Multi-dimensional spectra

The terms transformation, convolution, and correlation are used over and over again in NMR spectroscopy and imaging in different contexts and sometimes with different meanings. The transformation best known in NMR is the Fourier transformation in one and in more dimensions [Bral]. It is used to generate one- and multi-dimensional spectra from experimental data as well as ID, 2D, and 3D images. Furthermore, different types of multi-dimensional spectra are explicitly called correlation spectra [Eml]. It is shown below how these are related to nonlinear correlation functions of excitation and response. 

S. Laage, J.R. Sachleben, S. Steuemagel, R. PieratteUi, G. Pintacuda, L. Emsley, Fast acquisition of multi-dimensional spectra in sohd-state NMR enabled by ultra-fast MAS, J. Magn. Reson. 196 (2009) 133-141. 

Non-uniform/sparse data sampling. Sparse, non-uniform sampling (NUS) of the indirect dimensions of multi-dimensional spectra... 

During a period of evolution in which multiple-quantum states are present, the magnetization acquires a modulation due to the nutation frequency of the MQCs. Thus, signals which are due to single sites during other evolution periods will have the same spectral frequency in a multiple-quantum domain as those signals which had passed through the same MQC. This can therefore easily be read in a multi-dimensional spectrum as a correlation between these nuclear sites. In addition, a multiple-quantum domain as part of a multi-dimensional spectrum removes the auto-correlation peaks on... 

Figure 8.2.7 Reference deconvolution applied to the methanol peak (a) original peak (b) methanol peak deconvolved to a 2 Hz Lorenzian lineshape (c) gradient-shifted methanol peak (d) gradient-shifted methanol peak deconvolved to the same 2 Hz Lorenzian lineshape (e) comparison of the sub-spectrum of the methanol sample using the subtraction algorithm with (sharp peak) and without (flattened peak) reference deconvolution. Reprinted from Hou, T., MacNamara, E. and Raftery, D., NMR analysis of multiple samples using parallel coils improved performance using reference deconvolution and multi-dimensional methods , Anal. Chem. Acta, 400, 297-305, copyright (1999), with permission of Elsevier Science...

Let us consider the concept of "relaxation in more detail since no accurate definition for it has been given previously. The term "relaxation is often used for the process by which either an equilibrium or a steady state is achieved in the system, and the relaxation time is treated as the time to achieve complete or partial thermodynamic equilibrium. It is evident that, in this context, the difference between "equilibrium and "steady state is insignificant. The concept of "relaxation time is often used for the time during which a certain function characterizing the deviation from the equilibrium or the steady state diminishes by e ( 2.718) times compared with its initial value. It is evident, however, that this definition is only correct for one-dimensional linear systems. For multi-dimensional linear systems, a spectrum of relaxation times must be used. For non-linear systems, the application of these definitions is correct only in the neighbourhood of a singular point. 

Fig. 6.4. Schematic illustration of the multi-dimensional reflection principle in the adiabatic limit. The left-hand side shows the vibrationally adiabatic potential curves en(R). The independent part of the bound-state wavefunction in the ground electronic state is denoted by

The next milestone, in the history of NMR [Frel], was the extension of the NMR spectrum to more than one frequency coordinate. It is called multi-dimensional spectroscopy and is a form of nonlinear spectroscopy. The technique was introduced by Jean Jeener in 1971 [Jeel] with two-dimensional (2D) NMR. It was subsequently explored systematically by the research group of Richard Ernst [Em 1 ] who also introduced Fourier imaging [Kuml]. Today such techniques are valuable tools, for instance, in the structure elucidation of biological macromolecules in solution in competition with X-ray analysis of crystallized molecules as well as in solid state NMR of polymers (cf. Fig. 3.2.7) [Sch2]. 

Despite the extremely wide use of multi-dimensional NMR techniques for conformational studies of biological macromolecules, at present, only a small part of the information contained in the NMR spectrum can be used for structure determination [118,119]. The reason is that there are no well-established correlations between NMR chemical shifts and the structural parameters [118] and only a few useful correlations besides Karplus relations [120,121] for nuclear spin-spin coupling constants. As a consequence, a great deal of important information about the system is not available without turning to quantum chemical approaches for the theoretical interpretation. 

At the core of the analysis and methods that are discussed in this Chapter is the consistent consideration of the fact that the form of each resonance wavefunction is = fl I o+Xas (Eq. (4.1) of text), if necessary, the extension to multi-dimensional forms is obvious. Depending on the formalism, the coefficient a and the asymptotic part, Xas, are functions of either the energy (real or complex) or the time. The many-body square-integrable, %, represents the localized part of the decaying (unstable) state, i.e., the unstable wavepacket which is assumed to be prepared at f = 0. its energy, Eo, is real and embedded inside the continuous spectrum, it is a minimum of the average value of the corresponding state-specific effective Hamiltonian that keeps all particles bound. 

In the CESE procedure, the coordinates of the Hamiltonian (40) are left real. The total function space is formally divided info two multi-dimensional parts of N-elecfron wavefuncfions, say Q and P, nof necessarily orthonormal between them. The Q space is fixed by considering judiciously the electronic structure and spectrum of the system under study. The P space is not completely fixed. It contains parameterized configurations with real as well as complex orbitals representing contributions from fhe multichannel high Rydberg and scattering states. It is the variational optimization of this space via the diagonalization of fhe matrix... 

The N-dimensional phase portrait describing the well-stirred (that is, homogeneous) Belousov-Zhabotinskii system could be constructed from measurements of the time dependence of the concentration of all N chemical species in the reaction. Fortunately, such a difficult task is unnecessary - a multi-dimensional phase portrait can be constructed from measurements of a single variable by a procedure proposed by RUELLE [31], PACKARD et al. [32], and TAKENS [33]. The idea is a follows For almost every observable B(t) and time delay T an m-dimensional portrait constructed from the vectors B(tj ), B( tj +T),..., B( t +(m-l)T), where t =kAt, k= l, 2,..., >, will have the same properties (for example, the same spectrum of Lyapunov exponents) as one constructed from measurements of N independent variables, if m>2N-hl. In principle the choice of time delay T is arbitrary, but in practice time delays of from about one-tenth to two-thirds of a characteristic oscillation time are usually optimum [25]. 

So far, the explanation was for an example of diatomic molecules. For the polyatomic molecules composed of n-pieces of atoms, the idea of the correspondence of the appearance of the absorption spectrum and the photo-excitation is the same, but the potential surfaces are in n 1 dimensions. Since photodissociation can occur with multiple processes such as ABC AB + C, A + BC, multi-dimensional potential surfaces for each dissociating inter-atomic distance must be considered. 

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