Scalar coupling Hamiltonian

These relations turn out to be quite useftd. For example, when spins are weakly coupled, the Zeeman and scalar coupling portions of the Hamiltonian commute, so that these effects may be treated separately and in any order. 

Figure A3 2 Evaluation of the diagonal elements of the Hamiltonian for scalar coupling in a first-order, two-spin system (AX).

The time independent isotropic spin-spin scalar coupling Hamiltonian is... 

MQCs are not excited uniformly and the efficiency with which the various orders of MQC are excited depends specifically on the parameters of the spin system (dipolar couplings, scalar couplings, quadrupolar couplings, chemical shifts) in the spin system and the choice of the preparation time t. Many researchers have co-added spectra acquired with different preparation times to ensure that all transitions are observed with reasonable intensity. A number of broadband excitation techniques have been developed,13-15 where the value of t in the preparation sequence has been varied either in a pseudo-random or systematic fashion to achieve a more uniform excitation in the multiple quantum domain. An experimental search method has been used to optimise the delays in the preparation period of the MQ excitation sequence16 and Wimperis17 used average Hamiltonian theory to propose... 

The options couple and weak denote a scalar coupling interaction. Couple is adequate for a strongly coupled second order spin system and weak for a coupling that can be described by an AX approximation. Dipolar describes a dipolar coupling, only the energy conserving part of the dipolar Hamiltonian are implemented. The qpolar option is reserved for the coupling interaction of a quadrupolar nucleus. Before a nucleus can appear in the interaction statement it must be defined in the nucleus statement. 

For spin-spin (scalar) coupling between spins I and S, where I = j and S >, the interaction Hamiltonian involves the scalar coupling tensor J. The scalar relaxation of the nucleus I can arise from either a time-dependent 5 or a time-dependent J ( first kind ). If the relaxation time (T (S)) of the nucleus S is short compared with 1//, where J is the scalar coupling constant, the nucleus I sees the average of the spin-spin interaction and does not show the expected multiplet, but a single line. The scalar relaxation correlation time tg is then equal to T (S), and the scalar spin-spin contributions to the longitudinal and transverse relaxation times of the I nucleus are given by... 

The coupling constants of the hyperfme and the electron Zeeman interactions are scalar as long as radicals in isotropic solution are considered, leading to the Hamiltonian... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

While the Hamiltonian operator Hq for the hydrogen atom in the absence of the spin-orbit coupling term commutes with L and with S, the total Hamiltonian operator H in equation (7.33) does not commute with either L or S because of the presence of the scalar product L S. To illustrate this feature, we consider the commutators [L, L S] and [S, L S],... 

The hyperfine constant a in Eq. (1) was also taken to be a scalar quantity for the hydrogen atom however, it is in general a tensor because of the various directional interactions in a paramagnetic species. The hyperfine term in the spin Hamiltonian is more correctly written as S-a-I, where a is the hyperfine coupling tensor. 

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