Individual density matrix

Within the MC-AFDF ADMA method, the management of multiple index assignments ofbasis orbitals and individual density matrix elements requires a series of index conversion relations. These relations are briefly reviewed below, using the notations of the original reference [143]. 

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... 

Key Words Dynamic NMR, Kinetic Monte Carlo, Chemical exchange, Spin system, Spin set, Individual density matrix, Trajectory, Eigencoherence, Vector model, Mutual exchange, Nonmutual exchange. 

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. 

By knowing the trajectory of a spin set, its individual density matrix can be calculated at any time points. The key to the simulation is the determination of the propagating matrix (see Section 3.5). The FID and spectrum of a spin set upon the individual trajectory (one scan) can be determined from the actual values of the time-dependent density matrix. 

Precession (or propagation) describes the spontaneous evolution of the individual density matrix of a spin set in a time interval with no exchange point. These conditions make the operation called propagation conformer and time slice dependent, but it is independent of the rate coefficients of the exchange processes (unlike to the case in Equation (10)). 

Figure 11 Simulation of the fid of a spin set. (A) Individual density matrix is calculated at each exchange point. (B) Eigencoherence representation of the density matrix is propagated from the beginning of the time slice for each detection point.

The vector model of a single spin is the vector representation of the complex number in the individual density matrix of a single nucleus. This density matrix consists of only one complex number thus there is only one vector in the model. In the case of more than one nuclei, the density matrix is larger, there are more single quantum coherences and more vectors belong to one spin set in the model. Moreover, in case of a strongly coupled spin system, the density matrix has different numerical form for different basis sets of the vector space of the simulation (the basis can be one of the ) and

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The evolution of the density matrix in time requires the solution of the equation of motion, Eq. (10) or (20) in a basis set of N states, this represents a system of coupled linear differential equations for the individual density matrix elements. It is most natural to consider the equation in Liouville space, where ordinary operators (N x N matrices) are treated as vectors (of length N ) and superoperators such as and which act on operators to create new operators, become simple matrices (of size X N ). In the Liouville space notation, Eq. (9) would... 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

With reference to the individual AO basis sets

fragment density matrices P t((p (Kt)) obtained from parent molecules Ms of nuclear configurations Kt, on the one hand, and the macromolecular AO basis set cp (K) of the macromolecular density matrix P (cp (K)) associated with the macromolecular nuclear configuration K, on the other hand, the following mutual compatibility conditions are assumed ... 

For in-situ studies of reaction mechanisms using parahydrogen it is desirable to compare experimentally recorded NMR spectra with those expected theoretically. Likewise, it is advantageous to know, how the individual intensities of the intermediates and reaction products depend on time. For this purpose a computer simulation program DYPAS2 [45] has been developed, which is based on the density matrix formalism using superoperators, implemented under the C++ class library GAMMA. 

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° ... 

On the basis of these transformation relations, it is then possible to introduce the topological density matrix P(Qt, Q2) (Eq. 16), and consequently, also the individual similarity indices r/X(, rt. 

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). 

In the Monte Carlo simulation, a few hundreds (100-2000) of scans are calculated, and the Fourier transform of their sum gives the simulated spectrum. The trajectories of spin sets are individual which makes their scans different providing the variety of the samples necessary for the simulation, similar to the case of the single spin interpretation. The calculation of the scans remains independent of each other thus the calculation can be parallelised in the case of coupled spin systems as well.101 The density matrix introduced because of the coupling and the increased amount of calculations on the matrix elements emphasise the use of modern architectures in parallel computation.104... 

When V 0 transitions between L and R can take place, and their populations evolve in time. Defining the total L and R populations by our goal is to characterize the kinetics of the L R process. This is a reduced description because we are not interested in the dynamics of individual level /) and r), only in the overall dynamics associated with transitions between the L and R species. Note that reduction can be done on different levels, and the present focus is on Pl and Pr and the transitions between them. This reduction is not done by limiting attention to a small physical subsystem, but by focusing on a subset of density-matrix elements or, rather, their combinations. 

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