Interaction Hamiltonian hyperfine structure

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

The operators so and ss are compound tensor operators of rank zero (scalars) composed of vector (first-rank tensor) operators and matrix (second-rank tensor) operators. We will make use of this tensorial structure when it comes to selection rules for the magnetic interaction Hamiltonians and symmetry relations between their matrix elements. Similar considerations apply to the molecular rotation and hyperfine splitting interaction... 

The CN radical in its 21 ground state shows fine and hyperfine structure of the rotational levels which is more conventional than that of CO+, in that the largest interaction is the electron spin rotation coupling../ is once more a good quantum number, and the effective Hamiltonian is that given in equation (10.45), with the addition of the nuclear electric quadrupole term given in chapter 9. The matrix elements in the conventional hyperfine-coupled case (b) basis set were derived in detail in chapter 9,... 

For radicals with magnetic nuclei, the hyperfine structure of ESR spectra is produced by the interaction of the electron magnetic moment with the nuclear spin of those nuclei covered by the molecular orbital of the unpaired electron. This interaction splits further the two spin levels in a magnetic field. The hyperfine coupling is often given by the Hamiltonian HgN ... 

In microwave spectroscopy, pure rotational transitions are studied. If one or more nuclei in the molecule has a nuclear quadrupole moment, the quadrupole Hamiltonian [Eq. (6)] has to be included in the quantum mechanical treatment because the field gradient q [Eq. (5)] is dependent on the rotational wave function. The nuclear quadrupole interaction, which causes the rotational transitions to split into hyperfine structure, can usually be treated as a perturbation to the rotational Hamiltonian. 

The first term results from the Fermi contact interaction, while the second represents the long-range dipole-dipole interaction. In the equations above, ge is the free-electron g factor, /Xe the Bohr magneton, gi the nuclear gyromagnetic ratio, and /xi the nuclear moment. Moreover, the nucleus is located at position R, and the vector r has the nuclear position as its origin. Finally, p (r) = p (r) — p (r) is the electron spin density. The only nontrivial input into these equations is precisely this last quantity, i.e. Ps(r), which can be computed in the LSDA or another DFT approximation. The resulting Hamiltonian can be used to interpret the hyperfine structure measured in experiments. A recent application to metal clusters is reported in Ref. [118]. 

The principal advantage of ENDOR spectroscopy is the much finer energy scale upon which the state-to-state transitions are recorded. As discussed in 3, conventional cw-EMR spectroseopy detects the weak hyperfine interactions of the spin Hamiltonian as a perturbation of the electronic Zeeman effect in many practical situations, inhomogeneous broadening will wash out the hyperfine structure of the speetrum. In such cases of inhomogeneously broadened EMR spectra, interaction can only be deconvoluted for practical analysis via simulations (Hyde Fron-cisz, 1982). The ENDOR method, however, records a spectram that represents the... 

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]... 

The constant b therefore contains contributions from two quite different magnetic interactions, the Fermi contact and the electron-nuclear dipolar interactions. Interpretation of the magnitudes of these constants in terms of electronic structure theory always involves the separate assessment of these different effects, so that we prefer to use an effective Hamiltonian which separates them at the outset. Consequently the effective magnetic hyperfine Hamiltonian used throughout this book is... 

Magnetic tuning of diatomic bound-state and scattering properties relies on the Zee-man effect in the hyperline structure of alkali-metal atoms. The splitting into sublevels of the 5i/2 electronic ground state of such an atom which is exposed to a magnetic field B can be described by the following Hamiltonian comprised of hyperfine and Zeeman interactions [27] ... 

The real power of ESR spectroscopy for structural studies is based on the interaction of the unpaired electron spin with nuclear spins. This hyperfine interaction splits each energy level into sublevels and often allows the determination of the atomic or molecular structure of species containing unpaired electrons, and of the ligation scheme around paramagnetic transition metal ions. For a system with m nuclear spins (identified by index k) and a single electron spin, which may be larger than one-half as explained below, the hyperfine Hamiltonian is given in Eq. 2,... 

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). 

The information obtained from the spin Hamiltonian, the 3x3 matrices g, D, A, and P, is very sensitive to the geometric and electronic structure of the paramagnetic center. The electron Zeeman interaction reveals information about the electronic states the zero-field splitting describes the coupling between electrons for systems where S > Vi the hyperfine interactions contain information about the spin density distribution [8] and can be used to evaluate the distance and orientation between the unpaired electron and the nucleus the nuclear Zeeman interaction identifies the nucleus the nuclear quadrupole interaction is sensitive to the electric field gradient at the site of the nucleus and thus provides information on the local electron density. 

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