Chemical shift Hamiltonian

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

In strongly coupled systems, it is not possible to eliminate chemical shifts by refocusing nor is it possible to describe the evolution in terms of an effective Hamiltonian.44 A 90° or a 180° pulse leads to coherence transfer between various transitions, and a multitude of new effective precession frequencies may appear in the F1 dimension. A detailed analysis shows artefacts resulting of strong coupling induced by the 180° pulse applied on the H channel can be efficiently removed by applying a LPJF before acquisition.42 Likewise, artefacts present in HMBC with a terminal LPJF are suppressed by an LPJF in the beginning of the sequence as in conventional HMBC. 

The basic RFDR element consists of two 7i-pulses each centered in the middle of each rotor period in two consecutive rotor periods leading to a cyclic rf field. In the toggling frame of ideal (i.e., infinitely strong) 7i-pulses, the Hamiltonian of the chemical shift terms and the homonuclear dipolar coupling interaction is split up into three commuting parts ... 

The calculation of the effective Hamiltonian is greatly simplified by only considering the isotropic chemical shift difference, Aoj1so = cof° — a> f leading to... 

As demonstrated by Griffin, Levitt, and coworkers in the late 1980s [21, 93], it is also possible to recouple homonuclear dipolar couplings through interference between isotropic chemical shifts and the rotor revolution. This phenomenon, called rotational resonance, occurs when the spinning frequency is adjusted to a submultiple of the isotropic chemical shift difference, i.e., ncor = ct> so o) °. To understand this experiment, the dipolar coupling Hamiltonian in (10) is transformed... 

In the simplest setup, the two strong field components may be set identical to Ci = Cs = C. The relatively large CIX or CSX term averages isotropic and anisotropic chemical shift effects as well as the heteronuclear dipolar coupling interaction between 15N or 13C and H. The difference of - or the sum of - the B coefficients selects the form of the recoupled heteronuclear dipole-dipole coupling interaction, as expressed in terms of the effective Hamiltonian in the interaction frame of the rf irradiation... 

The total Hamiltonian that governs the spin system is the sum of contributions arising from the various interactions, including Zeeman interaction, chemical shift... 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

Molecules in ordinary liquids average out all anisotropic spin interactions due to isotropic Brownian motions, and their NMR spectra are governed by the Hamiltonian in units of Hz due to the Zeeman interaction, the isotropic chemical shift (a) and the isotropic indirect spin-spin coupling (7)... 

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

In an organic solid representative broadenings are 150 ppm for aromatic carbon chemical shift anisotropy and 25 kHz (full width at half-height) for a rather strong carbon-proton dipolar interaction. At a carbon Larmor frequency of 15 MHz, the shift anisotropy corresponds to 2.25 kHz. In high magnetic fields the forms of the respective Hamiltonians are... 

Rapid molecular motions in solutions average to zero the dipolar and quadrupolar Hamiltonian terms. Hence, weak interactions (chemical shift and electron-coupled spin-spin couplings) are the main contributions to the Zeeman term. The chemical shift term (Hs) arises from the shielding effect of the fields produced by surrounding electrons on the nucleus ... 

The main purpose of the sequences is to obtain an averaged Hamiltonian H in which the dipolar term is very small compared with the chemical shift term. The zeroth-order of the average dipolar Hamiltonian term is given by the following equation ... 

The electrons modify the magnetic field experienced by the nucleus. Chemical shift is caused by simultaneous interactions of a nucleus with surrounding electrons and of the electrons with the static magnetic field B0. The latter induces, via electronic polarization and circulation, a secondary local magnetic field which opposes B0 and therefore shields the nucleus under observation. Considering the nature of distribution of electrons in molecules, particularly in double bonds, it is apparent that this shielding will be spatially anisotropic. This effect is known as chemical shift anisotropy. The chemical shift interaction is described by the Hamiltonian... 

Each of the resonances which are separated by the chemical shifts may be further split into several lines by spin-spin interactions. These can be described by an isotropic coupling Hamiltonian of the type... 

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