absorption ocq(v) depicted in Fig. 8b. Hence, there is no practical need to account for the coupling between the harmonic reorientation and vibration of HB molecules for calculation of spectra in liquid water. However, the effect of such coupling becomes noticeable (being, however, a rather small) in the case of ice, where the absorption lines are much narrower. 

In order to pass on to pure elastic reorientations [we have in mind that the latter occur in the parabolic (and not in the hat) potential], we let/and w tend to infinity. Using (138a), (130b), and (130d), we find analogously to (150) and (151) the corresponding spectral function L ... 

Turning to a particular case of frozen translations, in which the spectral function is given by (153) and the effective reorientation frequency by (154d), one can see that, if the rotary constant c exceeds a certain critical value... 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

Another important linear parameter is the excitation anisotropy function, which is used to determine the spectral positions of the optical transitions and the relative orientation of the transition dipole moments. These measurements can be provided in most commercially available spectrofluorometers and require the use of viscous solvents and low concentrations (cM 1 pM) to avoid depolarization of the fluorescence due to molecular reorientations and reabsorption. The anisotropy value for a given excitation wavelength 1 can be calculated as... 

As seen from the above theoretical developments, accessing geometrical (and stereochemical) information implies at least an estimation of the dynamical part of the various relaxation parameters. The latter is represented by spectral densities which rest on the calculation of the Fourier transform of auto- or cross-correlation functions. These calculations require necessarily a model for describing molecular reorientation... 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

The measured spin relaxation parameters (longitudinal and transverse relaxation rates, Ri and P2> and heteronuclear steady-state NOE) are directly related to power spectral densities (SD). These spectral densities, J(w), are related via Fourier transformation with the corresponding correlation functions of reorientional motion. In the case of the backbone amide 15N nucleus, where the major sources of relaxation are dipolar interaction with directly bonded H and 15N CSA, the standard equations read [21] ... 

Up to this point only overall motion of the molecule has been considered, but often there is internal motion, in addition to overall molecular tumbling, which needs to be considered to obtain a correct expression for the spectral density function. Here we apply the model-free approach to treat internal motion where the unique information is specified by a generalized order parameter S, which is a measure of the spatial restriction of internal motion, and the effective correlation time re, which is a measure of the rate of internal motion [7, 8], The model-free approach only holds if internal motion is an order of magnitude (<0.3 ns) faster than overall reorientation and can therefore be separated from overall molecular tumbling. The spectral density has the following simple expression in the model-free formalism ... 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

In Fig. 3, the orientational diffusion time constants ror of the first solvation shell of the halogenie anions CD. Br, and D are presented as a function of temperature. From the observation that ror is shorter than rc, it follows that the orientational dynamics of the HDO molecules in the first solvation shell of the Cl ion must result from motions that do not contribute to the spectral diffusion, i.e. that do not affect the length of the O-H- -Cl hydrogen bond. Hence, the observed reorientation represents the orientational diffusion of the complete solvation structure. Also shown in Fig. 3 are fits to the data using the relation between ror and the temperature T that follows from the Stokes-Einstein relation for orientational diffusion ... 

Figure 26 Results of the decomposition of the measured transient spectra the peak positions (a), spectral widths (FWHM) (b), and reorientation times (c) of three prominent spectral components I—in, attributed to different local structures of water, as a function of temperature experimental points the lines are drawn as a guide for the eye.

It has been known for a long time that the kind of simplistic distance calibration suggested by Eq. [8] may be subject to systematic errors. First, the intensity of an NOE depends on the spectral density function for the reorientation of the vector between relaxing nuclei. This means that Eq. [8] is valid only if the reference distance and unknown distance are undergoing the same motions. As this is not likely to be the case, distance calibrations have attempted to allow for the possibility of systematic errors.23 Equation [8] also assumes that the dipolar relaxation can be considered in terms of isolated spins relaxing each other. In the presence of spin diffusion, this will lead to a systematic underestimation of distances.41 57 58... 

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