The Spectral Density Function

I FIGURE 7.1 The energetics and excess populations of the four allowed spin state combinations for a pair af equilibrium. 

Before considering what happens when we apply decoupling to this system, we acquaint ourselves with the spectral density function, J(v), and how it relates to the relative magnitude of Wq, Wjh/ Wjc/ and Wj. 

The spectral density function, l(v), is a mathematical function that describes how energy is spread as a function of frequency. This energy may be what the ancients referred to in their discussions of the ether that permeates all space, but that is more of a philosophical topic. In a liquids sample at a given temperature, the spectral 

Single quantum spin flip rate constant, W,. The kinetic rate constant controiiing the change in the spin state of a singie spin- /2 spin from either the a to the p state or from the a to the p state. 

---

The case of parallel orientations (e J. Oz) differs radically from the previously considered one, since the frequency dependence of the spectral density function is specified by the fractional power law 109... 

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 ... 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

If this observation corresponds to the true situation in solution then the internal motions are an order of magnitude faster than the rotational correlation time. Under such circumstances, the spectral density function used in these calculations is incorrect. This aspect requires further investigation, particularly once the data from dynamics calculations specifically including water become available. 

R is called the relaxation superoperator. Expanding the density operator in a suitable basis (e.g., product operators [7]), the a above acquires the meaning of a vector in a multidimensional space, and eq. (2.1) is thereby converted into a system of linear differential equations. R in this formulation is a matrix, sometimes called the relaxation supermatrix. The elements of R are given as linear combinations of the spectral density functions (a ), taken at frequencies corresponding to the energy level differences in the spin system. 

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... 

It is assumed that the noise voltage n(t) is the result of a real stationary process (Davenport and Root, 1958) with zero mean. Because it can be shown that the spectral density function S(f) is the Fourier transform of the autocorrelation function of the noise, it follows that the rms noise is given by... 

Figure 14. The spectral density function versus the frequency that corresponds to the fluorescence maximum in various solvents for coumarin 311. The points for each wavelength correspond to different solvents. Reprinted from Ref. 31 with permission, from J. Chem. Phys. 88, 2372 (1988). Copyright 1988, American Physical Society.

Both Ti and T2 relaxations of water protons are mainly due to fluctuating dipole-dipole interactions between intra- and inter-molecular protons [62]. The fluctuating magnetic noise from all the magnetic moments in the sample (these moments are collectively tamed the lattice) includes a specific range of frequency components which depends on the rate of molecular motion. The molecular motion is usually represented by the correlation time, xc, i.e., the average lifetime staying in a certain state. A reciprocal of the correlation time corresponds to the relative frequency (or rate) of the molecular motion. The distribution of the motional frequencies is known as the spectral density function. 

Transverse nuclear relaxation can also occur when the local fields at the nucleus fluctuate slowly, i.e. with an co frequency near zero (see also Section 3.4). Then the spectral density function will take the form... 

Fig. 3.14. Plot of the spectral density functions for dipolar relaxation in the presence of an axially symmetric g tensor. Conditions gn = 2.3, gj = 2.0, xc = 2 x 10-9 s, 6 — 0° (upper curve) and 0 = 90° (middle curve) compared with the Solomon behavior (lower curve).

Fig. 4.5. Plot of the spectral density functions of Eqs. (4.46) and (4.47) as a function of magnetic field (expressed as proton Larmor frequency log scale), rp = 2 x 10-9 s. The Solomon profiles obtained for xc = 2 x 10 9 s are also reported (dotted lines) for comparison purposes.

The MD simulations show that second shell water molecules exist and are distinct from freely diffusing bulk water. Freed s analytical force-free model can only be applied to water molecules without interacting force relative to the Gd-complex, it should therefore be restricted to water molecules without hydrogen bonds formed. Freed s general model [91,92] allows the calculation of NMRD profiles if the radial distribution function g(r) is known and if the fluctuation of the water-proton - Gd vector can be described by a translational motion. The potential of mean force in Eq. 24 is obtained from U(r) = -kBT In [g(r)] and the spectral density functions have to be calculated numerically [91,97]. 

G of some benzenoid hydrocarbons. We define the spectral density function corresponding to G as [9-13]... 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

By inserting the form of the spectral density function discussed above into the rate coefficient expression, the second line of Eq. (65) may be replaced by... 

Equation 37 has been used in an attempt130 to describe internal flexibility of the three hydroxymethyl groups of sucrose molecule in DzO solutions. The experimental data showed that the contribution of the overall motion to the spectral-density function of the hydroxymethyl group is similar to that of the ring carbons of sucrose. However, the presence of rapid internal motions about the three exocyclic bonds reduces the spectral density amplitudes. On the basis of the calculated order parameters in conjunction with model calculations, it was suggested130 that internal motions may be described as torsional librations. 

According to the fluctuational dissipation theorem, the spectral density function in terms of % reads [96,97] as follows ... 

---