Hamiltonian operator shielding

Now consider a d ion as an example of a so-called many-electron atom. Here, each electron possesses kinetic energy, is attracted to the (shielded) nucleus and is repelled by the other electron. We write the Hamiltonian operator for this as follows ... 

In the equation (12.83) for Hamiltonian, the shielding constants occur in the term I a H. The perturbation operator contains a lot of terms, but most of them, when inserted into the above formula, are unable to produce terms that behave like I a H. Only some very particular terms could produce such a dot product dependence. A minute of reflection leads directly to 83,84, 85 and Bio as the only terms of the Hamiltonian that have any chance of producing the dot product form. Therefore, using the definition of the reduced resolvent of Eq. (10.76), we have ... 

Up until this point, we have implicitly assumed time independence of the interactions and their corresponding Hamiltonian operators. This assumption is valid for a rigid stationary sample. However, when the sample is spun rapidly, each of the internal Hamiltonians becomes time-dependent. For example, Jfz,cs becomes time-dependent when there is chemical shielding anisotropy due to the fact that the orientations of the chemical shielding tensors relative to the applied magnetic field change as the sample... 

A transition metal with the configuration t/ is an example of a hydrogen-like atom in that we consider the behaviour of a single (d) electron outside of any closed shells. This electron possesses kinetic energy and is attracted to the shielded nucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq. (3.4). 

Table 1 Nonrelativistic one-electron magnetic terms in the Hamiltonian. Their derivatives with respect to /m and/or or [ip enter the expressions for the nuclear shielding and spin-spin coupling tensors via the perturbation operators ) ancj g is the spin-operator for an electron, va a distance vector with respect to nucleus A etc. ...

In this Hamiltonian, the a/ are the shieldings, or chemical shifts, of the nuclei, J / the indirect spin-spin coupling constants between pairs of nuclei, IZi the z component of the spin operator 7, y the gyromagnetic ratic and H0 the strength of the static applied field. Within the experimental accuracy of measurements achieved thus far, no additional terms are required in the Hamiltonian to attain an exact fit of theoretical and calcu-... 

The HCp operator represents the nonspherically symmetric components of the one-electron CF interactions, i.e. the perturbation of the Ln3+ 4fN electron system by all the other ions. The states arising from the 4fN configuration are well-shielded from the oscillating crystalline field (so that spectral lines are sharp) but a static field penetrates the ion and produces a Stark splitting of energy levels. The general form of the CF Hamiltonian Hcf is given by... 

Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a further degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfine terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the first-order and second-order ct at the nonrelativistic level (eqs. 5-7), third-order contributions to nuclear shielding (8) arise, that couple the one- and two-electron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted... 

This table contains the data obtained from the magnetic hyperfine structure and the Zeeman effect for molecules in a Z state or more generally in a state with Q = 0, i.e., the projection of the angular momentum onto the molecular axis is zero. For the magnetic hyperfine stracture one usually considers four terms the spin-rotation interaction for each nucleus and the scalar and tensorial spin-spin interaction of the two nuclear spins. For the Zeeman effect one takes into account the rotational Zeeman effect, the nuclear Zeeman effect with the scalar and tensorial shielding, and the scalar and tensorial magnetic susceptibility. The hamiltonian of these interactions can be written with the concept of spherical tensor operators [57Edm]... 

A number of static perturbations arise from internal interactions or fields, which are neglected in the nonrelativistic Born-Oppenheimer electronic Hamiltonian. The relativistic correction terms of the Breit-Pauli Hamiltonian are considered as perturbations in nonrelativistic quantum chemistry, including Darwin corrections, the mass-velocity correction, and spin-orbit and spin-spin interactions. Some properties, such as nuclear magnetic resonance shielding tensors and shielding polarizabilities, are computed from perturbation operators that involve both internal and external fields. 

Let us first consider the standard interpretation of the NMR spectrum. The parameters used to define it - the shielding constants and spin-spin coupling constants - are obtained from an effective NMR Hamiltonian This operator acts in the space defined by all possible... 

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