Single correlated motion

In Eq. (4-62) Wq is the Larmor precessional frequency, and Tc is the correlation time, a measure of the rate of molecular motion. The reciprocal of the correlation time is a frequency, and 1/Tc may receive additive contributions from several sources, in particular I/t, where t, is the rotational correlation time, t, is, approximately, the time taken for the molecule to rotate through one radian. Only a rigid molecule is characterized by a single correlation time, and the value of Tc for different atoms or groups in a complex molecule may provide interesting chemical information. 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

In the case of extreme narrowing in which fast isotropic molecular motions dominate as in the solution state, the spectral density is written by a single correlation time,... 

The previous approach is valid as long as the molecular reorientation can be described by a single correlation time. This excludes molecules involving internal motions and/or molecular shapes which cannot, to a first approximation, be assimilated to a sphere. Due to its shape, the molecule shown in Figure 15 cannot evidently fulfil the latter approximation and is illustrative of the potentiality of HOESY experiments as far as carbon-proton distances and the anisotropy of molecular reorientation are concerned.45 58... 

Equations (8.25) to (8.28) are no longer valid in the case of hindered rotations occurring in anisotropic media such as lipid bilayers and liquid crystals. In these media, the rotational motions of the probe are hindered and the emission anisotropy does not decay to zero but to a steady value rc0 (see Chapter 5). For isotropic rotations (rod-like probe), assuming a single correlation time, the emission anisotropy can be written in the following form ... 

The rotational mobility of human low-density (LDL) and very-low-density (VLDL) lipoproteins was studied as a function of viscosity and temperature in the range of —90 to — 50°C.(86)The rotational behavior for LDL is represented by a single correlation time, consistent with the overall rotation of a spherical rigid particle as the source of the phosphorescence depolarization. For VLDL, internal peptide motions dominate the depolarization profile. 

The relation between collective and self-motion in simple monoatomic liquids was theoretically deduced by de Gennes [233] applying the second sum rule to a simple diffusive process. Phenomenological approaches like those proposed by Vineyard [ 194] and Skbld [234] also relate pair and single particle motions and may be applied to non-exponential functions. The first clearly fails to describe the PIB results since it considers the same time dependence for both correlators. Taking into account the stretched exponential forms for Spair(Q.t) (Eq. 4.21) and Sseif(Q>0 (Eq 4.9), the Skold approximation ... 

As outlined above, the normalized spectral density is especially simple when a single correlation time is involved (for instance, when the molecular reorientation can be approximated by the motion of a sphere) ... 

Correlation Times for Backbone and Side-Chain Motions in Poly(but-1-ene sulfone) of P = 700 as a 25% w/v Solution in Chloroform-d, Deduced from the Simple Isotropic Single-T Motional Model... 

Because there is no general microscopic theory of liquids, the analysis of inelastic neutron scattering experiments must proceed on the basis of model calculations. Recently1 we have derived a simple interpolation model for single particle motions in simple liquids. This derivation, which was based on the correlation function formalism, depends on dispersion relation and sum rule arguments and the assumption of simple exponential decay for the damping function. According to the model, the linear response in the displacement, yft), satisfies the equation... 

Variations in broad line NMR are primarily sensitive to motion. Consequently, changes in Av and AM2 are indications of motional processes in the system. If, as a first approximation, the relaxation process can be expressed by a single correlation time, xc, the following relation between NMR parameters and tc can be established ... 

Thus, larger-scale cooperative chain motions can be observed using this technique. A Tj vs. temperature plot is generally analogous to a curve with the lower time values and minima shifted to lower temperatures (Fig. 4). In the case of rapid motions (to2i 4 1), Tj is identical to Tt and, assuming a single correlation time, both relaxation times are equal to 1/5 Ctc (Eqs. (15) and (19)). 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

However, because of the correlated motion of the electrons, many-electron processes will also occur. (Looking at the many-particle effects in this way, the photon operator is a single-particle operator and electron-electron interactions have to be incorporated explicitly into the wavefunction. It is, however, also possible to describe the combined action of the electrons as an induced field which adds to the external field of the photoprocess, i.e., the transition operator becomes modified. Generally, the influence of the electron-electron interaction can be represented by modifying the wavefunction or the operator or by modifying both the wavefunction and the operator [DLe55, CWe87].) Of all the possible processes, only the important two-electron processes restricted to electron emission will be considered here. In many cases they can be divided into two different classes (see Fig. 1.3) t... 

For a rigid spherical or nearly spherical molecule undergoing diffusive rotational motion in the extreme narrowing limit, a single correlation time, given by Eq. 16, is adequate to describe the overall motion. Equation 16 can be modified to take into account intramolecular interactions from other protons attached to other carbons in the molecule. Assuming that tc is the correlation time for each such... 

T2, and the nuclear Overhauser enhancement, NOE, comprise a set of parameters which characterize molecular motions. In the case of simple isotropic motion, the dependence is in terms of a single correlation time characterizing the exponential decay of the autocorrelation function. However, in many instances, the assumption of isotropic motion is not valid. For rigid systems, the relaxation behavior can then often be predicted by assuming simple anisotropic motion (1 ). Often, superposition of two or more independent motions must be used to satisfactorily interpret observed relaxation behavior. Recently, however, the wide-... 

Polymer Backbone Motion. Alternate descriptions of molecular motion utilize an effectively non-exponential autocorrelation function to describe polymer dynamics. One formalism is the use of a log-/2 distribution of correlation times in place of a single correlation time(14). Such a description may simulate the various time scales for overall and internal motions in polymers. 

---