Relaxation matrix derivation methods

This situation has led to a series of attempts to find approximations to the method of Yip and Case that would allow the method to be computationally more tractable. Baleja et al.66 used a complete relaxation matrix analysis, but took two steps to reduce calculation time. First, they used a cutoff of 8 A when considering multispin effects, and, second, they used numerical rather than analytical derivatives. 

The deconvolution method proposed earlier by our group, translated the 3D cross peak intensities into 2D NOE intensities, which can be used to derive accurate distance constraints that can be used in structure refinement by an iterative relaxation matrix approach (11). This approach avoids systematic errors while retaining computational efficiency. 

Various first-order properties can also be calculated by analytical derivative methods, for example dipole moments and spin densities. Moreover, these are all calculated from the same relaxed density matrix via the expression... 

Among the methods discussed in this book, FEP is the most commonly used to carry out alchemical transformations described in Sect. 2.8 of Chap. 2. Probability distribution and TI methods, in conjunction with MD, are favored if there is an order parameter in the system, defined as a dynamical variable. Among these methods, ABF, derived in Chap. 4, appears to be nearly optimal. Its accuracy, however, has not been tested critically for systems that relax slowly along the degrees of freedom perpendicular to the order parameter. Adaptive histogram approaches, primarily used in Monte Carlo simulations - e.g., multicanonical, WL and, in particular, the transition matrix method - yield superior results in applications to phase transitions,... 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

All the alternative variants of the MPn may be implemented using a relaxed density matrix or a unrelaxed density matrix, in analogy with the Cl solvation methods. In the first case the correlated electronic density is computed as a first derivatives of the free energy, while in the second case only the MPn perturbative wavefunction amplitudes are necessary. 

Recently a new method for studying the relaxation around photo-excited centers in solids was reported by Straus and Walder 120). The authors observed the effect of the spatial relaxation of the matrix around the excited center in a condensed phase on the bulk optical index of refraction and used this information to derive the change in the equilibrium distance (i.e., AQ). For details the reader is referred to ref. 120. The experiments were performed on Mn + in water (i.e., on the aqua-complex [Mn(H20)e] ). No luminescence was observed. Since the... 

Since convergence by steepest descents can be rather slow in the final stages of relaxation, the search for the minimum at this point can be switched to the Newton-Raphson method. In this procedure the second derivative at x is also calculated and compounded into a Hessian matrix H, whose elements are... 

The nOe data may be obtained by either one- or two-dimensional methods. These through-space effects may be used to estimate intemuclear distances. However, nOe cross-peak volumes are inversely proportional to the sixth power of the distance between the correlated protons only in the case of rigid spherical molecules whose tumbling is isotropic. The simplest method that has been proposed for the interpretation of nOe-derived distance data is the isolated spin pair approximation. The matrix treatment is more rigorous, although rel5dng also on several approximations. A set of equations describing the cross-relaxation pathways of all the protons in the molecule is cast into a matrix form and solved. 

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