Spectra density matrices

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

An alternative way to calculate the SLE spectrum is to expand the molecular density matrix to second order in the field and compute the time-dependent photon emission rate. The resulting expression is [23]... 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

The term density matrix has already occurred in Section 2.2. There the density matrix (p) generally used in simulations was consequently called average density matrix. This matrix gives the average state of the spin system, and the spectrum of the whole macroscopic system is determined from this quantity. 

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. 

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... 

In Eq. (10), E nt s(u) and Es(in) are the s=x,y,z components of the internal electric field and the field in the dielectric, respectively, and p u is the Boltzmann density matrix for the set of initial states m. The parameter tmn is a measure of the line-width. While small molecules, N<pure solid show well-defined lattice-vibrational spectra, arising from intermolecular vibrations in the crystal, overlap among the vastly larger number of normal modes for large, polymeric systems, produces broad bands, even in the crystalline state. When the polymeric molecule experiences the molecular interactions operative in aqueous solution, a second feature further broadens the vibrational bands, since the line-width parameters, xmn, Eq. (10), reflect the increased molecular collisional effects in solution, as compared to those in the solid. These general considerations are borne out by experiment. The low-frequency Raman spectrum of the amino acid cystine (94) shows a line at 8.7 cm- -, in the crystalline solid, with a half-width of several cm-- -. In contrast, a careful study of the low frequency Raman spectra of lysozyme (92) shows a broad band (half-width 10 cm- -) at 25 cm- -,... 

The following pages show the 15 Cartesian product operators for a spin system consisting of two /-coupled protons I (Ha) and S (Hb) (Fig. A.l). Each operator is represented in six ways the product operator symbol, an energy diagram with transitions, a vector diagram, a spectrum, a density matrix, and the coherence order. 

The set of coupled first-order density matrix equations in matrix form is exhibited in equation 19, where the proportionality constant C is given by equation 20 it simply weights the entire spectrum8. For calculational purposes we just set C equal to unity. Equation 20 does show that the signal/noise ratio for an NMR spectrum increases on cooling the sample. 

For calculating the c elements for an NMR spectrum subject to dynamic effects, all the NMR parameters should be known together with a range of trial values for the rate constants. Then, via an iterative procedure, comparison of calculated and observed NMR line shapes provides the rate constants. As of this writing software for solving these density matrix equations is readily available and easily managed using any current PC. 

Many workers have in fact used density matrix methods for the calculation of line shapes and intensities in multiple resonance experiments, and two excellent reviews of the background theory are available. (49, 50) In addition there is also a simple guide (51) to the actual use of the method which is capable of predicting the results of quite elaborate experiments. Major applications have included the calculation of the complete double resonance spectrum from an AX spin system which gives 12 transitions in all (52) an extremely detailed study of the relaxation behaviour of the AX2 systems provided by 1,1,2-trichloroethane and 2,2-dichloroethanol (53) the effects of gating and of selective and non-selective pulses on AB and AX spin systems and the importance of the time evolution of the off-diagonal elements of the density matrix in repetitively pulsed FT NMR and spin-echo work (54) the use of double resonance to sort out relaxation mechanisms and transient responses (55) the calculation of general multiple resonance spectra (56) and triple resonance studies of relaxation in AB and AX spin systems. (57)... 

In order to evaluate Eq. (39) we need some detailed knowledge of the density matrix p(t). This operator will contain information about the prior evolution in the applied magnetic field gradients as well as contain information about the relaxation processes and the NMR free precession spectrum. In order to handle this complexity it is very helpful to separate the prior evolution domain from the detection domain. 

Consider the free procession signal acquired as a function of time t after the spin-echo formation at time t. We shall presume that the sole term acting in the Hamiltonian after the echo center is due to the Larmor precession at frequency in the uniform field, where the use of the subscript i allows for the range of chemical shifts in the NMR spectrum. This precession contributes an evolution operator exp(iWo,t ->, ) to the density matrix so that we can rewrite Eq. (39) as... 

To calculate the nutation spectrum the density matrix formalism is used, in which the Hamiltonian is diagonalised either numerically (Kentgens et al. 1987) or analytically (Pandey et al. 1986, Janssen and Veeman 1988). 

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