Nuclear spin-electron orbit interaction

Electron orbital moment-nuclear spin interaction energy ... 

In case (a) coupling two main possibilities arise. The first, which is expected to arise very rarely, if at all, implies that the magnetic interaction of the nuclear spin magnetic moment with the electronic orbital and spin moments is sufficiently strong to force the nuclear spin to be quantised in the molecular axis system. The basis kets may be expressed in the form rj, A S, S, A, 2 2, Iz, 2 2, N, J) this scheme is known... 

The first three terms are present for both para- and ortho-H2 they represent the electron spin-rotation, electron spin-spin dipolar and spin orbit interactions respectively. The fourth term in (8.187) represents the magnetic hyperfine interactions, which we will come to a little later. We deal first, however, with the terms that do not involve nuclear spin interactions. 

The first two terms are Zeeman terms and the third represents the hyperfine interaction of the electron and nuclear spins, / b and are the Bohr and nuclear magnetons respectively, S is a fictitious effective spin (S = 2 for a simple Kramers doublet), and / is the nuclear spin tensor. The hyperfine tensor is further split into Fermi contact, dipolar, and orbital components according to ... 

The isotropic and anisotropic hyperfine coupling terms in a arise from interactions between electron and nuclear spins, and provide information about the nature of the orbital containing the unpaired electron and the extent to which it overlaps with orbitals on adjacent atoms. The anisotropic term can cause similar difficulties to the g tensor anisotropy in analysing spectra of polycrystalline powders extracting coupling constants from spectra of transition metal ions or radicals in zeolites can be difficult or impossible without computer simulation. 

Molecular orbitals must have some s character in order to contribute to the Fermi contact interaction, since only, v-atomic orbitals have nonzero density at the nucleus. When a nuclear spin interacts with a spin-sinulel electron pair in a... 

The terms in equation (4) are generally referred to as the orbital-dipolar interaction (o) between the orbital magnetic fields of the electrons and the nuclear spin dipole, the spin-dipolar interaction (D) between the spin magnetic moments of the electrons and nucleus and the Fermi contact interaction (c) between the electron and nuclear spins, respectively. Discussion of the mathematical forms of each of these three terms appears elsewhere. (3-9)... 

The first defect is in ignoring or overlooking the fact that orbital effects, discussed under Electron Spin-Nuclear Spin Interaction, can make major contributions to the observed shifts and must be accounted for in order to get the correct Fermi contact contributions. Many researchers have assumed the total isotropic term is from the Fermi contact interaction. 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

The interactions here describe molecular rotation, coupling of the spin and orbital angular momentum of any unpaired electrons, and hyperfine splittings, which result from the nuclei having spin. A more detailed discussion of these interactions can be found in (J). It should be recognized that analysis of molecular spectra can be very involved because of additional splittings in rotational levels due to electron and nuclear spins, i.e. relativistic effects. 

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... 

Spin density in s orbitals of an atom with a magnetic nucleus results in an isotropic hyperfine coupling (Fermi contact interaction of electron and nuclear spin). Proton s orbital spin densities are usually caused by spin polarization and are proportional to the spin density on the directly bound neighbor atom. Isotropic hyperfine couplings are thus related to the delocalization of the SOMO. 

The interaction between the spin magnetic moments of two unpaired electrons, or of an electron and a nucleus gives rise to an anisotropic coupling. The point-dipole approximation applies when the distance R between the electron spins or the electron and nuclear spins is large compared to the extension of the orbital(s) of the... 

The approximation holds when the distance r between the electron and nuclear spins is large compared to the extension of the orbital of the unpaired electron. At short distance the interaction energy has to be calculated by a quantum mechanical average. A procedure for the common situation of a hydrogen atom in a carbon-centred K-electron radical )CcrH was given in a classical paper [25],... 

Purely anisotropic contributions + Ay + A = 0) to the hyperfine coupling result from spin density in p, d, or / orbitals on the nucleus and from the dipole-dipole interaction T between the electron and nuclear spin. If the electron spin is confined to a region that is much smaller than the electron-nuclear distance r n, both spins can be treated as point dipoles and the magnitude of T is proportional to In this case, T has axial symmetry and its principal values are given by T =Ty= T and = IT. Furthermore, if the spin density in p, d, and/orbitals on... 

Often fliso is used to estimate the s-orbital spin population on the corresponding nucleus [9] (see, e.g., [10]). In Eq. (2) matrix T describes the anisotropic dipole-dipole coupling. The dominant contribution to dd, for nuclei other than protons, usually comes from the interaction of an electron spin in a p-, d-, or f-type orbital with the magnetic moment of the corresponding nucleus. By reference to suitable tables the dipole-dipole coupling can also be used to estimate the spin population in these orbitals [10]. For distances between the electron and nuclear spin greater than approximately 0.25 nm, the anisotropic part of the hyperfine interaction can be used to calculate the electron-nuclear distance and orientation with the electron-nuclear point-dipole formula ... 

Some of the terms included in the Breit-Pauli Hamiltonian also describe small interactions that can be probed experimentally by inducing suitable excitations in the electron or nuclear spin space, giving rise to important contributions to observable NMR and ESR parameters. In particular, for molecular properties for which there are interaction mechanisms involving the electron spin, also the spin-orbit interaction (O Eqs. 11.13 and O 11.14) becomes important The Breit-Pauli Hamiltonian in O Eqs. 11.5-11.22, however, only includes molecule-external field interactions through the presence of a scalar electrostatic potential 0 (and the associated electric field F) and the appearance of the magnetic vector potential in the mechanical momentum operator (O Eq. 11.23). In order to extract in more detail the interaction between the electronic structure of a molecule and an external electromagnetic field, we need to consider in more detail the form of the scalar and vector potentials. 

Rotational effects (fine structure) are also seen in high resolution electronic and vibrational spectra additional structure due to the interaction of electrons with nuclear electric and magnetic moments may also be observed and is called hyperfine structure. The simplest rotational spectra are associated with diatomic molecules with no electronic orbital or spin angular momentum (i.e. singlet sigma states) and these are considered first. 

The value of A depends on the nuclear moment, and the extent of interaction of the unpaired electron spin density with the nucleus A has a sign as well as magnitude. If A > 0, the state in which electron and nuclear spins align antiparallel is of lower energy. Measurements of the magnitude and sign of hyperfine couplings provide some of the most detailed experimental evidence for electronic structures of molecules. They have been used to verify the results of molecular orbital calculations. 

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