Individual density matrix average

Contents 1. Introduction 176 2. Static NMR Spectra and the Description of Dynamic Exchange Processes 178 2.1. Simulation of static NMR spectra 178 2.2. Simulation of DNMR spectra with average density matrix method 180 3. Calculation of DNMR Spectra with the Kinetic Monte Carlo Method 182 3.1. Kinetic description of the exchange processes 183 3.2. Kinetic Monte Carlo simulation of DNMR spectra for uncoupled spin systems 188 3.3. Kinetic Monte Carlo simulation of coupled spin systems 196 3.4. The individual density matrix 198 3.5. Calculating the FID of a coupled spin system 200 3.6. Vector model and density matrix in case of dynamic processes 205 4. Summary 211 Acknowledgements 212 References 212... 

However, the individual density matrix models the individual behaviour of the spin sets. An individual density matrix—like the average one—changes due to pulses, precession and exchange processes, and it can be exactly calculated taking these effects into account at any time point. While the average density matrix contains the average state of the spin system at a certain time point, the individual density matrix describes a possible state of the spin set. 

The spin density matrix Pj(t) which describes the properties of any spin system of a molecule A, is defined as follows. We assume that the density matrices Pj(0), j = 1, 2,..., S, which describe the individual components of the dynamic equilibrium at any arbitrary time zero, are known explicitly, and that at any time t such that t > t > 0 the pj(t ) matrices are already defined. Our reasoning is applied to a pulse-type NMR experiment, and we therefore construct the equation of motion in a static magnetic field. The p,(t) matrix is the weighted average over the states involved, according to equation (5). The state of a molecule A, formed at the moment t and persisting as such until t, is given by the solution of equation (35) with the super-Hamiltonian H° ... 

Figure 10 Comparison of individual (left) and average (right) density matrices in the case of a non-mutual exchange. (A) Definition of spin set or spin system (see Figure 9). (B) Basis functions (lines) and intramolecular single quantum coherences (arrows) defined by the spin set or spin system. (C) Elements corresponding to the intramolecular single quantum coherences in the density matrix.

In this case, the average density matrix is the weighted average of all possible individual density matrices where the weighting factors are the same as given in Equation (24) (T acquisition time should be replaced by the t actual time). 

Cortona developed a method to calculate the electronic structure of solids by calculating individually the electron density of atoms in a unit cell with a spherically averaged Hamiltonian as the local Hamiltonian. The tests of the method have been successful for alkali halides where the density around each nucleus can be well approximated by a spherical description. Goedecker proposed a scheme closely related to the divide-and-conquer approach. The local Hamiltonian is also constructed by truncation in the atomic orbital space. Instead of the matrix diagonalization for the local Hamiltonian described in equation (34) in the divide-and-conquer approach, Goedecker used an iterative diagonalization based on the Chebyshev polynomial approximation for the density matrix. Voter, Kress, and Silver s method is related to that of Goedecker with the use of a kernel polynomial method. 

Notice that while the trajectories jump between the electronic states at a given time, all density matrix elements are propagated continuously over the entire time according to Eqs. 17.14—17.15, or alternatively either Eq. 17.17 or 17.18. Although the individual trajectory is allowed to jump, the fraction of trajectories in a given state, which represents pu as an ensemble average, is also a continuous function of time. The phase of the electronic wavefunction is preserved and our procedure gives... 

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 . 

The matrix elements density distribution cok(Ei) associated with the individual zeroth-order state 4>k- These distributions may be converged rapidly by using either Lanczos or Chebychev expansions of the density operator S(E - H) with the state [Pg.3139]

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