Hamiltonian equations, electron spin resonance

The real power of electron spin resonance spectroscopy for structural studies is based on the interaction of the impaired electron spin with nuclear spins. This interaction splits the energy levels and often allows determination of the atomic or molecular structure of species containing unpaired electrons, and of the ligation scheme around paramagnetic transition-metal ions. The more complete Hamiltonian is given in equation 2 for a species containing one unpaired electron, where the summations are over all the nuclei, n, interacting with the electron spin. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The electron spin-spin interaction Hamiltonian and its case (a) matrix elements were previously derived, equation (9.124), for the analysis of the magnetic resonance spectrum of 3S SeO ... 

The rotational and Zeeman perturbation Hamiltonian (X) to the electronic eigenstates was given in equation (8.105). It did not, however, contain terms which describe the interaction effects arising from nuclear spin. These are of primary importance in molecular beam magnetic resonance studies, so we must now extend our treatment and, in particular, demonstrate the origin of the terms in the effective Hamiltonian already employed to analyse the spectra. Again the treatment will apply to any molecule, but we shall subsequently restrict attention to diatomic systems. 

Here p, is the reduced mass, and are the Hamiltonians defined in Equation 11.1 for each atom, and Vint is the effective interaction potential depending on the relative position of the atoms, r. For many applications, such as the description of broad scattering resonances and their associated Feshbach molecules, it is sufficient to include in Vint only the rotationally symmetric singlet and triplet Born-Oppenheimer potentials, Vs=o and V5=i, respectively. Their labels 5 = 0 and 5=1 refer to the possible values of the angular-momentum quantum number associated with the total spin of the two atomic valence electrons, S = si -E S2. In this approximation, the interaction part of Equation 11.4 can be represented by [8,29]... 

---