Scalar coupling matrix

These coupling matrix elements are scalars due to the presence of the scalar Laplacian V. in Eq. (25). These elements are, in general, complex but if we require the to be real they become real. The matrix unlike its... 

When performing 2D-NMR experiments one must keep in mind that the second frequency dimension (Fx) is digitized by the number of tx increments. Therefore, it is important to consider the amount of spectral resolution that is needed to resolve the correlations of interest. In the first dimension (F2), the resolution is independent of time relative to F. The only requirement for F2 is that the necessary number of scans is obtained to allow appropriate signal averaging to obtain the desired S/N. These two parameters, the number of scans acquired per tx increment and the total number of tx increments, are what dictate the amount of time required to acquire the full 2D-data matrix. 2D-homo-nuclear spectroscopy can be summarized by three different interactions, namely scalar coupling, dipolar coupling and exchange processes. 

Couple it with a model for the joint scalar dissipation rate that predicts the correct scalar covariance matrix, including the effect of the initial scalar length-scale distribution. 

It can be seen that, in the average density matrix formalism which is based on spin system model, the scalar couplings and the exchange processes are handled simultaneously. Thus they cannot be separated and a larger atomic basis (spin system) is required for their description. Meanwhile, the Monte Carlo method based on spin sets separates the two interactions, and thus spin systems can be reduced to smaller spin sets. 

Consider now the two-spin system, in which chemical shifts and scalar coupling come into play. In Chapter 6 we discussed the two-spin system in detail, both the weakly coupled AX system and the general case, AB. To illustrate the application of the density matrix, we concentrate first on the AX system and then indicate briefly how the results would be altered for AB. To simplify the notation, we call the nuclei I and S, rather than A and X, and use the common notation in which the spin operators and their components are designated, for example, Ix and Sx, rather than the more cumbersome 4(A) or /. Although the I-S notation is usually applied to heteronuclear spin systems, we use it here to include homonuclear systems (e.g., H-H) as well. 

The detection of NMR signals is based on the perturbation of spin systems that obey the laws of quantum mechanics. The effect of a single hard pulse or a selective pulse on an individual spin or the basic understanding of relaxation can be illustrated using a classical approach based on the Bloch equations. However as soon as scalar coupling and coherence transfer processes become part of the pulse sequence this simple approach is invalid and fails. Consequently most pulse experiments and techniques cannot be described satisfactorily using a classical or even semi-classical description and it is necessary to use the density matrix approach to describe the quantum physics of nuclear spins. The density matrix is the basis of the more practicable product operator formalism. 

The execution of a rf pulse or the evolution of chemical shift or J-scalar coupling is described by an operator. The operation of this operator on the expanded density matrix is exclusively related to the coefficients of the corresponding operator matrix cJexpanded-... 

Here, a few comments are in order. The matrix of derivative couplings F is antihermitian. The matrix of scalar couplings G is composed of an hermitian as well as an antihermitian part. Of course, the dressed kinetic energy operator —(1/2M)(V - - F) in our basic Eq. (10) is hermitian, as is also the case for the nonadiabatic couplings A in Eq. (9a). The latter follows immediately from the relation (lie). The notation (V F) is self evident from Eq. (lid). Since F is a vector matrix, it can be written as F = (Fi, F2,..., Fjv ), where the matrices Fq, are simply defined by their... 

The scalar potential given in Eq. (7) contains contributions which involve derivative coupling matrix elements between the electronic subspace of interest and the excited electronic states. These contributions have the effect of a small correction (proportional to n ) to the potential energy Vn x). Prom now on, we ignore these small corrections and take enm(x) = Pn (x)( nm ... 

The g-value of a free electron is a scalar, ge = 2.00232. In a radical species, g becomes a matrix because of the admixture of orbital angular momentum into S through spin-orbit coupling. The components of the g-matrix thus differ from ge to the extent that p-, d-, or f-orbital character has been incorporated, and they differ from one another, depending on which p-, d-, or f-orbitals are involved. 

---