Spin Hamiltonian nuclear-orbit interaction

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

We have chosen to use the hyperfine-coupled representation, where for 12CH, F is equal to J 1 /2. An appropriate basis set is therefore t], A N, A S, J, /, F), with MF also important when discussing Zeeman effects. As usual the effective zero-field Hamiltonian will be, at the least, a sum of terms representing the spin-orbit coupling, rigid body rotation, electron spin-rotation coupling and nuclear hyperfine interactions, i.e. 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

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... 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as... 

This operator accounts for the interaction between the electron spins and the magnetic field created by nuclear motion. As the nuclei are heavy, their angular velocity is approximately m/M times smaller than the angular velocity of the electrons. Consequently, except for light molecules, the spin-rotation interaction is very small compared to the spin-orbit interaction. The microscopic Hamiltonian from Kayama and Baird (1967) and Green and Zare (1977) has the case (a) form,... 

Table 3.1 Effective Nuclear Charge (Scaling Parameter) Z ff for Approximate Spin-Orbit Interaction Calculations Using the One-Electron Term in the Breit-Pauli Hamiltonian (Equation 3.8, developed by Koseki et...

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

Another recent development is the implementation of DK Hamiltonians which include spin-orbit interaction. An early implementation shared the restriction of the relativistic transformation to the kinetic energy and the nuclear potential with the efficient scalar relativistic variant electron-electron interaction terms were treated in nonrelativistic fashion. Further development of the DKH approach succeeded in including also the Hartree potential in the relativistic treatment. This resulted in considerable improvements for spin-orbit splitting, g tensors and molecular binding energies of small molecules of heavy main group and transition elements. Application of Hamiltonians which include spin-orbit interaction is still computationally demanding. On the other hand, the SNSO method is an approximation which seems to afford a satisfactory level of accuracy for a rather limited computational effort. 

Hq is the one-electron Hamiltonian including kinetic, nuclear attraction and RECP operators. The spin-orbit interaction operator is given by... 

The fine structure of atomic line spectra and the hyperfine splittings of electronic Zeeman spectra are non-symmetric for those atomic nuclei whose spin equals or exceeds unity, / > 1. The terms of the spin Hamiltonian so far mentioned, that is, the nuclear Zeeman, contact interaction, and the electron-nuclear dipolar interaction, each symmetrically displace the energy, and the observed deviation from symmetry therefore suggests that another form of interaction between the atomic nucleus and electrons is extant. Like the electronic orbitals, nuclei assume states that are defined by the total angular momentum of the nucleons, and the nuclear orbitals may deviate from spherical symmetry. Such non-symmetric nuclei possess a quadrupole moment that is influenced by the motion of the surrounding electronic charge distribution and is manifest in the hyperfine spectrum (Kopfer-mann, 1958). 

Advanced EMR methods may be used to conduct quantitative measurements of nuclear hyperfine interaction energies, and these data, in turn, may be used as a tool in molecular design because of their direct relation to the frontier orbitals. The Zeeman field dependence of hyperfine spectra enables one to greatly improve the quantitative analysis of hyperfine interaction and assign numeric values to the parametric terms of the spin Hamiltonian. Graphical methods of analysis have been demonstrated that reduce the associated error that comes from a multi-parameter fit of simulations based on an assumed model. The narrow lines inherent to ENDOR and ESEEM enable precise measures of peak position and high-resolution hyperfine analyses on even powder sample materials. In particular, ESEEM can be used to obtain very narrow lines that are distributed at very nearly the zero-field NQI transition frequencies because of a quantum beating process that is associated with... 

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