The Relaxation Matrix

Our discussion so far has focused on systems with only two quantum states, and our treatment of the time constants Ti and T2 has been entirely phenomenological. We now discuss the elements of the relaxation matrix R in a more general way and consider how they depend on the strengths and dynamics of interactions with the surroundings. 

If we know the averaged reduced density matrix for the ensemble at zero time (p(0)), we can obtain an estimate of p for a later time (t) by integrating the voti Neumann-Liouville equation (Eq. 10.24), using p(0) in the commutator  

The integrand in Eq. (10.47) includes a sum of terms of the form p )Vnm h)yjk t7), each of which involves the product of the density matrix at zero time with a correlation function or memory function, 

MrunjkihJi) = yrm h)V jk t2)-M ,njkih,h) is the ensemble average of the product of Vran time t wifli at time t2- If E m and V),t vary randomly, their average product should not depend on the particular times ti and t2, but only on the difference, t=t2 ti. The correlation function M mjk therefore can be written as a function of this single variable  

V m (Eq. 5.79). Panel B of Fig. 10.6 shows the autocorrelation function of the fluctuating quantity shown in panel /I. 

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The relaxation matrix of Eq. 5 is symmetric and contains two classes of elements the diagonal elements, p, values, which represent the self-relaxa-tion terms, and the off-diagonal elements, ay values, which are the cross-... 

Here, the W j represent the probability per unit time that the system performs a transition from state i to state j. In the present case of relaxation between the two states denoted as LS and HS, we may associate the matrix elements W 2 and 1 21 with the rate constants /clh and Ichl fc)r the conversion process LS HS and HS LS, respectively. The diagonal elements of the relaxation matrix W... 

The symbols Rauto and Rcross within the relaxation matrix are the auto- and cross-relaxation rates, respectively. and (l2Z) are the longitudinal magnetizations of spin 1 and 2, respectively, and the brackets indicate averaging over the whole ensemble of spins. Rcross in terms of the spectral densities is given by... 

The symbol (Oaa denotes the energy difference between the two eigenstates, converted into angular frequency. The first term on the right-hand side (rhs) of Eq. (18) vanishes for the populations (oo a = 0) and describes the preces-sional motion for coherences. Rota pp is an element of the relaxation matrix (also called relaxation supermatrix) describing various decay and transfer processes in the spin system. Under certain conditions (secular approximation), one neglects the relaxation matrix elements unless the condition = pp is fulfilled. 

In this equation, A is the relaxation matrix in eq. (14), A is the diagonal matrix of the eigenvalues, and U is the matrix with the eigenvectors of A as its columns. Note that this is identical to eq. (8), except that now all the frequencies are zero. 

The AECS scaling model is based on the results of the ECS approximation. The elements of the relaxation matrix / for an isotropic Q branch CARS experiment on N2 are given as ... 

Here p is the density matrix for all molecular states in the three-level system depicted in Fig. 4, and all incoherent relaxation terms caused, for example, by collisions, spontaneous emission, or decay in a (quasi)continuum are incorporated in the relaxation matrix rreiax. 

Powell et al. give an excellent review of several approaches to interpret the frequency dependence of Tle and T2e in these systems [71]. One convenient approach is that developed by Hudson and Lewis [72], who showed that the eigenvalues of the relaxation matrix R as defined in Bloch-Wangsness-Red-field (BWR) theory [73] are functions of rv and the experimental frequency co, and are related to the relaxation time T2ei of the i-th allowed electron spin transition by the expression ... 

The components of the relaxation matrix in the basis set adopted are real numbers. They may be computed from the correlation functions of the molecular rotations or, more precisely, from their Fourier transforms. 

In our further considerations we assume that the components of the relaxation matrix are known in advance. This is a realistic assumption provided that unsaturated NMR spectra are being dealt with. One should note that equation (35) can be used for the description of motion of the system during time spans which are much longer than the typical correlation times for molecular rotation in liquids (about 10-11—10—12 sec). The processes of intra- and inter-molecular exchange which are considered here are characterized by half-life times longer than 10 5 sec. It seems justified to consider these processes independently of molecular rotations, in spite of the fact that they all participate in the relaxation. 

Needless to say, the so-called combination transitions are also considered in this subspace. Lineshape equations for special forms of the relaxation matrix can also be written in terms of the Hilbert space. However, the notation becomes quite involved. This is probably the source of some erroneous simplifications which consist of neglecting combination transitions in the equations of lineshape. (50)... 

In a typical situation we are interested in the absorption mode of a dynamic spectrum,/abs (co), which equals the real part of the complex function f(co) given by equation (145). In most cases of unsaturated spectra the relaxation matrix which describes single-quantum transitions can be replaced by a constant — E/T2(effective) which is characteristic of the experimental conditions involved and reflects the inhomogeneity of the external magnetic field B0. The absorption mode spectrum is given by ... 

Here we have introduced the relaxation matrix operator... 

To be able to obtain cubic susceptibilities, the perturbation theory sequence of calculations with Eq. (4.343) must be carried out down to the third order in As the relaxation matrix R is tridiagonal, this can be done using the sweep method described in described in Section II.C.l. 

A similar formula describes relaxation in the ground state. However, in obtaining the Eqs. (5.22) and (5.23) we assumed that relaxation takes place in isotropic processes with respect to directions in space. In this case the relaxation matrix is diagonal and does not depend on Q, Q, i.e. = rK KK QQi [132, 134]. This is the case, e.g., of radia-... 

The relaxational matrix also defines the corrections to the relaxed hardness matrix (see Sect. 3.3). 

If we have tiie relaxation matrix and an approximate structure, we can back calculate the NOESY spectra. The problem with the relaxation matrix method is that some of the cross relaxation rates are not observed due to spectral overlap, dynamic averaging and exchange. Boelens et al. (1988 1989) attempted to solve the problem by supplementing the imobserved NOEs with those calculated from a model structure. From a starting structure, the authors use NOE build-ups, stereospedfic assignments and model-calculated order parameters to construct the relaxation matrix. An NOE matrix is then calculated. This NOE matrix is used to calculate the relaxation matrix and it is in turn used to calculate the new distances. The new distances are then used to calculate a new model structure. The new structure can be used again to construct a new NOE matrix and the process can be iterated to improve the structures. The procedure is called IRMA or iterated relaxation matrix analysis. 

In addition to corrections tising an appropriate spectral density function, in principle one also needs to consider an ensemble of structures. Bonvin et al. U993) used an ensemble iterative relaxation matrix approach in which the NOE is measured as an ensemble property. A relaxation matrix is built from an ensemble of structures, using averaging of contributions from different structures. The needed order parameters for fast motions were obtained fi um a 50-ps molecular dynamics calculation. The relaxation matrix is then used to refine individual structures. The new structures are used again to reconstruct the relaxation matrix, and a second new set of structures is defined. One repeats the process until the ensemble of structures is converged. The caveat espressed earlier that the accuracy of the result is limited by the accuracy of the spectral density function applies to all calculations of this typ . 

Here, V is the interaction of the electric field with die molecule, N is the concentration of the absorbing molecules, v is tire tlrermal velocity of molecules, and the term v V is responsible for the Doppler effect. The relaxation matrix, I, contains the rates of various radiative and non-radiatlve transitions. 

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